id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
dict |
|---|---|---|---|---|---|
1001.0573
|
# Theory of time-resolved spectral function in high-temperature
superconductors with bosonic modes
Jianmin Tao Jian-Xin Zhu Theoretical Division & CNLS, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545
###### Abstract
We develop a three-temperature model to simulate the time dependence of
electron and phonon temperatures in high-temperature superconductors
displaying strong anistropic electron-phonon coupling. This model not only
takes the tight-binding band structure into account, but also is valid in
superconducting state. Based on this model, we calculate the time-resolved
spectral function via the double-time Green’s functions. We find that the dip-
hump structure evolves with the time delay. More interestingly, new phononic
structures are obtained when the phonons are excited by a laser field. This
signature may serve as a direct evidence for electron-vibration mode coupling.
###### pacs:
74.25.Jb, 74.72.-h, 71.38.-k, 79.60.-i
The discovery of high-temperature superconductors (HTSC) has raised an
important issue on the mechanism leading to the formation of Cooper pairs,
which is still under debate. To address this issue, a number of spectroscopy
techniques ADamascelli03 ; JLee06 ; FCarbone08 have been used to study the
role and nature of bosonic modes, to which electrons are strongly coupled. In
particular, due to technological advances and the improved sample quality,
angle-resolved photoemission spectroscopy (ARPES) ADamascelli03 has been used
to probe details of the energy and momentum structure of single-particle
excitations via the measurement of photoemission intensity. However, different
interpretations of the same data may lead to completely different mechanisms
MRNorman01 ; AWSandvik04 . For example, the dip-hump structure observed in
ARPES HDing96 ; JCCampuzano99 ; AKaminski01 ; ALanzara01 ; PDJohnson01 ;
XJZhou03 ; TSato03 ; TCuk04 could be interpreted ZXShen97 as naturnally
occurring in the interacting system but having little effect on the
superconducting pairing mechanism. It could also be interpreted ZXShen02 in
terms of phonon modes that could drive $d$-wave pairing. Recent theoretical
analysis TPDevereaux04 on the spectral function of thermally excited
electrons in the cuprates has shown that the out-of-plane and out-of-phase
buckling mode strongly couples to the electronic states near the anti-nodal
$M$ points in the Brillouin zone, while the in-plane breathing mode couples
strongly to the electronic states near the $d$-wave nodal points. The
signature of the anistropic electron-phonon (el-ph) coupling seems to get
enhanced in the superconducting state, suggesting the significant role of the
el-ph interaction in the superconducting mechanism.
The information extracted from the conventional ARPES is limited. Time-
resolved ARPES offers the capability to simultaneously capture the single-
particle (frequency domain) and collective (time domain) information, thus
making it possible to directly probe the link between the collective modes and
single-particle states. In this setting, either electrons or lattice
vibrational modes can be selectively excited with an ultrafast laser pulse.
Recent applications of this technique include the studies of transient
electronic structure in Mott insulators LPerfetti06 ; FSchmitt08 and
high-$T_{c}$ cuprates LPerfetti07 with optical pump. Furthermore, direct
pumping of vibrational mode has also been realized in manganites MRini07 ,
though not yet in the cuprates. Motivated by the thrust of this experimental
technique, in this Letter, we aim to provide a theoretical underpinning of
transient electronic structure for HTSC, in a hope to better understand the
nature of bosonic modes in these systems. As such, the time evolution of the
el-ph coupling is investigated for both normal and superconducting states with
the time-resolved spectral function. Our theory consists of two parts. First,
we develop a three-temperature model to simulate the time dependence of the
electron and phonon temperatures. Then, based on the three-temperature model,
we calculate the time-resolved spectral function. Our results show a kink-
structure in the time dependence of the electronic temperature at the
superconducting transition temperature. Accordingly, we find from the spectral
density that the energy position of the phonon mode is offset by a time-
dependent gap function. More interestingly, new signatures of the el-ph
coupling, which are absent when electrons are excited, can be observed in the
case of selective excitation of phonons. This signature may serve as a direct
evidence for the electron-vibration mode coupling as opposed to the coupling
between electrons and spin fluctuations.
Three-temperature model: Consider a two-dimensional superconductor exposed to
a laser field. The model Hamiltonian for a vibrational mode $\nu$ can be
written as
$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{{\bf k}\sigma}\xi_{{\bf
k}}c^{\dagger}_{{\bf k}\sigma}c_{{\bf k}\sigma}+\sum_{{\bf k}}(\Delta_{{\bf
k}}c_{{\bf k}\uparrow}^{\dagger}c_{-{\bf k}\downarrow}^{\dagger}+{\rm
h.c.})+\sum_{{\bf q}}\hbar\Omega_{\nu{\bf q}}$ (1) $\displaystyle\times$
$\displaystyle\bigg{(}b_{\nu{\bf q}}^{\dagger}b_{\nu{\bf
q}}+\frac{1}{2}\bigg{)}+\frac{1}{\sqrt{N_{L}}}\sum_{{\bf k}{\bf
q}\sigma}g_{\nu}({\bf k},{\bf q})c^{\dagger}_{{\bf k}+{\bf q},\sigma}c_{{\bf
k}\sigma}A_{\nu{\bf q}}$ $\displaystyle+$ $\displaystyle H_{\rm field}(\tau),$
where $c^{\dagger}_{{\bf k}\sigma}$ ($b_{\nu{\bf q}}^{\dagger}$) and $c_{{\bf
k}\sigma}$ ($b_{\nu{\bf q}}$) are the creation and annihilation operators for
an electron with momentum $\mathbf{k}$ and spin $\sigma$ (phonon with momentum
$\mathbf{q}$ and vibrational mode $\nu$), $A_{\nu{\bf q}}=b_{-\nu{\bf
q}}^{\dagger}+b_{\nu{\bf q}}$, the quantity $\xi_{\mathbf{k}}$ is the normal-
state energy dispersion, $\mu$ the chemical potential, $\Delta_{\bf k}$ the
gap function, and $g_{\nu}$ the coupling matrix.
By performing the Bogoliubov-de Gennes transformation PGdeGennes66 , $c_{{\bf
k}\uparrow}=u_{\bf k}\alpha_{\bf k}-v_{\bf k}\beta_{\bf k}^{\dagger}$ and
$c_{-{\bf k}\downarrow}=u_{\bf k}\beta_{\bf k}+v_{\bf k}\alpha_{\bf
k}^{\dagger}$, we obtain $E_{e}=\sum_{\bf k}E_{\bf k}(\langle\alpha_{\bf
k}^{\dagger}\alpha_{\bf k}\rangle-\langle\beta_{\bf k}\beta_{\bf
k}^{\dagger}\rangle)$, where $E_{\bf k}=\sqrt{\xi_{\bf k}^{2}+\Delta_{\bf
k}^{2}}$. The time evolution of $\langle\alpha_{\bf k}^{\dagger}\alpha_{\bf
k}\rangle$ and $\langle\beta_{\bf k}\beta_{\bf k}^{\dagger}\rangle$ is
calculated using the Heisenberg equations-of-motion approach. They are found
to be
$\displaystyle\frac{\partial\langle\alpha_{\bf k}^{\dagger}\alpha_{\bf
k}\rangle}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{N_{L}}\sum_{\bf q}g_{\nu}^{2}(u_{\bf k}u_{{\bf
k}-{\bf q}}-v_{\bf k}v_{{\bf k}-{\bf q}})^{2}(\delta_{2}e^{\beta_{\rm
e}\Omega_{0}}-\delta_{1})$ (2) $\displaystyle\times$
$\displaystyle\bigg{[}e^{(\beta_{ph}-\beta_{e})\Omega_{0}}-1\bigg{]}(1-f_{\mathbf{k}})f_{\mathbf{k}-\mathbf{q}}N_{\Omega_{0}}\;,$
and $\partial\langle\beta_{\bf k}\beta_{\bf
k}^{\dagger}\rangle/\partial\tau=-\partial\langle\alpha_{\bf
k}^{\dagger}\alpha_{\bf k}\rangle/\partial\tau$, where
$f_{\mathbf{k}}=f(E_{\bf k})=1/(e^{\beta_{e}E_{\bf k}}+1)$,
$N_{\Omega_{0}}=N(\Omega_{0})=1/(e^{\beta_{ph}\Omega_{0}}-1)$,
$\delta_{1}=\delta(E_{{\bf k}-{\bf q}}-E_{\bf k}+\Omega_{0})$, and
$\delta_{2}=\delta(E_{{\bf k}-{\bf q}}-E_{\bf k}-\Omega_{0})$. Here we have
specifically set $\Omega_{\nu}=\Omega_{0}$. Differentiation of both sides of
the expression for $E_{e}$ and substitution of Eq. (2) leads to the rate of
energy exchange
$\displaystyle\frac{\partial E_{e}}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{N_{L}}\sum_{\bf q}g_{\nu}^{2}(u_{\bf k}u_{{\bf
k}-{\bf q}}-v_{\bf k}v_{{\bf k}-{\bf q}})^{2}\delta(E_{{\bf k}-{\bf q}}-E_{\bf
k}-\Omega_{0})$ (3) $\displaystyle\times$
$\displaystyle\Omega_{0}\bigg{[}e^{(\beta_{ph}-\beta_{e})\Omega_{0}}-1\bigg{]}f_{\mathbf{k}}(1-f_{\mathbf{k}-\mathbf{q}})N_{\Omega_{0}}\;.$
Recent experiment LPerfetti07 shows that there exists cold lattice, which is
negligibly coupled to the electrons but dissipates the energy of hot phonons
via anharmonic cooling. Considering this observation, we write a set of rate
equations for the electron, hot phonon, and lattice temperatures as
$\displaystyle\frac{\partial T_{e}}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{C_{e}}\frac{\partial
E_{e}}{\partial\tau}+\frac{P_{e}}{C_{e}},$ (4) $\displaystyle\frac{\partial
T_{ph}}{\partial\tau}$ $\displaystyle=$
$\displaystyle-\frac{1}{C_{ph}}\frac{\partial
E_{e}}{\partial\tau}+\frac{P_{ph}}{C_{ph}}-\frac{T_{ph}-T_{l}}{\tau_{\beta}},$
(5) $\displaystyle\frac{\partial T_{l}}{\partial\tau}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{C_{ph}}{C_{l}}\bigg{)}\frac{T_{ph}-T_{l}}{\tau_{\beta}},$
(6)
where $P_{e}$ is the power for pumping electrons and $P_{ph}$ the power for
pumping hot phonons. The specific heat of electrons can be calculated from the
Boltzmann entropy $S_{e}=-2k_{B}\sum_{\bf k}\\{[1-f(E_{\bf k})]{\rm
ln}[1-f(E_{\bf k})]+f(E_{\bf k}){\rm ln}f(E_{\bf k})\\}$ by
$C_{e}=T_{e}\partial S_{e}/\partial T_{e}$, while the specific heat of hot
phonons for one-vibrational mode can be calculated from the simple
relationship $C_{ph}=\hbar\Omega\partial N(\Omega)/\partial
T_{ph}|_{\Omega=\Omega_{0}}$. The results are given by
$\displaystyle C_{e}$ $\displaystyle=$ $\displaystyle\beta k_{B}\sum_{\bf
k}\bigg{(}-\frac{\partial f(E_{\bf k})}{\partial E_{\bf
k}}\bigg{)}\bigg{(}2E_{\bf
k}^{2}+\beta\Delta_{k}\frac{\partial\Delta_{k}}{\partial\beta}\bigg{)},~{}~{}~{}~{}$
(7) $\displaystyle C_{ph}$ $\displaystyle=$
$\displaystyle\frac{k_{B}}{4}\bigg{(}\frac{\hbar\Omega_{0}}{k_{B}T_{ph}}\bigg{)}^{2}\bigg{[}{\rm
coth}^{2}\bigg{(}\frac{\hbar\Omega_{0}}{2k_{B}T_{ph}}\bigg{)}-1\bigg{]},$ (8)
respectively, with $\coth x\equiv(e^{x}+e^{-x})/(e^{x}-e^{-x})$. Equations
(3)-(8) constitute our three-temperature model. The original version of this
model was phenomenologically proposed LPerfetti07 as an extension of the two-
temperature model PBAllen87 for the normal state. The present model has the
following advantages: (i) it incorporates the detailed band structure, (ii) it
is valid for both normal and superconducting states, and (iii) it includes the
anistropic effect on the el-ph coupling.
Figure 1: (Color) Time evolution of electron ($T_{e}$), phonon ($T_{ph}$), and
lattice ($T_{l}$) temperatures for selectively exciting electrons (a) and
phonons (b) with a laser pulse.
Now we apply our three-temperature model to simulate the time evolution of the
electron, hot phonon, and lattice temperatures for a $d$-wave superconductor.
We use a five-parameter tight-binding model MRNorman95 to describe the energy
dispersion: $\xi_{\mathbf{k}}=-2t_{1}(\cos k_{x}+\cos k_{y})-4t_{2}\cos
k_{x}\cos k_{y}-2t_{3}(\cos 2k_{x}+\cos 2k_{y})-4t_{4}(\cos 2k_{x}\cos
k_{y}+\cos k_{x}\cos 2k_{y})-4t_{5}\cos 2k_{x}\cos 2k_{y}-\mu$, where
$t_{1}=1$, $t_{2}=-0.2749$, $t_{3}=0.0872$, $t_{4}=0.0938$, $t_{5}=-0.0857$,
and $\mu=-0.8772$. Unless specified explicitly, the energy is measured in
units of $t_{1}$ and the time is measured in units of $\hbar/t_{1}$ hereafter.
(For $t_{1}=150$ meV, it corresponds to 1740 Kelvin in temperature and
$\hbar/t_{1}$ to $4.4$ femtosecond in time.) The $d$-wave gap function has the
form $\Delta_{k}=\Delta_{0}(T_{e})({\rm cos}k_{x}-{\rm cos}k_{y})/2$. The
temperature-dependent part is given by FGross86
$\Delta_{0}(T_{e})=\Delta_{00}{\rm
tanh}\\{(\pi/z)\sqrt{ar(T_{c}/T_{e}-1)}\\}$, where
$z=\Delta_{00}/(k_{B}T_{c})$ and $\tanh x=1/\coth x$. In our calculations, we
set $\Delta_{00}=0.2$, the critical temperature $T_{c}=0.06$, the specific
heat jump at $T_{c}$ is $r=\Delta C_{e}/C_{e}\sim 1.43$, and $a=2/3$. In this
work, we focus on the buckling phonon mode, for which TPDevereaux04 ; JXZhu06a
; JXZhu06b $g_{B_{1g}}=g_{0}\\{[{\rm cos}^{2}(q_{x}/2)+{\rm
cos}^{2}(q_{y}/2)]/2\\}^{-1/2}\\{\phi_{x}({\bf k})\phi_{x}({\bf k}+{\bf
q}){\rm cos}(q_{y}/2)-\phi_{y}({\bf k})\phi_{y}({\bf k}+{\bf q}){\rm
cos}(q_{x}/2)\\}$ with $\phi_{x}=(i/N_{\bf k})[\xi_{{\bf k}}t_{x,{\bf
k}}-t_{xy,{\bf k}}t_{y,{\bf k}}]$, $\phi_{y}=(i/N_{k})[\xi_{{\bf k}}t_{y,{\bf
k}}-t_{xy,{\bf k}}t_{x,{\bf k}}]$, $N_{\bf k}=[(\xi_{{\bf k}}^{2}-t_{xy,{\bf
k}}^{2})^{2}+(\xi_{{\bf k}}t_{x,{\bf k}}-t_{xy,{\bf k}}t_{y,{\bf
k}})^{2}+(\xi_{{\bf k}}t_{y,{\bf k}}-t_{xy,{\bf k}}t_{x,{\bf k}})^{2}]^{1/2}$,
$t_{\alpha,{\bf k}}=-2t_{1}{\rm sin}(k_{\alpha}/2)$, and $t_{xy,{\bf
k}}=-4t_{2}{\rm sin}(k_{x}/2){\rm sin}(k_{y}/2)$. To be consistent with
experiment LPerfetti07 ; TPDevereaux04 , we set $g_{0}=0.4$ and the mode
frequency $\Omega_{0}=0.3$. The relaxation time $\tau_{\beta}=200$. The pump
power is a Gaussian pulse of $P=P_{0}e^{-\tau^{2}/(2\sigma^{2})}$, for which,
the corresponding FWHM (full width at half maximum) is 2.35 $\sigma$. To pump
electrons, we set $P_{0}=0.15$ and $\sigma=1$. Considering that the energy
scale of phonons is much smaller than that of electrons, we set $P_{0}=0.006$
and $\sigma=100$ for pumping phonons. The ratio $C_{ph}/C_{l}$ in Eq. (6) is
set to be 0.2.
Figure 1 displays the temporal evolution of three respective temperatures for
pumping electrons (a) and hot phonons (b). Before pumping, all three types of
degrees of freedom (DoF) are in equilibrium, which is set at
$T_{e}=T_{ph}=T_{l}=0.01$. As shown in Fig. 1(a), when the pump pulse is
absorbed by the electrons, the electronic temperature rises steeply around
$\tau=0$, and reaches the maximum after a small time delay. It then begins to
drop at a time scale determined by the el-ph coupling strength, followed by a
slower relaxation. Interestngly, we also observe a small kink in $T_{e}$ at
$T_{e}=T_{c}$, which entirely arises from the breakup of the Cooper pairs,
resulting in the dramatic change in the rise rate of $T_{e}$. This kink was
not captured in the early model LPerfetti07 . Simultaneously, the hot phononic
temperature $T_{ph}$ rises smoothly via the energy exchange with electrons and
then drops by exchanging energy with the cold lattice, causing the slight
increase of $T_{l}$. When directly pumping phonons, the time dependence of the
temperatures is similarly observed, including the kink in $T_{e}$, as shown in
Fig. 1(b). Due to the large pumping width, the kink is more easily visible in
this case.
Time-resolved spectral function: The time-resolved spectral function is
defined as $A({\bf k},\omega)\equiv-\frac{2}{\pi}~{}{\rm Im}{\bf{\cal
G}}_{11}({\bf k},\omega)$. Here ${\bf{\cal G}}_{11}$ is the one-one component
of the retarded Green’s function ${\hat{\bf{\cal G}}}({\bf k},\omega)$, which
is related to the self-energy by ${\hat{\bf{\cal G}}}^{-1}({\bf
k},\omega)={\hat{\bf{\cal G}}}_{0}^{-1}({\bf k},\omega)-{\hat{\Sigma}}({\bf
k},\omega)$, with ${\hat{\bf{\cal G}}}_{0}^{-1}({\bf
k},\omega)=\omega{\hat{\sigma}}_{0}-\Delta_{\bf k}{\hat{\sigma}}_{1}-\xi_{\bf
k}{\hat{\sigma}}_{3}$. ${\hat{\sigma}}_{0,1,2,3}$ are the unit and Pauli
matrices. As is known TPDevereaux04 , in the equilibrium state with
$T_{e}=T_{ph}$, the self-energy can be evaluated more conveniently within the
imaginary-time Green’s function approach, in which a key step is to convert
the Bose-Einstein distribution to the Fermi distribution,
$n_{B}(i\omega_{n}\pm E_{{\bf k}-{\bf q}})=-n_{F}(\pm E_{{\bf k}-{\bf q}})$
(with $\omega_{n}=(2n+1)\pi T$, $T=T_{e}=T_{ph}$). For the current situation,
the temperatures of electrons and hot phonons are no longer tied to each other
and the above conversions are not valid any more. To avoid this restriction,
here we apply the double-time Green’s function approach DNZubarev60 to
calculate ${\hat{\Sigma}}({\bf k},\omega)$. The result is
$\displaystyle{\hat{\Sigma}}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{L}}\sum_{{\bf
q}}|g_{\nu}(\mathbf{k},-\mathbf{q})|^{2}$ (9)
$\displaystyle\\{[(\omega-\Omega_{0})\Phi_{1}-(\omega+\Omega_{0})\Phi_{2}+\Omega_{0}(\Phi_{3}+\Phi_{4})]{\hat{\sigma}}_{0}$
$\displaystyle+[2E_{{\bf k}-{\bf
q}}(\Phi_{1}-\Phi_{3})+2\Omega_{0}(\Phi_{3}-\Phi_{4})]{\hat{\sigma}}_{1}$
$\displaystyle+[\xi_{\bf k}(\Phi_{1}-\Phi_{2})+(\Omega_{0}\xi_{\bf k}/E_{\bf
k})(\Phi_{3}-\Phi_{4})]{\hat{\sigma}}_{3}\\},$
where $\Phi_{1}=N(\Omega_{0})/[(\omega-\Omega_{0}+E_{{\bf k}-{\bf
q}})(\omega-\Omega_{0}-E_{{\bf k}-{\bf q}})]$,
$\Phi_{2}=N(-\Omega_{0})/[(\omega+\Omega_{0}+E_{{\bf k}-{\bf
q}})(\omega+\Omega_{0}-E_{{\bf k}-{\bf q}})]$, $\Phi_{3}=f(-E_{{\bf k}-{\bf
q}})/[(\omega+\Omega_{0}-E_{{\bf k}-{\bf q}})(\omega-\Omega_{0}-E_{{\bf
k}-{\bf q}})]$, and $\Phi_{4}=f(E_{{\bf k}-{\bf
q}})/[(\omega+\Omega_{0}+E_{{\bf k}-{\bf q}})(\omega-\Omega_{0}+E_{{\bf
k}-{\bf q}})]$.
Figure 2: (Color) Time evolution of the density of states for selectively
exciting electrons (a) and phonons (b), respectively.
For simplicity, let us first look at the density of states (DOS), which is
defined by $\rho(\omega)=\frac{1}{N_{L}}\sum_{\bf k}A({\bf k},\omega)$. For
direct pumping of electrons, we calculate the DOS at time sequences
$\tau=-100$, $-2.7$, $2.8$, and $600$, respectively. The results are plotted
in Fig. 2 (a). At the initial time $\tau=-50$ where $T_{e}=0.01$ and the
system is at superconducting state [$\Delta_{0}(T_{e})=0.19$], we observe two
signatures of the el-ph coupling located at
$\pm(\Omega_{0}+\Delta_{0}(T_{e}))$, as shown in Fig. 2 (dip 1 and dip 2).
They are symmetric with respect to $\omega=0$, but dip 2 is much stronger. It
is related to the fact that the normal-state van Hove singularity is located
below the Fermi energy and as such the coherent peak at $-\Delta_{0}(T_{e})$
has stronger intensity than that at $\Delta_{0}(T_{e})$. In addition, dip 3
occurs near the characteristic energy $-(E_{M}+\Omega_{0})$, where $E_{M}$ is
the quasiparticle energy at $\mathbf{k}=(\pi,0)$ or equivalent wavevector
points in Brillouin zone. It arises from the van Hove singularity at
$\mathbf{k}=(\pi,0)$. The signature at the energy $E_{M}+\Omega_{0}$ is much
weaker also because of the van Hove singularity location in the normal state.
At $\tau=-2.7$ where $T_{e}=0.041$ and $\Delta_{0}(T_{e})\approx 0.1$, similar
behaviors are observed. However, at $\tau=2.8$, $T_{e}=1.05$ and thus only the
signature of the normal-state el-ph coupling (dip 3) can be observed. At
$\tau=600$, a similar structure of the normal-state el-ph coupling is
observed.
Figure 3: (Color) Time-resolved spectral function for selectively pumping
electrons (a1-a4 for $\tau=-100,-2.7,2.8,600$) and phonons (b1-b4 for
$\tau=-1000,-158,110,800$), respectively. The vertical axis is $k_{x}/\pi$ at
a fixed $k_{y}=0.75\pi$ and the horizontal axis is energy. The black, red, and
magenta lines represent the energy location, $-(E_{M}+\Omega_{0})$,
$-(\Delta_{0}(T_{e})+\Omega_{0})$, and $-(\Omega_{0}-\Delta_{0}(T_{e}))$,
corresponding to the dip or hump location discussed in Fig. 2.
For direct pumping of phonons, we choose different time sequences
$\tau=-1000$, $-158$, $110$, and $800$ to evaluate the DOS. The results are
displayed in Fig. 2(b). From Fig. 2(b), we observe that the dips suggesting
the el-ph couplig in both normal and supercondcting states becomes stronger
than those for selectively pumping electrons, as expected. In particular, we
observe two new small dips at $\omega=\pm 0.05$ for $\tau=-158$ (at which
$T_{e}=0.041$ and $T_{ph}=0.163$), both of which arise from the poles of
$\Sigma_{\bf k}(\omega)$ at $\omega=\pm(\Omega_{0}-\Delta_{0}(T_{e}))$, in
addition to those observed in the case of electrons being excited directly.
This is because the hot phononic temperature becomes very high while the
electronic temperature is still cold and the contributions from the terms as
weighted by $N(\pm\Omega_{0})$ are significantly enhanced. Figure. 2(b)
clearly shows that, as time elapses from $\tau=-1000$, $-158$, to $110$, the
peaks move toward the zero energy, while from $\tau=110$ to $800$, the
locations of these two peaks remain unchanged. This can be understood by
considering that, after pumping, $T_{e}$ rises with the time delay, while
$\Delta_{0}(T_{e})$ decreases and rapidly vanishes when $T_{e}$ reaches
$T_{c}$.
Finally we numerically evaluate the time-resolved spectral function for the
two excitations discussed above at those time sequences chosen for calculating
the DOS. In our calculations, we consider cuts along $k_{x}$-axis in the
Brillouin zone at a $k_{y}$ chosen near the zone boundary. Fig. 3 shows the
snapshots of the image of the spectral function $A(\mathbf{k},\omega)$ as a
function of $k_{x}$ and $\omega$. From Fig. 3(a1,a2), we observe kinks at
$\omega=-(\Delta_{0}(T_{e})+\Omega_{0})$ (red lines) and $-(E_{M}+\Omega_{0})$
(black lines) at $\tau=-100$ and $-2.7$. The red line is shifted in energy as
time elapses from $\tau=-100$ to $\tau=-2.7$ because the BCS gap
$\Delta_{0}(T_{e})$ drops significantly from 0.19 to 0.1. These kinks
correspond to the dip structures discussed in Fig. 2. As $\tau>-2.3$, the
superconducting gap vanishes, so do the kinks marked with red lines (see Fig.
3(a3-a4)). The kinks marked with black lines persist through the whole time
sequence, which can be ascribed to the el-ph coupling signature in the normal
state. Therefore we have every reason to regard these kinks marked with red
lines as the signature of the el-ph coupling in the superconducting state. For
hot phonon pumping (Fig. 3(b1-b4)), in addition to those signatures of the el-
ph coupling observed for electron pumping, new kink structure at
$-(\Omega_{0}-\Delta_{0}(T_{e}))$(marked with magenta line in Fig. 3(b2) is
also observed. These observations are consistent with our finding in the DOS.
In the viewpoint that the electronic spin fluctuations arise from the strong
correlation within the electronic DoF itself, and if one can assume that the
spin fluctuations will ride on the electrons and thus its effective
temperature will be tied to the electronic temperature, direct pumping of hot
phonons will provide a unique way to differentiating the bosonic modes being
of electronic or phononic origin, to which electronic quasiparticles are
strongly coupled in HTSC.
In conclusion, we have developed a three-temperature model to simulate the
time evolution of temperature for electrons, hot phonons, and the lattice. Our
model takes both quasiparticle excitation and relaxation into account and thus
is valid in both normal and superconducting states, because at the early stage
of excitation, the system can be still in the supercondcting state. Based on
this model, we derive a formula for the time-resolved spectral function,
allowing us to investigate the dynamics of the el-ph coupling for HTSC
materials.
Acknowledgments. We thank A. V. Balatsky, Elbert E. M. Chia, Hari Dahal, J. K.
Freericks, M. Graf, A. Piryatinski, A. J. Taylor, S. A. Trugman, and D.
Yarotski for valuable discussions. This work was supported by the National
Nuclear Security Administration of the U.S. DOE at LANL under Contract No. DE-
AC52-06NA25396, the U.S. DOE Office of Science, and the LDRD Program at LANL.
## References
* (1) A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).
* (2) J. Lee et al., Nature 442, 546 (2006).
* (3) F. Carbone et al., Proc. Natl. Acad. Sci. 105, 20161 (2008).
* (4) M. R. Norman et al., Phys. Rev. B 64, 184508 (2001).
* (5) A. W. Sandvik, D. J. Scalapino, and N. E. Bickers, Phys. Rev. B 69, 094523 (2004).
* (6) H. Ding et al., Phys. Rev. Lett. 76, 1533 (1996).
* (7) J. C. Campuzano et al., Phys. Rev. Lett. 83, 3709 (1999).
* (8) A. Kaminski et al., Phys. Rev. Lett. 86, 1070 (2001).
* (9) A. Lanzara et al., Nature (London) 412, 510 (2001).
* (10) P. D. Johnson et al., Phys. Rev. Lett. 87, 177007 (2001).
* (11) X. J. Zhou et al., Nature (London) 423, 398 (2003).
* (12) T. Sato et al., Phys. Rev. Lett. 91, 157003 (2003).
* (13) T. Cuk et al., Phys. Rev. Lett. 93, 117003 (2004).
* (14) Z. X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997).
* (15) Z.-X. Shen et al., Philos. Mag. B 82, 1349 (2002).
* (16) T.P. Devereaux et al., Phys. Rev. Lett. 93, 117004 (2004).
* (17) L. Perfetti et al., Phys. Rev. Lett. 97, 067402 (2006).
* (18) F. Schmitt et al., Science 321, 1649 (2008).
* (19) L. Perfetti et al., Phys. Rev. Lett. 99, 197001 (2007).
* (20) M. Rini et al., Nature 449, 72 (2007).
* (21) P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).
* (22) P. B. Allen, Phys. Rev. Lett. 59, 1460 (1987).
* (23) M. R. Norman et al., Phys. Rev. B 52, 615 (1995).
* (24) F. Gross et al., Z. Phys. B 64, 175 (1986).
* (25) J.-X. Zhu et al., Phys. Rev. Lett. 97, 177001 (2006).
* (26) J.-X. Zhu et al., Phys. Rev. B 73, 014511 (2006).
* (27) D. N. Zubarev, Sov. Phys. Usp. 3, 320 (1960).
|
arxiv-papers
| 2010-01-04T21:02:07 |
2024-09-04T02:49:07.481015
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianmin Tao and Jian-Xin Zhu",
"submitter": "Jianmin Tao",
"url": "https://arxiv.org/abs/1001.0573"
}
|
1001.0743
|
# Microscopic evaluation of the pairing gap
M Baldo1, U Lombardo2, S S Pankratov3, and E E Saperstein3 1 INFN, Sezione di
Catania, 64 Via S.-Sofia, I-95125 Catania, Italy baldo@ct.infn.it 2 INFN-LNS
and University of Catania, 44 Via S.-Sofia, I-95125 Catania, Italy
lombardo@lns.infn.it 3 Kurchatov Institute, 123182, Moscow, Russia
pankratov@pretty.mbslab.kiae.ru saper@mbslab.kiae.ru
###### Abstract
We discuss the relevant progress that has been made in the last few years on
the microscopic theory of the pairing correlation in nuclei and the open
problems that still must be solved in order to reach a satisfactory
description and understanding of the nuclear pairing. The similarities and
differences with the nuclear matter case are emphasized and described by few
illustrative examples. The comparison of calculations of different groups on
the same set of nuclei show, besides agreements, also discrepancies that
remain to be clarified. The role of the many-body correlations, like
screening, that go beyond the BCS scheme, is still uncertain and requires
further investigation.
###### pacs:
21.60.-n, 21.65.+f, 26.60.+c, 97.60.Jd
## 1 Introduction
Despite the pairing in nuclei was established more than fifty years ago, the
underlying interaction processes that are responsible of this remarkable and
important correlation are not yet understood completely. The main reason of
the difficulty is that the microscopic theory of nuclear pairing must rely on
the effective interaction among nucleons, that is not known from first
principles. Furthermore it is strong and therefore it requires the solution of
a complex many-body problem. Even in nuclear matter, where the problem is
hoped to be simplified to some extent, an accurate microscopic theory is not
available. Numerical Monte-Carlo ”exact” calculations are present in the
literature only for low-density pure neutron matter [1]. Unfortunately these
Monte-Carlo estimates are not in satisfactory agreement among each others [1]
and in any case they can hardly elucidate the microscopic mechanisms which are
at the basis of the onset of pairing.
Besides the many-body aspects of the problem, at least other two features of
the nuclear pairing have to be mentioned. One is related to the fact that the
pairing phenomenon occurs close to the Fermi surface, while the bare nucleon-
nucleon (NN) potential necessarily involves also scattering to high energy (or
momentum) due to its strong hard core component, which is one of the main
characteristics of the nuclear interaction. It looks therefore natural to
develop a procedure which removes the high energy states and ”renormalize” the
interaction into a region close to the Fermi energy. This can be done in
different ways, among which the most commonly used seems to be the
Renormalization Group (RG) Method [2]. A second feature is the relevance of
the single particle spectrum, not only because the density of states at the
Fermi surface plays of course a major role, but also because the whole single
particle spectrum has influence on the effective pairing interaction.
Despite these uncertainties of the microscopic theory of pairing in nuclear
matter, there is a commonly accepted point of view that the gap value $\Delta$
is quite small at the normal density $\rho_{0}$, much smaller than typical
values of heavy atomic nuclei. This is an indication that the nuclear surface
must play a major role in establishing the value of the pairing gap in finite
nuclei.
In the last few years relevant progresse have been made in the microscopic
calculations of pairing gap in nuclei [3, 4, 5, 6, 7, 8, 9, 10]. The main
established results is that the bare NN interaction, renormalized by
projecting out the high momenta, is a reasonable starting point that is able
to produce a pairing gap which shows a discrepancy with respect to the
experimental value not larger than a factor 2\. In view of the great
sensitivity of the gap to the effective interaction this result does not
appear obvious. The effective pairing interaction constructed within this
renormalization scheme [11] explains qualitatively also the surface relevance.
Indeed, this interaction at the surface can exceed the value inside by one
order of magnitude. Direct effect of the surface enhancement of the gap was
presented in [12, 13] for semi-infinite nuclear matter and in [14] for a
nuclear slab.
The role of the surface is also apparent in explaining the puzzling value of
the coherence length extracted from the pairing gap. In fact, for a pairing
gap of the order of 1 MeV, the coherence length is expected to be much larger
than the size of the nucleus. It can be argued that at the surface the pairing
gap is larger and the coherence length shorter, i.e. the size of the Cooper
pairs should shrink. This was explicitly shown in a recent paper by N. Pillet,
N. Sandulescu, and P. Schuck [15] on the basis of the HFB approach and the
effective D1S Gogny force. It was shown that Cooper pairs in nuclei
preferentially are located in the surface region, with a small size
($2-3\;$fm). In [16], it was examined to what extent the effect found in [15]
is general and independent of the specific choice of NN force. This
investigation was carried out for a slab of nuclear matter, and the pairing
characteristics obtained with the Gogny force were compared with those with
the realistic Paris and Argonne v18 forces. The results obtained with the two
realistic forces agree with each other within an accuracy of about 10% and
agree qualitatively with those of the Gogny force. In [17] an analogous
analysis, with the Paris potential, was made for the nucleus 120Sn which is a
standard polygon for examining nuclear pairing. Again the results turned out
to be very close to those of [15]. One of them is shown in figure 1, where the
“probability” distribution of Cooper pairs,
$p(R)=4\pi R^{2}\int\varkappa^{2}(R,{\bf r})\;d^{3}r,$ (1)
is displayed. Here ${\bf R},{\bf r}$ are c.m. and relative coordinates. In
Fig. 1 this quantity is normalized to the total number of Cooper pairs,
$N_{\rm Cp}=\int p(R)dR\simeq 10$. One can see that Cooper pairs, indeed, are
strongly concentrated on the surface.
Figure 1: Cooper pair distribution for the 120Sn nucleus. The self-consistent
mean field is found with the Energy Density Functional DF3 of [18] .
Besides the renormalization of the bare interaction of the high momentum
components, other physical effects should be included in a microscopic
approach, like the ones related to the effective mass, or more generally, the
single particle spectrum and the many-body renormalization of the pairing
interaction. The role of these and other features of the microscopic scheme
are still to be clarified.
In this paper we will try to summarize the recent achievements and the main
open problems that hinder the development of an accurate microscopic many-body
theory of the nuclear pairing.
## 2 Renormalization of the high energy components
The pairing gap in nuclei is of the order of 1-2 MeV, while the typical energy
scale of the bare NN interaction is of few hundreds MeV. This indicates the
difficulty of controlling the microscopic construction of the effective
pairing interaction $V_{\mbox{\scriptsize eff}}$ at the Fermi surface. In the
BCS approach a simplified estimate is commonly used, the so called weak
coupling limit. In this approximation the relationship between the gap
$\Delta_{\rm F}$ at the Fermi energy and the effective interaction has the
form
$\Delta_{\rm F}\,\approx\,2\varepsilon_{\rm F}\exp(1/\nu_{\rm
F}V_{\mbox{\scriptsize eff}})\;,$ (2)
where $\varepsilon_{\rm F}$ is the Fermi energy and $\nu_{\rm F}=m^{*}k_{\rm
F}/\pi^{2}$ is the density of state at the Fermi energy. The exponential
dependence makes the problem particularly delicate, since a small change in
the effective interaction can result in substantial change of the pairing gap.
In phenomenological theories, like the Finite Fermi System (FFS) theory or the
Hartree-Fock-Bogolyubov method with effective forces, the value of
$V_{\mbox{\scriptsize eff}}$ is considered as a phenomenological parameter (or
a set of parameters) to be used to fit the data. Despite the undoubt success
of the phenomenological approaches, the challenge of ab initio evaluation of
the pairing gap remains one of the most fundamental and not completely solved
problems in nuclear physics.
Let us first consider the problem of reducing the interaction to an effective
one close to the Fermi energy. In general language this can be seen as a
typical case that can be approached by an ”Effective Theory”, where the low
energy phenomena are decoupled from the high energy components. In this
procedure the resulting low energy effective interaction is expected to be
independent of the particular form of the high energy components. However the
procedure is not unique. The RG method has been particularly developed for the
reduction of the general NN interaction keeping the deuteron properties and NN
phase shifts up to the energy where they are well established. In this way a
potential, phase equivalent to a known realistic NN potential, can be
obtained, which contains only momenta up to a certain cut-off. The extension
of this procedure to the many-body problem appears in general to require the
introduction of strong three-body forces. It has been applied also to the
pairing problem. To illustrate the difficulty of decoupling low and high
momentum components for the pairing problem, we consider the simple case of
nuclear matter in the BCS approximation. The BCS equation for the gap
$\Delta(k)$ can be written as
$\Delta(k)=-\sum_{k^{\prime}}{V(k,k^{\prime})\over
2\sqrt{\varepsilon(k^{\prime})^{2}+\Delta(k^{\prime})^{2}}}\Delta(k^{\prime})\>,$
(3)
where $V$ is the free NN potential, $\varepsilon(k)=e(k)-\mu$, $e(k)$ is the
single particle spectrum and $\mu$, the chemical potential. It is possible to
project out the momenta larger than a cutoff $k_{c}$ by introducing the
interaction $V_{\mbox{\scriptsize eff}}(k,k^{\prime})$, which is restricted to
momenta $k<k_{c}$. It satisfies the integral equation
$V_{\mbox{\scriptsize
eff}}(k,k^{\prime})=V(k,k^{\prime})-\\!\\!\sum_{k^{\prime\prime}>k_{c}}\\!\\!\frac{V(k,k^{\prime\prime})V_{\mbox{\scriptsize
eff}}(k^{\prime\prime},k^{\prime})}{2E_{k^{\prime\prime}}}\>,$ (4)
where $E(k)=\sqrt{\varepsilon(k)^{2}+\Delta(k)^{2}}$. The gap equation,
restricted to momenta $k<k_{c}$ and with the original interaction $V$ replaced
by $V_{\mbox{\scriptsize eff}}$, is exactly equivalent to the original gap
equation. The relevance of this equation is that for a not too small cut-off
the gap $\Delta(k)$ can be neglected in $E(k)$ to a very good approximation
and the effective interaction $V_{\mbox{\scriptsize eff}}$ depends only on the
normal single particle spectrum above the cutoff $k_{c}$. In the RG approach
the low momenta interaction $V_{low-k}(k,k^{\prime})$ is constructed in such a
way to keep the half-on-shell two-body T-matrix in free space and therefore
the phase shifts. To the extent that the pairing gap is determined only by the
phase shifts, the two approaches are therefore equivalent.
## 3 The single particle spectrum and the effective mass
The pairing gap value is strongly affected by the density of state at the
Fermi level, as the weak coupling limit (2) indicates. In turn, the density of
state is proportional to the effective mass. As it is well known, one has to
distinguish between the so-called k-mass, which is generated by the momentum
dependence of the single particle potential, and the e-mass, which is
generated by the energy dependence of the single particle self-energy. The
total effective mass is just the product of these two effective masses. The
inclusion of an energy dependent single particle self-energy produces also the
so-called Z-factor, i.e. the quasi-particle strength.
In nuclear matter the four-dimensional gap equation incorporating the momentum
and energy dependent irreducible particle-particle vertex ${\cal
V}(k,\varepsilon;k^{\prime},\varepsilon^{\prime})$ and the self-energy
$\Sigma(k,\varepsilon)$, reads [19, 20, 21]
$\Delta(k,\varepsilon)=-\int\\!\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\int\\!\frac{d\varepsilon^{\prime}}{2\pi
i}\,{\cal
V}(k,\varepsilon;k^{\prime},\varepsilon^{\prime}){\Delta(k^{\prime},\varepsilon^{\prime})\over
D(k^{\prime},\varepsilon^{\prime})}\;,$ (5)
with
$\hskip-28.45274ptD(k,\varepsilon)=({\cal G}{\cal
G}^{s})^{-1}=[M(k,+\varepsilon)-\varepsilon-i0][M(k,-\varepsilon)+\varepsilon-i0]+\Delta^{2}(k,\varepsilon)$
(6)
and
$M(k,\varepsilon)={k^{2}\over 2m}+\Sigma(k,\mu+\varepsilon)-\mu\>,$ (7)
where we define for convenience the energy $\varepsilon$ relative to the
chemical potential $\mu$. ${\cal G}^{s}$ and ${\cal G}$ in (6) are,
correspondingly, the single particle Green functions with and without pairing.
For realistic systems, ${\cal V}$ and $\Sigma$ cannot be determined in exact
way and significant approximations have to be performed. The usual BCS
approximation amounts to replacing the interaction vertex by the (energy
independent) bare nucleon-nucleon potential $V$, and the nucleon self-energy
by some realistic s.p. spectrum. The latter is characterized by the single
particle potential, that can have, in nuclear matter, a momentum dependence
which persists even at high momenta. To illustrate this point we have reported
in Fig. 2 the effective mass obtained within the BHF approximation as a
function of momentum at different Fermi momenta. The effective mass has
usually a peak close to the Fermi momentum, but it is substantially different
from the bare one even at high momenta.
As shown in detail in Ref. [21], energy dependence of the self-energy can be
taken into account explicitly provided the interaction vertex remains static,
${\cal
V}(k,\varepsilon;k^{\prime},\varepsilon^{\prime})=\tilde{V}(k,k^{\prime})$. In
this case, the general gap equation (6) is reduced to the form
$\Delta(k)=-\sum_{k^{\prime}}{\tilde{V}(k,k^{\prime})Z(k^{\prime})\over
2\sqrt{M_{s}(k^{\prime})^{2}+\Delta(k^{\prime})^{2}}}\Delta(k^{\prime})\>,$
(8)
which reminds the BCS gap equation (3), with the “symmetrized” s.p. energy
$M_{s}(k)\equiv\mathrm{Re}\left({M(k,+e_{k})+M(k,-e_{k})\over 2}\right)$ (9)
appearing in the denominator and with the kernel modified by the spectral
factor
$Z(k)\equiv\sqrt{M_{s}(k)^{2}+\Delta(k)^{2}}\;{2\over\pi}\\!\int_{0}^{\infty}\\!\\!\\!d\varepsilon\,\mathrm{Im}{\left(1\over
D(k,\varepsilon)\right)}\>.$ (10)
Figure 2: Momentum dependence of the effective mass $m^{*}(k)$ at different
Fermi momentum values $k_{\rm F}$.
Figure 3: Effective mass $m^{*}_{\rm F}$ at the Fermi surface depending on
the Fermi momentum $k_{\rm F}$. Table 1: Comparison of the neutron effective
mass in the symmetric and asymmetric nuclear matter from the BHF calculation
with that of different kinds of the Skyrme force.
*
* | SKP [22] | SKM* [23] | SLy4 [24] | BHF
---|---|---|---|---
$(m^{*}/m)_{\rm snm}$ | 0.85 | 0.71 | 0.74 | 0.78
$\delta(m^{*}/m)_{\rm anm}$ | 0.22 | 0.12 | -0.04 | $-(0.03\div 0.05)$
In table 1 the neutron effective masses at the Fermi surface $m^{*}(k_{\rm
F})$ of symmetric and asymmetric, $(N-Z)/A=1/6$, nuclear matter found within
the BHF approach at the equilibrium density $\rho_{0}$ is compared with those
of Skyrme forces. In the second line, the difference $\delta(m^{*}/m)_{\rm
anm}=(m^{*}/m)_{\rm anm}-(m^{*}/m)_{\rm snm}$ is given. As one can see, the
SLy4 effective mass is rather close to the BHF one at the Fermi surface. Other
two kinds of Skyrme force give absolutely different isotopic asymmetry
dependence of $m^{*}$. The density dependence of the effective mass in
symmetric nuclear matter for BHF and the considered Skyrme forces is reported
in figure 3. Around saturation the trends are smooth, but at lower density the
BHF effective mass has a behavior that interpolates between different Skyrme
forces, and therefore it cannot be reproduced by a given Skyrme force.
To illustrate the relevance of the high momenta components $k>k_{c}$ on
determining the effective pairing interaction $V_{\rm eff}$ around the Fermi
momentum, we present in figure 4 the value of $V_{\rm eff}$ at the Fermi
momentum obtained by solving Eq. (4). We take $k_{c}=\sqrt{2}k_{F}$ and
consider three cases : i) the free spectrum, ii) the spectrum obtained from
the BHF calculation, and iii) the spectrum with a constant effective mass,
taken at the Fermi momentum and from the BHF results. The calculations are
performed in the density range relevant for the bulk and surface regions of
finite nuclei.
Figure 4: Effective pairing interaction for the free single particle spectrum
(dotted line), the BHF spectrum (full line) and the constant effective mass
spectrum (dashed line).
The substantial difference between cases i) and iii) shows the relevance of
the high momentum components of the single particle spectrum. On the other
hand, the strong overlap of the curves corresponding to the cases ii) and iii)
suggests that the details of the momentum dependence of the effective mass are
less relevant. This will be helpful for the analysis in finite nuclei, where
the approximation of a constant effective mass will be made. The corresponding
pairing gaps at the Fermi momenta are reported in figure 5.
Figure 5: Pairing gap at the Fermi momentum for the three cases of free
single particle spectrum considered in Fig. (4).
## 4 The many-body problem
In atomic nuclei, a new branch of low-laying collective excitations appear,
the surface vibrations (”phonons”). They could be interpreted as a Goldstone
mode corresponding to spontaneous breaking of translation invariance in nuclei
[25]. The ghost dipole $1_{1}^{-}$-phonon is the head of this branch. In the
self-consistent FFS theory or other self-consistent approaches, say, the
SHF+RPA (or SHFB+qRPA) method, the constraint $\omega_{1_{1}^{-}}=0$ is
fulfilled identically and the next members of the branch of natural parity
states, $2_{1}^{+},3_{1}^{-}$ and so on, have small excitation energies, less
of a typical distance between neighboring shells,
$\omega_{0}\simeq\varepsilon_{\rm F}/A^{1/3}$. Up to now, only
phenomenological approaches were used to describe the surface vibrations and
their role in the gap equation via the induced interaction. The latter was
examined by the Milano group in a series of papers using various approaches,
with the application to the nucleus 120Sn. In Ref. [4] the study was performed
within the Nuclear Field Theory (see [26] and References therein) in the two-
phonon approximation. One and two-phonon terms of the induced interaction were
taken into account, as well as the corresponding corrections to the single-
particle states. The latter includes both shifts of the single-particle
energies and spread of their strength which is analogous to the $Z$-factor in
(8). The ”direct” term of the gap, $\Delta_{\rm dir}\simeq 0.7\;$MeV, was
found in [4] with the Argonne v14 NN-potential and the mean field generated by
the SLy4 Skyrme force with the effective mass $m^{*}\simeq 0.7m$. It is one
half of the experimental gap (in [4], which is estimated as $\Delta_{\rm
exp}\simeq 1.4\;$MeV, while the simplest 3-point formula yields $\Delta_{\rm
exp}^{(\rm 3p)}\simeq 1.3\;$MeV). By including all the corrections due to low-
laying surface vibrations, the complete value $\Delta=\Delta_{\rm dir+ind}$
turns out to be in agreement with the experiment.
In [27] the induced interaction itself was analyzed in detail, again
concentrating on low-lying ($\omega_{L}<5\;$MeV) surface vibrations. The
$V_{low-k}$ potential was used as the free NN-interaction, instead of Argonne
v14 in [4], and again the experimental value of the gap was obtained as a sum
of practically equal ”direct” and ”induced” contributions multiplied by the
$Z$-factor representing the quasiparticle strength at the Fermi surface. The
latter was estimated on the basis of calculations in [4] as $Z=0.7$.
An essentially different approach was used in [5] where all collective states
with spin values $J\leq 5$ and excitation energies $\omega_{J}<30\;$MeV were
included into the induced interaction term. The surface vibrations discussed
above are only a part of them. Both the natural and unnatural parity states
contribute to the sum. The qRPA method with the SLy4 Skyrme force was used for
every $J^{\pi}$-channel. The fragmentation and self-energy effects were
approximately taken into account with the following relation for the total
effective pairing interaction:
$<12|V_{\rm dir+ind}|34>=Z<12|V_{\rm dir}+V_{\rm ind}|34>\,,$ (11)
where $Z=0.7$ value again was used. A short notation $|12>$ is used in (11)
for the two-particle basis states. To make easier the analysis and comparison
with other calculations, separate calculations of the gap from $V_{\rm dir}$
(Argonne v14) and $V_{\rm ind}$, both without the $Z$-factor, were made in
[5]. They give $\Delta_{\rm dir}=1.04\;$MeV and $\Delta_{\rm ind}=1.11\;$MeV.
Using the recipe of Eq. (11), yields the average gap $\Delta_{\rm
F}=1.47\;$MeV which is again close to the experiment.
In our opinion, each method used in these papers has some weak points which
could be criticized. In the first case, the contribution of so-called tadpole
diagrams [25, 28] was neglected, see also discussion in [29]. In the last
case, the use of Skyrme force in the particle-hole spin and spin-isospin
channels (unnatural parity states) is questionable. Indeed, the Skyrme
parameters were fitted only to phenomena which are related to ”scalar”
Landau–Migdal amplitudes, $F,F^{\prime}$. The combinations which determine the
spin-dependent amplitudes $G,G^{\prime}$ were not checked up to now by an
attempt to describe corresponding phenomena, e.g. magnetic moments,
$M1$-transitions, and so on. This problem is discussed also in [5], and some
complementary calculation was made with $G=G^{\prime}=0$. It resulted in a
very strong induced interaction yielding very big gap $\Delta_{\rm
F}=2.12\;$MeV. This could be considered as an estimate of the uncertainty of
such calculations with the use of phenomenological forces. Indeed, even the
amplitudes $F,F^{\prime}$ which are known sufficiently well in vicinity of the
Fermi surface could change significantly when high energy excitations are
considered. Another questionable point is the use in this calculation scheme
of the $Z$-factor found only from the low-lying surface vibrations. The high
energy response function included into the $V_{\rm ind}$ will contribute to
the $Z$-factor as well. This contribution comes mainly from the spin-isospin
channel and could be estimated as $Z_{\rm nm}\simeq 0.8$ [30].
We see that the problem of the screening effect is, indeed, very difficult,
and some approximations must be introduced in the calculations. In our
opinion, the fact that essentially different methods were used in [4] and [5]
with different results for $\Delta_{\rm ind}$ shows by itself that the problem
of finding contribution of the induced interaction into the pairing gap in
atomic nuclei is far from being solved.
## 5 Solution of the “ab initio” gap equation in finite nuclei
The explicit form of the gap equation (5) in the coordinate representation for
a non-uniform system is as follows [31]:
$\displaystyle\Delta({\bf r}_{1},{\bf r}_{2},\varepsilon)=\int{\cal V}({\bf
r}_{1},{\bf r}_{2},{\bf r}_{3},{\bf
r}_{4};E=2\mu,\varepsilon,\varepsilon^{\prime})\times$ (12)
$\displaystyle{}\times{\cal G}({\bf r}_{3},{\bf
r}_{5},\varepsilon^{\prime}){\cal G}^{s}({\bf r}_{4},{\bf
r}_{6},-\varepsilon^{\prime})\Delta({\bf r}_{5},{\bf
r}_{6},\varepsilon^{\prime}){d\varepsilon^{\prime}\over{2\pi i}}d{\bf
r}_{3}d{\bf r}_{4}d{\bf r}_{5}d{\bf r}_{6}.$
As in Sections 2-4, single-particle energies
$\varepsilon,\varepsilon^{\prime}$ are counted off the chemical potential
$\mu$. Dealing with nuclear matter, we set $\mu=\mu_{0}\simeq-16$ MeV (the
leading term in the Weizsaecker mass formula), whereas we have $\mu\simeq-8$
MeV for stable atomic nuclei, as the 120Sn nucleus considered below. As it was
discussed above, in this Section we limit ourselves with the simplest
Brueckner-like approach in which the irreducible vertex ${\cal V}$ coincides
with the free $NN$-potential, ${\cal V}=V$, which is independent of energy. In
this case, the gap $\Delta$ is also independent of energy; hence, the product
of two Green’s functions in (12) can be integrated with respect to
$\varepsilon^{\prime}$:
$A^{s}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3},{\bf
r}_{4})=\int{d\varepsilon^{\prime}\over{2\pi i}}{\mathcal{G}}({\bf r}_{1},{\bf
r}_{2},\varepsilon^{\prime}){\mathcal{G}}^{s}({\bf r}_{3},{\bf
r}_{4},-\varepsilon^{\prime}).$ (13)
The gap equation (12) can be written in a compact form as
$\Delta=VA^{s}\Delta.$ (14)
Below we deal with the BCS gap equation, not the general one, Eq. (12). This
explains why the term ab initio in the title of the Section is in quotes. In
fact, we speak just about the first step into the problem, i.e. the solution
of the BCS-like gap equation with the free $NN$-potential as the pairing
interaction. The mean field potential (or more general, the mass operator)
used in this solution is taken as a phenomenological input. To go beyond the
BCS approximation it is necessary, within an ab initio method, to calculate,
first, the mass operator and, second, corrections to the interaction block
$\mathcal{V}$ in the gap equation (12). The latter includes the induced
interaction discussed above (see also Ref. [32]) and three-body forces [33].
It turns out that, even at the level of this simplest ab initio calculations,
serious contradictions are still present.
The integral equation (14) can be reduced to the form adopted in the
Bogoliubov method,
$\Delta=-{V}\varkappa,$ (15)
where the abnormal density matrix $\varkappa=A^{s}\Delta$ can be expressed in
terms of $u$ , $v$-functions,
$\varkappa({\bf r}_{1},{\bf r}_{2})=\sum_{i}u_{i}({\bf r}_{1})v_{i}({\bf
r}_{2}),$ (16)
which satisfy the system of Bogoliubov equations. The summation in (16) is
over the complete set of Bogoliubov functions with the eigen energies
$E_{i}>0$.
The Milano group was the first who, in a series of papers [3, 4], [5] and
Refs. therein, solved the gap equation (15) with the realistic Argonne
$NN$-force v14 for the nucleus 120Sn. The latter was chosen for a definite
reason. Indeed, the chain of semi-magic tin isotopes is a traditional polygon
for examining nuclear pairing [22], [18]. The nucleus under discussion is in
the middle of the chain and the number of neutrons participating in the
pairing, those above the closed shell $N=50$, is sufficiently big to use the
approximation of the ”developed” pairing, used, in fact, in (15). This
approximation implies neglecting the particle number fluctuations [34] typical
of the BCS-like theories. The set of Bogoliubov equations was solved directly
in the basis {$\lambda$} of the states with a fixed limiting energy
$E_{\max}$. Such direct method is difficult because of a slow convergence of
sums over intermediate states $\lambda$ in the gap equation. These sums are
analogous to integrals in the momentum space in the gap equation for infinite
nuclear matter, see Section 2. In [3] the value of $E_{\max}{=}600\;$MeV was
used, and in [4, 5], $E_{\max}{=}800\;$MeV. The analysis of [7] showed that
the use of such big value of $E_{\max}$ permits to find $\Delta$ only with
accuracy of 10%. In [3] the Shell Model basis was used with the Saxon-Woods
potential and the bare mass, $m^{*}=m$, and the value $\Delta{=}2.2\;$MeV was
obtained which is by a factor one and half greater than the experimental one
(${\simeq}1.3\div 1.4\;$MeV). Evidently, this contradiction forced the authors
to use in further works the self-consistent basis of the Skyrme-Hartree-Fock
(SHF) method with the density depending effective mass $m^{*}(\rho)\neq m$. In
particular, the popular Sly4 force was used, which is characterized by a small
effective mass, equal to $m^{*}\simeq 0.7m$ in nuclear matter at the normal
nuclear density. Solving the BCS gap equation with such a basis the value of
$\Delta$ $\simeq 0.7\;$MeV was obtained in [4] and $\simeq 1.0\;$MeV in [5]. A
close value for the gap in the BCS equation was found in [7] for the nuclear
slab with parameters which mimic 120Sn nucleus. This calculation was done with
Argonne v18 potential which differs only slightly from the v14 one, but the
single-particle basis with $m^{*}=m$ was used. Remind that in [4, 5]
corrections to the BCS gap due to induced interaction were considered bringing
the results rather close to the experimental data as discussed in Section 4.
The crucial dependence of the gap on $m^{*}$, which is easily seen in weak
coupling limit (2) for the gap in nuclear matter, is, of course, the main
reason of so strong variation of the BCS gap from [3] to [5]. In finite
nuclei, this dependence is weaker than in nuclear matter, as the surface
region plays here the main role and one has $m^{*}(\rho)\to m$ in this region.
However, the $m^{*}$ effect remains strong.
Recently, results from the ab initio BCS equation (15) for a number of semi-
magic nuclei were published [9, 10]. They are based on the soft realistic
low-k force discussed above which was calculated starting from the Argonne v18
potential, with the same self-consistent Sly4 basis, i.e. the same effective
mass, as in [5]. For the nucleus 120Sn under consideration the value
$\Delta{\simeq}1.6\;$MeV was obtained. This value exceeds the experimental
one, which raises some questions. Indeed, although there are discrepancies in
absolute value of the corrections to the BCS gap (see for instance [5] and
[35]), their sign is more or less definite. All calculations of these
corrections, to our knowledge, increase $\Delta$ considerably. Hence the BCS
equation has to lead to the gap value smaller than the experimental one. In
addition, there is a direct contradiction between the results of the Milano
group and the ones of Duguet with co-authors, despite the BCS problem was
solved with very similar inputs. Evidently, there is some difference in the
method of including the effective mass, which is hidden in [9, 10] under the
renormalization of v18 into $V_{low-k}$.
In [16] an attempt was taken to clarify the reasons of this contradiction. The
BCS gap equation (15) was solved for the same 120Sn nucleus with the separable
form of Paris potential, which simplifies calculations. Our experience of
calculations for nuclear slab [6, 7] shows that the difference between the
Paris potential and the Argonne v18 one for the gap value is of the order of
0.1 MeV which is significantly smaller than the deviation under discussion.
For projecting out the high momenta contribution, the so-called Local
Potential Approximation (LPA) method was used. This new version of the local
approximation was introduced by our group for semi-infinite nuclear matter and
nuclear slab system, see the reviews [29, 36]. In general, this method is
analogous to the renormalization scheme for solving the gap equation in
infinite nuclear matter described in Section 2. To solve the gap equation in
the form (14) for finite systems, we split the complete Hilbert space $S$ of
the pairing problem into the model subspace $S_{0}$, including the single-
particle states with energies less than a fixed value $E_{0}$, and the
subsidiary one, $S^{\prime}$. Correspondingly, the two-particle propagator
(13) is the sum
$A^{s}=A_{0}^{s}+A^{\prime}\,.$ (17)
The notation becomes obvious if one expands (13) in the basis of single-
particle functions $\phi_{\lambda}({\bf r})$,
$A^{s}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3},{\bf
r}_{4})=\sum_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}A^{s}_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}\;\phi^{*}_{\lambda_{1}}({\bf
r}_{1})\phi^{*}_{\lambda_{2}}({\bf r}_{2})\phi_{\lambda_{3}}({\bf
r}_{3})\phi_{\lambda_{4}}({\bf r}_{4})\,.$ (18)
The model space propagator $A^{s}_{0}$ includes the terms of the sum (18) with
single-particle energies $\varepsilon_{\lambda}<E_{0}$, $A^{\prime}$ being the
remaining part.
The gap equation is reduced to the one in the model space,
$\Delta=V_{\mbox{\scriptsize eff}}A^{s}_{0}\Delta\,,$ (19)
with the effective pairing interaction obeying the integral equation in the
subsidiary space,
$V_{\mbox{\scriptsize eff}}=V+VA^{\prime}V_{\mbox{\scriptsize eff}}\,.$ (20)
The LPA method concerns the solution of (20). Solving this equation directly
in coordinate space is rather complicated, not much simpler than the gap
equation (14) in the complete Hilbert space. The problem could be simplified
with the use of the LPA. It turned out that, with very high accuracy, for each
point ${\bf R}$, one can use the formulas for the infinite system in the
potential field $U({\bf R})$ (it explains the term LPA). This simplifies
equation for $V_{\mbox{\scriptsize eff}}$ significantly, in comparison with
the initial equation for $\Delta$. As a consequence, the subspace $S^{\prime}$
could be chosen as large as necessary. Validity of the LPA could be justified
by finding that, beginning from some value of $E_{0}$, the result for the gap
doesn’t change under additional increase of $E_{0}$. It turns out that for a
nuclear slab, the value of $E_{0}{\simeq}20{\div}30\;$MeV is sufficient [6,
7]. For finite nuclei, it should be taken a little bigger, $E_{0}=40\;$MeV
[8]. In the slab calculations, we used the bare nucleon mass, $m^{*}=m$.In
this case, the reduction of high momenta components of the NN-force in (20)
is, in fact, quite similar to the renormalization procedure resulting in a
low-k interaction [37]. The difference is that in the LPA the renormalization
is coordinate dependent.
Figure 6: Coordinate dependence of the effective mass for different Skyrme
forces in the 120Sn nucleus.
The gap equation in the model space $S_{0}$ was solved in the
$\lambda$-representation with the use of different single-particle bases
$\phi_{\lambda}$. A discretization method of the continuum spectrum was used
with the wall radius $L{=}16\;$fm. Increasing the radius to $L{=}24\;$fm does
not practically influence the results. The radial eigen-functions $R_{nlj}(r)$
were found with a step $h{=}0.05\;$fm. We have used the Shell-Model basis with
the Saxon–Woods potential with a standard set of parameters and also with
several self-consistent basis obtained with different methods : the
Generalized Energy Density Functional method by Fayans et al. [18] with the
functional DF3, and the SHF method as well with different kinds of Skyrme
forces. The bare mass, $m^{*}{=}m$, is used in the first method, just as in
the Shell Model, whereas in the SHF method the effective mass is not equal to
$m$ and is density dependent. To clarify the role of the effective mass, we
chose two kinds of the Skyrme force, SKP and SKM*, for which the difference
between $m^{*}$ and $m$ is quite moderate, and the popular SLy4 force, with
significant difference of $m^{*}$ from $m$, see Fig. 6. We see that the SKP
effective mass deviates significantly from that for symmetric nuclear matter,
Fig. 3. This is due to the strong isospin asymmetry effect ($(N-Z)/A=1/6$ for
120Sn nucleus) for this kind of Skyrme force. For SLy4 and SKM* this effect is
rather modest. Remind that the Sly4 basis was used in calculations of $\Delta$
by Milano group and by Duguet with coauthors as well. As to the calculation of
the effective interaction in the subspace $S^{\prime}$, we first put
$m^{*}{=}m$. In table 2 the diagonal matrix elements $\Delta_{\lambda\lambda}$
of the neutron gap in the 120Sn nucleus, for 5 levels nearby the Fermi level,
are given for each basis under consideration. The quantity $\Delta_{\rm F}$ is
the corresponding Fermi-average value: $\Delta_{\rm
F}=\sum_{\lambda}{(2j+1)\Delta_{\lambda\lambda}}/\sum_{\lambda}(2j+1)$. As we
see, in all cases except of the last one the found value of the gap exceeds
the experimental one, 1.4 MeV, significantly. This indicates the necessity to
take into account the difference between the effective and bare masses in the
ab initio BCS gap equation.
Table 2: Diagonal matrix elements $\Delta_{\lambda\lambda}$ (MeV) with the
Paris potential for several kinds of the self-consistent basis.
*
* $\lambda$ | SW | DF3 | SKP | SKM* | SLy4
---|---|---|---|---|---
3$s_{1/2}$ | 1.52 | 1.64 | 1.55 | 1.55 | 1.17
2$d_{5/2}$ | 1.60 | 1.73 | 1.68 | 1.64 | 1.24
2$d_{3/2}$ | 1.64 | 1.80 | 1.71 | 1.68 | 1.26
1$g_{7/2}$ | 1.85 | 2.11 | 2.02 | 1.91 | 1.37
2$h_{11/2}$ | 1.58 | 1.79 | 1.69 | 1.64 | 1.18
$\Delta_{\rm F}$ | 1.65 | 1.85 | 1.76 | 1.71 | 1.25
Figure 7: Anomalous density for the 120Sn nucleus calculated with different
self-consistent mean fields.
In Fig. 7, for each kind of mean field, the anomalous density
$\nu(R)=\varkappa(R,r=0)$ is displayed. Here the notation ${\bf R}=({\bf
r}_{1}+{\bf r}_{2})/2,{\bf r}={\bf r}_{2}-{\bf r}_{1}$ is used. As the gap
$\Delta$ is proportional to this quantity, $\nu(R)$ depends on the effective
mass in the same way as the matrix elements $\Delta_{\lambda\lambda}$ in table
2. Indeed, the anomalous densities for the effective forces DF3, SKP and SKM*
are rather close to each other, whereas for the SLy4 force with a small
effective mass the anomalous density is suppressed significantly. There is a
pronounced surface maximum of $\nu(R)$ for all the cases. This leads to the
surface enhancement of the pairing gap found in [14] for the slab and in [15],
for spherical nuclei.
Until now, in the case of the Skyrme forces, we took into account the
difference between the effective and bare masses only inside the model space,
whereas we put $m^{*}{=}m$ for calculating $V_{\mbox{\scriptsize eff}}$. At
this point, there is a principal difference between our LPA method and
calculations of [4, 5] and [9, 10], where the SLy4 effective mass was used for
all the $\lambda$-states. For a closer comparison, we made a modification of
the LPA method, which permits to take into account the density dependent
effective mass $m^{*}_{n}(\rho_{n},\rho_{p})$ for a part of the space
$S^{\prime}$, including momenta $k<\Lambda$, where $\Lambda$ is a parameter.
Following the idea of LPA, with the potentials $U_{n}(R),U_{p}(R)$ , it is
natural to determine at each point $R$ the densities of each type $\tau=n,p$
of nucleons with quasi-classical formulas: $\rho_{\tau}(R)=(k_{\rm
F}^{\tau}(R))^{3}/3\pi^{2}$, $k_{\rm
F}^{\tau}(R)=[2m^{*}_{\tau}(\rho_{n}(R),\rho_{p}(R))(\mu_{\tau}-U_{\tau}(R))]^{1/2}$,
where $\mu_{n},\mu_{p}$ are the chemical potentials of neutrons and protons in
the nucleus under consideration. For the functional SLy4 we made several
alternative calculations with different values of $\Lambda$. They are denoted
as SLy4-1 ($\Lambda{=}3\;$fm-1), SLy4-2 ($\Lambda{=}4\;$fm-1) and SLy4-3
($\Lambda{=}6.2\;$fm-1). The first two versions mimic calculations of [9, 10],
the latter, of [4, 5]. The obtained gap values are given in table 3.
Table 3: Diagonal matrix elements $\Delta_{\lambda\lambda}$ with the Paris
potential for the Sly4 basis depending on the way of account for the effective
mass in equation for $V_{\mbox{\scriptsize eff}}$.
* $\lambda$ | SLy4 | Sly4-1 | Sly4-2 | Sly4-3
---|---|---|---|---
3$s_{1/2}$ | 1.17 | 1.07 | 0.88 | 0.76
2$d_{5/2}$ | 1.24 | 1.13 | 0.93 | 0.80
2$d_{3/2}$ | 1.26 | 1.15 | 0.95 | 0.83
1$g_{7/2}$ | 1.37 | 1.23 | 0.99 | 0.85
2$h_{11/2}$ | 1.18 | 1.08 | 0.88 | 0.75
$\Delta_{\rm F}$ | 1.25 | 1.14 | 0.92 | 0.80
Table 4: The same as in table 3, but for the Argonne v18 potential.
* $\lambda$ | SLy4 | Sly4-1 | Sly4-2 | Sly4-3
---|---|---|---|---
3$s_{1/2}$ | 1.23 | 1.10 | 0.83 | 0.56
2$d_{5/2}$ | 1.32 | 1.18 | 0.89 | 0.61
2$d_{3/2}$ | 1.34 | 1.20 | 0.92 | 0.63
1$g_{7/2}$ | 1.48 | 1.31 | 0.96 | 0.64
2$h_{11/2}$ | 1.27 | 1.13 | 0.85 | 0.57
$\Delta_{\rm F}$ | 1.34 | 1.19 | 0.89 | 0.60
Let us now repeat calculations with the SLy4 basis, but for Argonne force v18.
Results are given in table 4\. Comparison with table 3 shows that the
difference between gap values for the Paris and Argonne force is of the same
order ($\simeq 0.1\;$MeV) as for the slab calculations [7], with the exception
of the SLy4-3 version with the cutoff for $m^{*}$ in the equation for
$V_{\mbox{\scriptsize eff}}$ equal to $\Lambda_{3}=6.2\;$fm-1. In this case,
the difference is $\simeq 0.2\;$MeV. It is worth of noticing that the sign of
the difference changes depending on $\Lambda$. Namely the Argonne gap exceeds
the Paris one for SLy4 ($\Lambda=0$) and SLy4-1 ($\Lambda_{1}=3\;$fm-1)
versions, but becomes smaller for SLy4-2 ($\Lambda_{2}=4\;$fm-1) and SLy4-3
runs. Such behavior is qualitatively clear.Indeed, the Paris potential is much
harder of the Argonne one, therefore the relative contribution to
$V_{\mbox{\scriptsize eff}}$ of the momentum region, say, between
$\Lambda_{2}$ and $\Lambda_{3}$, is less than for the Argonne potential.
Therefore, for the gap equation itself, the role of the corresponding
suppression of $V_{\mbox{\scriptsize eff}}$ due to putting $m^{*}<m$ is less
in the Paris case than in the Argonne one.
It is rather difficult to make a direct comparison of table 4 with results of
[9]. In the first column, the effective mass $m^{*}\neq m$ is introduced only
in the model space, for $\varepsilon_{\lambda}<E_{0}=40\;$MeV. In the free
space, outside the nucleus, it corresponds to the momentum cutoff
$\Lambda=1.4\;$fm-1; inside the nucleus we have $\Lambda\simeq 2\;$fm-1.We see
that the ”average” value of $\Lambda$ is less a bit of $\Lambda=2\;$fm-1 in
[9]. In the second column (SLy4-1) we deal with $\Lambda=3\;$fm-1. Thus, we
should attribute to the gap value of [9] ($\Delta\simeq 1.6\;$MeV) an average
value of these two columns, $\Delta\simeq 1.25\;$MeV, which is noticeably
smaller. For comparison with [4, 5] we should be guided by the last column
(SLy4-3), as the corresponding value $\Lambda=6.2\;$fm-1 was chosen to
reproduce $E_{\rm max}=800\;$MeV from these calculations. Again, this
correspondence is not literal as it takes place only outside the nucleus where
we have $m^{*}=m$. Inside, due to $m^{*}\neq m$, the same value of $E_{\rm
max}$ should correspond to smaller value of $\Lambda\simeq 5.5\;$fm-1. Again
we should take a value between those of the 3-rd column and the last one,
closer to the latter. In any case, it will be closer to the result of [4]
($\simeq 0.7\;$MeV) than of [5] ($\simeq 1\;$MeV).
The main observation, common to table 3 (Paris) and table 4 (Argonne), is a
drastic dependence of the gap on the $m^{*}(k)$ behavior in the subsidiary
space. In fact, a set of calculations with different cutoff $\Lambda$
imitates, very roughly, the $k$-dependence of the effective mass.In the case
that the asymptotic limit $m^{*}(k)\to m$ occurs sufficiently far away, the
pairing gap in nuclei does depend on the effect of $m^{*}\neq m$ at high
momenta. It is absolutely lost in calculations of $\Delta$ with low-k force
found for small cutoff values. Evidently, this is the main reason why gap
values found in [9] by solving the BCS gap equation are so high, keeping in
mind corrections due to the induced interaction.
Table 5: The same as in table 4, but for the SKM* Skyrme force.
* $\lambda$ | SKM* | SKM*-1 | SKM*-2 | SKM*-3
---|---|---|---|---
3$s_{1/2}$ | 1.58 | 1.52 | 1.40 | 1.29
2$d_{5/2}$ | 1.71 | 1.63 | 1.49 | 1.38
2$d_{3/2}$ | 1.75 | 1.68 | 1.55 | 1.43
1$g_{7/2}$ | 2.01 | 1.92 | 1.76 | 1.62
2$h_{11/2}$ | 1.72 | 1.64 | 1.51 | 1.40
$\Delta_{\rm F}$ | 1.78 | 1.71 | 1.57 | 1.44
To confirm the leading role of the effective mass in the problem, we made a
series of analogous calculations for the SKM* force for which the effective
mass in the nucleus under consideration is much closer to m, than for the SLy4
one, see Fig. 6. The results are given in table 5 with the notation similar to
that of table 4. We see that even for the SKM*-3 version with
$\Lambda=6.2\;$fm-1 the result does not leave any room for contribution of the
induced interaction.
## 6 Discussion and conclusions
In this paper we briefly reviewed the status of the microscopic theory of
nuclear pairing. Although to date there is no consistent theory of nuclear
matter pairing, progress has been made in developing approximated methods
yielding comparable results on the density dependence of the pairing gap.
However, the results cannot be applied to finite nuclei directly because the
nuclear surface plays a leading role in nuclear pairing and the standard LDA
fails at the nuclear surface. As to finite nuclei, some progress also exists
especially in solving the simplest, in fact BCS-type, ”ab initio” gap
equation, where the free NN potential is used as the effective pairing
interaction. Even this equation is not completely microscopic as far as the
phenomenological mean field is used. The inherent technical problems were
overcome, first, by the Milan group [3, 4, 5] and more recently by other teams
[9, 10] and [16]. In the first and the last cases, the nucleus 120Sn was
considered as ”testing sample”, whereas in [9] several isotopic and isotonic
chains were considered. The Argonne v14 force was used in [3, 4, 5], the low-k
force with the cutoff $\Lambda=2\;$fm-1 in [9], and the Paris potential, in
[16]. In this paper we performed calculations analogous to [16] for the
Argonne v18 potential which differ only a bit from the v14 one. All the
results differ from the experimental gap in 120Sn, $\Delta_{\rm exp}\simeq
1.3\div 1.4\;$MeV, not more than by a factor two which shows relevance of the
ab initio BCS gap equation as a starting point for the microscopic theory of
nuclear pairing. However, more detailed comparison shows that there are
contradictions even at this ”first level” of the problem. Indeed, the BCS gap
is $\simeq 1\;$MeV in [5] and is $\simeq 1.6\;$MeV in [9], whereas inputs look
quite similar. Namely, both the calculations use the SLy4 Skyrme force with
the effective mass $m^{*}\simeq 0.7m$; the Argonne v14 force is used in [5]
and the low-k force in [9], but the latter could be obtained from the first
one with the RGM procedure. The only important difference of the inputs is the
size of the momentum space where the effective mass contributes: $k<k_{\rm
max}\simeq 6\;$fm-1 in [5] and only $k<\Lambda=2\;$fm-1 in [9]. Indeed, the
RGM equation is defined for the free NN scattering where the equality
$m^{*}=m$ is postulated. In fact, we deal with different $k$-dependence of the
effective mass. The equality $m^{*}\simeq 0.7m$ takes place for all momenta in
[5] and only for $k<\Lambda$, in [9]. This reason of the contradictory results
of [5] and [9] was discussed in [10] and was confirmed with the analysis of
[16] for the Paris force and in this paper, for the Argonne v18 force. We use
the so-called LPA method developed by us previously in which high momenta are
excluded via some renormalization procedure which recall the RGM one but is
coordinate dependent and permits to introduce in the high momentum space the
effective mass into the equation for the effective pairing interaction. For
the same SLy4 basis, we changed ”by hands” the size of the space where the
equality $m^{*}\simeq 0.7m$ takes place, putting $m^{*}=m$ outside. Changing
this dividing point we evolve from the situation close to that of [9] to the
one of [5], although the correspondence, of course, is not literal. In the
first limit, we obtained the value of $\Delta_{\rm F}\simeq 1.25\;$MeV which
is noticeably smaller than that in [9]. In the second one, we obtained the
result closer to that of [4] than of [5]. But, in our opinion, fixing some
numerical contradictions is not of primary importance. This disagreement can
be resolved. Much more important is the very high sensitivity of the pairing
gap to the $k$-dependence of the effective mass found in our analysis. This
dependence could be hardly guessed phenomenologically by a lucky Skyrme-type
ansatz. Indeed, any such ansatz deals with a number, not a function! Therefore
an additional microscopic ingredient, namely a theory of the $k$-dependent
effective mass, is necessary even at this first level of the problem.
The next open problem is the inclusion of the many-body corrections to the BCS
gap equation. The induced pairing interaction due to exchange by virtual
surface vibrations and other particle-hole excitations is the main of them [4,
5]. Up to now, this problem was studied only within phenomenological
approaches and is far from being solved. Any attempt to attack it from first
principles hits the absence of the general microscopic nuclear theory.
## 7 Acknowledgments
We thank H.-J. Schulze for useful discussions. Two of the authors (S.S.P. and
E.E.S.) thank INFN (Sezione di Catania) for hospitality during their stay in
Catania. This research was partially supported by the Grants of the Russian
Ministry for Science and Education NSh-3004.2008.2 and 2.1.1/4540, the joint
Grant of RFBR and DFG, Germany, No. 09-02-91352-NNIO_a, 436 RUS
113/994/0-1(R), by the RFBR grants 07-02-00553-a, 09-02-01284-a and
09-02-12168-ofi_m.
## References
## References
* [1] Gezerlis A and Carlson J 2009 arXiv: 0911.3907, and reference therein.
* [2] Bogner S K, Furnstahl R J, Ramanan S, and Schwenk A 2007 Nucl. Phys. A 784 79 ; Hebeler K, Schwenk A, and Friman B, 2007 Phys. Lett. B 648 176
* [3] Barranco F, Broglia R A, Esbensen H and Vigezzi E 1997 Phys. Lett. B 390 13
* [4] Barranco F, Broglia R A, Colo G, et al., 2004 Eur. Phys. J. A 21, 57
* [5] Pastore A, Barranco F, Broglia R A, and Vigezzi E 2008 Phys. Rev. C 78, 024315
* [6] Pankratov S S, Baldo M, Lombardo U, Saperstein E E, and Zverev M V 2006 Nucl. Phys. A 765 61
* [7] Pankratov S S, Baldo M, Lombardo U, Saperstein E E, and Zverev M V 2008 Nucl. Phys. A 811 127
* [8] Pankratov S S, Baldo M, Zverev M V, Lombardo U, Saperstein E E, and Tolokonnikov S V 2009 JETP Lett 90 560
* [9] Duguet T and Lesinski T 2008 Eur. Phys. J. Special Topics 156 207
* [10] Hebeler K, Duguet T and Lesinski T, and Schwenk A arXiv:0904.3152v1 [nucl-th] 21 Apr 2009
* [11] Baldo M, Lombardo U, Saperstein E E, and Zverev M V 1998 Nucl. Phys. A 628 503
* [12] Baldo M, Lombardo U, Saperstein E E, and Zverev M V 1999 Phys. Lett. B 459 437
* [13] Farine M, and Schuck P 1999 Phys. Lett. B 459 444
* [14] Baldo M, Farine M, Lombardo U, Saperstein E E, Schuck P, and Zverev M V 2003 Eur. Phys. J. A 18 17
* [15] Pillet N, Sandulescu N, and Schuck P, 2007 Phys. Rev. C 76 024310
* [16] Pankratov S S, Saperstein E E, Zverev M V, Baldo M, and Lombardo U 2009 Phys. Rev. C 79 024309
* [17] Pankratov S S, Baldo M, Lombardo U, Saperstein E E, Zverev M V, and Tolokonnikov S V Proceedings of the conference NSRT09, Dubna, 2009, to be published.
* [18] Fayans S A, Tolokonnikov S V, Trykov E L, and Zawischa D 2000 Nucl. Phys. A 676 49
* [19] Lombardo U and Schulze H-J “Superfluidity in Neutron Star Matter” in Physics of Neutron Star Interiors, Lecture Notes in Physics vol. 578, pp. 30-54, Eds. D. Blaschke, N. K. Glendenning, and A. Sedrakian (Springer, 2001); Dean D J and Hjorth-Jensen M 2003 Rev. Mod. Phys. 75 607 and Refs. therein
* [20] P. Nozières, Theory of Interacting Fermi Systems (Benjamin, New York, 1966)
* [21] Baldo M and Grasso A 2000 Phys. Lett. B 485, 115 ; 2001 Phys. At. Nucl. 64 611; Lombardo U, Schuck P, and Zuo W 2001 Phys. Rev. C 64 021301(R); Baldo M, Lombardo U, Schulze H-J, and Zuo Wei 2002 Phys. Rev. C 66, 054304; Muther H and Dickhoff W H 2005 Phys. Rev C 72 054316 ; Bozek P 2003 Phys. Lett. B 551, 93
* [22] Dobachewsky J, Flocard H, Treiner J 1984 Nucl. Phys. A 422 103
* [23] Bartel J, Quentin P, Brack M, Guet C, and H aakansson H.-B. 1982 Nucl. Phys. A 386 79
* [24] Chabanat E, Bonche P, Haensel P, Meyer J, Schaeffer R 1997 Nucl. Phys. A 627 710
* [25] Khodel V A and Saperstein E E 1982 Phys. Rep. 92 183
* [26] Bertsch G F, Bortignon P F, and Broglia R A 1983 Rev. Mod. Phys. 55 287
* [27] Barranco F, Bortignon P F, Broglia R A, Colo G, Schuck P, Vigezzi E, and Viñas X 2005 Phys. Rev. C 72 054314
* [28] Kamerdzhiev S and Saperstein E E 2008 Eur. Phys. J. A 37, 333
* [29] Baldo M, Lombardo U, Saperstein E E, and Zverev M V 2004 Phys. Rep. 391 261
* [30] Saperstein E E and Tolokonnikov S V 1998 JETP Lett. 68 553
* [31] Migdal A B Theory of finite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967).
* [32] L.G. Cao, U. Lombardo and P. Schuck 2006 Phys. Rev. C 74 064901
* [33] W. Zuo and U. Lombardo 2009, Phys. Atomic Nuclei 72, 1359
* [34] Bohr A and Mottelson B R, Nuclear Structure (Benjamin, New York, 1971.), Vol. 1
* [35] Avdeenkov A V, and Kamerdjev S P 1999 JETP Lett. 69 669
* [36] Saperstein E E, Pankratov S S, Zverev M V, Baldo M, and Lombardo U 2009 Phys. At. Nucl. 72 1167
* [37] Bogner S K, Kuo T T S and Schwenk A 2003 Phys. Rep. 386 1
|
arxiv-papers
| 2010-01-05T18:47:15 |
2024-09-04T02:49:07.490850
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.Baldo, U.Lombardo, S.S.Pankratov, E.E.Saperstein",
"submitter": "Sergey Pankratov",
"url": "https://arxiv.org/abs/1001.0743"
}
|
1001.0746
|
2010669-680Nancy, France 669
Ryan Williams
# Alternation-Trading Proofs, Linear Programming, and Lower Bounds
(Extended Abstract)
R. R. Williams IBM Almaden Research Center
650 Harry Road, San Jose, CA, USA 95120 ryanwill@us.ibm.com
http://www.cs.cmu.edu/ ryanw/
###### Abstract.
A fertile area of recent research has demonstrated concrete polynomial time
lower bounds for solving natural hard problems on restricted computational
models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path,
$\text{MOD}_{6}\text{-SAT}$, Majority-of-Majority-SAT, and Tautologies, to
name a few. The proofs of these lower bounds follow a certain proof-by-
contradiction strategy that we call alternation-trading. An important open
problem is to determine how powerful such proofs can possibly be.
We propose a methodology for studying these proofs that makes them amenable to
both formal analysis and automated theorem proving. We prove that the search
for better lower bounds can often be turned into a problem of solving a large
series of linear programming instances. Implementing a small-scale theorem
prover based on this result, we extract new human-readable time lower bounds
for several problems. This framework can also be used to prove concrete
limitations on the current techniques.
###### Key words and phrases:
time-space tradeoffs, lower bounds, alternation, linear programming
###### 1991 Mathematics Subject Classification:
F.2.3, I.2.3
This material is based on work supported in part by NSF grant CCR-0122581
while the author was a student at Carnegie Mellon University, and NSF grant
CCF-0832797 while the author was a member of the Institute for Advanced Study.
## 1\. Introduction
Many known lower bounds for natural problems follow a type of algorithmic
argument that we call a resource-trading proof. Such a proof assumes that a
hard problem can be solved by a “good” algorithm, and tries to derive a
contradiction by combining two essential components. One is a speedup lemma,
which simulates all good algorithms super-efficiently on some “interesting”
computational model, trading time for some resource. The second component is a
slowdown lemma, which uses the assumed good algorithm for the hard problem to
simulate computations from the “interesting” model by good algorithms, thereby
trading the “interesting” resource for more time. Clever combinations of
speedup and slowdown lemmas are used to contradict a known result, in
particular some complexity hierarchy theorem. That is, by assuming a “good”
algorithm for a hard problem, we derive something like ${\sf
TIME}[n^{2}]\subseteq{\sf TIME}[n]$, a contradiction.
As an example, one can prove a time-space tradeoff for satisfiability (SAT) as
follows. Assume SAT has an algorithm running in $n^{c}$ time and $\text{\rm
poly}(\log n)$ space, for some $c>1$. One speedup lemma is that computations
running in $n^{a}$ time and $\text{\rm poly}(\log n)$ space can be simulated
by an alternating machine that switches from co-nondeterministic mode to
nondeterministic mode once (i.e., a ${\sf\Pi}_{2}$ machine), and runs in
$n^{a/2+o(1)}$ time. This speedup lemma trades time for alternations. The
relevant slowdown lemma is: if SAT has an $n^{c}$ time, $\text{\rm poly}(\log
n)$ space algorithm, then (by a strengthening of the Cook-Levin theorem) every
language in ${\sf NTIME}[t]$ has $t^{c+o(1)}$ time, $\text{\rm poly}(\log t)$
space algorithms. Consequently, an alternating machine running in $t$ time and
making $k-1$ alternations has $t^{c^{k}+o(1)}$ time, $\text{\rm poly}(\log t)$
space algorithms. Combining these speedup and slowdown lemmas, we derive
${\sf\Sigma}_{2}{\sf TIME}[t]\subseteq{\sf DTISP}[t^{c^{2}+o(1)},\text{\rm
poly}(\log t)]\subseteq{\sf\Pi}_{2}{\sf TIME}[t^{c^{2}/2}],$
where the first inclusion holds by slowdown and the second holds by speedup.
Now observe that the alternating time hierarchy is contradicted when
$c^{2}<2$. This proof is the $n^{\sqrt{2}-\varepsilon}$ time lower bound of
Lipton and Viglas [LV].
Some of the best known separations in complexity theory use resource-trading
proofs. Hopcroft, Paul, and Valiant [HPV] showed that ${\sf
SPACE}[n]\nsubseteq{\sf DTIME}[o(n\log n)]$ for multitape Turing machines, by
proving the “speedup lemma” that ${\sf DTIME}[t]\subseteq{\sf SPACE}[t/\log
t]$ and invoking diagonalization. Their result was later extended to general
models [PR, Halpern]. Paul, Pippenger, Szemeredi, and Trotter [PPST] proved
that ${\sf NTIME}[n]\neq{\sf DTIME}[n]$ for multitape Turing machines. The key
component in the proof is the “speedup lemma” ${\sf
DTIME}[t]\subseteq{\sf\Sigma}_{4}{\sf TIME}[t/\log^{*}t]$ for multitape TMs.
Despite their age, the above separations still constitute the best known
progress on ${\sf P}$ vs $\sf PSPACE$ and ${\sf P}$ vs ${\sf NP}$,
respectively.
In more recent years, resource-trading proofs have established time-space
lower bounds for ${\sf NP}$-complete problems and problems higher in the
polynomial hierarchy [K84, F, LV, FvM, FLvMV, W06, W08]. For instance, the
best known time lower bound for solving SAT with $n^{o(1)}$-space algorithms
is $n^{2\cos(\pi/7)-o(1)}\geq n^{1.801}$, obtained with a resource-trading
proof [W08]. (Note if one could improve the $1.801$ exponent to arbitrary
constants, one would separate ${\sf LOGSPACE}$ from ${\sf NP}$.) For
nondeterministic algorithms using $n^{o(1)}$ space, the best known time lower
bound for solving the ${\sf coNP}$-complete Tautology problem was
$n^{\sqrt{2}-o(1)}$ for several years [FvM]. Certain time-space lower bounds
for probabilistic and quantum computations also follow the resource-trading
paradigm [AKRRV, DvM, Viola, vMW]. Resource-trading proofs are also abound in
the multidimensional “hybrid” Turing machine model, which has read-only random
access to its input and an $n^{o(1)}$ read-write store, as well as read-write
two-way access to a $d$-dimensional tape for some $d\geq 1$. This is the most
powerful (and physically realistic) model known where we still know non-
trivial time lower bounds for problems such as SAT. Multidimensional TMs have
a long history; e.g., [Loui, PR, K83, MS, vMR, W06] proved lower bounds for
them. (For a more complete literature review, please see the full version of
the paper.)
### 1.1. Main Results
We introduce a methodology for reasoning about resource-trading proofs that is
also practically implementable for finding short proofs. Informally, the “hard
work” in these proofs can be replaced by solving a series of linear
programming problems. This perspective not only aids us practically in the
search for new lower bounds, but also allows us to show non-trivial
limitations on what can be proved.
This methodology is applied to several lower bound problems. In all cases
considered here, the resource being “traded” is alternations, so we call the
proofs alternation-trading.
Deterministic Time-Space Lower Bounds. Aided by results of a computer program,
we show that any SAT algorithm running in $t(n)$ time and $s(n)$ space
satisfies $t\cdot s\geq\Omega(n^{2\cos(\pi/7)-o(1)})$. Previously, the best
known result was $t\cdot s\geq\Omega(n^{1.573})$ [FLvMV]. It has been
conjectured that the current framework sufficed to prove a $n^{2-o(1)}$ time
lower bound for SAT, against algorithms using $n^{o(1)}$ space. We prove that
it is not possible to obtain $n^{2}$ with the framework, formalizing a
conjecture of [FLvMV].***That is, we formalize the statement: “…some
complexity theorists feel that improving the golden ratio exponent beyond 2
would require a breakthrough” in Section 8 of [FLvMV]. A computer search over
proofs of short length suggests that the best known $n^{2\cos(\pi/7)-o(1)}$
lower bound [W08] is already optimal for the framework. We also prove lower
bounds on $\text{\rm QBF}_{k}$ (quantified Boolean formulas with at most $k$
quantifier blocks), showing that the problem requires
$\Omega(n^{k+1-\delta_{k}})$ time for $n^{o(1)}$ space algorithms, where
$\delta_{k}<0.2$ and $\lim_{k\rightarrow\infty}\delta_{k}=0$.†††Note the
$\text{\rm QBF}_{k}$ results appeared in the author’s PhD thesis in 2007 but
have been unpublished to date.
Nondeterministic Time-Space Lower Bounds. Adapting our ideas to proving lower
bounds for Tautologies, a computer program found a very short proof improving
upon Fortnow and Van Melkebeek’s lower bound. Longer proofs suggested an
interesting pattern. Joint work with Diehl and Van Melkebeek on this
observation resulted in an $n^{4^{1/3}-o(1)}\geq n^{1.587}$ time lower bound
[DvM2]. Computer search suggests that this lower bound is best possible for
the framework. We prove that it is not possible to obtain an $n^{\phi}$ time
lower bound, where $\phi=1.618\ldots$ is the golden ratio. This is surprising
since we have known for some time that an $n^{\phi}$ lower bound is provable
for deterministic algorithms [FvM].
Multidimensional Turing Machine Lower Bounds. Here our method uncovers
peculiar behavior in the best lower bound proofs, regardless of the dimension.
Studying computer search results, we extract an $\Omega(n^{r_{d}})$ time lower
bound for the $d$-dimensional case, where $r_{d}\geq 1$ is the root of a
particular quintic $p_{d}(x)$ with coefficients depending on $d$. For example,
$r_{1}\approx 1.3009$, $r_{2}\approx 1.1887$, and $r_{3}\approx 1.1372$.
Again, our search suggests this is best possible, and we can prove it is not
possible to improve the bound for $d$-dimensional TMs to $n^{1+1/(d+1)}$ with
the current tools.
These limitations also hold for other ${\sf NP}$ and ${\sf coNP}$-hard
problems; the only property required is that all languages in ${\sf NTIME}[n]$
(respectively, ${\sf coNTIME}[n]$) have sufficiently efficient reductions to
the problem. Also our linear programming approach is not limited to the above,
and can be applied to the league of lower bounds discussed in Van Melkebeek’s
surveys [VMsurvey, VMsurvey2].
### 1.2. Some Remarks on the Reduction to Linear Programming
The key to our formulation is to separate the discrete choices in an
alternation-trading proof from the real-valued choices. The discrete choices
consist of the sequence of lemmas to apply in each step, and what sort of
hierarchy theorem to use in the contradiction. We present several
simplifications that greatly reduce the number of discrete choices, without
loss of generality. The real-valued choices are the running time exponents
that arise from the choices of time bounds and rule applications. We prove
that once the discrete choices are made, the remaining real-valued problem can
be expressed as an instance of linear programming. This makes it possible to
search for new proofs via computer, and it also gives us a formal handle on
the limitations of these proofs.
One cannot easily search over all possible proofs, as the number of discrete
choices is still about $2^{n}/n^{3/2}$ for proofs of $n$ lines (proportional
to the $n$th Catalan number). Nevertheless it is still feasible to try all
$24+$ line proofs. These proof searches reveal patterns, indicating that
certain strategies will be most successful in proving lower bounds; in each
case we study, the resulting strategies differ. Following the strategies, we
establish new lower bound proofs. The patterns also suggest how to show
limitations on the proof systems.
Note: Due to space limitations, we can only describe how our methods apply to
SAT time-space lower bounds. Please see the full version of the paper for
proofs and more details.
## 2\. Preliminaries
We assume familiarity with Complexity Theory, especially the notion of
alternationWe use big-$\Omega$ notation in the infinitely often sense, so
statements like “SAT is not in $O(n^{c})$ time” are equivalent to “SAT
requires $\Omega(n^{c})$ time.” All functions are assumed constructible within
the appropriate bounds. Our default computational model is the random access
machine, broadly construed: particular variants do not affect the results.
${\sf DTISP}[t(n),s(n)]$ is the class of languages accepted by a RAM running
in $t(n)$ time and $s(n)$ space, simultaneously. For convenience, we set ${\sf
DTS}[t(n)]:={\sf DTISP}[t(n)^{1+o(1)},n^{o(1)}]$ to omit negligible $o(1)$
factors.
In order to properly formalize alternation-trading proofs, we introduce
notation for alternating complexity classes that include input constraints
between alternations. Let us start with an example of the notation, then give
a general definition. Define $(\exists~{}f(n))^{b}{\sf DTS}[n^{a}]$ to be the
class of languages recognized by a machine which, on an input $x$ of length
$n$, writes a $f(n)^{1+o(1)}$ bit string $y$ nondeterministically, copies at
most $n^{b+o(1)}$ bits $z$ from the pair $\langle x,y\rangle$ (in
$O(n^{b+o(1)})$ time), then feeds $z$ as input to a machine $M$ running in
$n^{a+o(1)}$ time and $n^{o(1)}$ space. Note the runtime of $M$ is measured
with respect to the initial input length $n$, not the latter input length
$n^{b+o(1)}$ of $z$.
We generalize this definition as follows. Let $\mathcal{C}$ be a complexity
class. For $i=1,\ldots,k$, let $Q_{i}\in\\{\exists,\forall\\}$ and
$a_{i},b_{i}\geq 0$. Define
$(Q_{1}~{}n^{a_{1}})^{b_{2}}(Q_{2}~{}n^{a_{2}})\cdots^{b_{k}}(Q_{k}~{}n^{a_{k}})^{b_{k+1}}\mathcal{C}$
to be the class of languages recognized by a machine $M$ that, on input $x$ of
length $n$, has the following general behavior on input $x$:
Set $z_{0}:=x$.
---
For $i=1,\ldots,k$,
If $Q_{i}=\exists$, switch to existential mode.
If $Q_{i}=\forall$, switch to universal mode.
Guess an $n^{a_{i}+o(1)}$ bit string $y$ (universally or existentially).
Copy at most $n^{b_{i+1}+o(1)}$ bits $z_{i}$ from the pair $\langle
z_{i-1},y\rangle$.
End for
Run a machine recognizing a language in class $\mathcal{C}$ on the input
$z_{k}$.
When an input constraint $b_{i}$ is unspecified, its default value is
$\max\\{a_{i},1\\}$. We say that the existential and universal modes of an
alternating computation are quantifier blocks, to reflect the complexity class
notation. It is crucial to observe that the time bound in the $i$th quantifier
block is measured with respect to $n$, the input to the first quantifier
block.
Notice that by simple properties of nondeterminism and conondeterminism, we
can combine adjacent quantifier blocks that are of the same type, e.g.,
$(\exists n^{a})^{a}(\exists n^{b})^{b}{\sf DTS}[n^{c}]=(\exists
n^{\max\\{a,b\\}})^{b}{\sf DTS}[n^{c}]$. This useful property is exploited in
alternation-trading proofs.
### 2.1. A Short Introduction to Alternation-Trading Proofs
Here we give a brief overview of the tools used in alternation-trading proofs.
In this extended abstract we focus on deterministic time lower bounds for
satisfiability for algorithms using $n^{o(1)}$ workspace; the other lower
bound problems use similar tools.
It is known that satisfiability of Boolean formulas in conjunctive normal form
(SAT) is a complete problem under tight reductions for a small
nondeterministic complexity class. The class ${\sf NQL}$, called
nondeterministic quasilinear time, is defined as
${\sf NQL}:=\bigcup_{c\geq 0}{\sf NTIME}[n\cdot(\log n)^{c}]={\sf
NTIME}[n\cdot poly(\log n)].$
###### Theorem 2.1 ([Cook, Schnorr, Tourlakis, FLvMV]).
SAT is ${\sf NQL}$-complete under quasilinear time $O(\log n)$ space
reductions, for both multitape and random access machine models. Moreover,
each bit of the reduction can be computed in $O(poly(\log n))$ time and
$O(\log n)$ space in both machine models.‡‡‡In the multitape Turing machine
model we assume that the tape heads are already oriented on the appropriate
cells, otherwise it may take linear time to find the appropriate cells on a
tape.
Let $\mathcal{C}[t(n)]$ represent a time $t(n)$ complexity class under one of
the three models:
* •
deterministic RAM using time $t$ and $t^{o(1)}$ space,
* •
co-nondeterministic RAM using time $t$ and $t^{o(1)}$ space,
* •
$d$-dimensional Turing machine using time $t$.
Theorem 2.1 implies that if ${\sf NTIME}[n]\nsubseteq\mathcal{C}[t]$, then SAT
$\notin\mathcal{C}[t/\text{\rm poly}(\log t)]$.
###### Corollary 2.2.
If ${\sf NTIME}[n]\nsubseteq\mathcal{C}[t(n)]$, then
SAT$~{}\notin\mathcal{C}[t(n)/\log^{k}t(n)]$ for some $k>0$.
Hence we wish to prove ${\sf NTIME}[n]\nsubseteq\mathcal{C}[n^{c}]$ for large
$c>1$. To prove time-space lower bounds, we work with $\mathcal{C}[n^{c}]={\sf
DTS}[n^{c}]={\sf DTISP}[n^{c},n^{o(1)}]$. Van Melkebeek and Raz [vMR] observed
that a similar corollary holds for any problem $\Pi$ such that SAT reduces to
$\Pi$ under highly efficient reductions, e.g. Vertex Cover, Hamilton Path,
3-SAT, and Max-2-Sat. Therefore similar time lower bounds hold for these
problems as well.
#### Speedups, Slowdowns, and Contradictions.
Now that our goal is to prove ${\sf NTIME}[n]\nsubseteq{\sf DTS}[n^{c}]$, how
can we do this? In an alternation-trading proof, we attempt to establish a
contradiction from assuming ${\sf NTIME}[n]\subseteq{\sf DTS}[n^{c}]$, by
applying two lemmas which complement one another. A speedup lemma takes a
${\sf DTS}[t]$ class and places it in an alternating class with runtime
$o(t)$. A slowdown lemma takes an alternating class with runtime $t$ and
places it in a class with one less alternation and runtime $O(t^{c})$. The
Speedup Lemma dates back to Nepomnjascii [Nep] and Kannan [K84].
###### Lemma 2.3 (Speedup Lemma).
Let $a\geq 1$, $e\geq 0$ and $0\leq x\leq a$. Then
${\sf
DTISP}[n^{a},n^{e}]\subseteq(Q_{1}~{}n^{x+e})^{\max\\{1,x+e\\}}(Q_{2}~{}\log
n)^{\max\\{1,e\\}}{\sf DTISP}[n^{a-x},n^{e}],$
for $Q_{i}\in\\{\exists,\forall\\}$ where $Q_{1}\neq Q_{2}$. In particular,
${\sf DTS}[n^{a}]\subseteq(Q_{1}~{}n^{x})^{\max\\{1,x\\}}(Q_{2}~{}\log
n)^{1}{\sf DTS}[n^{a-x}].$
###### Proof 2.4.
Let $M$ use $n^{a}$ time and $n^{e}$ space. Let $y$ be an input of length $n$.
A complete description (i.e. configuration) of $M(y)$ at any step can be
described in $O(n^{e}+\log n)$ space. To simulate $M$ in
$(\exists~{}n^{x+e})^{\max\\{1,x+e\\}}(\forall~{}\log n)^{\max\\{1,e\\}}{\sf
DTISP}[n^{a-x},n^{e}]$, the algorithm $N(y)$ existentially guesses a sequence
of configurations $C_{1},\ldots,C_{n^{x}}$ of $M(x)$. Then $N(y)$ appends the
initial configuration $C_{0}$ of $M(y)$ to the beginning of the sequence, and
an accepting configuration $C_{n^{x}+1}$ to the end. $N(y)$ universally
guesses a $i\in\\{0,\ldots,n^{x}\\}$, erases all configurations except $C_{i}$
and $C_{i+1}$, then simulates $M(y)$ starting from $C_{i}$, accepting if and
only if $C_{i+1}$ is reached within $n^{a-x}$ steps. It is easy to see the
simulation is correct. The input constraints on the quantifier blocks are
satisfied since after the universal guess, the input is only $y$, $C_{i}$, and
$C_{i+1}$, which is of size $n+2n^{e+o(1)}\leq n^{\max\\{1,e\\}+o(1)}$. ∎
Observe in the above alternating simulation, the input to the final ${\sf
DTISP}$ computation is linear in $n+n^{e}$, regardless of the choice of $x$.
This is a subtle property that is exploited heavily in alternation-trading
proofs. The Slowdown Lemma is the following simple result:
###### Lemma 2.5 (Slowdown Lemma).
Let $a\geq 1$, $e\geq 0$, $a^{\prime}\geq 0$, and $b\geq 1$. If ${\sf
NTIME}[n]\subseteq{\sf DTISP}[n^{c},n^{e}]$, then for both
$Q\in\\{\exists,\forall\\}$,
$(Q~{}n^{a^{\prime}})^{b}{\sf DTIME}[n^{a}]\subseteq{\sf
DTISP}[n^{c\cdot\max\\{a,a^{\prime},b\\}},n^{e\cdot\max\\{a,a^{\prime},b\\}}].$
In particular, if ${\sf NTIME}[n]\subseteq{\sf DTS}[n^{c}]$, then
$(Q~{}n^{a^{\prime}})^{b}{\sf DTIME}[n^{a}]\subseteq{\sf
DTS}[n^{c\cdot\max\\{a,a^{\prime},b\\}}].$
|
arxiv-papers
| 2010-01-05T19:19:21 |
2024-09-04T02:49:07.500407
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ryan Williams",
"submitter": "Ryan Williams",
"url": "https://arxiv.org/abs/1001.0746"
}
|
1001.0792
|
# PSR J1907+0602: A Radio-Faint Gamma-Ray Pulsar Powering a Bright TeV Pulsar
Wind Nebula
A. A. Abdo22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA 33affiliation: National Research Council Research
Associate, National Academy of Sciences, Washington, DC 20001, USA
11affiliation: Corresponding authors: A. A. Abdo, aous.abdo@nrl.navy.mil; M.
S. E. Roberts, malloryr@gmail.com; P. M. Saz Parkinson, pablo@scipp.ucsc.edu;
K. S. Wood, Kent.Wood@nrl.navy.mil. , M. Ackermann44affiliation: 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 , M. Ajello44affiliation: 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 , L.
Baldini55affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,
I-56127 Pisa, Italy , J. Ballet66affiliation: Laboratoire AIM, CEA-
IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA Saclay, 91191
Gif sur Yvette, France , G. Barbiellini77affiliation: Istituto Nazionale di
Fisica Nucleare, Sezione di Trieste, I-34127 Trieste, Italy 88affiliation:
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy , D.
Bastieri99affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Padova, I-35131 Padova, Italy 1010affiliation: Dipartimento di Fisica “G.
Galilei”, Università di Padova, I-35131 Padova, Italy , B. M.
Baughman1111affiliation: Department of Physics, Center for Cosmology and
Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA ,
K. Bechtol44affiliation: 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 , R. Bellazzini55affiliation: Istituto Nazionale di Fisica Nucleare,
Sezione di Pisa, I-56127 Pisa, Italy , B. Berenji44affiliation: 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 , R. D. Blandford44affiliation:
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 , E. D.
Bloom44affiliation: 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 , E. Bonamente1212affiliation: Istituto Nazionale di Fisica Nucleare,
Sezione di Perugia, I-06123 Perugia, Italy 1313affiliation: Dipartimento di
Fisica, Università degli Studi di Perugia, I-06123 Perugia, Italy , A. W.
Borgland44affiliation: 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 , J. Bregeon55affiliation: Istituto Nazionale di Fisica Nucleare, Sezione
di Pisa, I-56127 Pisa, Italy , A. Brez55affiliation: Istituto Nazionale di
Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy , M.
Brigida1414affiliation: Dipartimento di Fisica “M. Merlin” dell’Università e
del Politecnico di Bari, I-70126 Bari, Italy 1515affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy , P.
Bruel1616affiliation: Laboratoire Leprince-Ringuet, École polytechnique,
CNRS/IN2P3, Palaiseau, France , T. H. Burnett1717affiliation: Department of
Physics, University of Washington, Seattle, WA 98195-1560, USA , S.
Buson1010affiliation: Dipartimento di Fisica “G. Galilei”, Università di
Padova, I-35131 Padova, Italy , G. A. Caliandro1414affiliation: Dipartimento
di Fisica “M. Merlin” dell’Università e del Politecnico di Bari, I-70126 Bari,
Italy 1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy , R. A. Cameron44affiliation: 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 , F. Camilo1818affiliation: Columbia
Astrophysics Laboratory, Columbia University, New York, NY 10027, USA , P. A.
Caraveo1919affiliation: INAF-Istituto di Astrofisica Spaziale e Fisica
Cosmica, I-20133 Milano, Italy , J. M. Casandjian66affiliation: Laboratoire
AIM, CEA-IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA
Saclay, 91191 Gif sur Yvette, France , C. Cecchi1212affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy
1313affiliation: Dipartimento di Fisica, Università degli Studi di Perugia,
I-06123 Perugia, Italy , Ö. Çelik2020affiliation: NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA 2121affiliation: Center for Research and
Exploration in Space Science and Technology (CRESST), NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA 2222affiliation: University of
Maryland, Baltimore County, Baltimore, MD 21250, USA , A.
Chekhtman22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA 2323affiliation: George Mason University, Fairfax,
VA 22030, USA , C. C. Cheung2020affiliation: NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA , J. Chiang44affiliation: 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 , S. Ciprini1212affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy
1313affiliation: Dipartimento di Fisica, Università degli Studi di Perugia,
I-06123 Perugia, Italy , R. Claus44affiliation: 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 , I. Cognard2424affiliation: Laboratoire
de Physique et Chemie de l’Environnement, LPCE UMR 6115 CNRS, F-45071 Orléans
Cedex 02, and Station de radioastronomie de Nançay, Observatoire de Paris,
CNRS/INSU, F-18330 Nançay, France , J. Cohen-Tanugi2525affiliation:
Laboratoire de Physique Théorique et Astroparticules, Université Montpellier
2, CNRS/IN2P3, Montpellier, France , L. R. Cominsky2626affiliation: Department
of Physics and Astronomy, Sonoma State University, Rohnert Park, CA
94928-3609, USA , J. Conrad2727affiliation: Department of Physics, Stockholm
University, AlbaNova, SE-106 91 Stockholm, Sweden 2828affiliation: The Oskar
Klein Centre for Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden
2929affiliation: Royal Swedish Academy of Sciences Research Fellow, funded by
a grant from the K. A. Wallenberg Foundation , S. Cutini3030affiliation:
Agenzia Spaziale Italiana (ASI) Science Data Center, I-00044 Frascati (Roma),
Italy , A. de Angelis3131affiliation: Dipartimento di Fisica, Università di
Udine and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Gruppo
Collegato di Udine, I-33100 Udine, Italy , F. de Palma1414affiliation:
Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico di Bari,
I-70126 Bari, Italy 1515affiliation: Istituto Nazionale di Fisica Nucleare,
Sezione di Bari, 70126 Bari, Italy , S. W. Digel44affiliation: 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 , B. L. Dingus3232affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA , M.
Dormody3333affiliation: Santa Cruz Institute for Particle Physics, Department
of Physics and Department of Astronomy and Astrophysics, University of
California at Santa Cruz, Santa Cruz, CA 95064, USA , E. do Couto e
Silva44affiliation: 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 , P. S. Drell44affiliation: 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 , R. Dubois44affiliation: 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 , D. Dumora3434affiliation: Université de
Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan,
33175, France 3535affiliation: CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux
Gradignan, UMR 5797, Gradignan, 33175, France , C. Farnier2525affiliation:
Laboratoire de Physique Théorique et Astroparticules, Université Montpellier
2, CNRS/IN2P3, Montpellier, France , C. Favuzzi1414affiliation: Dipartimento
di Fisica “M. Merlin” dell’Università e del Politecnico di Bari, I-70126 Bari,
Italy 1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy , S. J. Fegan1616affiliation: Laboratoire Leprince-Ringuet,
École polytechnique, CNRS/IN2P3, Palaiseau, France , W. B. Focke44affiliation:
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 , P.
Fortin1616affiliation: Laboratoire Leprince-Ringuet, École polytechnique,
CNRS/IN2P3, Palaiseau, France , M. Frailis3131affiliation: Dipartimento di
Fisica, Università di Udine and Istituto Nazionale di Fisica Nucleare, Sezione
di Trieste, Gruppo Collegato di Udine, I-33100 Udine, Italy , P. C. C.
Freire3636affiliation: Arecibo Observatory, Arecibo, Puerto Rico 00612, USA
6363affiliation: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany , Y. Fukazawa3737affiliation: Department of Physical
Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan ,
S. Funk44affiliation: 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 , P. Fusco1414affiliation: Dipartimento di Fisica “M. Merlin”
dell’Università e del Politecnico di Bari, I-70126 Bari, Italy
1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126
Bari, Italy , F. Gargano1515affiliation: Istituto Nazionale di Fisica
Nucleare, Sezione di Bari, 70126 Bari, Italy , D. Gasparrini3030affiliation:
Agenzia Spaziale Italiana (ASI) Science Data Center, I-00044 Frascati (Roma),
Italy , N. Gehrels2020affiliation: NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA 3838affiliation: University of Maryland, College
Park, MD 20742, USA , S. Germani1212affiliation: Istituto Nazionale di Fisica
Nucleare, Sezione di Perugia, I-06123 Perugia, Italy 1313affiliation:
Dipartimento di Fisica, Università degli Studi di Perugia, I-06123 Perugia,
Italy , G. Giavitto3939affiliation: Istituto Nazionale di Fisica Nucleare,
Sezione di Trieste, and Università di Trieste, I-34127 Trieste, Italy , B.
Giebels1616affiliation: Laboratoire Leprince-Ringuet, École polytechnique,
CNRS/IN2P3, Palaiseau, France , N. Giglietto1414affiliation: Dipartimento di
Fisica “M. Merlin” dell’Università e del Politecnico di Bari, I-70126 Bari,
Italy 1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
70126 Bari, Italy , F. Giordano1414affiliation: Dipartimento di Fisica “M.
Merlin” dell’Università e del Politecnico di Bari, I-70126 Bari, Italy
1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126
Bari, Italy , T. Glanzman44affiliation: 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 , G. Godfrey44affiliation: 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 , I. A. Grenier66affiliation:
Laboratoire AIM, CEA-IRFU/CNRS/Université Paris Diderot, Service
d’Astrophysique, CEA Saclay, 91191 Gif sur Yvette, France , M.-H.
Grondin3434affiliation: Université de Bordeaux, Centre d’Études Nucléaires
Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France 3535affiliation:
CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797,
Gradignan, 33175, France , J. E. Grove22affiliation: Space Science Division,
Naval Research Laboratory, Washington, DC 20375, USA , L.
Guillemot3434affiliation: Université de Bordeaux, Centre d’Études Nucléaires
Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France 3535affiliation:
CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797,
Gradignan, 33175, France , S. Guiriec4040affiliation: University of Alabama in
Huntsville, Huntsville, AL 35899, USA , Y. Hanabata3737affiliation: Department
of Physical Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima
739-8526, Japan , A. K. Harding2020affiliation: NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA , E. Hays2020affiliation: NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA , R. E. Hughes1111affiliation:
Department of Physics, Center for Cosmology and Astro-Particle Physics, The
Ohio State University, Columbus, OH 43210, USA , M. S. Jackson2727affiliation:
Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm,
Sweden 2828affiliation: The Oskar Klein Centre for Cosmoparticle Physics,
AlbaNova, SE-106 91 Stockholm, Sweden 4141affiliation: Department of Physics,
Royal Institute of Technology (KTH), AlbaNova, SE-106 91 Stockholm, Sweden ,
G. Jóhannesson44affiliation: 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 , A. S. Johnson44affiliation: 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 , T. J. Johnson2020affiliation: NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA 3838affiliation:
University of Maryland, College Park, MD 20742, USA , W. N.
Johnson22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA , S. Johnston4242affiliation: Australia Telescope
National Facility, CSIRO, Epping NSW 1710, Australia , T. Kamae44affiliation:
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 , H.
Katagiri3737affiliation: Department of Physical Sciences, Hiroshima
University, Higashi-Hiroshima, Hiroshima 739-8526, Japan , J.
Kataoka4343affiliation: Department of Physics, Tokyo Institute of Technology,
Meguro City, Tokyo 152-8551, Japan 4444affiliation: Waseda University, 1-104
Totsukamachi, Shinjuku-ku, Tokyo, 169-8050, Japan , N. Kawai4343affiliation:
Department of Physics, Tokyo Institute of Technology, Meguro City, Tokyo
152-8551, Japan 4545affiliation: Cosmic Radiation Laboratory, Institute of
Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan , M.
Kerr1717affiliation: Department of Physics, University of Washington, Seattle,
WA 98195-1560, USA , J. Knödlseder4646affiliation: Centre d’Étude Spatiale des
Rayonnements, CNRS/UPS, BP 44346, F-30128 Toulouse Cedex 4, France , M. L.
Kocian44affiliation: 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 , M. Kuss55affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Pisa, I-56127 Pisa, Italy , J. Lande44affiliation: 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 , L. Latronico55affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy , M.
Lemoine-Goumard3434affiliation: Université de Bordeaux, Centre d’Études
Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France
3535affiliation: CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan,
UMR 5797, Gradignan, 33175, France , F. Longo77affiliation: Istituto Nazionale
di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste, Italy 88affiliation:
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy , F.
Loparco1414affiliation: Dipartimento di Fisica “M. Merlin” dell’Università e
del Politecnico di Bari, I-70126 Bari, Italy 1515affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy , B.
Lott3434affiliation: Université de Bordeaux, Centre d’Études Nucléaires
Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France 3535affiliation:
CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797,
Gradignan, 33175, France , M. N. Lovellette22affiliation: Space Science
Division, Naval Research Laboratory, Washington, DC 20375, USA , P.
Lubrano1212affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Perugia, I-06123 Perugia, Italy 1313affiliation: Dipartimento di Fisica,
Università degli Studi di Perugia, I-06123 Perugia, Italy , A.
Makeev22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA 2323affiliation: George Mason University, Fairfax,
VA 22030, USA , M. Marelli1919affiliation: INAF-Istituto di Astrofisica
Spaziale e Fisica Cosmica, I-20133 Milano, Italy , M. N.
Mazziotta1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Bari, 70126 Bari, Italy , J. E. McEnery2020affiliation: NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA , C. Meurer2727affiliation: Department
of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden
2828affiliation: The Oskar Klein Centre for Cosmoparticle Physics, AlbaNova,
SE-106 91 Stockholm, Sweden , P. F. Michelson44affiliation: 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 , W. Mitthumsiri44affiliation: 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 , T.
Mizuno3737affiliation: Department of Physical Sciences, Hiroshima University,
Higashi-Hiroshima, Hiroshima 739-8526, Japan , A. A. Moiseev2121affiliation:
Center for Research and Exploration in Space Science and Technology (CRESST),
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 3838affiliation:
University of Maryland, College Park, MD 20742, USA , C. Monte1414affiliation:
Dipartimento di Fisica “M. Merlin” dell’Università e del Politecnico di Bari,
I-70126 Bari, Italy 1515affiliation: Istituto Nazionale di Fisica Nucleare,
Sezione di Bari, 70126 Bari, Italy , M. E. Monzani44affiliation: 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 , A. Morselli4747affiliation:
Istituto Nazionale di Fisica Nucleare, Sezione di Roma “Tor Vergata”, I-00133
Roma, Italy , I. V. Moskalenko44affiliation: 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 , S. Murgia44affiliation: 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 , P. L. Nolan44affiliation: 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 , J. P.
Norris4848affiliation: Department of Physics and Astronomy, University of
Denver, Denver, CO 80208, USA , E. Nuss2525affiliation: Laboratoire de
Physique Théorique et Astroparticules, Université Montpellier 2, CNRS/IN2P3,
Montpellier, France , T. Ohsugi3737affiliation: Department of Physical
Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan ,
N. Omodei55affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Pisa, I-56127 Pisa, Italy , E. Orlando4949affiliation: Max-Planck Institut für
extraterrestrische Physik, 85748 Garching, Germany , J. F.
Ormes4848affiliation: Department of Physics and Astronomy, University of
Denver, Denver, CO 80208, USA , D. Paneque44affiliation: 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 , D. Parent3434affiliation:
Université de Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan, UMR
5797, Gradignan, 33175, France 3535affiliation: CNRS/IN2P3, Centre d’Études
Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France , V.
Pelassa2525affiliation: Laboratoire de Physique Théorique et Astroparticules,
Université Montpellier 2, CNRS/IN2P3, Montpellier, France , M.
Pepe1212affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Perugia, I-06123 Perugia, Italy 1313affiliation: Dipartimento di Fisica,
Università degli Studi di Perugia, I-06123 Perugia, Italy , M. Pesce-
Rollins55affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,
I-56127 Pisa, Italy , F. Piron2525affiliation: Laboratoire de Physique
Théorique et Astroparticules, Université Montpellier 2, CNRS/IN2P3,
Montpellier, France , T. A. Porter3333affiliation: Santa Cruz Institute for
Particle Physics, Department of Physics and Department of Astronomy and
Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064,
USA , S. Rainò1414affiliation: Dipartimento di Fisica “M. Merlin”
dell’Università e del Politecnico di Bari, I-70126 Bari, Italy
1515affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126
Bari, Italy , R. Rando99affiliation: Istituto Nazionale di Fisica Nucleare,
Sezione di Padova, I-35131 Padova, Italy 1010affiliation: Dipartimento di
Fisica “G. Galilei”, Università di Padova, I-35131 Padova, Italy , P. S.
Ray22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA , M. Razzano55affiliation: Istituto Nazionale di
Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy , A.
Reimer5050affiliation: Institut für Astro- und Teilchenphysik and Institut für
Theoretische Physik, Leopold-Franzens-Universität Innsbruck, A-6020 Innsbruck,
Austria 44affiliation: 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 , O. Reimer5050affiliation: Institut für Astro- und Teilchenphysik and
Institut für Theoretische Physik, Leopold-Franzens-Universität Innsbruck,
A-6020 Innsbruck, Austria 44affiliation: 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 , T. Reposeur3434affiliation: Université
de Bordeaux, Centre d’Études Nucléaires Bordeaux Gradignan, UMR 5797,
Gradignan, 33175, France 3535affiliation: CNRS/IN2P3, Centre d’Études
Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France , S.
Ritz3333affiliation: Santa Cruz Institute for Particle Physics, Department of
Physics and Department of Astronomy and Astrophysics, University of California
at Santa Cruz, Santa Cruz, CA 95064, USA , M. S. E. Roberts22affiliation:
Space Science Division, Naval Research Laboratory, Washington, DC 20375, USA
2323affiliation: George Mason University, Fairfax, VA 22030, USA
5151affiliation: Eureka Scientific, Oakland, CA 94602, USA 11affiliation:
Corresponding authors: A. A. Abdo, aous.abdo@nrl.navy.mil; M. S. E. Roberts,
malloryr@gmail.com; P. M. Saz Parkinson, pablo@scipp.ucsc.edu; K. S. Wood,
Kent.Wood@nrl.navy.mil. , L. S. Rochester44affiliation: 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 , A. Y. Rodriguez5252affiliation:
Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, 08193 Barcelona,
Spain , R. W. Romani44affiliation: 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 , M. Roth1717affiliation: Department of
Physics, University of Washington, Seattle, WA 98195-1560, USA , F.
Ryde4141affiliation: Department of Physics, Royal Institute of Technology
(KTH), AlbaNova, SE-106 91 Stockholm, Sweden 2828affiliation: The Oskar Klein
Centre for Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden , H.
F.-W. Sadrozinski3333affiliation: Santa Cruz Institute for Particle Physics,
Department of Physics and Department of Astronomy and Astrophysics, University
of California at Santa Cruz, Santa Cruz, CA 95064, USA , D.
Sanchez1616affiliation: Laboratoire Leprince-Ringuet, École polytechnique,
CNRS/IN2P3, Palaiseau, France , A. Sander1111affiliation: Department of
Physics, Center for Cosmology and Astro-Particle Physics, The Ohio State
University, Columbus, OH 43210, USA , P. M. Saz Parkinson3333affiliation:
Santa Cruz Institute for Particle Physics, Department of Physics and
Department of Astronomy and Astrophysics, University of California at Santa
Cruz, Santa Cruz, CA 95064, USA 11affiliation: Corresponding authors: A. A.
Abdo, aous.abdo@nrl.navy.mil; M. S. E. Roberts, malloryr@gmail.com; P. M. Saz
Parkinson, pablo@scipp.ucsc.edu; K. S. Wood, Kent.Wood@nrl.navy.mil. , J. D.
Scargle5353affiliation: Space Sciences Division, NASA Ames Research Center,
Moffett Field, CA 94035-1000, USA , C. Sgrò55affiliation: Istituto Nazionale
di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy , E. J.
Siskind5454affiliation: NYCB Real-Time Computing Inc., Lattingtown, NY
11560-1025, USA , D. A. Smith3434affiliation: Université de Bordeaux, Centre
d’Études Nucléaires Bordeaux Gradignan, UMR 5797, Gradignan, 33175, France
3535affiliation: CNRS/IN2P3, Centre d’Études Nucléaires Bordeaux Gradignan,
UMR 5797, Gradignan, 33175, France , P. D. Smith1111affiliation: Department of
Physics, Center for Cosmology and Astro-Particle Physics, The Ohio State
University, Columbus, OH 43210, USA , G. Spandre55affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy , P.
Spinelli1414affiliation: Dipartimento di Fisica “M. Merlin” dell’Università e
del Politecnico di Bari, I-70126 Bari, Italy 1515affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy , M. S.
Strickman22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA , D. J. Suson5555affiliation: Department of
Chemistry and Physics, Purdue University Calumet, Hammond, IN 46323-2094, USA
, H. Tajima44affiliation: 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 , H. Takahashi3737affiliation: Department of Physical Sciences, Hiroshima
University, Higashi-Hiroshima, Hiroshima 739-8526, Japan , T.
Tanaka44affiliation: 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 , J. B. Thayer44affiliation: 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 , J. G. Thayer44affiliation: 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 , G. Theureau2424affiliation: Laboratoire
de Physique et Chemie de l’Environnement, LPCE UMR 6115 CNRS, F-45071 Orléans
Cedex 02, and Station de radioastronomie de Nançay, Observatoire de Paris,
CNRS/INSU, F-18330 Nançay, France , D. J. Thompson2020affiliation: NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA , L.
Tibaldo99affiliation: Istituto Nazionale di Fisica Nucleare, Sezione di
Padova, I-35131 Padova, Italy 66affiliation: Laboratoire AIM, CEA-
IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA Saclay, 91191
Gif sur Yvette, France 1010affiliation: Dipartimento di Fisica “G. Galilei”,
Università di Padova, I-35131 Padova, Italy , O. Tibolla5656affiliation: Max-
Planck-Institut für Kernphysik, D-69029 Heidelberg, Germany , D. F.
Torres5757affiliation: Institució Catalana de Recerca i Estudis Avançats
(ICREA), Barcelona, Spain 5252affiliation: Institut de Ciencies de l’Espai
(IEEC-CSIC), Campus UAB, 08193 Barcelona, Spain , G. Tosti1212affiliation:
Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia,
Italy 1313affiliation: Dipartimento di Fisica, Università degli Studi di
Perugia, I-06123 Perugia, Italy , A. Tramacere44affiliation: 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 5858affiliation: Consorzio
Interuniversitario per la Fisica Spaziale (CIFS), I-10133 Torino, Italy , Y.
Uchiyama5959affiliation: Institute of Space and Astronautical Science, JAXA,
3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan 44affiliation: 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 , T. L.
Usher44affiliation: 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 , A. Van Etten44affiliation: 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 , V. Vasileiou2020affiliation: NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA 2121affiliation: Center for Research and Exploration
in Space Science and Technology (CRESST), NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA 2222affiliation: University of Maryland, Baltimore
County, Baltimore, MD 21250, USA , C. Venter2020affiliation: NASA Goddard
Space Flight Center, Greenbelt, MD 20771, USA 6060affiliation: North-West
University, Potchefstroom Campus, Potchefstroom 2520, South Africa , N.
Vilchez4646affiliation: Centre d’Étude Spatiale des Rayonnements, CNRS/UPS, BP
44346, F-30128 Toulouse Cedex 4, France , V. Vitale4747affiliation: Istituto
Nazionale di Fisica Nucleare, Sezione di Roma “Tor Vergata”, I-00133 Roma,
Italy 6161affiliation: Dipartimento di Fisica, Università di Roma “Tor
Vergata”, I-00133 Roma, Italy , A. P. Waite44affiliation: 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 , P. Wang44affiliation: 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 , K.
Watters44affiliation: 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 , B. L. Winer1111affiliation: Department of Physics, Center for Cosmology
and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA
, M. T. Wolff22affiliation: Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA , K. S. Wood22affiliation: Space Science Division,
Naval Research Laboratory, Washington, DC 20375, USA 11affiliation:
Corresponding authors: A. A. Abdo, aous.abdo@nrl.navy.mil; M. S. E. Roberts,
malloryr@gmail.com; P. M. Saz Parkinson, pablo@scipp.ucsc.edu; K. S. Wood,
Kent.Wood@nrl.navy.mil. , T. Ylinen4141affiliation: Department of Physics,
Royal Institute of Technology (KTH), AlbaNova, SE-106 91 Stockholm, Sweden
6262affiliation: School of Pure and Applied Natural Sciences, University of
Kalmar, SE-391 82 Kalmar, Sweden 2828affiliation: The Oskar Klein Centre for
Cosmoparticle Physics, AlbaNova, SE-106 91 Stockholm, Sweden , M.
Ziegler3333affiliation: Santa Cruz Institute for Particle Physics, Department
of Physics and Department of Astronomy and Astrophysics, University of
California at Santa Cruz, Santa Cruz, CA 95064, USA
###### Abstract
We present multiwavelength studies of the 106.6 ms $\gamma$-ray pulsar PSR
J1907+06 near the TeV source MGRO J1908+06. Timing observations with Fermi
result in a precise position determination for the pulsar of R.A. =
19h07m54$\fs$7(2), decl. = +06∘02′16(2)′′ placing the pulsar firmly within the
TeV source extent, suggesting the TeV source is the pulsar wind nebula of PSR
J1907+0602. Pulsed $\gamma$-ray emission is clearly visible at energies from
100 MeV to above 10 GeV. The phase-averaged power-law index in the energy
range $E>0.1$ GeV is $\Gamma=1.76\pm 0.05$ with an exponential cutoff energy
$E_{c}=3.6\pm 0.5$ GeV. We present the energy-dependent $\gamma$-ray pulsed
light curve as well as limits on off-pulse emission associated with the TeV
source. We also report the detection of very faint (flux density of
$\simeq\,3.4\,\mu$Jy) radio pulsations with the Arecibo telescope at 1.5 GHz
having a dispersion measure DM = 82.1 $\pm$ 1.1 cm-3pc. This indicates a
distance of $3.2\pm 0.6$ kpc and a pseudo-luminosity of
$L_{1400}\,\simeq\,0.035$ mJy kpc2. A Chandra ACIS observation revealed an
absorbed, possibly extended, compact ($\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}4^{\prime\prime}$) X-ray source
with significant non-thermal emission at R.A. = 19h07m54$\fs$76, decl. =
+06°02′14.6′′ with a flux of $2.3^{+0.6}_{-1.4}\times 10^{-14}{\rm erg}\,{\rm
cm}^{-2}{\rm s}^{-1}$. From archival ASCA observations, we place upper limits
on any arcminute scale 2–10 keV X-ray emission of $\sim 1\times 10^{-13}{\rm
erg}\,{\rm cm}^{-2}{\rm s}^{-1}$. The implied distance to the pulsar is
compatible with that of the supernova remnant G40.5$-$0.5, located on the far
side of the TeV nebula from PSR J1907+0602, and the S74 molecular cloud on the
nearer side which we discuss as potential birth sites.
pulsars: individual: PSR J1907+0602 — gamma rays: observations
## 1 Introduction
The TeV source MGRO J1908+06 was discovered by the Milagro collaboration at a
median energy of 20 TeV in their survey of the northern Galactic Plane (Abdo
et al., 2007) with a flux $\sim 80$% of the Crab at these energies. It was
subsequently detected in the 300 GeV – 20 TeV range by the HESS (Aharonian et
al., 2009) and VERITAS (Ward, 2008) experiments. The HESS observations show
the source HESS J1908+063 to be clearly extended, spanning $\sim$0.3∘ of a
degree on the sky with hints of energy-dependent substructure. A decade
earlier Lamb & Macomb (1997) cataloged a bright source of GeV emission from
the EGRET data, GeV J1907+0557, which is positionally consistent with MGRO
J1908+06. It is near, but inconsistent with, the third EGRET catalog (Hartman
et al., 1999) source 3EG J1903+0550 (Roberts et al., 2001). The Large Area
Telescope (LAT) (Atwood et al., 2009) aboard the Fermi Gamma-Ray Space
Telescope has been operating in survey mode since soon after its launch on
2008 June 11, carrying out continuous observations of the GeV sky. The Fermi
Bright Source List (Abdo et al., 2009b), based on 3 months of survey data,
contains 0FGL J1907.5+0602 which is coincident with GeV J1907+0557. The 3EG
J1903+0550 source location confidence contour stretches between 0FGL
J1907.5+0602 and the nearby source 0FGL J1900.0+0356, suggesting it was a
conflation of the two sources.
The Fermi LAT collaboration recently reported the discovery of 16 previously-
unknown pulsars by using a time differencing technique on the LAT photon data
above 300 MeV (Abdo et al., 2009a). 0FGL J1907.5+0602 was found to pulse with
a period of 106.6 ms, have a spin-down energy of $\sim 2.8\times 10^{36}$ erg
s-1, and was given a preliminary designation of PSR J1907+06. In this paper we
derive a coherent timing solution using 14 months of data which yields a more
precise position for the source, allowing detailed follow-up at other
wavelengths, including the detection of radio pulsations using the Arecibo
305-m radio telescope. Energy resolved light curves, the pulsed spectrum, and
off-pulse emission limits at the positions of the pulsar and PWN centroid are
presented. We then report the detection of an X-ray counterpart with the
Chandra X-ray Observatory and an upper limit from ASCA. Finally, we discuss
the pulsar’s relationship to the TeV source and to the potential birth sites
SNR G40.5$-$0.5 and the S74 Hii region.
We present a coherent timing solution with a precisely determined position,
energy resolved light curves, pulsed spectra, and upper limits to any off-
pulse emission associated with MGRO J1908+06. We then present results of a
search for radio and X-ray counterparts, and discuss the pulsar’s relationship
to other sources.
## 2 Gamma-ray Pulsar Timing and Localization
The discovery and initial pulse timing of PSR J1907+06 was reported by Abdo et
al. (2009a). The source position used in that analysis (R.A. = 286.965∘, Decl.
= 6.022∘) was derived from an analysis of the measured directions of LAT-
detected photons in the on-pulse phase interval from observations made from
2008 August 4 through December 25. Here, we make use of a longer span of data
and also apply improved analysis methods to derive an improved timing
ephemeris for the pulsar as well as a more accurate source position.
For the timing and localization analysis, we selected “diffuse” class photons
(events that passed the tightest background rejection criteria (Atwood et al.,
2009)) with zenith angle $<105^{\circ}$ as is standard practice and chose the
minimum energy and extraction radius to optimize the significance of
pulsations. We accepted photons with $E>200$ MeV from within a radius of 0.7∘
of the nominal source direction. We corrected these photon arrival times to
terrestrial time (TT) at the geocenter using the LAT Science Tool
111http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/index.html
gtbary in its geocenter mode.
We fitted a timing model using Tempo2(Hobbs et al., 2006) to 23 pulse times of
arrival (TOAs) covering the interval 2008 June 30 to 2009 September 18. We
note that during the on-orbit checkout period (before 2008 August 4) several
instrument configurations were tested that affected the energy resolution and
event reconstruction but had no effect on the LAT timing. To determine the
TOAs, we generated pulse profiles by folding the photon times according to a
provisional ephemeris using polynomial coefficients generated by Tempo2 in its
predictive mode (assuming a fictitious observatory at the geocenter). The TOAs
were measured by cross correlating each pulse profile with a kernel density
template that was derived from fitting the full mission dataset (Ray et al.,
2009). Finally, we fitted the TOAs to a timing model that included position,
frequency, and frequency derivative. The resulting timing residuals are
$0.4\,ms$ and are shown in Figure 1. The best-fit model is displayed in Table
LABEL:tab:ephemeris. The numbers in parentheses are the errors in the last
digit of the fitted parameters. The errors are statistical only, except for
the position error, as described below. The derived parameters of $\dot{E}$,
$B$, and $\tau_{c}$ are essentially unchanged with respect to those reported
by Abdo et al. (2009a), but the position has moved by 1.2′.
The statistical error on the position fit is $<1$′′; however, this is an
underestimate of the true error. For example, with only one year of data,
timing noise can perturb the position fit. We have performed a Monte Carlo
analysis of these effects by simulating fake residuals using the fake plugin
for Tempo2. We generated models with a range of frequency second derivatives
($\pm 2\times 10^{-22}\mathrm{s}^{-3}$, the allowed magnitude for $\ddot{\nu}$
in our fits) to simulate the effects of timing noise and fitted them to timing
models. Based on these simulations, we assigned an additional systematic error
on the position of 2′′, which we added in quadrature to the statistical error
in Table LABEL:tab:ephemeris. As a result of the improved position estimate
provided by this timing analysis, we have adopted a more precise name for the
pulsar of PSR J1907+0602.
## 3 Detection of Radio Pulsations
To search for radio pulsations, we observed the timing position of PSR
J1907+0602 with the L-wide receiver on the Arecibo 305-m radio telescope. On
2009 August 21 we made a 55-minute pointing with center frequency 1.51 GHz and
total bandwidth of 300 MHz, provided by three Wideband Arecibo Pulsar
Processors (WAPPs, Dowd et al. (2000)), each individually capable of
processing 100 MHz. We divided this band into 512-channel spectra accumulated
every 128 $\mu$s. The small positional uncertainty of PSR J1907+0602 derived
from the LAT timing means that a single Arecibo pointing covers the whole
region of interest.
After excising strong sources of radio-frequency interference with rfifind,
one of the routines of the PRESTO signal analysis package (Ransom et al.,
2002), we performed a search by folding the raw data with the Fermi timing
model into 128-bin pulse profiles. We then used the PRESTO routine prepfold to
search trial dispersion measures between 0 and 1000 pc cm-3. We found a pulsed
signal with a signal-to-noise ratio $S/N=9.4$222This was estimated using
another software package, SIGPROC (a package developed by Duncan Lorimer, see
http://sigproc.sourceforge.net/), which processes the bands separately and
produced S/N of 4.2, 6.3 and 5.5 for the WAPPs centered at 1410, 1510 and 1610
MHz. Although S/N = 9.4 is close to the detection threshold for pulsars in a
blind search, it is much more significant in this case because of the reduced
number of trials in this search relative to a blind search. and duty cycle of
about 0.03 at a dispersion measure $DM=82.1\pm 1.1$ cm-3 pc. This value was
estimated by dividing the detection data into 3 sub-bands and making TOAs for
each sub-band and fitting for the DM with tempo.
We applied the same technique for 4 different time segments of 12.5 minutes
each and created a time of arrival for each of them. We then estimated the
barycentric periodicity of the detected signal from these times of arrival.
This differs from the periodicity predicted by the LAT ephemeris for the time
of the observation by $(-0.000005\pm 0.000020)$ ms, i.e., the signals have the
same periodicity.
Subsequent radio observations showed that the phase of the radio pulses is
exactly as predicted by the LAT ephemeris, apart from a constant phase offset
(depicted in Figure 2)
A confirmation observation with twice the integration time (1.8 hr) was made
on 2009 September 4. The radio profile is shown in the bottom panel of Figure.
2 with an arbitrary intensity scale. The pulsar is again detected with S/N of
3.4, 5.1, 7.3 and 8.6 at 1170, 1410, 1510 and 1610 MHz. The higher S/N at the
higher frequencies suggest a positive spectral index, similar to what has been
observed for PSR J1928+1746 (Cordes et al., 2006). However, this might instead
be due to scattering degrading the S/N at the lowest frequencies— for the band
centered at 1610 MHz the pulse profile is distinctively narrower (about 2% at
50% power) than at 1410 or 1510 MHz (about 3%). At 1170 MHz the profile is
barely detectable but very broad. This suggests an anomalously large
scattering timescale for the DM of the pulsar. Observations at higher
frequencies will settle the issue of the positive spectral index. For the 300
MHz centered at 1410 MHz, where the detection is clear, we obtain a total S/N
of 12.4.
With an antenna $T_{\mathrm{sys}}=33$ K (given by the frequency-dependent
antenna temperature of 25–27 K off the plane of the Galaxy plus 6 K of
Galactic emission in the specific direction of the pulsar Haslam et al.,
1982), Gain = 10.5 K Jy-1 and 2 polarizations, and an inefficiency factor of
12% due to the 3-level sampling of the WAPP correlators, we obtain for the
first detection a flux density at 1.4 GHz of $S_{1400}\,\simeq\,4.1\,\mu$Jy
and for the second detection $S_{1400}\,\simeq\,3.1\,\mu$Jy. These values are
consistent given the large relative uncertainties in the S/N estimates and the
varying effect of radio frequency interference; at this DM, scintillation is
not likely to cause a large variation in the flux density.
The time-averaged flux density is $\simeq\,3.4\,\mu$Jy. Using the NE2001 model
for the electron distribution in the Galaxy (Cordes & Lazio, 2002), we obtain
from the pulsar’s position and DM a distance of 3.2 kpc with a nominal error
of 20%(Cordes & Lazio, 2002). The time-averaged flux density thus corresponds
to a pseudo-luminosity $L_{1400}\simeq 0.035$ mJy kpc2. This is fainter than
the least luminous young pulsar in the ATNF catalog (PSR J0205+6449, with a
1.4 GHz pseudo-luminosity of 0.5 mJy kpc2). It is, however, more luminous than
the radio pulsations discovered through a deep search of another pulsar first
discovered by Fermi, PSR J1741$-$2054 which has $L_{1400}\sim 0.025$ mJy kpc2
(Camilo et al., 2009). These two detections clearly demonstrate that some
pulsars, as seen from the Earth, can have extremely low apparent radio
luminosities; i.e., similarly deep observations of other $\gamma$-ray selected
pulsars might detect additional very faint radio pulsars. We note that these
low luminosities, which may well be the result of only a faint section of the
radio beam crossing the Earth, are much lower than what has often been termed
“radio quiet” in population synthesis models used to estimate the ratio of
“radio-loud” to “radio quiet” $\gamma$-ray pulsars (eg. Gonthier et al.,
2004).
## 4 Energy-Dependent Gamma-ray Pulse Profiles
The pulse profile and spectral results reported in this paper use the survey
data collected with the LAT from 2008 August 4 through 2009 September 18. We
selected “diffuse” class photons (see §2) with energies $E>100$ MeV and, to
limit contamination from photons from Earth’s limb, with zenith angle
$<105^{\circ}$.
To explore the dependence of the pulse profile on energy, we selected an
energy-dependent region of interest (ROI) with radius $\theta=0.8\times
E^{-0.75}$ degrees, but constrained not to be outside the range
[$0.35^{\circ},1.5^{\circ}$]. We chose the upper bound to minimize the
contribution from nearby sources and Galactic diffuse emission. The lower
bound was selected in order to include more photons from the wings of the
point spread function (PSF) where the extraction region is small enough to
make the diffuse contribution negligible. Figure 2 shows folded light curves
of the pulsar in 32 constant-width bins for different energy bands. We use the
centroid of the 1.4 GHz radio pulse profile to define phase 0.0. Two rotations
are shown in each case. The top panel of the figure shows the folded light
curve for photons with $E>0.1$ GeV. The $\gamma$-ray light curve shows two
peaks, P1 at phase 0.220 $\pm$ 0.002 which determines the offset with the
radio peak, $\delta$. The second peak in the $\gamma$-ray, P2, occurs at phase
0.580 $\pm$ 0.003. The phase separation between the two peaks is
$\Delta=0.360\pm 0.004$. The radio lead $\delta$ and gamma peak separation
$\Delta$ values are in good agreement with the correlation predicted for outer
magnetosphere models, (Romani & Yadigaroglu, 1995) and observed for other
young pulsars (Figure 3 of Abdo et al. (2009c)).
Pulsed emission from the pulsar is clearly visible for energies $E>$ 5 GeV
with a chance probability of $\sim 4\times 10^{-8}$. Pulsed emission is
detected for energies above 10 GeV with a confidence level of 99.8%. We have
measured the integral and widths of the peaks as a function of energy and have
found no evidence for significant evolution in shape or P1/P2 ratio with
energy. We note that the pulsar is at low Galactic latitude
($b\sim-0.89^{\circ}$ ) where the Galactic $\gamma$-ray diffuse emission is
bright ( it has not been subtracted from the light curves shown.)
Figure 3 shows the observed LAT counts map of the region around PSR
J1907+0602. We defined the “on” pulse as pulse phases 0.12 $\leq\phi\leq$ 0.68
and the “off” pulse as its complement (0.0 $\leq\phi<$ 0.12 and 0.68
$<\phi\leq$ 1.0). We produced on-pulse (left panel) and off-pulse (right
panel) images, scaling the off-pulse image by 1.27. The figure indicates the
complexity of the region that must be treated in spectral fitting. Besides the
pulsar there are multiple point sources, Galactic, and extragalactic diffuse
contributions.
## 5 Energy Spectrum
The phase-averaged flux of the pulsar was obtained by performing a maximum
likelihood spectral analysis using the Fermi LAT science tool gtlike. Starting
from the same data set described in §4, we selected photons from an ROI of 10
degrees around the pulsar position. Sources from a preliminary version (based
on 11 months of data) of the first Fermi LAT $\gamma-$ray catalog (Abdo et
al., 2009c) that are within 15 degree ROI around the pulsar were modeled in
this analysis. Spectra of sources farther away than 5∘ from the pulsar were
fixed at the cataloged values. Sources within 5∘ degrees of the pulsar were
modeled with a simple power law. For each of the sources in the 5∘ degree
region around the pulsar, we fixed the spectral index at the value in the
catalog and fitted for the normalization. Two sources that are at a distance
$>5^{\circ}$ showed strong emission and were treated the same way as the
sources within 5∘. The Galactic diffuse emission (gll_iemv02) and the
extragalactic diffuse background (isotropic_iem_v02) were modeled as
well333Descriptions of the models are available at
http://fermi.gsfc.nasa.gov/.
The assumed spectral model for the pulsar is an exponentially cut-off power
law: $dN/dE=N_{o}\mbox{ }(E/E_{o})^{-\Gamma}\exp(-E/E_{c}).$ The resulting
spectrum gives the total emission for the pulsar assuming that the
$\gamma$-ray emission is 100% pulsed. The unbinned gtlike fit, using P6_v3
instrument response functions (Atwood et al., 2009), for the energy range
$E\geq 100$ MeV gives a phase-averaged spectrum of the following form:
$\frac{dN}{dE}=(7.06\pm 0.43_{stat.}\,+(^{+0.004}_{-0.064})_{sys.})\times
10^{-11}E^{-\Gamma}e^{-E/E_{\mathrm{c}}}\,\gamma\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{MeV}^{-1}$
(1)
where the photon index $\Gamma=1.76\pm
0.05_{stat.}+\,(^{+0.271}_{-0.287})_{sys.}$ and the cutoff energy
$E_{c}=3.6\pm 0.5_{stat.}\,+(^{+0.72}_{-0.36})_{sys.}$ GeV.
The integrated energy flux from the pulsar in the energy range $E\geq 100$ MeV
is $F_{\gamma}=(3.12\pm 0.15_{stat.}\,+(^{+0.16}_{-0.15})_{sys.})\times
10^{-10}\mathrm{ergs\;cm}^{-2}\mathrm{s}^{-1}$. This yields a $\gamma$-ray
luminosity of $L_{\gamma}=4\pi f_{\Omega}F_{\gamma}d^{2}=3.8\times
10^{35}f_{\Omega}d_{3.2}^{2}\mathrm{ergs\;s}^{-1}$ above 100 MeV, where
$f_{\Omega}$ is an effective beaming factor and $d_{3.2}=d/(3.2){\rm kpc}$.
This corresponds to an efficiency of
$\eta=L_{\gamma}/\dot{E}=0.13F_{\gamma}d_{3.2}^{2}$ for conversion of spin-
down power into $\gamma-$ray emission in this energy band.
We set a 2$\sigma$ flux upper limit on $\gamma$-ray emission from the pulsar
in the off-pulse part of $F_{off}<8.31\times 10^{-8}\;{\rm cm^{-2}}\;{\rm
s^{-1}}$. In addition to the $\gamma$-ray spectrum from the point-source
pulsar PSR J1907+0602, we measured upper limits on $\gamma$-ray flux from the
extended source HESS J1908+063 in the energy range 0.1–25 GeV. We performed
binned likelihood analysis using the Fermi Science Tool gtlike. In this
analysis we assumed an extended source with gaussian width of 0.3∘ and
$\gamma$-ray spectral index of $-2.1$ at the location of the HESS source. The
upper limits suggest that the spectrum of HESS J1908+063 has a low energy
turnover between 20 GeV and 300 GeV. Figure 4 shows the phase-averaged
spectral energy distribution for PSR J1907+0602 (green circles). On the same
figure we show data points from HESS for the TeV source HESS J1908+063 (blue
circles) and the 2$\sigma$ upper limits from Fermi for emission from this TeV
source. Figure 5 shows an off pulse residual map of the region around PSR
J1907+0602. The timing position of the pulsar is marked by the green cross.
The 5$\sigma$ contours from Milagro (outer) and HESS (inner) are overlaid. As
can be seen from the residual map, there is no gamma-ray excess at the
location of either the pulsar or the PWN.
## 6 X-ray Counterpart
A 23 ks ASCA GIS exposure of the EGRET source GeV J1907+0557 revealed an $\sim
8^{\prime}\times 15^{\prime}$ region of possible extended hard emission
surrounding two point-like peaks lying $\sim 15^{\prime}$ to the southwest of
PSR J1907+0602 (Roberts et al., 2001) and no other significant sources in the
$44^{\prime}$ ASCA FOV. A 10 ks Chandra ACIS-I image of the ASCA emission
(ObsID 7049) showed it to be dominated by a single hard point source, CXOU
J190718.6+054858 with no compact nebular structure and just a hint of the
several arcminute-scale emission seen by ASCA. CXOU J190718.6+054858 seemed to
turn off for $\sim 2$ ks during the Chandra exposure, suggesting that it may
be a binary of some sort or else a variable extragalactic source. There is no
obvious optical counterpart in the digital sky survey optical or 2MASS near
infrared images, nor in a I band image taken with the 2.4m Hiltner telescope
at MDM (Jules Halpern, private communication). This strongly suggests that it
is not a nearby source. An absorbed power law fits the spectrum of this source
well, with absorption $n_{H}=1.8^{+1.3}_{-0.9}\times 10^{22}{\rm cm}^{-2}$
(90% confidence region), a photon spectral index $\Gamma=0.9^{+0.6}_{-0.4}$,
and an average 2–10 keV flux of $4.4^{+0.7}_{-1.8}\times 10^{-13}{\rm
erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$ (68% confidence region). The fit absorption
is similar to the estimated total Galactic absorption from the HEASARC $nH$
tool of $1.6\times 10^{22}{\rm cm}^{-2}$ based on the Dickey and Lockman
(1990) HI survey (Dickey & Lockman, 1990), suggesting that an $n_{H}$ of $\sim
2\times 10^{22}{\rm cm}^{-2}$ is a reasonably conservative estimate of
interstellar absorption for sources deep in the plane along this line of
sight.
The timing position of LAT PSR J1907+0602 is in the central $20^{\prime}$ of
the ASCA GIS FOV (Figure 6). There is no obvious emission in the ASCA image at
the pulsar position. Using the methodology of Roberts et al. (2001), a 24
pixel radius extraction region ($\sim 6^{\prime}$), and assuming an absorbed
power law spectrum with $n_{H}=2\times 10^{22}{\rm cm}^{-2}$ and $\Gamma=1.5$,
we place a 90% confidence upper limit on the 2–10 keV flux $F_{x}<5\times
10^{-14}{\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$. This suggests that for any
reasonable absorption, the total unabsorbed X-ray flux from the pulsar plus
any arcminute-scale nebula is less than $10^{-13}{\rm erg}\,{\rm
cm}^{-2}\,{\rm s}^{-1}$.
PSR J1907+0602 was well outside of the FOV of the first Chandra observation,
and so we proposed for an observation centered on the pulsar. We obtained a 19
ks exposure with the ACIS-S detector (ObsID 11124). The time resolution of the
ACIS-S detector on board Chandra does not allow for pulse studies. The only
source within an arcminute of the timing position and the brightest source in
the FOV of the S3 chip is shown in Figure 7. It is well within errors of the
timing position. Examination of the X-ray image in different energy bands
showed virtually no detected flux below $\sim 1$keV and significant flux above
2.5 keV, suggesting a non-thermal emission mechanism for much of the flux. A
comparison of the spatial distribution of counts between 0.75 keV and 2 keV to
those between 2keV and 8keV shows some evidence for spatial extent beyond the
point spread function for the harder emission but not for the softer emission.
This would be consistent with an interpretation as predominantly absorbed but
thermal emission from a neutron star surface surrounded by non-thermal
emission from a compact pulsar wind nebula, which is the typical situation for
young pulsars (see Kaspi et al., 2006, and references therein). We plot the
Chandra 0.75-2keV, 2-8keV, and 0.75-8 keV images with an ellipse showing the
timing position uncertainty, and a circle with a radius of
$0.8^{\prime\prime}$. From a modeled PSF, we estimate 80% of the counts should
be contained within this circle. While this seems to be the case for the soft
image, only roughly half the counts in the harder image are contained within
that radius. With only $\sim 12$ source counts in the 0.75-2 keV image within
$6^{\prime\prime}$ and $\sim 30$ source counts in the 2-8 keV image,
quantitative statements about source size and spectrum are difficult to make.
We obtain a best fit position for the nominal point source of R.A. =
19h07m54$\fs$76, decl. = +06°02′14.6′′ and estimated error of 0.7′′ (an
additional 0.1′′ centroid fitting uncertainty added to the nominal Chandra
0.6′′ uncertainty). Using a $6^{\prime\prime}$ radius extraction region and an
annulus between 6′′ and 24′′ for background, we extracted source and
background spectra and fit them within XSPEC (Figure 8). A simple power law
plus absorption model fit the data well in the energy range 2-10 KeV, with
best fit values $n_{H}=1.3\times 10^{22}{\rm cm}^{-2}$ and $\Gamma=1.6$, with
a total flux of $2.3^{+0.6}_{-1.4}\times 10^{-14}{\rm erg}\,{\rm cm}^{-2}{\rm
s}^{-1}$. The low count rates and covariance between the absorption and photon
index meant the spectral parameters could not be simultaneously meaningfully
constrained. Fixing the spectral index $\Gamma=1.6$, a typical value for
compact pulsar wind nebulae (Kaspi et al., 2006), we obtain a 90% confidence
region for the absorption of $0.7-2.5\times 10^{22}{\rm cm}^{-2}$, consistent
with a source a few kilo parsecs or more away and with CXOU J190718.6+054858
discussed above. We note that with such an absorption a significant thermal
component in the below 2 keV emission is neither required nor ruled out by the
spectral fitting.
## 7 Discussion
The dispersion measure from the radio detection suggests a distance of 3.2
kpc, with a nominal error of 20%. However, there are many outliers to the DM
error distribution, although the largest fractional errors tend to be from
pulsars at high Galactic latitudes or very low DMs (Deller et al., 2009;
Chatterjee et al., 2009). For PSR J1907+0602, at a latitude $b=-0.9^{\circ}$
with a moderate DM, the distance estimate is likely to be reasonable. Since
the apparent $\gamma$-ray pulsed efficiency in the Fermi pass-band is well
above the median for other gamma-ray pulsars in Abdo et al. (2009d) (13%
compared to 7.5%), it is worth checking secondary distance indicators to see
if the DM measure could be a significant overestimate of the true distance. We
can use the X-ray observations of PSR J1907+0602 to do this. Several authors
have noted a correlation between the X-ray luminosity of young pulsars and
their spin-down power (eg. Saito 1998, Possenti et al. 2002, Li, Lu and Li
2007). Most of these have the problem of using X-ray fluxes derived from the
literature using a variety of instruments with no uniform way of choosing
spectral extraction regions. This can be especially problematic with Chandra
data, since faint, arcminute scale emission can easily be overlooked. We
compare our ASCA GIS upper limits to Figure 1 of Cheng, Taam and Wang (2004)
who used only ASCA GIS data to derive their X-ray luminosity relationships. We
see that the typical X-ray luminosity in the ASCA band for a pulsar with
$\dot{E}=2\times 10^{36}{\rm erg}\,{\rm s}^{-1}$ is $L_{x}\sim
10^{33}-10^{34}{\rm erg}\,{\rm s}^{-1}$ with all of the pulsars used in their
analysis with $\dot{E}>10^{36}{\rm erg}\,{\rm s}^{-1}$ having
$L_{x}>10^{32}{\rm erg}\,{\rm s}^{-1}$. From these values and the ASCA upper
limit, we derive a lower limit for the distance to LAT PSR J1907+0602 of $\sim
3$ kpc. From Figure 2 of Li, Lu and Li (2007), who used XMM-Newton and Chandra
derived values, we see we can expect the luminosity to be between $\sim
10^{31.5}-10^{34.5}{\rm erg}\,{\rm s}^{-1}$. From our detection with Chandra,
we again estimate a lower distance limit of $\sim 3$ kpc. The “best guess”
estimate from their relationship would result in a distance of $\sim 13$kpc.
We note that if we assume the pulsed emission to be apparently isotropic (i.e.
$f_{\Omega}=1$ as simple outer gap models suggest should approximately be the
case, see Watters et al. (2009)), a distance of 9 kpc would result in 100%
$\gamma$-ray efficiency.
The derived timing position of PSR J1907+0602 is well inside the extended HESS
source, although $\sim 14^{\prime}$ southwest of the centroid. The TeV source
is therefore plausibly the wind nebula of PSR J1907+0602. The physical size of
this nebula is then $\lower 2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}40$ pc,
and the integrated luminosity above 1 TeV is $\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}40\%$ that of the Crab, and in the
MILAGRO band ($\sim 20$ TeV) at least twice that of the Crab. There is a hint
of some spatial dependence of the TeV spectrum in the HESS data, with the
harder emission ($>2.5$ TeV) peaking nearer the pulsar than the softer
emission (Aharonian et al., 2009). If confirmed, this would be consistent with
the hardening of the TeV emission observed towards PSR B1823$-$13, thought to
be the pulsar powering HESS J1825$-$137 (Aharonian et al., 2006). This latter
pulsar has a spin period, characteristic age, and spin-down energy similar to
PSR J1907+0602, and is also located near the edge of its corresponding TeV
nebula. We also note that HESS J1825$-$137 subtends $\sim 1^{\circ}$ on the
sky and has a flux level above 1 TeV of around 20% of the Crab. While the
overall spectrum of HESS J1825$-$137 is somewhat softer than the spectrum of
HESS J1908+063, near the pulsar its spectrum is similarly hard. At a distance
of $\sim 4$ kpc, HESS J1825$-$137 has a luminosity similar to the Crab TeV
nebula, but with a much larger physical size of $\sim 70$pc. Given the
distance implied above and a flux above 1 TeV $\sim 17$% of the Crab, HESS
J1908+063 is similar in size and luminosity to HESS J1825$-$137.
At 20 TeV, HESS J1908+063 has a flux $\sim$80% of the Crab, and so at a
distance $\lower 2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}1.5$ times that of
the Crab, is much more luminous at the highest energies. This is because there
is no sign of a high-energy cutoff or break, as is seen in many other TeV
nebulae. Aharonian et al. (2009) place a lower limit of 19.1 TeV on any
exponential cutoff to the spectrum. This implies that either the spectrum is
uncooled due to a very low nebular magnetic field ($\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}3\mu$G, see, eg. de Jager (2008)),
an age much less than the characteristic age of 19.5 kyr, or else there is a
synchrotron cooling break below the HESS band.
Our upper limits above a few GeV (Figure 4) requires there to be a low energy
turnover between 20 GeV and 300 GeV. Given the nominal PWN spectrum, we
constrain the overall PWN flux to be $\leq\,25\%$ of that of the pulsar. If
only the HESS band is considered, and assuming the DM distance, the TeV
luminosity $L_{\mathrm{PWN}}=5-8\%\dot{E}$. However, since the TeV emission is
generally thought to come from a relic population of electrons the luminosity
is likely a function of the spin-down history of the pulsar rather than the
current spin-down luminosity (eg. de Jager, 2008). These numbers support
consistency of the association of the TeV source with the pulsar, in the weak
sense of not being discrepant with other similar systems.
### 7.1 On the possible association with SNR G40.5$-$0.5
The bulk of HESS J1908+063 is between PSR J1907+0602 and the young radio SNR
G40.5$-$0.5, suggesting a possible association. The distance estimate ($\sim
3.4$ kpc Yang et al., 2006) and age (Downes et al., 1980) estimates of SNR
G40.5$-$0.5 are also consistent with those of PSR J1907+0602. If we use the
usually assumed location for SNR G40.5-0.5 given by Langston et al. (2000)
(RA=19h07${}^{\mathrm{m}}11\fs 9$,
Dec=6${}^{\circ}35^{\prime}15^{\prime\prime}$), we get an angular separation
of $\sim 35^{\prime}$ between the timing position for the pulsar and the SNR.
However, this position for the SNR is from single dish observations that were
offset towards one bright side of the nominal shell. We use the VLA Galactic
Plane Survey 1420 MHz image (Stil et al., 2006) of this region to estimate the
SNR center to be RA=19h07${}^{\mathrm{m}}08\fs 6$,
Dec=6${}^{\circ}29^{\prime}53^{\prime\prime}$ (Figure 9) which, for an assumed
distance of 3.2 kpc, would give a separation of $\sim$28 pc. Given the
characteristic age of 19.5 kyr years, this would require an average transverse
velocity of $\sim 1400$ km/s. While velocities about this high are seen in
some cases (eg. PSR B1508+55 has a transverse velocity of $\sim 1100$ km/s,
Chatterjee et al. 2005 ), it is several times the average pulsar velocity and
many times higher than the local sound speed. We note that pulsars with a
braking index significantly less than $n=3$ assumed in the derivation of the
characteristic age could have ages as much as a factor of two greater (see eg.
Kaspi et al., 2001), and thus a space velocity around half the above value may
be all that is required. But with any reasonable assumption of birthplace,
distance, and age, if the pulsar was born in SNR G40.5$-$0.5, any associated
X-ray or radio PWN should show a bow-shock and trail morphology, with the
trail likely pointing back towards the SNR center. Unfortunately, the
compactness and low number of counts in our Chandra image precludes any
definite statement about the PWN morphology. One arrives at a different, and
lower, minimum velocity if one assumes the pulsar was born at the center of
the TeV PWN and moved to its present position, but the resulting velocity
would still require a bow shock.
One can also get a pulsar offset towards the edge of a relic PWN if there is a
significant density gradient in the surrounding ISM. A gradient will cause the
supernova blast wave to propagate asymmetrically. Where the density is higher,
the reverse shock propagating back to the explosion center will also be
asymmetric. This will tend to push the PWN away from the region of higher
density (Blondin et al., 2001; Ferreira & de Jager, 2008). This has been
invoked to explain the offsets in the Vela X and HESS J1825$-$137 nebulae as
well as several others. Infrared and radio imaging of the region shows that
HESS J1908+063 borders on a shell of material surrounding the S74 HII region,
also known as the Lynds Bright Nebula 352. Russeil (2003) gives a kinematic
distance of $3.0\pm 0.3$ kpc for this star forming region, compatible with the
pulsar distance. In this scenario, the pulsar would not have to be highly
supersonic to be at the edge of a relic nebula, and would not have to be
traveling away from the center of the TeV emission.
A third, hybrid possibility is that SNR G40.5$-$0.5 is only a bright segment
of a much larger remnant, whose emission from the side near the pulsar is
confused with that from the molecular cloud. The asymmetry would be explained
by the difference in propagation speed in the lower density ISM away from the
molecular cloud.
Our current Chandra data are insufficient to distinguish between the above
scenarios. However, there is also the possibility of a compact cometary radio
nebula, such as is seen around PSR B1853+01 in SNR W44 (Frail et al., 1996)
and PSR B0906$-$49 (Gaensler et al., 1998). In addition, sensitive long
wavelength radio imaging could reveal any larger, faint SNR shells. Imaging
with the EVLA and LOFAR of this region is therefore highly desirable. The
connection between the pulsar and the TeV nebula could be further strengthened
by a confirmation of the spatio-spectral dependence of the nebula where the
spectrum hardens nearer to the pulsar.
The Fermi LAT Collaboration acknowledges the generous support of a number of
agencies and institutes that have supported the Fermi LAT Collaboration. 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 and the Swedish National
Space Board in Sweden. The Arecibo Observatory is part of the National
Astronomy and Ionosphere Center, which is operated by Cornell University under
a cooperative agreement with the National Science Foundation. The National
Radio Astronomy Observatory is a facility of the National Science Foundation
Operated under cooperative agreement by Associated Universities, Inc. Support
for this work was provided by the National Aeronautics and Space
Administration through Chandra Award Number GO6-7136X issued by the Chandra
X-Ray Observatory Center, which is operated by the Smithsonian Astrophysical
Observatory for and on behalf of NASA under contract NAS8-03060. This research
has made use of software provided by the Chandra X-Ray Center in the
application package CIAO. This research has made use of data obtained from the
High Energy Astrophysics Science Archive Research Center (HEASARC), provided
by NASA’s Goddard Space Flight Center.
## References
* Abdo et al. (2009a) Abdo, A. A. et al. 2009a, Science, 325, 840, (Blind Search Pulsars)
* Abdo et al. (2009b) —. 2009b, ApJS, 183, 46, (Bright Source List)
* Abdo et al. (2009c) —. 2009c, in prep (1st Fermi LAT Gamma-Ray Catalog)
* Abdo et al. (2009d) —. 2009d, Submitted to ApJ (Fermi Catalog of Gamma-ray Pulsars)
* Abdo et al. (2007) —. 2007, ApJ, 664, L91, (The Milagro Collaboration)
* Aharonian et al. (2009) Aharonian, F. et al. 2009, A&A, 499, 723
* Aharonian et al. (2006) —. 2006, A&A, 460, 365
* Atwood et al. (2009) Atwood, W. B. et al. 2009, ApJ, 697, 1071, (LAT)
* Blondin et al. (2001) Blondin, J. M., Chevalier, R. A., & Frierson, D. M. 2001, ApJ, 563, 806
* Camilo et al. (2009) Camilo, F. et al. 2009, ApJ, in press (arXiv:0908.2626) - Radio detection of two gamma-ray pulsars
* Chatterjee et al. (2009) Chatterjee, S. et al. 2009, ApJ, 698, 250
* Chatterjee et al. (2005) —. 2005, ApJ, 630, L61
* Cordes et al. (2006) Cordes, J. M. et al. 2006, ApJ, 637, 446
* Cordes & Lazio (2002) Cordes, J. M., & Lazio, T. J. W. 2002, ArXiv e-prints, (arXiv:astro-ph/0207156)
* de Jager (2008) de Jager, O. C. 2008, ApJ, 678, L113
* Deller et al. (2009) Deller, A. T., Tingay, S. J., Bailes, M., & Reynolds, J. E. 2009, ApJ, 701, 1243
* Dickey & Lockman (1990) Dickey, J. M., & Lockman, F. J. 1990, ARA&A, 28, 215
* Dowd et al. (2000) Dowd, A., Sisk, W., & Hagen, J. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 202, IAU Colloq. 177: Pulsar Astronomy - 2000 and Beyond, ed. M. Kramer, N. Wex, & R. Wielebinski, 275
* Downes et al. (1980) Downes, A. J. B., Salter, C. J., & Pauls, T. 1980, A&A, 92, 47
* Ferreira & de Jager (2008) Ferreira, S. E. S., & de Jager, O. C. 2008, A&A, 478, 17
* Frail et al. (1996) Frail, D. A., Giacani, E. B., Goss, W. M., & Dubner, G. 1996, ApJ, 464, L165+
* Gaensler et al. (1998) Gaensler, B. M., Stappers, B. W., Frail, D. A., & Johnston, S. 1998, ApJ, 499, L69+
* Gonthier et al. (2004) Gonthier, P. L., Van Guilder, R., & Harding, A. K. 2004, ApJ, 604, 775
* Hartman et al. (1999) Hartman, R. C. et al. 1999, ApJS, 123, 79
* Haslam et al. (1982) Haslam, C. G. T., Salter, C. J., Stoffel, H., & Wilson, W. E. 1982, A&AS, 47, 1
* Hobbs et al. (2006) Hobbs, G., Edwards, R., & Manchester, R. 2006, Chinese Journal of Astronomy and Astrophysics Supplement, 6, 189
* Kaspi et al. (2001) Kaspi, V. M., Roberts, M. E., Vasisht, G., Gotthelf, E. V., Pivovaroff, M., & Kawai, N. 2001, ApJ, 560, 371
* Kaspi et al. (2006) Kaspi, V. M., Roberts, M. S. E., & Harding, A. K. 2006, in Compact stellar X-ray sources, ed. W. H. G. Lewin & M. van der Klis, 279–339
* Lamb & Macomb (1997) Lamb, R. C., & Macomb, D. J. 1997, ApJ, 488, 872
* Langston et al. (2000) Langston, G., Minter, A., D’Addario, L., Eberhardt, K., Koski, K., & Zuber, J. 2000, AJ, 119, 2801
* Ransom et al. (2002) Ransom, S. M., Eikenberry, S. S., & Middleditch, J. 2002, AJ, 124, 1788
* Ray et al. (2009) Ray, P. S., et al. 2009, ApJ, in prep (Precise Timing of Fermi Gamma-Ray Pulsars)
* Roberts et al. (2001) Roberts, M. S. E., Romani, R. W., & Kawai, N. 2001, ApJS, 133, 451
* Romani & Yadigaroglu (1995) Romani, R. W., & Yadigaroglu, I.-A. 1995, ApJ, 438, 314
* Russeil (2003) Russeil, D. 2003, A&A, 397, 133
* Stil et al. (2006) Stil, J. M. et al. 2006, AJ, 132, 1158
* Ward (2008) Ward, J. E. 2008, in American Institute of Physics Conference Series, Vol. 1085, American Institute of Physics Conference Series, ed. F. A. Aharonian, W. Hofmann, & F. Rieger, 301–303
* Watters et al. (2009) Watters, K. P., Romani, R. W., Weltevrede, P., & Johnston, S. 2009, ApJ, 695, 1289
* Yang et al. (2006) Yang, J., Zhang, J.-L., Cai, Z.-Y., Lu, D.-R., & Tan, Y.-H. 2006, Chinese Journal of Astronomy and Astrophysics, 6, 210
Table 1: Measured and Derived timing parameters of PSR J1907+0602 Fit and
data-set
---
Pulsar name | J1907+0602
MJD range | 54647–55074
Number of TOAs | 23
Rms timing residual ($\mu s$) | 404
Measured Quantities
Right ascension, $\alpha$ | 19:07:54.71(14)
Declination, $\delta$ | +06:02:16.1(23)
Pulse frequency, $\nu$ (s-1) | 9.3780713067(19)
First derivative of pulse frequency, $\dot{\nu}$ (s-2) | $-$7.6382(4)$\times 10^{-12}$
Second derivative of pulse frequency, $\ddot{\nu}$ (s-3) | 2.5(6)$\times 10^{-22}$
Epoch of frequency determination (MJD) | 54800
Dispersion measure, DM (cm-3pc) | 82.1(11)
Derived Quantities
Characteristic age (kyr) | 19.5
Surface magnetic field strength (G) | $3.1\times 10^{12}$
$\dot{E}$ (erg s-1) | $2.8\times 10^{36}$
Assumptions
Time units | TDB
Solar system ephemeris model | DE405
Figure 1: Post-fit timing residuals for PSR J1907+0602. The reduced chi-square
of the fit is 0.5. Figure 2: Folded light curves of PSR J1907+0602 in 32
constant-width bins for different energy bands and shown over two pulse
periods with the 1.4 GHz radio pulse profile plotted in the bottom panel. The
top panel of the figure shows the folded light curve for photons with $E>0.1$
GeV. The other panels show the pulse profiles in exclusive energy ranges:
$E>3.0$ GeV (with $E>5.0$ GeV in black) in the second panel from the top; 1.0
to 3.0 GeV in the next panel; 0.3 to 1.0 GeV in the fourth panel; and 0.1 to
0.3 GeV in the fifth panel. Figure 3: The observed Fermi-LAT counts map of
the region around PSR J1907+0602. Left: “on” pulse image, right: “off” pulse
image. The open cross-hair marks the location of the pulsar. Color scale shows
the counts per pixel. Figure 4: Phase-averaged spectral energy distribution
for PSR J1907+0602 (green circles). Blue circles are data from HESS for HESS
J1908+063 TeV source. 2 $\sigma$ upper limits from Fermi for emission from
this TeV source are shown in blue. The black line shows the spectral model for
the pulsar (equation 1). The upper limits suggest that the spectrum of HESS
J1908+063 has a low energy turnover between 20 GeV and 300 GeV. Figure 5:
Residual map of the region around PSR J1907+0602 in the off-pulse. The timing
position of the pulsar is marked by the cross. The 5 $\sigma$ contours from
Milagro (outer) and HESS (inner) are overlaid. Figure 6: ASCA GIS 2-10 keV
image of the region around PSR J1907+0602. The green contours are the 4-7
$\sigma$ significance contours from HESS. Figure 7: Chandra ACIS images of PSR
J1907+0602. The blue ellipse shows the uncertainty in the timing position. The
green circle of radius 0.8′′ is twice the FWHM of the 5keV PSF at this
position, and should contain roughly 80% of the counts. The image at 0.75-2
keV (Left), 2-8 keV (Center) and 0.75-8 keV (right) is shown. Color scale
shows the counts per pixel. Figure 8: Chandra X-ray spectrum of PSR
J1907+0602. Figure 9: VGPS 1420 MHz image of region in Galactic coordinates
showing relationship between SNR G40.5$-$0.5, HESS J1907+063 (blue contours
representing the 4,5,6 and 7$\sigma$ significance levels), the star forming
region S74, and PSR J1907+0602.
|
arxiv-papers
| 2010-01-06T01:28:44 |
2024-09-04T02:49:07.508244
|
{
"license": "Public Domain",
"authors": "Fermi LAT Collaboration, the Fermi Pulsar Timing Consortium: A. A.\n Abdo, et al",
"submitter": "Aous Abdo",
"url": "https://arxiv.org/abs/1001.0792"
}
|
1001.0794
|
-titleNew technologies for probing the diversity of brown dwarfs and exoplanets 11institutetext: Institut d’Astrophysique de Paris, Université Pierre et Marie Curie, UMR7095 CNRS, 98bis bd. Arago, 75014 Paris, France 22institutetext: Observatoire de Haute Provence, CNRS/OAMP, 04870 St Michel l’Observatoire, France 33institutetext: Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290 Sauverny, Switzerland
# Consequences of spectrograph illumination for the accuracy of radial-
velocimetry
Boisse I 11 iboisse@iap.fr Bouchy F 11 2 2 Chazelas B 33 Perruchot S 22
Pepe F 33 Lovis C 33 Hébrard G 11
###### Abstract
For fiber-fed spectrographs with a stable external wavelength source,
scrambling properties of optical fibers and, homogeneity and stability of the
instrument illumination are important for the accuracy of radial-velocimetry.
Optical cylindric fibers are known to have good azimuthal scrambling. In
contrast, the radial one is not perfect. In order to improve the scrambling
ability of the fiber and to stabilize the illumination, optical double
scrambler are usually coupled to the fibers. Despite that, our experience on
SOPHIE and HARPS has lead to identified remaining radial-velocity limitations
due to the non-uniform illumination of the spectrograph. We conducted tests on
SOPHIE with telescope vignetting, seeing variation and centering errors on the
fiber entrance. We simulated the light path through the instrument in order to
explain the radial velocity variation obtained with our tests. We then
identified the illumination stability and uniformity has a critical point for
the extremely high-precision radial velocity instruments (ESPRESSO@VLT,
CODEX@E-ELT). Tests on square and octagonal section fibers are now under
development and SOPHIE will be used as a bench test to validate these new feed
optics.
## 1 Introduction
High-precision radial-velocimetry (RV) is still a main technique in the search
and characterization of planetary systems. Up to date, most of the known
extrasolar planets have been detected with this method. It will remain at the
forefront of exoplanet science for the coming years thanks to its efficiency
in finding low-mass planets since HARPS has reached an RV accuracy less than
1ms-1 (ref. RefM ). In the aim to reach Earth-type planets and, also measure
the expansion of the universe, new spectrographs in the visible are now in
study to reach a precision of only few cms-1 in the near future (ESPRESSO@VLT
RefPa1 , CODEX@E-ELT RefPa2 ).
To reach this level of accuracy, we must avoid a certain number of
limitations. Among these, one can quote the stellar noise (activity,
pulsation, surface granulation), the contamination by external sources (moon,
close-by objects, background continuum), and what light may encounter during
its path (e.g. atmospheric dispersion). As part of instrumental factors, CCD
cosmetics, charge transfer inefficiency (ref. RefB1 ), wavelength calibration
(refs. RefL and RefC ) should be mentioned. For an exhaustive discussion of
these limitations, we refer to RefP . The authors highlight the impact of the
spectrograph illumination as one of the main limiting factors, and the need to
analyze and quantify it. In this proceeding, we then focus on the limitation
due to the injection of light in the fiber-fed spectrograph. This work is
based on our experiment on two high-resolution, high-precision fiber-fed
spectrographs, SOPHIE@1.93m, Observatoire de Haute-Provence (OHP), France
(refs. RefB2 and RefS ) and HARPS@3.6m, European Southern Observatory
(ESO)-La Silla, Chile (ref. RefM ).
Up to day, SOPHIE has a Doppler accuracy of about 5 ms-1, as most of high-
accuracy spectrographs involved in exoplanet survey. In service since November
2006, we identified that instrumental limitations mainly come from the old
Cassegrain Fiber Adaptater (developed for ELODIE spectrograph). In order to
reach the so far unique HARPS level of accuracy, we conducted tests that
illustrate the incomplete fiber scrambling. Detected with SOPHIE, these
effects were also characterized on HARPS at a lower level. In the development
of the future instruments, the optimization of SOPHIE is a bench test for the
improvement of the spectrograph illumination uniformity and stability.
The proceeding is structured as followed. In a the second section, we expose
the properties of the fiber-fed spectrographs and the light injection. The
third section reviews the different effects that are detected in the RV due to
the incomplete and imperfect scrambling obtained with the standard fibers. In
the last section, we describe solutions proposed to optimize the fibers
scrambling.
## 2 Fiber-fed spectrographs
HARPS and SOPHIE are fiber-fed spectrographs with simultaneous reference
calibrations. The stellar light collected by the telescope are lead to the
instrument through a standard step-index multi-mode cylindrical optical fiber.
SOPHIE has two observing mode using two different fibers, the High-Resolution
(HR) and the High-Efficiency mode (HE). In HR mode, the spectrograph is fed by
a 40.5-$\mu$m slit superimposed on the output of the 100-$\mu$m fiber,
reaching a spectral resolution of $\lambda/\Delta\lambda$ = 75,000. In HE
mode, the spectrograph is directly fed by the 100-$\mu$m fiber with a
resolution power of 40,000. Both SOPHIE fibers have an sky acceptance of
3-arcsec. HARPS has 70-$\mu$m fiber with a sky acceptance of 1-arcsec and a
spectral resolution of 110,000 (see Table 1).
Non-uniform illumination of the slit or output fiber at the spectrograph
entrance decreases the radial-velocity precision. Indeed, variations in
seeing, focus and image shape at the fiber entrance may induced non-uniform
illumination inside the pupil of the spectrograph. The optical aberrations
lead to variations in the centroids of the stellar lines on the focal plane.
Table 1: Comparison of parameters of fiber-fed spectrographs SOPHIE and HARPS. Instrument | Doppler prec. | 1 pixel | Resol. $\Delta\lambda/\lambda$
---|---|---|---
HARPS | $\leqslant$ 1ms-1 | 800 ms-1 | 110,000
SOPHIE (HR) | $\sim$ 4 ms-1 | 1400 ms-1 | 75,000
SOPHIE (HE) | $\sim$ 10 ms-1 | 1400 ms-1 | 40,000
Optical fibers are used to lead the light from the telescope to the entrance
of the spectrograph. They have the properties to scrambler the atmospheric
effects and guiding and centering errors discussed previously. But the
scrambling ability of one multimode fiber is not perfect as shown in Fig. 1.
With an input off axes image, the azimuthal scrambling is good whereas it
remains some effects in the radial one. This pattern observed is bigger in far
field than in near field. Near field and far field are respectively defined
as, the brightness distribution across the output face of the fiber, and, as
the angular distribution of light of fiber output beam. Therefore, changes in
the input beam cause only circular symmetric changes in the fiber output
pattern. In contrast, imperfect radial scrambling allows small zonal errors to
remain.
Figure 1: Illumination at the output of one cylindrical optical fiber in the
near and the far field. Sources are off axes. A high degree of azimuthal
scrambling is observed. In contrast, it remains some effects in the radial
one.
HARPS and the SOPHIE HR mode are equipped with optical double scramblers. It
is used to increase and improve the uniformity and stabilization of the
illumination of the spectrograph entrance (refs. RefBr and RefH ). The near
field of a circular fiber is observed to be better scramble than the far
field. In spite of only one fiber guiding the light from the telescope to the
instrument, double scrambler is composed of two fibers coupled with two
doublets. The system is designed to inverse near field and far field in order
to induce a better radial scrambling, although it causes flux losses (ref.
RefB0 ).
## 3 RV effects of fiber imperfect scrambling
### 3.1 Centering/Guiding on the fiber entrance
The degree of radial scrambling describes the stability of the output beam as
the input image is moved from the center to the edge of the fiber. It is
possible to detect in RV some variations due to the insufficiency of radial
scrambling. The light from a stable star is moved with the guiding/centering
system from an edge to the other of the fiber. We compute the RV with the data
reduction pipeline for different positions of the star. The test was done for
the two modes of SOPHIE and for HARPS. The results are computed in Table 2.
Table 2: Variation observed in RV as the input image (star) is moved from the center to the edge of the fiber. HARPS and SOPHIE with High-Resolution fiber used a double scrambler instead of the High-Efficiency mode of SOPHIE. Instrument | fib. diam. | RV effect [ms-1]
---|---|---
HARPS | 1” | $\sim$ 3
SOPHIE (HR) | 3” | $\sim$ 13
SOPHIE (HE) | 3” | $\sim$ 36
Because the scrambling of the fiber is not perfect, the movement of the input
image corresponds to a shift on the detector. We observed first that, wider is
the fiber, more important is the RV variations. Moreover, the value of meter
per second per pixel vary as a function of the resolution power. Moving the
input image from the center to the edge is then expected to have a lower
effect for a higher spectral resolution. In addition, we expect that the
double scrambler improve the radial scrambling, i.e. decrease the RV effect.
### 3.2 Vignetting telescope
The pupil of the telescope, i.e. the entrance of the telescope or the far
field of the input light is known to be more stable than the telescope image.
We would like to test the effect in RV of variations of the far field. For
that, we vignette the telescope on day sky with the dome. We computed the RV
with the pipeline SOPHIE, estimating that the variation of Barycentric Earth
RV is not significant during our measurements. We made this test with the HE
mode in order that the variation of the far-field of the telescope are
projected on the grating (i.e. without double scrambler). We remarked that the
way of the RV variations depends on exterior or interior vignetting (Fig. 2).
Figure 2: Schema illustrating how the vignetting of the telescope is
translated at the output of the fiber HE in the far-field, i.e. at the
entrance of the spectrograph. For external occultation, the RV decrease
whereas, for internal occultation the RV increase.
### 3.3 Optical path simulations
We simulated the optical path in the SOPHIE spectrograph. We observed a
displacement of the spectrum in function of wavelength on the focal plane when
the input beam at the entrance of the spectrograph is center-illuminated or
center-darked, corresponding respectively when vignetting external and
internal part of the pupil in HE mode (cf. Sect.3.2). Variation of the slit or
the optical fiber illumination are directly translated on the spectrum in Fig.
3. We observed that the shift is more significant for an external occultation.
Because the effect is not symmetric along an order and not monitored by the
calibration lamp, the final computed RV vary. So, variations of the far-field
pattern projected onto the grating can introduce radial-velocity shifts at the
detector lowering the final precision in RV.
Figure 3: Displacement of the position of a nine wavelength as a function of a
internal (green) or an external (red) occultation of the entrance of the
spectrograph on the CCD detector.
Figure 4: When the seeing is good, the telescope image is less wide. With the
double scrambler in HR mode, near field becomes far-field and is projected on
the collimator at the entrance of the spectrograph. The pattern is comparable
to that observed in HE when vignetting telescope (Fig. 2)
### 3.4 RV effects due to seeing variations
Variations of the far-field of the input beam in HE mode is equivalent to a
variation of the telescope image in HR mode, due to the double scrambler.
Variations in seeing induce a variation of the input image as illustrate in
Fig. 4. The input beam at the entrance of the spectrograph is center-
illuminated (external occultation) when a target is observed with a good
seeing in HR mode. The previous tests and simulations explain a systematic
effect observed with SOPHIE named as ”seeing effect”. For a sample of stars
observed at high signal to noise ratio (SNR) in HR mode, we observed a
decrease in RV correlated with an estimation of seeing. The value of the
seeing is not monitored with SOPHIE. We estimated it with calculating the
relative flux by unit of exposure time :
$S=\frac{SNR^{2}}{T_{exp}10^{-M_{V}/2.5}}$ (1)
with SNR, the signal to noise ratio of the spectra, Texp the time exposure of
the measurement, MV the visual magnitude of the target. The seeing decrease
when the seeing estimation $S$ increase. With a lower seeing, the input image
is smaller than the fiber width, and as shown in the simulations, induced a
variation of the far field input in the spectrograph, due to the double
scrambler. We expect and observed a decrease of RV when the value of the
seeing decrease as shown in Fig. 5.
Figure 5: Radial-velocity measurements as a function of a seeing index (cf.
text). The seeing decrease when the seeing estimation $S$ increase. As the
seeing is lower, the RV decreases.
Moreover, when the input image is smaller than the diameter of the fiber, we
are more sensible to effect of guiding and centering system. The guiding and
centering system may induced variation of the position of the input image on
the slit or the entrance of the fiber, and so introduce RV variations. These
RV variation are quite random, and add some noise.
Whereas the seeing effect is directly related to the optical path in the
spectrograph, it may be quantified. We are modeling this variations with our
data in order to remove this noise with a software tool.
We note that the effect on HARPS is expected to be lower. The 1” width of the
fibers is smaller than the SOPHIE and roughly equivalent to the typical seeing
at La Silla. Furthemore the quality image provided by the 3.6-m telescope is
close to 0.7 arcsec. Hence, HARPS is less sensible to the variations of
seeing. Moreover, the resolution power of the instrument is better and then
the incomplete radial scrambling has a lower impact in RV as shown in Sect.
3.1. Furthermore the optical aberrations on HARPS are smaller than in SOPHIE.
## 4 Perspectives and conclusions
The CODEX experiment calls for a Doppler precision as low as 1 cms-1. This
work shows that the future instruments need for better scrambling to reach
this accuracy. New types of fibers for feeding the telescope or new double
scrambler are needed to improve the precision and to minimize the noise due to
variations of the illumination at the entrance of the spectrograph. To remove
the spherical symmetry of cylindrical fibers, octagonal and square section
optical fibers are then in study at Geneva, OHP-LAM, and ESO Garching. As
first results, very good scrambling properties are observed in the near field,
whereas strange patterns in far field are not well understood at the moment.
It needs future tests to improve the measurements, to study its best used, on
all the path or in a double scrambling mode. SOPHIE is then going to be a
bench test for the future instruments.
We note that additional solution is being installed on HARPS to reduce the
guiding and centering limitations. A tip-tilt mirror generate small-amplitude
and high frequencies variations in order to induce a spread of the light.
A new guiding camera is now in operation on SOPHIE and allows a better guiding
and centering and also allows to monitor the exact seeing.
## References
* (1) Bouchy & Connes A&A, 1999, Volume136, 193
* (2) Bouchy, F., Isambert, J., Lovis, C. Boisse, I. et al. EAS Publications Series, Volume37, 2009, pp.247-253
* (3) Bouchy, F., Hébrard, G., Udry, S., Delfosse, X., Boisse, I. et al. A&A, Volume505, 2009, pp.853
* (4) Brown, T. M. 1990, ASPC, 8, 335B
* (5) Lovis, C. & Pepe, F. 2007, A&A, 468, 1115L
* (6) Hunter, T.D. & Ramsey, L.W. 1992, PASP, 104, 1244
* (7) Mayor, M., Udry, S., Lovis, C. et al. 2009, A&A, 493, 639
* (8) Pasquini, L., Manescau, A., Avila, G. et al. 2009, svlt.conf, 411
* (9) Pasquini, L., Avila, G., Dekker, H. et al. 2008, SPIE.7014E, 51P
* (10) Pepe, F. & Lovis, C., Phys. Scr. T130, (2008) 014007
* (11) Perruchot, S., Kohler, D., Bouchy, F. et al. 2008, SPIE.7014E, 17P
* (12) Wildi, F., Pepe, F., Lovis, C., Chazelas, B. et al. 2009, SPIE.7440E, 19W
|
arxiv-papers
| 2010-01-06T00:39:48 |
2024-09-04T02:49:07.517931
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. Boisse, F. Bouchy, B. Chazelas, S. Perruchot, F. Pepe, C. Lovis, G.\n Hebrard",
"submitter": "Isabelle Boisse",
"url": "https://arxiv.org/abs/1001.0794"
}
|
1001.0824
|
2010513-524Nancy, France 513
NEELESH KHANNA
SURENDER BASWANA
# Approximate shortest paths avoiding a failed vertex : optimal size data
structures for unweighted graphs
N. KHANNA Oracle India Pvt. Ltd, Bangalore-560029, India.
neelesh.khanna@gmail.com and S. BASWANA Indian Institute of Technology
Kanpur, India. sbaswana@cse.iitk.ac.in
###### Abstract.
Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path P
between any two vertices $u,v\in V$ is said to be $t$-approximate shortest
path if its length is at most $t$ times the length of the shortest path
between $u$ and $v$. We consider the problem of building a compact data
structure for a given graph $G$ which is capable of answering the following
query for any $u,v,z\in V$ and $t>1$.
report $t$-approximate shortest path between $u$ and $v$ when vertex $z$ fails
We present data structures for the single source as well all-pairs versions of
this problem. Our data structures guarantee optimal query time. Most
impressive feature of our data structures is that their size nearly match the
size of their best static counterparts.
###### Key words and phrases:
Shortest path, distance, distance queries, oracle
###### 1991 Mathematics Subject Classification:
E.1 [Data Structures]:Graphs and Networks; G.2.2[Discrete Mathematics]:Graph
Theory - Graph Algorithms
Part of this work was done while the authors were at Max-Planck Institute for
Computer Science, Saarbruecken, Germany during the period May-July 2009.
## 1\. Introduction
The shortest paths problem is a classical and well studied algorithmic problem
of computer science. This problem requires processing of a given graph
$G=(V,E)$ on $n=|V|$ vertices and $m=|E|$ edges to compute a data structure
using which shortest path or distance between any two vertices can be
efficiently reported. Two famous and thoroughly studied versions of this
problem are single source shortest paths (SSSP) problem and all-pairs shortest
paths (APSP) problem.
Most of the applications of the shortest paths problem involve real life
graphs and networks which are prone to failure of nodes (vertices) and links
(edges). This has motivated researchers to design dynamic solution for the
shortest paths problem. For this purpose, one has to first develop a suitable
model for the shortest paths problem in dynamic networks. In fact two such
models exists, and each of them has its own algorithmic objectives.
The shortest paths problem in the first model is described as follows : There
is an initial graph followed by an on-line sequence of insertion and deletion
of edges interspersed with shortest path (or distance) queries. Each query has
to be answered with respect to the graph which exists at that moment
(incorporating all the updates preceding the query on the initial graph). A
trivial solution of this problem is to recompute all-pairs shortest paths from
scratch after each update. This is certainly a wasteful approach since a
single update usually does not cause a huge change in the all-pairs distance
information. Therefore, the algorithmic objective here is to maintain a data
structure which can answer distance query efficiently and can be updated after
any edge insertion or deletion in an efficient manner. In particular, the time
required to update the data structure has to be substantially less than the
running time of the best static algorithm. Many novel algorithms have been
designed in the last ten years for this problem and its variants (see [6] and
the references therein).
On one hand the first model is important since it captures the worst possible
hardness of any dynamic graph problem. On the other hand, it can also be
considered as a pessimistic model for real life networks. It is true that the
networks are never immune to failures. But in addition to it, it is also rare
to have networks which may have arbitrary number of failures in normal
circumstances. It is essential for network designers to choose suitable
technology to make sure that the failures are quite infrequent in the network.
In addition, when a vertex or edge fails (goes down), it does not remain
failed/down indefinitely. Instead, it comes up after some finite time due to
simultaneous repair mechanism going on in the network. These aspects can be
captured in the second model which takes as input a graph and a number
$\ell\ll n$. This model assumes that there will be at most $\ell$ vertices or
edges which may be inactive at any time, though the corresponding set of
failed vertices or edges may keep changing as the time progresses : the old
failed vertices become active while some new active vertices may fail. The
algorithmic objective in this model is to preprocess the given graph to
construct a compact data structure which for any subset $S$ of at most $\ell$
vertices may answer the following query for any $u,v\in V$.
Report the shortest-path (or distance) from $u$ to $v$ in $G\backslash S$.
It is desired that each query gets answered in optimal time : retrieval of
distance in $O(1)$ time and the shortest path in time which is of the order of
the number of its edges. The ultimate research goal would be to understand the
complexity of the above problem for any given value $\ell$. In this pursuit,
the first natural step would be to efficiently solve and thoroughly understand
the complexity of the problem for the case $\ell=1$, that is, the shortest
paths problem avoiding any failed vertex. Interestingly, this problem appears
as a sub problem in many other related problems, namely, Vickrey pricing of
networks [9], most vital node of a shortest path [11], the replacement path
problem [12], and shortest paths avoiding forbidden subpaths [1].
The first nontrivial and quite significant breakthrough on the all-pairs
version of this problem was made by Demetrescu et al. [7]. They designed an
$O(n^{2}\log n)$ space data structure, namely distance sensitivity oracle,
which is capable of reporting the shortest path between any two vertices
avoiding any single failed vertex. The preprocessing time of this data
structure is $O(mn^{2})$. Recently, Bernstein and Karger [4] improved the
preprocessing time to $O(mn\log n)$. Though ${\Theta}(n^{2}\log n)$ space
bound of this all-pairs distance sensitivity oracle is optimal up to
logarithmic factors, it is too large for many real life graphs which appear in
various large scale applications [13]. In most of these graphs usually $m\ll
n^{2}$, hence a table of $\Theta(n^{2})$ size may be too large for practical
purposes. However, it is also known [7] that even a data structure which
reports exact distances from a fixed source avoiding a single failed vertex
will require $\Omega(n^{2})$ space in the worst case. So approximation seems
to be the only way to design a small space compact data structure for the
problem of shortest paths avoiding a failed vertex. A path between $u,v\in V$
is said to be $t$-approximate shortest path if its length is at most $t$ times
that of the shortest path between the two. The factor $t$ is usually called
the stretch. We would like to state here that many algorithms and data
structures have been designed in the last fifteen years for the static all-
pairs approximate shortest paths (see [2, 13] and references therein). The
prime motivation underlying these algorithms has been to achieve sub-quadratic
space and/or sub-cubic preprocessing time for the static APSP problem.
However, no data structure was designed in the past for approximate shortest
paths avoiding any failed vertex.
In this paper, we present really compact data structures which are capable of
reporting approximate shortest paths between two vertices avoiding any failed
vertex in undirected graphs. The most impressive feature of our data
structures is their nearly optimal size. In fact their size almost matches the
size of their best static counterparts.
### 1.1. New Results
Single source approximate shortest paths avoiding any failed vertex.
First we address weighted graphs. For the weighted graphs, we present an
$O(m\log n)$ time constructible data structure of size $O(n\log n)$ which can
report 3-approximate shortest path from the source to any vertex $v\in V$
avoiding any $x\in V$. We then consider the case of undirected unweighted
graphs. For these graphs, we present an $O(n\frac{\log n}{\epsilon^{3}})$
space data structure which can even report $(1+\epsilon)$-approximate shortest
path for any $\epsilon>0$.
All-pairs approximate shortest paths avoiding any failed vertex.
Among the existing data structures for static all-pairs approximate shortest
paths, the approximate distance oracle of Thorup and Zwick [13] stands out due
to its amazing features. Thorup and Zwick [13] showed that an undirected graph
can be preprocessed in sub-cubic time to build a data structure of size
$O(kn^{1+1/k})$ for any $k>1$. This data structure, despite of its sub-
quadratic size, is capable of reporting $(2k-1)$-approximate distance between
any two vertices in $O(k)$ time (and the corresponding approximate shortest
path in optimal time), and hence the name oracle. Moreover, the size-stretch
trade off achieved by this data structure is essentially optimal. It is a very
natural question to explore whether it is possible to design all-pairs
approximate distance oracle which may handle single vertex failure. We show
that it is indeed possible for unweighted graphs. For this purpose, we
suitably modify the approximate distance oracle of Thorup and Zwick [13] using
some new insights and our single source data structure mentioned above. These
modifications make the approximate shortest-paths oracle of Thorup and Zwick
handle vertex failure easily, and (surprisingly) still preserving the old
(optimal) trade-off between the space and the stretch. For precise details,
see Theorem 5.4.
For the algorithmic details missing in this extended abstract due to page
limitations, we suggest the reader to refer to the journal version [10]. Our
data structures can be easily adapted for handling edge failure as well
without any increase in space or time complexity.
## 2\. Preliminaries
We use the following notations and definitions in the context of a given
undirected graph $G=(V,E)$ with $n=|V|$, $m=|E|$ and a weight function
$\omega:E\rightarrow\textbf{R}^{+}$.
* •
$T_{r}$ : single source shortest path tree rooted at $r$.
* •
$\textbf{P}(x,y)$ : the shortest path between $x$ and $y$.
* •
$\delta(x,y)$ : the length of the shortest path between $x$ and $y$.
* •
$\textbf{P}(x,y,z)$ : the shortest path between $x$ and $y$ avoiding vertex
$z$.
* •
$\delta(x,y,z)$ : the length of the shortest path between $x$ and $y$ avoiding
vertex $z$.
* •
$T_{r}(x)$ : the subtree of $T_{r}$ rooted at $x$.
* •
$G_{r}(x)$ : the subgraph induced by the vertices of set $T_{r}(x)$ and
augmented by vertex $r$ and edges from $r$ as follows. For each $v\in
T_{r}(x)$ with neighbors outside $T_{r}(x)$, keep an edge $(r,v)$ of weight =
$\min_{(u,v)\in E,u\notin T_{r}(x)}(\delta(r,u)+\omega(u,v))$.
* •
$P::Q$ : a path formed by concatenating path $Q$ at the end of path $P$ with
an edge $(u,v)\in E$, where $u$ is the last vertex of $P$ and $v$ is the first
vertex of $Q$.
* •
$E(X)$ : the set of edges from $E$ with at least one endpoint in $X$.
Our algorithms will also use a data structure for answering lowest common
ancestor (LCA) queries on $T_{r}$. There exists an $O(n)$ time computable data
structure which occupies $O(n)$ space and can answer any LCA query in $O(1)$
time (see [3] and references therein).
## 3\. Single source 3-approximate shortest paths avoiding a failed vertex
We shall first solve a simpler sub-problem where the vertex which may fail
belong to a given path $P\in T_{r}$. Then we use divide and conquer strategy
wherein we decompose $T_{r}$ into a set of disjoint paths and for each such
path, we solve this sub-problem.
### 3.1. Solving the Sub-Problem : the failures of a vertex from a given path
$\textbf{P}(r,t)$
Given the shortest path tree $T_{r}$, let $\textbf{P}(r,t)=\langle
r(=x_{0}),x_{1},...,x_{k}(=t)\rangle$ be any shortest path present in $T_{r}$.
We shall design an $O(n)$ space data structure which will support retrieval of
a 3-approximate shortest path from $r$ to any $v\in V$ when some vertex from
$\textbf{P}(r,t)$ fails. The preprocessing time of our algorithm will be
$O(m+n\log n)$ which matches that of Dijkstra’s algorithm. The algorithm is
inspired by the algorithm of Nardelli et al. [11] for computing the most vital
vertex on a shortest path.
Figure 1. Partitioning of the shortest path tree $T_{r}$ at
$x_{i}\in\textbf{P}$
Consider vertex $x_{i}$ lying on the path $\textbf{P}(r,t)$. We partition the
tree $T_{r}\backslash\\{x_{i}\\}$ into the following 3 parts (see Figure 1).
$U_{i}$ : the tree $T_{r}$ after removing the subtree $T_{r}(x_{i})$
$D_{i}$ : the subtree of $T_{r}$ rooted at $x_{i+1}$
$O_{i}$ : the portion of $T_{r}$ left after removing $U_{i}$, $x_{i}$, and
$D_{i}$. Note that a vertex of the tree $T_{r}$ is either a vertex of the path
$\textbf{P}(r,t)$ or it belongs to some $O_{i}$ for some $i$. We build the
following two data-structures of total $O(n)$ size.
a data structure to retrieve 3-approximate shortest path from $r$ to any $v\in
D_{i}$.
a data structure to retrieve 3-approximate shortest path from $r$ to any $v\in
O_{i}$.
#### 3.1.1. Data structure for 3-approximate shortest paths to vertices of
$D_{i}$ when $x_{i}$ has failed
Consider the vertex $x_{i+1}$ and any other vertex $y\in D_{i}$. Note that the
shortest path $\textbf{P}(x_{i+1},y)$ remains intact even after removal of
$x_{i}$, and its length is certainly less than $\delta(r,y)$. Based on this
simple observation one can intuitively see that in order to travel from $r$ to
$y$ when $x_{i}$ fails, we may travel along shortest route to $x_{i+1}$ (that
is $\textbf{P}(r,x_{i+1},x_{i})$) and then along $\textbf{P}(x_{i+1},y)$.
Using triangle inequality and the fact that the graph is undirected, the
length of this path $\textbf{P}(r,x_{i+1},x_{i})::\textbf{P}(x_{i+1},y)$ can
be approximated as follows.
$\displaystyle\delta(r,x_{i+1},x_{i})+\delta(x_{i+1},y)$ $\displaystyle\leq$
$\displaystyle\delta(r,y,x_{i})+\delta(y,x_{i+1},x_{i})+\delta(x_{i+1},y)$
$\displaystyle\leq$ $\displaystyle\delta(r,y,x_{i})+2\delta(x_{i+1},y)$
$\displaystyle\leq$ $\displaystyle\delta(r,y,x_{i})+2\delta(r,y)\leq
3\delta(r,y,x_{i})$
Therefore, in order to support retrieval of 3-approximate shortest path to any
$v\in D_{i}$ in optimal time, it suffices to store the path
$\textbf{P}(r,x_{i+1},x_{i})$.
In order to devise ways of efficient computation and compact storage of
$\textbf{P}(r,x_{i+1},x_{i})$ for a given $i$, we use the following lemma
about the structure of the path $\textbf{P}(r,x_{i+1},x_{i})$.
###### Lemma 3.1.
The shortest path $\textbf{P}(r,x_{i+1},x_{i})$ is of the form $P_{1}::P_{2}$
where $P_{1}$ is a shortest path from $r$ in the subgraph induced by
$U_{i}\cup O_{i}$, and $P_{2}$ is a path present in $D_{i}$.
It follows that in order to compute $\textbf{P}(r,x_{i+1},x_{i})$, first we
need to compute shortest paths from $r$ in the subgraph induced by $U_{i}\cup
O_{i}$. Let $\delta_{i}(r,v)$ denote the distance from $r$ to $v\in U_{i}\cup
O_{i}$ in this subgraph. Note that $\delta_{i}(r,v)$ for $v\in U_{i}$ and the
corresponding shortest path is the same as in the original graph, and is
already present in $T_{r}$. For computing shortest paths from $r$ to vertices
of $O_{i}$, we build a shortest path tree (denoted as $T_{r}(O_{i})$) from $r$
in the subgraph induced by vertices $O_{i}\cup\\{r\\}$ and the following
additional edges. For each $z\in O_{i}$ with at least one neighbor in $U_{i}$,
we add an edge $(r,z)$ with weight = $min_{(u,z)\in E,u\in
U_{i}}(\delta(r,u)+\omega(u,z))$. Applying Lemma 3.1, let $(y_{i},z_{i})$ be
the edge of ${\mathbf{P}}(r,x_{i+1},x_{i})$ joining the sub path present in
$U_{i}\cup O_{i}$ with the sub path present in $D_{i}$. This edge can be
identified using the fact that this is the edge which minimizes
$\delta_{i}(r,y)+\omega(y,z)+\delta(x_{i+1},z)$ over all $z\in D_{i},y\in
U_{i}\cup O_{i},(y,z)\in E$. The vertex $x_{i+1}$ stores the path
$\textbf{P}(r,x_{i+1},x_{i})$ implicitly by keeping the edge $(y_{i},z_{i})$
and the tree $T_{r}(O_{i})$. The shortest path $\textbf{P}(r,x_{i+1},x_{i})$
can be retrieved in optimal time using the trees $T_{r}$, $T_{r}(O_{i})$, and
the edge $(y_{i+1},z_{i+1})$. Due to mutual disjointness of $O_{i}$’s, the
overall space requirement of the data structure for retrieving
$\textbf{P}(r,x_{i+1},x_{i})$ for all $i\leq k$ will be $O(n)$.
#### 3.1.2. Data structure for 3-approximate shortest paths to vertices of
$O_{i}$ when $x_{i}$ has failed
In order to compute 3-approximate shortest path to $O_{i}$ upon failure of
$x_{i}$, we shall use the approximate shortest paths to $D_{i}$ as computed
above. Here we use an interesting observation which states that if we have a
data structure to retrieve $\alpha$-approximate shortest paths from $r$ to
vertices of $D_{i}$ when $x_{i}$ fails, then we can use it to have a data-
structure to retrieve $\alpha$-approximate shortest paths to vertices of
$O_{i}$ as well. To prove this result, this is how we proceed. Consider the
subgraph induced by $O_{i}$ and augmented with vertex $r$ and some extra edges
which are defined as follows.
* •
For each $o\in O_{i}$ having neighbors from $U_{i}$, keep an edge $(r,o)$ and
assign it weight = $\min_{(u,o)\in E,u\in U_{i}}(\delta(r,u)+\omega(u,o))$.
* •
For each $o\in O_{i}$ having neighbors from $D_{i}$, keep an edge $(r,o)$ and
assign it weight = $\min_{(u,o)\in E,u\in
D_{i}}(\hat{\delta}(r,u,x_{i})+\omega(u,o))$, where $\hat{\delta}(r,u,x_{i})$
is the $\alpha$-approximate distance to $u$ upon failure of $x_{i}$. (In the
present situation we have $\alpha=3$.)
Let us denote this graph as $G_{r}(O_{i})$. Observation 3.2 is based on the
following lemma which is easy to prove.
###### Lemma 3.2.
The Dijkstra’s algorithm from $r$ in the graph $G_{r}(O_{i})$ computes
$\alpha$-approximate shortest paths from $r$ to all $v\in O_{i}$ avoiding
$x_{i}$.
If we can design a data structure for retrieving $(1+\epsilon)$-approximate
shortest paths from $r$ to vertices of $D_{i}$ upon failure of $x_{i}$, then
it can also be used to design a data structure which can support retrieval of
$(1+\epsilon)$-approximate shortest paths to all vertices of the graph upon
failure of $x_{i}$. We compute and store the shortest path tree rooted at $r$
in the graph $G_{r}(O_{i})$. This tree along with the tree $T_{r}$ and the
data structure described in the previous sub-section suffice for retrieval of
3-approximate shortest paths to $o\in O_{i}$ upon failure of $x_{i}$.
Query answering: Suppose the oracle receives a query asking for approximate
shortest path from $r$ to $v$ avoiding $x_{i}\in{\textbf{P}}(r,t)$. It first
invokes lowest common ancestor (LCA) query between $v$ and $x_{i}$ on $T_{r}$.
If $LCA(v,x_{i})\not=x_{i}$, the shortest path from $r$ to $v$ remains
unaffected and so it reports the path $\textbf{P}(r,t)$. Otherwise, it
determines if $v\in D_{i}$ or $v\in O_{i}$. Depending upon the two cases, it
reports the approximate shortest path between $r$ and $v_{i}$ using one of the
two data structures described above.
###### Theorem 3.3.
An undirected weighted graph $G=(V,E)$, a source $r\in V$, and a shortest path
$P\in T_{r}$ can be processed in $O(m+n\log n)$ time to build a data structure
of $O(n)$ space which can report 3-approximate shortest path from $r$ to any
$v\in V$ avoiding any single failed vertex from $P$.
### 3.2. Handling the failure of any vertex in $T_{r}$
We follow divide and conquer strategy based on the following simple lemma.
###### Lemma 3.4.
There exists an $O(n)$ time algorithm to compute a path $P$ in $T_{r}$ whose
removal splits $T_{r}$ into a collection of disjoint subtrees
$T_{r}(v_{1}),...T_{r}(v_{j})$ such that
* •
$|T_{r}(v_{i})|<n/2$ for each $i\leq j$.
* •
$P\cup_{i}T_{r}(v_{i})=T$ and $P\cap T_{r}(v_{i})=\emptyset~{}~{}~{}\forall
i$.
First we compute the path $P\in T_{r}$ as mentioned in Lemma 3.4. We build the
data structure for handling failure of any vertex from $P$ by executing the
algorithm of Theorem 3.3. Let $v_{1},...,v_{j}$ be the roots of the sub trees
of $T_{r}$ connected to the path $P$ with an edge. For each $1\leq i\leq j$,
we solve the problem recursively on the subgraph $G_{r}(v_{i})$, and build the
corresponding data structures. Lemma 3.4 and Theorem 3.3 can be used in
straight forward manner to prove the following theorem.
###### Theorem 3.5.
An undirected weighted graph $G=(V,E)$ can be processed in $O(m\log
n+n\log^{2}n)$ time to build a data structure of size $O(n\log n)$ which can
answer, in optimal time, any 3-approximate shortest path query from a given
source $r$ to any vertex $v\in V$ avoiding any single failed vertex.
## 4\. Single source (1+$\epsilon$)-approximate shortest paths avoiding a
failed vertex
In this section, we shall present a compact data structure for single source
$(1+\epsilon)$-approximate shortest paths avoiding a failed vertex in an
unweighted graph. Let $level(v)$ denote the level (or distance from $r$) of
vertex $v$ in the tree $T_{r}$. Let $U_{x},D_{x},O_{x}$ denote the partitions
of the tree $T_{r}$ formed by deletion of vertex $x$, with the same meaning as
that of $U_{i},D_{i},O_{i}$ defined for $x_{i}$ in the previous section. On
the basis of Observation 3.2, our objective is to build a compact data
structure which will support retrieval of $(1+\epsilon)$-approximate shortest-
paths to vertices of $D_{x}$ upon failure of $x$ for any $x\in V$. Let ${\tt
uchild}(x)$ denote the root of the subtree corresponding to $D_{x}$ (it is
similar to $x_{i+1}$ in case of $D_{i}$). For reporting approximate distance
between $r$ and $v\in D_{x}$ when $x$ fails, the data structure of previous
section reports path of length $\delta(r,{\tt uchild}(x),x)+\delta({\tt
uchild}(x),v)$ which is bounded by $\delta(r,v,x)+2\delta({\tt uchild}(x),v)$.
It should be noted that the approximation factor associated with it is already
bounded by $(1+\epsilon)$ for any $\epsilon>0$ if the following condition
holds.
C : ${\tt uchild}(x)$ is close to $v$, that is, $\delta({\tt
uchild}(x),v)\leq\frac{\epsilon}{2}\delta(r,v)$.
We shall build a supplementary data structure which will ensure that whenever
the condition C does not hold, there will be some ancestor $w$ of $v$ lying on
$\textbf{P}(x,v)$, called a special vertex, satisfying the following two
properties.
1. (1)
$\delta(w,v)\ll\delta(r,v)$, that is $w$ is much closer to $v$ than $r$.
2. (2)
vertex $w$ stores approximate shortest path to $r$ avoiding $x$ (with the
approximation factor arbitrarily close to 1).
We shall refer to such vertices $w$ as special-vertices.
### 4.1. Constructing the set of special vertices
Let $h$ be the height of BFS tree rooted at $r$. Let $L$ be a set of integers
such that $L=\\{i|\lfloor{(1+{\epsilon})}^{i}\rfloor<h\\}$. For a given $i\in
L$, we define a subset $S_{i}$ of special vertices as $S_{i}=\\{u\in
V|level(u)=\lfloor{(1+{\epsilon})}^{i}\rfloor\wedge|T_{r}(u)|\geq{\epsilon}level(u)\\}$.
We define the set of special vertices as $S=\cup_{\forall i\in L}S_{i}$. In
addition, we also introduce the following terminologies.
* •
$S(v)$: the nearest ancestor of $v$ which belongs to set $S$.
* •
$V(u)$: For a vertex $u\in S$, $V(u)$ denotes the set of vertices $v\in V$
with $S(v)=u$. In essence, the vertex $u$ will serve as the special vertex for
each vertex from $V(u)$. For failure of any vertex $x\in\textbf{P}(r,u)$, each
vertex of set $V(u)$ will query the data structure stored at $u$ for retrieval
of approximate shortest path/distance from the source.
We now state two simple lemmas based on the above construction.
###### Lemma 4.1.
Let $v\in V\backslash S$, then
$\delta(v,S(v))\leq\big{(}\frac{2{\epsilon}}{1+{\epsilon}}\big{)}level(v)$ if
${\epsilon}<1$
###### Lemma 4.2.
Let $u$ be a vertex at level $\ell$ and $u\in S$. Then
$V(u)\geq{\epsilon}\ell$.
If we can ensure that the data structure for a special vertex $u$ (for
retrieving approximate shortest paths from $r$ upon failure of any
$x\in\textbf{P}(r,u)$) is of size $O(level(u))$, then it would follow from
Lemma 4.2 that the space required by our supplementary data structure will be
linear in $n$.
### 4.2. The data structure for a special vertex
Consider a special vertex $v$ with $level(v)=\lfloor(1+\epsilon)^{i}\rfloor$
We shall now describe a compact data structure stored at $v$ which will
facilitate retrieval of approximate shortest path from $r$ to $v$ upon failure
of any vertex $x\in\textbf{P}(r,v)$.
Let $v^{\prime}$ be the special vertex which is present at level
$\lfloor(1+\epsilon)^{i-1}\rfloor$ and is ancestor of $v$. The data structure
stored at $v$ will be defined in a way that will prevent it from storing
information that is already present in the data structure of some special
vertex lying on $\textbf{P}(r,v^{\prime})$.
If $x\in\textbf{P}(v^{\prime},v)$, then the data structure described in the
previous section itself stores a path which is $(1+2\epsilon)$-approximation
of $\textbf{P}(r,v,x)$.
Let us now consider the nontrivial case when
$x\in\textbf{P}(r,v^{\prime}),x\not=v^{\prime}$. In order to discuss this
case, we would like to introduce the notion of detour. To understand it, let
us visualize the paths $\textbf{P}(r,v,x)$ and $\textbf{P}(r,v)$
simultaneously. Since $\textbf{P}(r,v,x)$ and $\textbf{P}(r,v)$ have the same
end-points and $x$ doesn’t lie on $\textbf{P}(r,v,x)$, there must be a middle
portion of $\textbf{P}(r,v,x)$ which intersects $\textbf{P}(r,v)$ at exactly
two vertices, and the remaining portion of $\textbf{P}(r,v,x)$ overlaps with
$\textbf{P}(r,v)$. This middle portion is called a detour. We now define it
more formally. Let $a$ and $b$ be two vertices on the shortest path
$\textbf{P}(r,v)$. We use $a\prec b$ to denote that vertex $a$ is closer to
$r$ than vertex $b$. The notation $a\preceq b$ would mean that either $a\prec
b$ or $a=b$. . So here is the definition of detour (and the underlying
observation).
###### Definition 4.3.
Let $x\in\textbf{P}(r,y)$. When $x$ fails, the path $\textbf{P}(r,y,x)$ will
be of the form of $\textbf{P}(r,a)::p_{a,b}::\textbf{P}(b,y)$, where $r\preceq
a\prec x\prec b\preceq y$ and the path $p_{a,b}$ is such that
$p_{a,b}\cap\textbf{P}(a,b)=\\{a,b\\}$. In other words, $p_{a,b}$ meets
$\textbf{P}(a,b)$ only at the end points. We shall call $p_{a,b}$ as the
detour associated with the shortest path $\textbf{P}(r,y,x)$.
Let $p_{a,b}$ represent the detour w.r.t. to $\textbf{P}(r,v,x)$. The handling
of failure of vertices $x\in\textbf{P}(r,v)$ which lie above $v^{\prime}$
would depend upon the detour $p_{a,b}$. This detour can be of any of the
following types (see Figure 2 for illustration).
* •
I : $b\preceq v^{\prime}$.
* •
II : $v^{\prime}\prec b$.
Figure 2. $p_{a,b}$ is shortest detour of $\textbf{P}(r,v,x)$. (i) : detour of
type I, (ii) : detour of type II
Handling detours of type I is relatively easy. Let $w$ be the farthest
ancestor of $v$ such that $w\in S$ and level of $w$ is greater or equal to the
level of $b$. In this case, $v$ stores the corresponding detour implicitly by
just keeping a pointer to the vertex $w$.
Handling detours of type II is slightly tricky since we can’t afford to store
each of them explicitly. However, we shall employ the following observation
associated with the detours of type II to guarantee low space requirement. Let
${\alpha}_{1}$, ${\alpha}_{2}$,…,${\alpha}_{t}$ be the vertices on
$\textbf{P}(r,v)$ (in the increasing order of their levels) such that the
shortest detour corresponding to $\textbf{P}(r,v,{\alpha}_{i})$ is of type II
$\forall i$, then
$\delta(r,v,{\alpha}_{1})\geq\delta(r,v,{\alpha}_{2})\geq\cdots\geq\delta(r,v,{\alpha}_{t})$
It follows from the above observation that if
$\delta(r,v,\alpha_{i})\leq(1+{\epsilon})\delta(r,v,\alpha_{j})$ for any
$i<j$, then $\textbf{P}(r,v,{\alpha}_{i})$ may as well serve as
$(1+{\epsilon})$-approximate shortest path from $r$ to $v$ avoiding
$\alpha_{j}$. In other words, we need not store the detour associated with
$\textbf{P}(r,v,{\alpha}_{j})$ in such situation. Using this observation, we
shall have to explicitly store only $O(\log_{1+{\epsilon}}n)$ detours of type
II. Moreover, we do not store explicitly detours of type II whose length is
much larger than $level(v)$. Specifically, if
$\textbf{P}(r,v,x)\geq\frac{1}{{\epsilon}}level(v)$, then $v$ will merely
store pointer to the path $\textbf{P}(r,{\tt uchild}(x),x)::\textbf{P}({\tt
uchild}(x),v)$. This ensures that each detour of type II which $v$ has to
explicitly store will have length $O(\frac{1}{{\epsilon}}level(v))$.
It follows from the above description that for a special vertex $v$ and
$x\in\textbf{P}(r,v)$, the data structure associated with $v$ stores
$(1+2\epsilon)$-approximation of the path $\textbf{P}(r,v,x)$. Moreover, the
total space required by the data structure associated with all the special
vertices will be $O(n\frac{\log n}{\epsilon^{3}})$. This supplementary data
structure combined with the data structure of previous section can report
$(1+6\epsilon)$-approximation of $\textbf{P}(r,z,x)$ for any $z,x\in V$.
###### Theorem 4.4.
Given an undirected unweighted graph $G=(V,E)$, source $r\in V$, and any
$\epsilon>0$, we can build a data structure of size $O(n\frac{\log
n}{\epsilon^{3}})$ that can report $(1+\epsilon)$-approximate shortest path
from $r$ to any $z\in V$ avoiding any failed vertex in optimal time.
## 5\. All-pairs $(2k-1)(1+\epsilon)$-approx. distance oracle avoiding a
failed vertex
We start with a brief description of the approximate distance oracle of Thorup
and Zwick [13]. The key idea to achieve sub-quadratic space is to store
distance from each vertex to only a small set of vertices. For retrieving
approximate distance between any two vertices $u,v\in V$, it is ensured that
there is a third vertex $w$ which is close to both of them, and whose distance
from both of them is known. To realize this idea, Thorup and Zwick [13]
introduced two novel structures called ball and cluster which are defined for
any two subsets $A,B$ of vertices as follows. (here $\delta(v,B)$ denotes the
distance between $v$ and its nearest vertex from $B$).
$Ball(v,A,B)=\\{w\in A|\delta(v,w)<\delta(v,B)\\}\hskip
28.45274ptC(w,A,B)=\\{v\in V|\delta(v,w)<\delta(v,B)\\}$
Construction of $(2k-1)$-approximate distance oracle of Thorup and Zwick [13]
employs a $k$-level hierarchy ${\mathbf{A}}_{k}=\langle A_{0}\supseteq
A_{1}\supseteq A_{2}...\supseteq A_{k-1}\supset A_{k}\rangle$ of subsets of
vertices as follows.
$A_{0}=V$, $A_{k}=\emptyset$, and $A_{i+1}$ for any $i<k-1$ is formed by
selecting each vertex from $A_{i}$ independently with probability $n^{-1/k}$.
The data structure associated with the $(2k-1)$-approximate distance oracle of
Thorup and Zwick [13] stores for each vertex $v\in V$ the following
information :
* •
the vertices of set $\cup_{i<k}Ball(v,A_{i},A_{i+1})$ (and their distances).
* •
the vertex from $A_{i}$ nearest to $v$ (to be denoted as $p_{i}(v)$).
Due to randomization underlying the construction of ${\mathbf{A}_{k}}$, the
expected size of $Ball(v,A_{i},A_{i+1})$ is $O(n^{1/k})$, and hence the space
required by the oracle is $O(kn^{1+1/k})$. We shall now outline the ideas in
extending the $(2k-1)$-approximate distance oracle to handle single vertex
failure. Kindly refer to the extended version [10] of this paper for complete
details.
### 5.1. Overview of all-pairs approx. distance oracles avoiding a failed
vertex
Firstly the notations used by the static approximate distance oracle of [13],
in particular ball and cluster, get extended for single vertex failure in a
natural manner as follows. (here $\delta(v,B,x)$ is the distance between $v$
and its nearest vertex from $B$ in $G\backslash\\{x\\}$).
$Ball^{x}(v,A,B)=\\{w\in A|\delta(v,w,x)<\delta(v,B,x)\\}$
$C^{x}(w,A,B)=\\{v\in V|\delta(v,w,x)<\delta(v,B,x)\\}$
Let $p_{i}^{x}(v)$ denote the vertex from $A_{i}$ which is nearest to $v$ in
$G\backslash\\{x\\}$. Along the lines of the static approximate distance
oracle of Thorup and Zwick [13], the basic operation which the approximate
distance oracle avoiding a failed vertex should support is the following :
Report distance (exact or approximate) between $v$ and $w\in A_{i}$ if $w\in
Ball^{x}(v,A_{i},A_{i+1})$ for any given $v,x\in V$.
However, it can be observed that we would have to support this operation
implicitly instead of explicitly keeping $Ball^{x}(v,A_{i},A_{i+1})$ for each
$v,x,i$. Our starting point is the simple observation that clusters and balls
are inverses of each others, that is, $w\in Ball^{x}(v,A_{i},A_{i+1})$ is
equivalent to $v\in C^{x}(w,A_{i},A_{i+1})$. Now we make an important
observation. Consider the subgraph ${\mathbf{G}_{i}}(w)$ induced by the
vertices of set $\cup_{x\in V}C^{x}(w,A_{i},A_{i+1})$. This subgraph preserves
the path $\textbf{P}(w,v,x)$ for each $x,v\in V$ if $w\in
Ball^{x}(v,A_{i},A_{i+1})$. So it suffices to keep a single source
(approximate) shortest paths oracle on ${\mathbf{G}_{i}}(w)$ with $w$ as the
root. Keeping this data structure for each $w\in A_{i}$ provides an implicit
compact data structure for supporting the basic operation mentioned above.
Using Theorem 4.4, it can be seen that the space required at a level $i$ will
be of the order of $\sum_{w\in A_{i}}|\cup_{x\in V}C^{x}(w,A_{i},A_{i+1})|$,
but it is not clear whether we can get an upper bound of the order of
$n^{1+1/k}$ on this quantity. Here, as a new tool, we introduce the notion of
$\epsilon$-truncated balls and clusters.
###### Definition 5.1.
Given a vertex $x$, any subsets $A,B$, and $\epsilon>0$
$Ball^{x}(v,A,B,\epsilon)=\left\\{w\in
A|\delta(v,w,x)<\frac{\delta(v,B,x)}{1+\epsilon}\right\\}$
Instead of dealing with the usual balls (and clusters) under deletion of
single vertex, we deal with $\epsilon$-truncated balls (and clusters) under
deletion of single vertex. We note that the inverse relationship between
clusters and balls gets seamlessly extended to $\epsilon$-truncated balls and
clusters under single vertex failure as well. That is,
$\sum_{w\in A_{i}}|\cup_{x\in V}C^{x}(w,A_{i},A_{i+1},\epsilon)|=\sum_{v\in
V}|\cup_{x\in V}Ball^{x}(v,A_{i},A_{i+1},\epsilon)|$
So it suffices to get an upper bound on the size of the set $\cup_{x\in
V}Ball^{x}(v,A_{i},A_{i+1},\epsilon)$ for any vertex $v\in V$. The following
lemma states a very crucial property of $\epsilon$-truncated balls which leads
to prove the existence of a small set $S$ of $O(\frac{1}{\epsilon^{2}}\log n)$
vertices such that
$\cup_{x\in V}Ball^{x}(v,A_{i},A_{i+1},\epsilon)\subseteq\cup_{x\in
S}Ball^{x}(v,A_{i},A_{i+1})\cup Ball(v,A_{i},A_{i+1})$ (1)
###### Lemma 5.2.
In a given graph $G=(V,E)$, let $v$ be any vertex and let $u=p_{i+1}(v)$. Let
$x_{1}$ and $x_{2}$ be any two vertices on the $\textbf{P}(v,u)$ path with
$x_{1}$ appearing closer to $v$ on this path and
$\delta(v,A_{i+1},x_{1})\leq(1+\epsilon)\delta(v,A_{i+1},x_{2})$. Then
$Ball^{x_{1}}(v,A_{i},A_{i+1},\epsilon)\subseteq Ball(v,A_{i},A_{i+1})\cup
Ball^{x_{2}}(v,A_{i},A_{i+1})$
###### Proof 5.3.
Let $w$ be any vertex in $A_{i}$. It suffices to show the following. If $w$
does not belong to $Ball(v,A_{i},A_{i+1})\cup Ball^{x_{2}}(v,A_{i},A_{i+1})$,
then $w$ does not belong to $Ball^{x_{1}}(v,A_{i},A_{i+1},\epsilon)$. The
proof is based on the analysis of the following two cases.
Case 1 : The vertex $x_{2}$ is present in $\textbf{P}(v,w,x_{1})$.
Since, $w\notin Ball(v,A_{i},A_{i+1})$, therefore, $\delta(v,w)$ is at least
$\delta(v,u)$. Hence using triangle inequality,
$\delta(v,x_{2})+\delta(x_{2},w)\geq\delta(v,u)$. Now
$\delta(v,u)=\delta(v,x_{2})+\delta(x_{2},u)$ (since $x_{2}$ lies on
$P(v,u)$). Hence $\delta(x_{2},w)\geq\delta(x_{2},u)$. Moreover, since $x_{1}$
does not appear on $\textbf{P}(x_{2},u)$, so
$\delta(x_{2},u)=\delta(x_{2},u,x_{1})$. So
$\delta(x_{2},w,x_{1})\geq\delta(x_{2},u,x_{1})$ (2)
Now it is given that $x_{2}\in\textbf{P}(v,w,x_{1})$, so
$\textbf{P}(v,w,x_{1})$ must be of the form
$\textbf{P}(v,x_{2},x_{1})::\textbf{P}(x_{2},w,x_{1})$, the length of which is
at least $\delta(v,x_{2},x_{1})+\delta(x_{2},u,x_{1})$ using Equation 2. The
latter quantity is at least $\delta(v,u,x_{1})$ which by definition is at
least $\delta(v,A_{i+1},x_{1})$. Hence $w\notin
Ball^{x_{1}}(v,A_{i},A_{i+1})$, and therefore, $w\notin
Ball^{x_{1}}(v,A_{i},A_{i+1},\epsilon)$.
Case 2 : The vertex $x_{2}$ is not present in $\textbf{P}(v,w,x_{1})$.
In this case,
$\delta(v,w,x_{1})=\delta(v,w,\\{x_{1},x_{2}\\})\geq\delta(v,w,x_{2})$. The
value $\delta(v,w,x_{2})$ is in turn at least $\delta(v,A_{i+1},x_{2})$ since
$w\notin Ball^{x_{2}}(v,A_{i},A_{i+1})$. It is given that
$\delta(v,A_{i+1},x_{2})\geq\frac{\delta(v,A_{i+1},x_{1})}{1+\epsilon}$, hence
conclude that
$\delta(v,w,x_{1})\geq\frac{\delta(v,A_{i+1},x_{1})}{1+\epsilon}$. So $w\notin
Ball^{x_{1}}(v,A_{i},A_{i+1},\epsilon)$.
We shall now outline the construction of a small set $S$ of vertices which
will satisfy Equation 1. Let $u=p_{i+1}(v)$ and let
$\textbf{P}(v,u)=v(=x_{0}),x_{1},...,x_{\ell}(=u)$. Observe that $\cup_{x\in
V}Ball^{x}(v,A_{i},A_{i+1},\epsilon)=\cup_{1\leq
j\leq\ell}Ball^{x_{j}}(v,A_{i},A_{i+1},\epsilon)$. For any node
$x\in{\mathbf{P}}(u,v)$, let $value(x)=\delta(v,A_{i+1},x)$, and let $h$ be
the maximum $value$ of any node on this path. The set $S$ is initially empty.
Let $\alpha(1)$ be the largest index from $[1,\ell]$ such that
$value(x_{i})\geq h/(1+\epsilon)$. It can be seen that for all $j<\alpha(1)$,
$\delta(v,A_{i+1},x_{j})\leq(1+\epsilon)\delta(v,A_{i+1},x_{\alpha(1)})$.
Therefore, it follows from Lemma 5.2 that for each vertex
$x\in\\{x_{1},...,x_{\alpha(1)}\\}$,
$Ball^{x}(v,A_{i},A_{i+1},\epsilon)\subseteq
Ball^{x_{\alpha(1)}}(v,A_{i},A_{i+1})\cup Ball(v,A_{i},A_{i+1})$. So we insert
$x_{\alpha(1)}$ to $S$. Similarly $\alpha(2)\in[\alpha(1)+1,\ell]$ be the
greatest integer such that $value(x_{\alpha(2)})\geq h/(1+\epsilon)^{2}$. We
add $x_{\alpha(2)}$ to $S$, and so on. It can be seen that the set $S$
constructed in this manner will satisfy Equation 1 and its size will be
$O(\log_{1+\epsilon}h)=O(\frac{\log n}{\epsilon})$.
It can be shown using elementary probability theory that for each $x\in V$,
the set $Ball^{x}(v,A_{i},A_{i+1})$ has size $O(n^{1/k}\log n)$ with high
probability. Therefore, the construction of the set $S$ outlined above implies
the following crucial bound for each $v\in V,i<k-1$ which helps us design all-
pairs approximate distance oracle avoiding a failed vertex.
$|\cup_{x\in
V}Ball^{x}(v,A_{i},A_{i+1},\epsilon)|=O\left(n^{1/k}\frac{\log^{2}n}{\epsilon}\right)$
Using this equation, and owing to inverse relationship between clusters and
balls, it follows that $\sum_{w\in A_{i}}|\cup_{x\in
V}C^{x}(w,A_{i},A_{i+1},\epsilon)|=O\left(n^{1+1/k}\frac{\log^{2}n}{\epsilon}\right)$.
Our all-pairs approximate distance oracle avoiding any failed vertex will keep
the following data structures.
* •
Let $p_{i}^{x}(v,\epsilon)$ denote a vertex $w$ from $A_{i}$ with
$\delta(v,w,x)\leq(1+\epsilon)\delta(v,p_{i}^{x}(v),x)$. We keep a data
structure $\textbf{N}_{i}$ $\forall i<k$, using which we can retrieve
$p_{i}^{x}(v,\epsilon)$. This data-structure is obtained by suitable
augmentation of our single source $(1+\epsilon)$-approximate oracle.
* •
For each $w\in A_{i}$, we keep our single source $(1+\epsilon)$-approximate
oracle in $\textbf{G}_{i}(w,\epsilon)$ which is the subgraph induced by
$\cup_{x\in V}C^{x}(w,A_{i},A_{i+1},\epsilon)$.
It follows that the overall space required by the data structure will be
$O(kn^{1+1/k}\frac{\log^{3}n}{\epsilon^{4}})$. The query algorithm and the
analysis on the stretch of the approximate distance reported by the oracle are
similar in spirit to that of Thorup and Zwick [13] (see [10] for details).
###### Theorem 5.4.
Given an integer $k>1$ and a fraction $\epsilon>0$, an unweighted graph
$G=(V,E)$ can be processed to construct a data structure which can answer
$(2k-1)(1+\epsilon)$-approximate distance query between any two nodes $u\in V$
and $v\in V$ avoiding any single failed vertex in $O(k)$ time. The total size
of the data structure is $O(kn^{1+1/k}\frac{\log^{3}n}{\epsilon^{4}})$.
Future work. (i) Can we design a data structure for single source
$(1+\epsilon)$-approx. shortest paths avoiding a failed vertex for weighted
graphs ? Such a data structure will immediately extend our all-pairs approx.
distance oracle avoiding a failed vertex to weighted graphs.
(ii) How to design approx. distance oracles avoiding two or more failed
vertices ? Recent work of Duan and Pettie [8], and Chechik et al. [5] provides
additional motivation for this.
## References
* [1] M. Ahmed and A. Lubiw. Shortest paths avoiding forbidden subpaths. In STACS ’09: Proceedings of 26th International Symposium on Theoretical Aspects of Computer Science, pages 63–74, Freiburg, Germany, 2009\. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany.
* [2] S. Baswana and T. Kavitha. Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In FOCS ’06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 591–602, Washington, DC, USA, 2006. IEEE Computer Society.
* [3] M. A. Bender and M. Farach-Colton. The lca problem revisited. In LATIN ’00: Proceedings of the 4th Latin American Symposium on Theoretical Informatics, pages 88–94, London, UK, 2000. Springer-Verlag.
* [4] A. Bernstein and D. Karger. A nearly optimal oracle for avoiding failed vertices and edges. In STOC ’09: Proceedings of the 41st annual ACM symposium on Theory of computing, pages 101–110, New York, NY, USA, 2009. ACM.
* [5] S. Chechik, M. Langberg, D. Peleg, and L. Roditty. Fault-tolerant spanners for general graphs. In STOC ’09: Proceedings of the 41st annual ACM symposium on Theory of computing, pages 435–444, New York, NY, USA, 2009. ACM.
* [6] C. Demetrescu and G. F. Italiano. A new approach to dynamic all pairs shortest paths. J. ACM, 51(6):968–992, 2004.
* [7] C. Demetrescu, M. Thorup, R. A. Chowdhury, and V. Ramachandran. Oracles for distances avoiding a failed node or link. SIAM J. Comput., 37(5):1299–1318, 2008.
* [8] R. Duan and S. Pettie. Dual-failure distance and connectivity oracles. In SODA ’09: Proceedings of the Nineteenth Annual ACM -SIAM Symposium on Discrete Algorithms, pages 506–515, Philadelphia, PA, USA, 2009\. Society for Industrial and Applied Mathematics.
* [9] J. Hershberger and S. Suri. Vickrey prices and shortest paths: What is an edge worth? In FOCS ’01: Proceedings of the 42nd IEEE symposium on Foundations of Computer Science, page 252, Washington, DC, USA, 2001. IEEE Computer Society.
* [10] N. Khanna and S. Baswana. Approximate shortest paths avoiding a failed vertex : optimal data structures for unweighted graphs. http://www.cse.iitk.ac.in/$\sim$sbaswana/publications/algorithmica-09.pdf.
* [11] E. Nardelli, G. Proietti, and P. Widmayer. Finding the most vital node of a shortest path. Theor. Comput. Sci., 296(1):167–177, 2003.
* [12] L. Roditty. On the k-simple shortest paths problem in weighted directed graphs. In SODA ’07: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages 920–928, Philadelphia, PA, USA, 2007\. Society for Industrial and Applied Mathematics.
* [13] M. Thorup and U. Zwick. Approximate distance oracles. J. ACM, 52(1):1–24, 2005.
|
arxiv-papers
| 2010-01-06T07:02:30 |
2024-09-04T02:49:07.524700
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Neelesh Khanna Surender Baswana",
"submitter": "Neelesh Khanna",
"url": "https://arxiv.org/abs/1001.0824"
}
|
1001.0844
|
# Dynamics of entanglement in the
transverse Ising model
Zhe Chang111e-mail: changz@mail.ihep.ac.cn and Ning Wu222e-mail:
wun@mail.ihep.ac.cn
Institute of High Energy Physics
Chinese Academy of Sciences
P. O. Box 918(4), 100049 Beijing, China
###### Abstract
We study the evolution of nearest-neighbor entanglement in the one dimensional
Ising model with an external transverse field. The system is initialized as
the so called “thermal ground state” of the pure Ising model. We analyze
properties of generation of entanglement for different regions of external
transverse fields. We find that the derivation of the time at which the
entanglement reaches its first maximum with respect to the reciprocal
transverse field has a minimum at the critical point. This is a new indicator
of quantum phase transition.
PACS numbers: 03.67.Bg, 75.10.Pq
## 1 Introduction
Entanglement has been recognized as an important resource for quantum
information and computation [1], and has been the subject of intense research
over the past few years [2]. Recently, there has been extensive analysis of
entanglement in quantum spin models [3]. Various models have been considered
for entanglement generation, and their static [4, 5] as well as dynamical
properties [6, 7, 8] have been investigated. The reason for this is that, many
of these models can be realized in cold atomic gas in an optical lattice [9,
10].
A special one among these models is the anisotropic XY model. It can be solved
exactly by means of the Jordan-Wigner transformation [11]. A lot of works have
been done on the dynamics of entanglement in this model. In Ref.[6], the time
evolution of initial Bell states was studied. Ref.[7] investigated the
dynamics of entanglement of a system prepared in a thermal equilibrium state,
and the system starts evolving after the transverse field was turned off.
In this paper, we analyze a spin-1/2 transverse Ising chain with a particular
separable initial state, the so-called “thermal ground state” [4] for the case
of zero field. We study the generation of nearest-neighbor entanglement as a
function of the external transverse field, and find the time at which the
entanglement reaches its first maximum. It turns out that the derivation of
this time with respect to the reciprocal transverse field has a minimum at the
critical point. This is a new indicator of quantum phase transition.
## 2 Static correlations
The anisotropic XY model is described by the Hamiltonian
$\displaystyle
H=-\frac{\lambda}{4}\sum^{N}_{i=1}[(1+\gamma)\sigma^{x}_{i}\sigma^{x}_{i+1}+(1-\gamma)\sigma^{y}_{i}\sigma^{y}_{i+1}]-\frac{1}{2}\sum^{N}_{i=1}\sigma^{z}_{i}~{},$
(1)
where $\sigma^{\alpha}_{i}$ are the Pauli matrix of the spin at site $i$ and
we assume periodic boundary conditions. We have neglected a common factor and
absorbed the external field into the reciprocal field $\lambda$. The
anisotropy parameter $\gamma$ connects the quantum Ising model for $\gamma=1$.
In the range $0<\gamma\leq 1$, the model belongs to the Ising universality
class and undergoes a quantum phase transition at the critical point
$\lambda_{c}=1$. We will consider the case of $\gamma=1$ in this paper, namely
the transverse Ising model
$\displaystyle
H=-\frac{1}{2}\sum^{N}_{i=1}(\lambda\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{z}_{i})~{}.$
(2)
The Hamiltonian defined in Eq.(2) can be mapped into a one-dimensional
spinless fermion system with creation and annihilation operators
$c^{\dagger}_{i}$ and $c_{i}$ via the Jordan-Wigner transformation
$\frac{\sigma^{x}_{i}-i\sigma^{y}_{i}}{2}=\prod_{j<i}(1-2c^{\dagger}_{j}c_{j})c_{i},~{}\frac{\sigma^{x}_{i}+i\sigma^{y}_{i}}{2}=c^{\dagger}_{i}\prod_{j<i}(1-2c^{\dagger}_{j}c_{j}),~{}\frac{\sigma^{z}_{i}}{2}=c^{\dagger}_{i}c_{i}-\frac{1}{2}$.
By making use of the transformation
$\displaystyle\eta_{k}=\frac{1}{\sqrt{N}}\sum_{l}e^{ikl}(\alpha_{k}c_{l}+i\beta_{k}c^{\dagger}_{l})$
(3)
further, with $\alpha_{k}=\frac{\Lambda_{k}-(1+\lambda\cos
k)}{\sqrt{2[\Lambda^{2}_{k}-(1+\lambda\cos
k)\Lambda_{k}]}},~{}\beta_{k}=\frac{\lambda\sin
k}{\sqrt{2[\Lambda^{2}_{k}-(1+\lambda\cos k)\Lambda_{k}]}}$, we get the
fermionic Hamiltonian finally
$\displaystyle
H=\sum_{k}\Lambda_{k}\left(\eta^{\dagger}_{k}\eta_{k}-\frac{1}{2}\right)~{},$
(4)
where the spectrum is $\Lambda_{k}=\sqrt{1+\lambda^{2}+2\lambda\cos k}$.
The evolution of the system is governed by Eq.(2). In the Heisenberg picture,
the time evolution of the spinless fermion operator $c_{i}(t)$ is [6]
$\displaystyle c_{i}(t)=\frac{1}{N}\sum_{k,l}[\cos k(l-i)A(k,t)c_{l}+\sin
k(l-i)B(k,t)c^{\dagger}_{l}]~{},$ (5)
where $A(k,t)=e^{i\Lambda_{k}t}-2i\beta^{2}_{k}\sin\Lambda_{k}t$,
$B(k,t)=-2i\alpha_{k}\beta_{k}\sin\Lambda_{k}t$.
The dynamics of the system also depends on the initial state. We choose the
“thermal ground state” $\rho_{0}=\frac{1}{2}(|N^{+}\rangle\langle
N^{+}|+|N^{-}\rangle\langle N^{-}|)$ of the pure Ising model (no transverse
field is applied) as the initial state. Here
$|N^{+}\rangle=|\rightarrow\rangle_{1}...|\rightarrow\rangle_{N}$, and
$|N^{-}\rangle=|\leftarrow\rangle_{1}...|\leftarrow\rangle_{N}$ are the two
degenerate ground states of the Ising model with all spins pointing to the
positive (negative) $x$ direction. Note that $|N^{+}\rangle$ and
$|N^{-}\rangle$ are the ground states of the Hamiltonian (2) with
$\lambda\to\infty$. When $\lambda$ is finite, they are even not eigenstates of
the Hamiltonian. Thus, the evolution from this initial state must be
nontrivial for finite $\lambda$. Physically, this can be viewed as an
instantaneous, i.e., idealized sudden quench [8] in the reciprocal field
$\lambda_{1}\to\lambda_{2}$ with $\lambda_{1}\to\infty$ and
$\lambda_{2}=\lambda$.
Due to the translational symmetry, we need only to consider averages of the
form $\langle c_{1}...\rangle_{0}$. Namely, we choose site-1 as “the first
site”, where $\langle...\rangle_{0}$ denote the average over the initial state
$\rho_{0}$. We rewrite $|N^{\pm}\rangle$ in the fermionic representation,
$\displaystyle|N^{\pm}\rangle=\left(\frac{1}{\sqrt{2}}\right)^{N}(1\pm
c^{\dagger}_{1})(1\pm c^{\dagger}_{2})...(1\pm
c^{\dagger}_{N})|0_{1},0_{2},...,0_{N}\rangle~{},$ (6)
where $|0_{1},0_{2},...,0_{N}\rangle$ denotes the vacuum of the fermions, or
the state for all spins down. Here we have to pay attention to the order of
operators in this equation. We write $(1\pm c^{\dagger}_{1})$ before operators
on all other sites in order to be compatible with the correlation $\langle
c_{1}...\rangle_{0}$. In other words, if another site $i$ is chosen as “the
first site”, we should write the state as
$|N^{\pm}\rangle=(\frac{1}{\sqrt{2}})^{N}(1\pm c^{\dagger}_{i})(1\pm
c^{\dagger}_{i+1})...(1\pm
c^{\dagger}_{i-1})|0_{i},0_{i+1},...,0_{i-1}\rangle$ to calculate averages
like $\langle c_{i}...\rangle_{0}$, due to the periodic boundary conditions.
The only single-site averages are $\langle c_{1}\rangle_{0}$ and $\langle
c^{\dagger}_{1}c_{1}\rangle_{0}$ (or their complex conjugate). It is easy to
see that $\langle N^{+}|c_{1}|N^{+}\rangle=\frac{1}{2}=-\langle
N^{-}|c_{1}|N^{-}\rangle$, so $\langle c_{1}\rangle_{0}=0$. As to $\langle
c^{\dagger}_{1}c_{1}\rangle_{0}$, we have $\langle
N^{+}|c^{\dagger}_{1}c_{1}|N^{+}\rangle=\frac{1}{2}\langle
0_{1}|(1+c_{1})c^{\dagger}_{1}c_{1}(1+c^{\dagger}_{1})|0_{1}\rangle=\frac{1}{2}$.
Similarly, $\langle N^{-}|c^{\dagger}_{1}c_{1}|N^{-}\rangle=\frac{1}{2}$. So
that, $\langle c^{\dagger}_{1}c_{1}\rangle_{0}=\frac{1}{2}$. We can also see
this point from $0=\langle\frac{\sigma^{z}_{1}}{2}\rangle_{0}=\langle
c^{\dagger}_{1}c_{1}-\frac{1}{2}\rangle_{0}$.
Next we consider two-point correlations of the form $\langle
c_{1}c_{l}\rangle_{0},~{}l\geq 2$. For $l=2$, we have
$\displaystyle\langle N^{+}|c_{1}c_{2}|N^{+}\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)^{2}\langle
0_{1},0_{2}|(1+c_{2})(1+c_{1})c_{1}c_{2}(1+c^{\dagger}_{1})(1+c^{\dagger}_{2})|0_{1},0_{2}\rangle$
$\displaystyle=$ $\displaystyle\left(\frac{1}{2}\right)^{2}\langle
0_{1}|(1+c_{1})c_{1}(1-c^{\dagger}_{1})|0_{1}\rangle\cdot\langle
0_{2}|(1+c_{2})c_{2}(1+c^{\dagger}_{2})|0_{2}\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{4}~{}.$
Similarly $\langle N^{-}|c_{1}c_{2}|N^{-}\rangle=-\frac{1}{4}$. So that
$\langle c_{1}c_{2}\rangle_{0}=-\frac{1}{4}$. We can show that $\langle
c^{\dagger}_{1}c_{2}\rangle_{0}=\frac{1}{4}$ in the same manner. For $l>2$,
when $c_{l}$ or $c^{\dagger}_{l}$ crosses $(1\pm c^{\dagger}_{2})$ to its
right side, this will make it be $(1\mp c^{\dagger}_{2})$. But $\langle
0_{2}|(1\pm c_{2})(1\mp c^{\dagger}_{2})|0_{2}\rangle=0$. So we have obtained
all the two-point correlations
$\displaystyle\langle
c^{\dagger}_{i}c_{j}\rangle_{0}=\frac{1}{4}(\delta_{i+1,j}+\delta_{i-1,j}+2\delta_{i,j}),$
$\displaystyle\langle
c_{i}c_{j}\rangle_{0}=\frac{1}{4}(-\delta_{i+1,j}+\delta_{i-1,j}),$
$\displaystyle\langle
c^{\dagger}_{i}c^{\dagger}_{j}\rangle_{0}=\frac{1}{4}(\delta_{i+1,j}-\delta_{i-1,j}).$
(7)
By the same means one can check that the averages involving three fermion
operators all vanish.
## 3 Reduced density matrix and evolution of concurrence
Making use of the periodic boundary conditions, we can focus on the reduced
density matrix $\rho^{12}(t)$ at time $t$ of the first and second spins. In
the
$\\{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\\}$
basis, the matrix elements reads,
$\displaystyle\rho^{12}_{11}(t)$ $\displaystyle=$
$\displaystyle\langle\uparrow\uparrow|\rho^{12}(t)|\uparrow\uparrow\rangle$
$\displaystyle=$
$\displaystyle\langle\uparrow\uparrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\uparrow\uparrow\rangle+\langle\uparrow\downarrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\uparrow\downarrow\rangle$
$\displaystyle+\langle\downarrow\uparrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\downarrow\uparrow\rangle+\langle\downarrow\downarrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\downarrow\downarrow\rangle$
$\displaystyle=$ $\displaystyle
tr_{12}(\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2})$ $\displaystyle=$
$\displaystyle tr(\rho(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2})$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{1}c_{1}c^{\dagger}_{2}c_{2}\rangle_{t}.$
Other matrix elements can be obtained similarly. Thus, we have
$\displaystyle\rho^{12}(t)=\left(\begin{array}[]{cccc}\langle
c^{\dagger}_{1}c_{1}c^{\dagger}_{2}c_{2}\rangle_{t}&-\langle
c^{\dagger}_{1}c_{1}c_{2}\rangle_{t}&\langle
c_{2}^{\dagger}c_{2}c_{1}\rangle_{t}&\langle c_{2}c_{1}\rangle_{t}\\\ \\\
-\langle c^{\dagger}_{2}c^{\dagger}_{1}c_{1}\rangle_{t}&\langle
c^{\dagger}_{1}c_{1}c_{2}c^{\dagger}_{2}\rangle_{t}&-\langle
c_{1}c^{\dagger}_{2}\rangle_{t}&\langle
c_{1}c_{2}c^{\dagger}_{2}\rangle_{t}\\\ \\\ \langle
c^{\dagger}_{1}c_{2}^{\dagger}c_{2}\rangle_{t}&-\langle
c_{2}c_{1}^{\dagger}\rangle_{t}&\langle
c_{1}c^{\dagger}_{1}c^{\dagger}_{2}c_{2}\rangle_{t}&\langle
c_{1}c^{\dagger}_{1}c_{2}\rangle_{t}\\\ \\\ \langle
c^{\dagger}_{1}c^{\dagger}_{2}\rangle_{t}&\langle
c_{2}c^{\dagger}_{2}c^{\dagger}_{1}\rangle_{t}&\langle
c^{\dagger}_{2}c_{1}c^{\dagger}_{1}\rangle_{t}&\langle
c_{1}c^{\dagger}_{1}c_{2}c^{\dagger}_{2}\rangle_{t}\\\ \end{array}\right),$
(15)
All the elements can be equally evaluated in the Heisenberg picture, e.g.,
$\rho^{12}_{11}(t)=\langle
c^{\dagger}_{1}c_{1}c^{\dagger}_{2}c_{2}\rangle_{t}=\langle
c^{\dagger}_{1}(t)c_{1}(t)c^{\dagger}_{2}(t)c_{2}(t)\rangle_{0}$. We have
mentioned that all the static averages involving three fermion operators equal
zero. From Eq.(5), the fermion operators in the Heisenberg picture are merely
some linear superpositions of the same set of operators in the Schrodinger
picture. So all matrix elements involving three operators in Eq.(8) also
vanish. Using Wick’s theorem, one gets all non-vanishing elements of
$\rho^{12}(t)$ as following
$\displaystyle\rho^{12}_{11}(t)=\langle
c^{\dagger}_{1}(t)c_{1}(t)\rangle_{0}\langle
c^{\dagger}_{2}(t)c_{2}(t)\rangle_{0}-\langle
c^{\dagger}_{1}(t)c^{\dagger}_{2}(t)\rangle_{0}\langle
c_{1}(t)c_{2}(t)\rangle_{0}-\langle c^{\dagger}_{1}(t)c_{2}(t)\rangle\langle
c^{\dagger}_{2}(t)c_{1}(t)\rangle_{0},$
$\displaystyle\rho^{12}_{22}(t)=\rho^{12}_{33}(t)=\langle
c^{\dagger}_{1}(t)c_{1}(t)\rangle_{0}-\rho^{12}_{11}(t),$
$\displaystyle\rho^{12}_{44}(t)=1-\rho^{12}_{11}(t)-2\rho^{12}_{22}(t),$
$\displaystyle\rho^{12}_{14}(t)=\rho^{12*}_{41}(t)=\langle
c_{2}(t)c_{1}(t)\rangle_{0},$
$\displaystyle\rho^{12}_{23}(t)=\rho^{12*}_{32}(t)=-\langle
c_{1}(t)c^{\dagger}_{2}(t)\rangle_{0}.$ (16)
We see that if we can calculate $\langle
c^{\dagger}_{1}(t)c_{1}(t)\rangle_{0},~{}\langle
c^{\dagger}_{1}(t)c_{2}(t)\rangle_{0}$ and $\langle
c_{1}(t)c_{2}(t)\rangle_{0}$, then the reduced density matrix is totally
determined.
Using Eq.(5) and (7), we find, in the thermodynamic limit, that
$\displaystyle\langle c^{\dagger}_{1}(t)c_{1}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int^{\pi}_{-\pi}dk\sin
k(A^{*}(k)-A(k))(B(k)-B^{*}(k))+\frac{1}{4\pi}\int^{\pi}_{-\pi}dk(|A(k)|^{2}+|B(k)|^{2})$
$\displaystyle+\frac{1}{4\pi}\int^{\pi}_{-\pi}dk\cos k(|A(k)|^{2}-|B(k)|^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\pi}_{0}dk\frac{\lambda\sin^{2}k\sin^{2}\Lambda_{k}t}{\Lambda^{2}_{k}}+\frac{1}{2},$
$\displaystyle\langle c^{\dagger}_{1}(t)c_{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int^{\pi}_{-\pi}dk(1+\cos
2k)(|A(k)|^{2}-|B(k)|^{2})+\frac{1}{4\pi}\int^{\pi}_{-\pi}dk\cos
k(|A(k)|^{2}+|B(k)|^{2})$ $\displaystyle+\frac{1}{8\pi}\int^{\pi}_{-\pi}dk\sin
2k[A(k)-A^{*}(k)]B^{*}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\pi}_{0}dk\frac{\lambda\cos
k\sin^{2}k\sin^{2}\Lambda_{k}t}{\Lambda^{2}_{k}}+\frac{1}{4},$
$\displaystyle\langle c_{1}(t)c_{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int^{\pi}_{-\pi}dk\sin
2kA(k)B(k)+\frac{1}{8\pi}\int^{\pi}_{-\pi}dk[B(k)^{2}-A(k)^{2}](1-\cos 2k)$
(17) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int^{\pi}_{0}dk\sin^{2}k[\frac{i\sin
2\Lambda_{k}t}{\Lambda_{k}}-\cos
2\Lambda_{k}t-2\frac{\lambda\sin^{2}\Lambda_{k}t}{\Lambda_{k}^{2}}(\cos
k+\lambda)]$
Note that $\langle c^{\dagger}_{1}(t)c_{2}(t)\rangle$ is indeed real, so that
we have $\rho^{12}_{23}(t)=\rho^{12}_{32}(t)$.
For bipartite entanglement, a commonly used measure for arbitrary states of
two qubits is the so-called concurrence [12]. The concurrence is defined as
$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\rm{max}\\{0,2\lambda_{max}(t)-tr\sqrt{\rho^{12}(t)\tilde{\rho}^{12}(t)}\\},$
(18) $\displaystyle\tilde{\rho}^{12}(t)$ $\displaystyle=$
$\displaystyle\sigma_{y}\otimes\sigma_{y}\rho^{12*}(t)\sigma_{y}\otimes\sigma_{y},$
(19)
where $\lambda_{max}$ is the largest eigenvalue of the matrix
$\sqrt{\rho^{12}(t)\tilde{\rho}^{12}(t)}$.
In the present case,
$\displaystyle\rho^{12}(t)=\left(\begin{array}[]{cccc}\rho^{12}_{11}(t)&0&0&\rho^{12}_{14}(t)\\\
\\\ 0&\rho^{12}_{22}(t)&\rho^{12}_{23}(t)&0\\\ \\\
0&\rho^{12}_{23}(t)&\rho^{12}_{22}(t)&0\\\ \\\
\rho^{12*}_{14}(t)&0&0&\rho^{12}_{44}(t)\\\ \end{array}\right),$ (27)
the four eigenvalues of $\sqrt{\rho^{12}(t)\tilde{\rho}^{12}(t)}$ are of the
form
$\displaystyle\lambda_{1}(t)$ $\displaystyle=$
$\displaystyle||\rho^{12}_{14}(t)|+\sqrt{\rho^{12}_{11}(t)\rho^{12}_{44}(t)}|,$
$\displaystyle\lambda_{2}(t)$ $\displaystyle=$
$\displaystyle||\rho^{12}_{14}(t)|-\sqrt{\rho^{12}_{11}(t)\rho^{12}_{44}(t)}|,$
$\displaystyle\lambda_{3}(t)$ $\displaystyle=$
$\displaystyle|\rho^{12}_{22}(t)+\rho^{12}_{23}(t)|,$
$\displaystyle\lambda_{4}(t)$ $\displaystyle=$
$\displaystyle|\rho^{12}_{22}(t)-\rho^{12}_{23}(t)|.$ (28)
Figure 1: The evolution of bipartite entanglement for the initial state being
the “thermal ground state” of the Ising model.
In Fig.1, we give a plot of the evolution of entanglement as a function of
time $t$ and parameter $\lambda$. At the beginning, the reduced density matrix
is
$\displaystyle\rho^{12}(0)=\frac{1}{4}\left(\begin{array}[]{cccc}1&0&0&1\\\
0&1&1&0\\\ 0&1&1&0\\\ 1&0&0&1\\\ \end{array}\right).$ (33)
So that the concurrence equals zero. At any later fixed instant, the
entanglement as a function of $\lambda$ may have different behaviors in
different time intervals. For times before about $t=2.5$ (Fig.2 (a)), the
entanglement maintains its magnitude for large $\lambda$ with slow
oscillations. As $\lambda$ increases, the height of each peak deceases
gradually, and finally vanishes as $\lambda\to\infty$. This is easy to
understand. For $\lambda\to\infty$, the Hamiltonian reduces to the pure Ising
model, and the initial state is just the eigenstate of the Ising model. Thus,
only a phase factor contributes to the times evolving state, which does not
change the entanglement. Physically, this corresponds to
$\lambda_{1}=\lambda_{2}=\infty$, hence no quench takes place. For the narrow
band $2.47<t<2.69$, there is no entanglement at all for all $\lambda$. For
times greater than around $t=3.0$ (Fig.2 (b)), the entanglement emerges
abruptly, but only has non-vanishing values for very short interval of
$\lambda$. As the time increases, the generations of entanglement are
different from each other for different $\lambda$. For $\lambda$ is less than
about $\lambda=0.8$ (Fig.3 (a)), the concurrence will encounter more than one
maximums. In a time region between the first two maximums, no entanglement
appear. Although weak oscillations emerge after this region, the entanglement
can last for a fairly long time with a not too small magnitude. For
$0.80<\lambda<1.07$, there is only one maximum as the time increases. After a
short time, the entanglement vanishes. For $\lambda>1$ (Fig. 3 (b)), strong
oscillations emerge. The entanglement reaches its maximum fast, then suddenly
decreases to zero. Such a behavior will repeat again at later times, and
finally the entanglement vanishes completely.
Figure 2: (a): The concurrence at different (b): The concurrence at different
times
times $t=0.5,1.0,1.5,2.0$ as a function of $\lambda$, $t=4.0,5.0,6.0,7.0$ as a
function of $\lambda$,
in the region of $t<2.5$, in the region of $t>3.0$.
Figure 3: (a): The dynamics of entanglement (b): The dynamics of entanglement
for
for different reciprocal fields $\lambda=0.2,0.4,0.6,0.8$, different
reciprocal fields $\lambda=1.5,3.0$,
in the region of $\lambda<0.8$, in the region of $\lambda>1.07$. Figure 4:
The time at which the entanglement reaches its first maximum and its
derivative.
For any $\lambda$, there is a time $T_{m}(\lambda)$ at which the entanglement
reaches its first maximum (Fig.4). This time is a monotonically decreasing
function of $\lambda$. Taking its derivative with respect to $\lambda$, we
find that the derivative $\frac{dT_{m}(\lambda)}{d\lambda}$ has a minimum at
the critical point $\lambda=1$. This interesting property maybe an indicator
of the quantum phase transition at the critical point.
## 4 Brief discussions and conclusions
In this paper, we have studied the dynamics of the transverse Ising chain
prepared in the “thermal ground state” of the pure Ising model. For different
values of the reciprocal field, the evolution of the entanglement has dramatic
distinctions. We also obtained the times at which the entanglement reaches its
first maximum, which is a monotonically decreasing function of the parameter
$\lambda$. The derivative of this quantity with respect to the reciprocal
field gets its minimum at the critical field. This is a new indicator of the
quantum phase transition. Let us finally remark that the initial state we
choose is simple to be prepared experimentally. One can first cool the pure
Ising system down to near absolute zero, then turn on some desired transverse
field at $t=0$ to generate the wanted entanglement.
## Acknowledgments
We would like to thank X. Li for useful discussions. The work was supported
partially by the NSF of China under Grant No. 10875129.
## References
* [1] M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ (Cambridage University Press, Cambridge, 2000).
* [2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
* [3] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
* [4] T. J. Osborne and M. A. Nielsen, Phys, Rev. A 66, 032110 (2002).
* [5] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature (London) 416, 608 (2002).
* [6] L. Amico, A. Osterloh, F. Plastina, R. Fazio, and G. M. Palma, Phys, Rev. A 69, 022304 (2004).
* [7] A. Sen(De), U. Sen, and M. Lewenstein, Phys, Rev. A 72, 052319 (2005).
* [8] H. Wichterich and S. Bose, Phys, Rev. A 79, 060302(R) (2009).
* [9] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys, Rev. Lett. 81, 3108 (1998).
* [10] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, and I. Bloch, Nature (London) 425, 937 (2003).
* [11] E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).
* [12] W. K. Wootters, Phys, Rev. Lett. 80, 2245.
|
arxiv-papers
| 2010-01-06T09:35:15 |
2024-09-04T02:49:07.532964
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang and Ning Wu",
"submitter": "Ning Wu",
"url": "https://arxiv.org/abs/1001.0844"
}
|
1001.0891
|
††thanks: tanzg@iopp.ccnu.edu.cn
# Some Characteristic Parameters of Proton from the Bag Model
Z. G. Tan Department of Electronics and Communication Engineering, Changsha
University,Changsha, 410003,P.R.China L. Y. Huang Department of Electronics
and Communication Engineering, Changsha University,Changsha, 410003,P.R.China
C. B. Yang Institute of Particle Physics, Hua-Zhong Normal University, Wuhan,
430079, China
###### Abstract
We treat the mass of a proton as the total static energy which can be
separated into two parts that come from the contribution of quarks and gluons
respectively. We adopt the essential of the bag model of hadron to discuss the
structure of a proton and find that the calculated temperature, proton radius,
the bag constant are compare well with QCD results if a proton is a thermal
equilibrium system of quarks and gluons.
###### pacs:
12.39.Ba, 14.20.Dh
## I Introduction
Exploring proton structure c1 ; c2 ; c3 is still one of most important
subjects for more profound enhancement of human knowledge on strong
interactions. It is also very helpful for people to search for a new matter
state–quark gluon plasma (QGP) which is the deconfined state of strongly
interacting matter. Most theoretical investigations focus on Quantum
Chromodynamics (QCD) c4 . However,there are a few phenomenological models
about nucleon structure and interactions. The classical string model describes
mesons as string segments executing longitudinal expansion and contraction c5
. The bag model describes quarks being confined inside a hadron c6 . In high
energy collisions, the string model represents the process of particle
production in the fragmentation of a stretching string by creating pairs of
quark and anti-quark. It worked well c7 for elementary collisions where
strings can be formed among the few initial partons and break up to form the
soft final state hadrons. However, in relativistic heavy ion collisions
(RHIC), there are thousands initial partons. It is intractable to pair partons
and have a string for each pair. Even if the strings are formed, they must be
modified by the presence of many other color charges. The independent
fragmentation approach c7 , though valid for high $Q^{2}$ partonic processes
in the vacuum, cannot explain experimentally observed large $p/\pi$ ratio in
central $Au+Au$ collisions at RHIC c8 . Another possible hadronization
mechanism, the deconfined quark (fled out from the bag) recombination model,
has been able to reproduce spectra for almost all stable particles for
different colliding systems. It provides a natural explanation for the
baryon/meson ratio and the nuclear suppression factor observed at RHICc9 .
A natural mechanism for quark confinement is given by the bag model c6 . While
the bag model has a few different versions, we shall in this paper keep the
essential characteristics of the phenomenology of quark confinement. The
gluons are mediate bosons which transfer the interactions between quarks .
Their effect can be replaced by the bag pressure which confines the quarks in
a hadron. On the other hand, if all interactions among partons in the bag are
neglected, we assume that the partons might be treated as a thermally
equilibrated system with a given volume. Then properties of a hadron can be
investigated and some characteristic parameters for the hadron can be obtained
in the bag model.
The organization of this paper is as follows. In Sec.II, we will discuss some
features about a thermally equilibrated QGP system. Then in Sec.III, we get an
estimate of the maximum kinetic energy of a confined quark in a spherical
cavity of radius $R$. Combining the discussions in section II and III by
assuming its origin from the contribution of gluons, the magnitude of the
hadronic bag pressure is discussed in Sec.IV. The last section is for
conclusions and discussions.
## II Free equilibrated QGP
Let us first consider a thermally equilibrated quark-gluon plasma (QGP) system
at first. When its temperature $T$ and volume $V$ are given, the total energy
and particle number can be calculated:
$\displaystyle E$ $\displaystyle=$
$\displaystyle\sum_{i=-N_{f}}^{N_{f}}\frac{g_{i}}{(2\pi\hbar)^{3}}\int
f_{i}(T)p^{0}\,d\Gamma$ (1) $\displaystyle=$
$\displaystyle\sum_{i=-N_{f}}^{N_{f}}\frac{g_{i}V}{2\pi^{2}\hbar^{3}}\int
f_{i}(T)p^{0}|{\bf p}|^{2}\,dp\,,$ $\displaystyle N$ $\displaystyle=$
$\displaystyle\sum_{i=-N_{f}}^{N_{f}}\frac{g_{i}}{(2\pi\hbar)^{3}}\int
f_{i}(T)\,d\Gamma$ (2) $\displaystyle=$
$\displaystyle\sum_{i=-N_{f}}^{N_{f}}\frac{g_{i}V}{2\pi^{2}\hbar^{3}}\int
f_{i}(T)|{\bf p}|^{2}\,dp\,,$
where $N_{f}$ is the number of flavors and $g_{i}=N_{c}N_{s}$ is the
degeneracy number for a parton and equals to the product of quantum numbers of
quark’s color and spin. $f_{i}$ is the distribution function which is of
Fermi-Dirac for quarks and Bose-Einstein for gluons
$\displaystyle f_{i}=$
$\displaystyle\frac{1}{1+e^{(p^{0}\mp\mu_{q})/T}}\qquad\mbox{'-'\, for quark,
}$ ’+’ for anti-quark $\displaystyle f_{i}=$
$\displaystyle\frac{1}{e^{p^{0}/T}-1}\ \qquad\qquad\mbox{ for gluon}\,.$ (4)
Here $\mu_{q}$ is the quark’s chemical potential. For the case when the number
density of the quarks is the same as that of the anti-quarks, $\mu_{q}=0$.
For a massless quark gas with zero chemical potential $\mu_{q}=0$, Eqs.(1,2)
can be solved analytically. For example for the case with only two flavors, we
get the densities for energy and quark number as ($\hbar=1$)
$\displaystyle\epsilon=\frac{E}{V}$ $\displaystyle=$
$\displaystyle\frac{7}{4}(g_{q}+g_{\bar{q}})\frac{\pi^{2}}{30}T^{4}+g_{g}\frac{\pi^{2}}{30}T^{4}$
(5) $\displaystyle=$ $\displaystyle\frac{37}{30}\pi^{2}T^{4},$ $\displaystyle
n=\frac{N}{V}$ $\displaystyle=$
$\displaystyle\frac{3\zeta(3)}{2\pi^{2}}(g_{q}+g_{\bar{q}})T^{3}+\frac{g_{g}\Gamma(3)\zeta(3)}{2\pi^{2}}T^{3}$
(6) $\displaystyle=$ $\displaystyle\frac{34\times 1.202}{\pi^{2}}T^{3}.$
Now if we treat a proton as a free equilibrated QGP system as use the above
results, we can get the relation of the proton radius and temperature. The
result is shown in Fig. (1).
Figure 1: The relation between the radius of a proton and its temperature when
take its mass as the total energy. The numerical simulative formula is given
on the legend as well.
From Fig.(1), we can see, the radius of a proton is decreased with its
internal temperature. The numerical simulative formula is
$RT^{4/3}=0.0524$ (7)
If the temperature is 100 MeV, the corresponding radius of proton is about 1
fm. If a proton is compressed to have a radius of 0.6 fm, the inner
temperature will be about 170 MeV, which is close to the critical temperature,
so that a proton may break and quarks may flee out from the bag.
## III Quarks confined in a hadron bag
Let’s give some discussions about quark wave-function from theory. We assume
the quarks are confined in a spherical cavity of radius R, they are free
fermions inside it but cannot fly out because all contributions from gluons
are attributed to the bag constant. Then the surface of the sphere bag becomes
the maximum range that quarks can arrive. On the bag boundary where the
current of fermion must be zero. A hadron’s wave-function is then product of
those for quarks.
The Dirac equation for a free massless fermion in the bag is
$(i\gamma^{\mu}\partial_{\mu}-m)\psi=0\quad\mbox{with}\quad m=0,$ (8)
where $\partial_{\mu}=(p^{0},\bf{p})$. We will in this paper use the Dirac
representation
$\gamma^{0}=\left(\begin{array}[]{cc}I&0\\\ 0&-I\end{array}\right),$
and
$\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right),$
where $I$ is a $2\times 2$ unit matrix and $\sigma^{i}$ are the Pauli
matrices. We write the four-component wave function for the massless fermion
$\psi$ as
$\psi=\displaystyle{\psi_{+}\choose\psi_{-}}$
where both $\psi_{+}$ and $\psi_{-}$ are two dimensional spinors. Eq.(8)
becomes
$\left(\begin{array}[]{cc}p^{0}&-{\bf\sigma\cdot p}\\\ {\bf\sigma\cdot
p}&-p^{0}\end{array}\right)\displaystyle{\psi_{+}\choose\psi_{-}}=0$ (9)
The lowest energy solution for the above equation is the $s_{1/2}$ state given
byc10
$\displaystyle\psi_{+}({\bf r},t)$ $\displaystyle=$
$\displaystyle\mathcal{N}e^{-ip^{0}t}j_{0}(p^{0}r)\chi_{+}$
$\displaystyle\psi_{-}({\bf r},t)$ $\displaystyle=$
$\displaystyle\mathcal{N}e^{-ip^{0}t}{\bf\sigma\cdot\hat{r}}j_{1}(p^{0}r)\chi_{-}\,,$
where $j_{l}$ is the spherical Bessel function which can be expressed by an
elementary function
$j_{l}(x)=(-1)^{l}x^{l}\left(\frac{1}{x}\frac{d}{dx}\right)^{l}\frac{\sin
x}{x},$ (10)
$\chi_{\pm}$ are two-dimensional spinors, and $\mathcal{N}$ is a normalization
constant. The confinement of the quarks is equivalent to the requirement that
the normal component of the vector current
$J_{\mu}=\bar{\psi}\gamma_{\mu}\psi$ vanishes at the surface. This condition
is the same as the requirement that the scalar density $\bar{\psi}\psi$ of the
quark vanishes at the bag surface $r=R$. This leads to
$\left.\bar{\psi}\psi\right|_{r=R}=[j_{0}(p^{0}R)]^{2}-{\bf\sigma\cdot\hat{r}\sigma\cdot\hat{r}}[j_{1}(p^{0}R)]^{2}=0$
or
$[j_{0}(p^{0}R)]^{2}-[j_{1}(p^{0}R)]^{2}=0$ (11)
From Eq.(10), solution of the above equation is given by
$p^{0}_{m}R=2.04,\quad\mbox{or}\quad p^{0}_{m}=\frac{2.04}{R}.$ (12)
This result means that in order to keep the bag from being broken, the kinetic
energy of any quark can not larger than $p^{0}_{m}$ determined by the radius
of the bag.
## IV Bag Pressure
Take $p^{0}_{m}$ as the upper limit, we can separate the energy of quarks from
the total of a proton
$E_{q}=\frac{(g_{q}+g_{\bar{q}})V}{2\pi^{2}\hbar^{3}}\int_{0}^{p^{0}_{m}}\frac{p^{3}dp}{1+e^{p/T(R)}}.$
(13)
For simplicity, we have neglected the chemical potential, and treat the quarks
as massless. Then $g_{q}=g_{\bar{q}}=N_{c}N_{s}N_{f}=3\times 2\times 2=12$.
The energy from gluon contribution provides the pressure effect directed from
the region outside the bag
$B=\frac{M-E_{q}}{V}.$ (14)
Here $M$ is the mass of the hadron.
From Eq.(14), we can easily learn that the the bag pressure will change with
radius as shown in Fig.(2). We also give the numerical formula
$B^{1/4}=0.17R^{-0.65}.$ (15)
The average kinetic energy of each quark can be calculated as
$\bar{E}_{q}=\frac{E_{q}}{N_{q}}=\frac{\int_{0}^{p^{0}_{m}}\frac{p^{3}dp}{1+e^{p/T(R)}}}{\int_{0}^{p^{0}_{m}}\frac{p^{2}dp}{1+e^{p/T(R)}}}.$
(16)
The energy carried by the valence quarks in a proton is then
$3\times\bar{E}_{q}$. So that contributions from gluons and sea quarks to the
energy is $M-3\bar{E}_{q}$. The bag pressure decreases with the radium. This
may be used to explain why the resonance particle (which has large radius thus
small bag pressure) is usually unstable because their bags are more fragile.
Figure 2: The bag pressure change with the radium of proton. The open circle
is calculated with Eq.(14), and the line is its numerical simulation results,
while the star is with the scenario that three quarks are surrounded by
gluons.
## V Conclusion
In this work we have discussed some feature of a proton from the bag model.
Especially, we got the relations between temperature and the radius of the
proton when a proton is treated as a non-interacting thermal equilibrium QGP
system. The bag pressure comes from the contribution of gluons.
###### Acknowledgements.
We are grateful to the financial support from China ChangSha University under
grant No.SF080101. We also thank Prof. A. Bonasera for stimulating discussions
and comments.
## References
* (1) Marc Vanderhaeghen, Nucl. Phys.A 805:210-220,2008.
* (2) J. Sowinski, Nucl.Phys.A 790:485-488,2007,
* (3) A. Bhattacharya, S. N. Banerjee, B.Chakraborti, S. Banerjee, Nucl. Phys. Proc. Suppl. 142 (2005) 13-15
* (4) F. Wilczek, Ann. Rev. Nucl. and Part. Sci. 32, 177(1982)
* (5) X. Artru and G. Mennessier, Nucl. Phys. B70, 93(1974),B.Andersson, G. Gustafson and B. Söderberg, Z. Phys. C20,317(1983)
* (6) C. D. Detar and J. F. Donoghue, Ann. Rev. Nucl. Part. Sci. 33,235(1983)
* (7) Torbjörn Sjöstrand, Leif Lönnblad, Stephen Mrenna, Peter Skrands, JHEP 0605(2006) 026
* (8) R. C. Hwa and C. B. Yang, J. Phys. G30(2004) S1117-S1120
* (9) C. B. YANG, Int. J. Mod. Phys. E 16, No. 10, 3148(2007)
* (10) C. Y. Wong, Introduction to High-Energy Heavy ion Collisions, World Scientific Co., Singapore,1994.
* (11) Z. G. Tan, S. Terranova and A. Bonasera, Int. J. Mod. Phys. E17 No 8(2008)1577-1589;
* (12) Z. G. Tan, A. Bonasera, C. B. Yang, D. M. Zhou and S. Terranova. Int. J. Mod. Phys. E16, Nos.7&8 (2007) 2269-2275
|
arxiv-papers
| 2010-01-06T13:28:08 |
2024-09-04T02:49:07.539380
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z.G. Tan, L.Y. Huang, C.B. Yang",
"submitter": "Z.G. Tan",
"url": "https://arxiv.org/abs/1001.0891"
}
|
1001.0919
|
# Sustained ferromagnetism induced by H-vacancies in graphane
Julia Berashevich and Tapash Chakraborty Department of Physics and Astronomy,
University of Manitoba, Winnipeg, Canada, R3T 2N2
###### Abstract
The electronic and magnetic properties of graphane with H-vacancies were
investigated using the quantum-chemistry methods. The hybridization of the
edges is found to be absolutely crucial in defining the size of the bandgap,
which is increased from 3.04 eV to 7.51 eV when the hybridization is changed
from the $sp^{2}$ to the $sp^{3}$ type. The H-vacancy defects also influence
the size of the gap that depends on the number of defects and their
distribution between the two sides of the graphane plane. Further, the
H-vacancy defects induced on one side of the graphane plane and placed on the
neighboring carbon atoms are found to be the source of ferromagnetism which is
distinguished by the high stability of the state with a large spin number in
comparison to that of the singlet state and is expected to persist even at
room temperatures. However, the ferromagnetic ordering of the spins is
obtained to be limited by the concentration of H-vacancy defects and ordering
would be preserved if number of defects do not exceed eight.
## I Introduction
Graphene is the carbon-based wonder material which has gained wide attention
due to its many unique electronic and magnetic properties. Despite the high
mobility of the charge carriers in graphene resulting from its zero effective
mass review ; novoselov ; ando , the absence of gap hinders its application in
nanoelectronics. The magnetic properties of graphene arising from spin
ordering of the localized states at the zigzag edges mine or by the presence
of defects rossier ; pereira ; epl ; peres ; zhang might facilitate its
application in carbon-based spintronics. If the localized states occupy the
same sublattice then they can induce the sublattice imbalance and according to
Lieb’s theorem lieb that can lead to the ground state being ferromagnetic
rossier ; pereira . The room-temperature ferromagnetism in graphene has been
obtained experimentally wang . However, there are some issues involved in
maintaining the ferromagnetic state whose stability depends on the
concentration of the localized states, distance between states and size of the
graphene flakes through the size of the band gap rossier ; pereira ; epl ;
peres ; zhang . Therefore, disappearance of the gap in bulk graphene brings
some inconsistencies in applying Lieb’s theorem epl .
Recently discovered graphane sofo ; exp ; exp1 ; flores – hydrogenated
graphene – has brought new impetus in the investigation of carbon-based
materials due to the predicted advantages in its application in
nanoelectronics and spintronics. Termination of the carbon atoms by hydrogens
leads to the generation of $sp^{3}$ carbon network removing the $\pi$ bands
from its band structure thereby generating a gap. It was theoretically
predicted that fully hydrogenated graphane is non-magnetic and a wide band gap
semiconductor sofo ; leb . However, H-vacancy defects in graphane generate
localized states characterized by non-zero magnetic moments (each defect has
$\mu=1.0\mu_{B}$ sahin , where $\mu_{B}$ is the Bohr magneton [Fig. 1]). As
graphane is characterized by a wide gap and the value of the charge transfer
integral ($t_{\sigma}\sim$-7.7 eV ui ) is higher than that in graphene
($t_{\pi}\sim$-2.4 eV ui ), according to the Hubbard model rossier ; pereira
these should stabilize ferromagnetism in graphane (for example through an
increase of the critical value of the on-site repulsion term peres ).
We report here on our investigation of the electronic and magnetic properties
of graphane and their modification once the H-vacancy defects are introduced
in the lattice. Our study is performed via the quantum-chemistry methods using
the spin-polarized density functional theory with the semilocal gradient
corrected functional (UB3LYP/6-31G) in the Jaguar 7.5 program jaguar . The
H-vacancies are introduced in the originally optimized structure of defect-
free graphane in the chair conformation (for the size of the graphane flake
see Fig. 1), whose carbon atoms at the edges are terminated by two hydrogen
atoms thereby preserving the $sp^{3}$ network over the whole lattice.
Figure 1: The spin density for a single H-vacancy defect in graphane plotted
with isovalues of $\pm 0.001$ e/Å3.
## II Single H-vacancy defect
The size of the band gap of graphane nanoribbons decreases exponentially with
increasing nanoribbon width as a result of vanishing of the confinement effect
li similarly to that in graphene graphene . Therefore, for graphane flakes
confined by the edges from all sides of size 10Å$\times$16Å suggesting strong
confinement effect, the large band gap in comparison to that obtained for
nanoribbons li ; leb ; sofo is expected. Defect-free graphane flakes of size
10Å$\times$16Å with edges possessing $sp^{3}$ hybridization are found to be
characterized by degenerate bands and by a band gap of 7.51 eV (the highest
and lowest molecular orbitals are HOMO=-6.09 eV and LUMO=1.42 eV,
respectively), as shown in Fig. 2 (a). For comparison we have examined flakes
of size 18Å$\times$16Å and found a decrease in gap to 7.15 eV (HOMO=-5.87 eV
and LUMO=1.27 eV). The edges in the $sp^{2}$ hybridization, for which the edge
carbon atoms are terminated by a single hydrogen, possess the localized
states. In this case the orbital degeneracy is lifted and the gap decreases to
3.04 eV (HOMO=$-4.64$ eV and LUMO=$-1.58$ eV). In the available experiment exp
; exp1 , only a transformation of graphene from the conductor to an insulator
due to its hydrogenation was reported, but the size of the gap and the type of
edge hybridization were not indicated. Since we found that the gap is
sensitive to edge hybridization and can be increased from 3.04 eV to a maximum
of 7.51 eV by its transformation from $sp^{2}$ to $sp^{3}$ type, this issue
should be the top priority for further experimental investigations.
Figure 2: Energetics of the bands in graphane with $sp^{3}$ hybridized edges
(solid lines) and the defect levels ($\pi$ and $\pi^{*}$) induced by
H-vacancies (dashed lines): (a) defect-free graphane; (b) graphane containing
a single H-vacancy; (c) graphane containing two H-vacancies distributed
between the two sides of the graphane plane ($AB$-distribution) and separated
by a distance of $d=3a_{\rm C-C}$ (antiferromagnetic spin ordering); (d)
graphane containing two H-vacancies located on one side of the graphane plane
($AA$-distribution) and separated by the distance of $d=4a_{\rm C-C}$
(ferromagnetic spin ordering). The spin density plotted with isovalues of $\pm
0.001$ e/Å3 is also presented.
A single H-vacancy defect in the graphane lattice generates an unsaturated
dangling bond on the carbon atom – $\pi$ unpaired electron (perpendicular
$p_{z}$ orbital). Moreover, bonding of the carbon atom carrying the defect
with its neighbors is changed from the $sp^{3}$ hybridization to $sp^{2}$,
thus providing a modification of the bond length from 1.55 Å to 1.52 Å. The
perpendicular $p_{z}$ orbital and the C-C bonds possessing $sp^{2}$
hybridization participate in the formation of the localized state
characterized by an unpaired spin (see the spin density in Fig. 1). Therefore,
this localized state is spin-polarized and generates a defect level in the
band gap (see the bands in Fig. 2(b)). For the $\alpha$-spin state, the defect
level ($\pi$) appears close to the valence band (HOMO=$-4.55$ eV) thereby
suppressing the size of the gap to $\Delta_{\alpha}$=6.06 eV, while for the
$\beta$-spin state the $\pi^{*}$ level occurs closer to the conduction band
(LUMO=$-0.84$ eV) and the gap is $\Delta_{\beta}$=5.29 eV.
## III Different distribution of the H-vacancy defects
For several H-vacancy defects, ordering of spins of the localized states and
the size of the bandgap are defined by the distance between the defects and
the distribution of those defects between the sides of the graphane plane. The
side dependence is related to the sublattice symmetry. For the chair
conformation of graphane, the carbon atoms belonging to different sublattices
are terminated by the hydrogen atoms from different sides of the plane. It was
already known that in graphene lee , when the localized states occupy the same
sublattice and if their spins have antiparallel alignment, then the
contribution of the $\pi$ states to the total energy diminishes as a result of
the destructive interference between the spin-up and spin-down tails.
Therefore, according to Lieb’s theorem lieb for the localized states
occupying the same sublattice a ferromagnetic ordering of their spins would be
energetically favored, but for the states on different sublattices, one
expects the antiferromagnetic ordering rossier ; pereira ; epl ; peres .
For vacancies equally distributed between both sides of the graphane plane
($AB$-distribution), the energetically favorable spin ordering is the
antiparallel alignment of spins between one side (A sublattice) and the other
(B sublattice). Thus, for even number of vacancies in the $AB$-distribution,
all spins are paired but the band degeneracy can be slightly lifted. In Fig.
2(c) we present the bands for two H-vacancy defects in the $AB$-distribution
separated by a distance $d=3a_{\rm C-C}$, where $a_{\rm C-C}$ is the length of
the C-C bond. Therefore for the $\alpha$\- and $\beta$-spin states, the
obtained band gap of the size $\Delta_{\alpha,\beta}=5.41$ eV is defined by
the energy gap between the $\pi$ (HOMO=$-4.56$ eV) and the $\pi^{*}$
(LUMO=$-0.85$ eV) defect levels. For odd number of defects, one localized
state would have unpaired spin which generates an extra level $\pi$ for the
$\alpha$\- and $\pi^{*}$ for the $\beta$-spin states. However, when several
H-vacancy defects are located on the same side of the graphane plane the
parallel alignment of their spins would be preferred because they belong to
the same sublattice ($AA$-distribution). In Fig. 2(d) we show the energetics
of the bands for graphane with two H-vacancy defects separated by a distance
of $d=4a_{\rm C-C}$ in its triplet state. Each spin state induces a defect
level in the valence band of the $\alpha$-spin state ($\pi$ states) and in the
conduction band of the $\beta$-spin state ($\pi^{*}$ states). Therefore, the
size of the bandgap for the $\alpha$-state is defined by the conduction band
and the defect level $\pi$ in the valence band, while for the $\beta$-spin
state by the valence band and the defect level $\pi^{*}$ in the conduction
band ($\Delta_{\alpha}$=6.08 eV, $\Delta_{\beta}$=5.24 eV) that is similar to
the case of the single H-vacancy (see Fig. 2(b)). However, the state
characterized by antiparallel alignment of two spins (the singlet state),
possesses the $\pi$ and $\pi^{*}$ defect levels for both the $\alpha$\- and
$\beta$-spin states and the size of the bandgap is defined by the energy gap
between these defect levels, $\pi$ and $\pi^{*}$, i.e.,
$\Delta_{\alpha}\simeq\Delta_{\beta}$ (for example see Fig. 2(c)).
The destructive and constructive contributions of the spin tails of the
localized states decrease with increasing distance between the defects lee .
Therefore, we have calculated the difference in the total energy between the
triplet and singlet states ($E_{(\frac{2}{2};\frac{1}{2})}$) depending on the
distance between the two H-vacancy defects. If $E_{(\frac{2}{2};\frac{1}{2})}$
energy is negative the ferromagnetic ordering of spins is energetically
preferred, but an antiferromagnetic ordering otherwise. We have calculated the
two components: the energy $E_{(\frac{2}{2};\frac{1}{2})}^{0}$ is considered
before relaxation of the lattice induced by the presence of defects and the
$E_{(\frac{2}{2};\frac{1}{2})}$ component after relaxation. These energies for
the $AA$\- and $AB$-distributions and splitting of the $\pi$ levels in the
valence band ($\varepsilon_{1}-\varepsilon_{2}$) are presented in Table 1. For
the $AB$-distribution the distance between the two defects should be $>a_{\rm
C-C}$ because for $d=a_{\rm C-C}$ the spins of the two localized states are
paired which leads to transformation of a single bond in $sp^{3}$
hybridization between the nearest-neighbor carbon atoms to a double bond in
$sp^{2}$ hybridization ($\sigma$ and $\pi$ bonds) of length 1.36 Å. Such a
defect forms the $\pi$ and $\pi^{*}$ defect levels which are energetically
close to the edges of the conduction and valence bands of graphane. Therefore,
the size of the band gap defined by the defect levels is 6.38 eV, that is much
larger than that for the levels formed by the non-bonded perpendicular $p_{z}$
orbital ($\Delta$=3.71 eV in Fig. 2(c)).
Table 1: Difference in the total energy between the triplet and singlet states for the non-optimized plane of graphane with two defects $E_{(\frac{2}{2};\frac{1}{2})}^{0}$ and after its full relaxation $E_{(\frac{2}{2};\frac{1}{2})}$. ($\varepsilon_{1}-\varepsilon_{2}$) is the energy splitting of the $\pi$ orbitals for the relaxed lattice of graphane with two defects. distance | $E_{(\frac{2}{2};\frac{1}{2})}^{0}$ (eV) | $E_{(\frac{2}{2};\frac{1}{2})}$ (eV) | ($\varepsilon_{1}-\varepsilon_{2}$) (eV)
---|---|---|---
$AA$-distribution (ferromagnetic ordering)
$d=2a_{\rm C-C}$ | $-1.52$ | $-1.23$ | 2.82$\times 10^{-1}$
$d=4a_{\rm C-C}$ | $-0.26$ | $-1.32\times 10^{-2}$ | 1.52$\times 10^{-1}$
$d=6a_{\rm C-C}$ | 7.11$\times 10^{-5}$ | $-1.15\times 10^{-2}$ | 1.53$\times 10^{-2}$
$d=8\ a_{\rm C-C}$ | 1.76$\times 10^{-5}$ | -7.93$\times 10^{-3}$ | 6.52$\times 10^{-3}$
$AB$-distribution (antiferromagnetic ordering)
$d=a_{\rm C-C}$ | 1.34 | 2.98 | -
$d=3a_{\rm C-C}$ | 4.30$\times 10^{-3}$ | 1.18$\times 10^{-2}$ | -
$d=5a_{\rm C-C}$ | $-1.08\times 10^{-4}$ | 6.13$\times 10^{-3}$ | -
A significant energy difference between the triplet and the singlet states was
found only for $d\leq 2a_{\rm C-C}$ in the $AA$-distribution for which the
relaxation of the graphane lattice stabilizes the state with ferromagnetic
spin ordering, and for $d<2a_{\rm C-C}$ in the $AB$-distribution. As a result,
for the nearest location of the defects, the ordering of spins is according to
Lieb’s theorem lieb . When the defects are spatially separated ($d>2a_{\rm
C-C}$) the energy difference between the state with a large spin number and
the singlet state diminishes because of decoupling of the magnetic moments of
the localized states and the random spin distribution with a minimum number of
unpaired spins would be energetically favored. Therefore, for even number of
defects the system would prefer to remain in the singlet state independent of
the distribution of defects over the sublattices, while for odd number of
defects the triplet state is preferred.
We have calculated the fluctuation of the gap size with increasing number of
defects for the system in its singlet state when the size of the gap is
defined by the energy difference between the induced defect levels, $\pi$ and
$\pi^{*}$, formed by the perpendicular $p_{z}$ orbitals. For the
$AB$-distribution ($d>a_{\rm C-C}$), the size of the gap can fluctuate from
1.2 to 3.7 eV depending on the distance between the defects and their
locations. The change in the gap size is related to the degree of the broken
sublattice symmetry. Moreover, with increasing concentration of the
H-vacancies ($N>8$) we found that the number of localized states can be
smaller than the number of defects. For the $AA$-distribution the size of the
gap is found to gradually decrease from 3.75 to 2.72 eV with increasing number
of defects from $N=2$ to $N=18$. Additionally, with a growing number of
defects a significant distortion of the planarity of graphane, such as
buckling of the lattice inherent for the $AA$-distribution, was observed.
## IV Stable ferromagnetism
A prediction of the ordering of the $AA$-distributed localized states on the
graphane surface is controversial. According to results in zhou for
semihydrogenated graphene, i.e. graphene hydrogenated from one side (the so
called graphone), the non-hydrogenated side of graphane is possessing the
localized states and the spins all of them are ferromagnetically ordered. We
believe that this behavior is highly debatable because it is known for the
carbon-like materials, such as diamond and graphitic structures, that the
total magnetization is suppressed by increasing the vacancy concentrations,
and particularly for graphitic structures this occurs more rapidly zhang . In
contrast, authors of Ref. sahin did not consider the ferromagnetic ordering
of the states in defected graphane at all because they found that for the
vacancy defects located on the neighbors ($d=2a_{\rm C-C}$), the spins are
paired, indicating a nonmagnetic state. When the distance exceeds $2a_{\rm
C-C}$ for which pairing of the spins can not occur, the interaction between
the localized states vanish, thereby favoring the antiferromagnetic ordering
of spins. However, we believe that pairing of the spins of the localized
states located on the neighboring carbon atoms is unlikely, considering the
significant distance between the vacancies ($\sim$ 2.55 Å). Moreover, pairing
may occur only when the spins are antiferromagnetically ordered (that for the
states localized on the same sublattice is against the Lieb’s theorem lieb )
and it implies the formation of the bond, which again seems unlikely because
of the large distance between the localized states (the typical $C-C$ bond
length in organic compounds is $\sim$ 1.4 Å against $\sim$ 2.55 Å for the
states separated by $d=2a_{\rm C-C}$ in graphane). Therefore, the question of
ordering of spins of the localized states in their $AA$ distribution is
unclear as yet which we set out to investigate here.
We found that for the $AA$-distribution of the localized states formed by the
H-vacancies, the ferromagnetic ordering of their spins is possible when the
vacancies are placed on the neighboring carbon atoms (see
$E_{(\frac{2}{2};\frac{1}{2})}^{0}$ for the $AA$-distribution in Table 1),
thereby generating a state characterized by a large spin number (see an
example for graphene in Ref. yazyev ). Just as in sahin it was noticed that
increasing the distance between the vacancies leads to vanishing of
$\pi$-$\pi$ interaction between localized states resulting in the
antiferromagnetic ordering of spins. Therefore, we investigating here the
stability of the ferromagnetic ordering of the spins of the localized states
placed on the neighboring carbon atoms ($d=2a_{\rm C-C}$) and formation of
domains of defects depending on the concentration of the defects in domain and
number of domains.
We found that with increasing concentration of defects ($N$) the stability of
the state with a large spin number, i.e., the difference in total energy
between the state with a large spin number and the singlet states, can
increase. Therefore, for two parallel lines of defects
$E_{(\frac{4}{2};\frac{1}{2})}^{0}=-3.32$ eV for $N=4$ (Fig. 3(a) for the spin
density distribution of $N=4$), $E_{(\frac{6}{2};\frac{1}{2})}^{0}=-5.03$ eV
for $N=6$ and $E_{(\frac{8}{2};\frac{1}{2})}^{0}=-6.56$ eV for $N=8$. However,
for $N=8$ the states characterized by the lower spin number are close in
energy to the state with the larger spin number
($E_{(\frac{8}{2};\frac{6}{2})}^{0}=-0.035$ eV,
$E_{(\frac{8}{2};\frac{4}{2})}^{0}=-0.065$ eV and
$E_{(\frac{8}{2};\frac{2}{2})}^{0}=-0.12$ eV) thereby destabilizing it and
limiting the number of defects having ferromagnetically ordered spins. A
further increase of the defect concentration ($N>8$) leads to significant
suppression of the energy difference between the ferromagnetic and singlet
states. Thus, $E_{(\frac{10}{2};\frac{1}{2})}^{0}=-0.25$ eV for $N=10$,
$E_{(\frac{12}{2};\frac{1}{2})}^{0}=-0.17$ eV for $N=12$ and
$E_{(\frac{14}{2};\frac{1}{2})}^{0}=-0.14$ eV for $N=14$. Therefore, just as
for graphene rossier ; pereira ; epl ; peres , there is a critical value of
defect concentration above which the ferromagnetic ordering of the spins of
the localized states occupying the same sublattice no longer exists. Another
destabilizing factor for ferromagnetism in graphene containing many
H-vacancies in the $AA$-distribution is the lattice relaxation leading to the
buckling of the graphane structure ($E_{(\frac{4}{2};\frac{1}{2})}=-1.49$ eV
for $N=4$ and $E_{(\frac{8}{2};\frac{1}{2})}=1.29$ eV for $N=8$). There is
also a significant decrease in stability when the defects are divided in
groups, because the state characterized by antiferromagnetic ordering of spins
between the groups is close in energy to that of the state with ferromagnetic
ordering of all spins (see the spin density distribution for $N=4$ in Fig.
3(b)). Finally, since graphane is a wide gap semiconductor and possess no
localized states at the edges which could interact ($\pi-\pi$ interaction)
with states formed by the H-vacancy defects, the increasing size of the
graphane flakes leading to suppression of the graphane gap should not
drastically alter the interaction of the localized states and their ordering,
which was the case in graphene epl .
Figure 3: Spin density for the $AA$-distribution of H-vacancies (a) 8
H-vacancies ($E_{(\frac{8}{2};\frac{1}{2})}^{0}=-6.36$ eV), (b) 4 H-vacancies
in the singlet state ($E_{(\frac{4}{2};\frac{1}{2})}^{0}=4.81\times 10^{-5}$
eV). Spin densities are plotted with isovalues of $\pm 0.005$ e/Å3.
To summarize, for the $AA$-distribution of the H-vacancies the size of the
band gap can be tuned by the level of hydrogenation – the gap slowly decreases
with increasing number of defects, while for the $AB$-distribution the size of
the gap fluctuates in the range of 1.2 – 3.7 eV depending on the level of the
broken sublattice symmetry. Moreover, formation of the H-vacancy defects
redistributed over one side of the graphane plane ($AA$-distribution) and
located on the neighboring carbon atoms belonging to the same sublattice will
generate a stable state characterized by a large spin number (ferromagnetic
ordering). For a better stabilization of this state, the reorganization of the
graphane lattice in response to the occurrence of defects should be minimized.
Deformation of the graphane lattice can be minimized for free standing
graphane in the low-temperature regime and through interaction of graphane
with a substrate under the condition that the $sp^{3}$ hybridization of
graphane lattice is preserved. If the rigidity of graphane on a substrate
could be achieved, graphane containing H-vacancies on one side of the plane
can even become a room-temperature ferromagnet, that obviously has enormous
potentials for application in nanoelectronics and spintronics. Because each
defect forms a perpendicular $p_{z}$ orbital possessing unpaired spin and,
therefore, its contribution to the magnetization is 1$\mu_{B}$, the magnitude
of magnetization of such room temperature ferromagnet can ideally be regulated
by the number of H-vacancy defects. However, the number of defects to achieve
the stable ferromagnetism at room temperature should be limited because above
a critical value the ferromagnetic ordering of spins would be unstable.
The work was supported by the Canada Research Chairs Program.
## References
## References
* (1) Abergel D S L, Apalkov V, Berashevich J, Ziegler K, Chakraborty T Advances in Physics. in press.
* (2) Novoselov K S, Geim, A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197\.
* (3) Ando T in Nano-Physics & Bio-Electronics: A New Odyssey, edited by Chakraborty T, Peeters F, and Sivan U (Elsevier, Amsterdam, 2002), Chap 1.
* (4) Berashevich J, Chakraborty T 2009 Phys. Rev. B 80 033404\.
* (5) Lieb E H 1989 Phys. Rev. Lett. 62 1201\.
* (6) Palacois J J, Fernanández-Rossier J, Brey L 2008 Phys. Rev. B 77 195428\.
* (7) Pereira V M, Guinea F, Lopes dos Santos J M B, Peres N M R, Castro Neto A H 2006 Phys. Rev. Lett. 96 036801\.
* (8) Huang B-L, Mou C-Y 2009 EPL 88 68005\.
* (9) Peres N M R, Araújo M A N, Bozi D 2004 Phys. Rev. B 70 195122\.
* (10) Zhang Y, Talapatra S, Kar S, Vajtai R, Nayak S K, Ajayan P M 2007 Phys. Rev. Lett. 99 107201\.
* (11) Wang Y, Huang Y, Song Y, Zhang X, Ma Y, Liang J, Chen Y 2009 Nano Lett. 9 220\.
* (12) Li Y, Zhou Z, Shen P, Chen Z 2009 J. Phys. Chem. C 113 15043\.
* (13) Han M Y, Özyilmaz B, Zhang Y, and Kim P 2007 Phys. Rev. Lett. 98 206805\.
* (14) Tang M S, Wang C Z, Chan C T, Ho K M 1996 Phys.Rev. B 53 979\.
* (15) Sofo J O, Chaudhari A S, Barber G D 2007 Phys. Rev. B 75 153401\.
* (16) Ryu S, Han M Y, Maultzsch J, Heinz T F, Kim P, Steigerwald M L, Brus L E 2008 Nano Lett. 8 4597\.
* (17) Elias D C, Nair R R, Mohiuddin T M G, Morozov S V, Blake P, Halsall M P, Ferrari A C, Boukhvalov D W, Katsnelson M I, Geim A K, Novoselov K S 2009 Science 323 610\.
* (18) Flores M Z S, Autreto P A S, Legoas S B, Galvao D S 2009 Nanotechnology 20 465704\.
* (19) Leb‘egue S, Klintenberg M, Eriksson O, Katsnelson M I 2009 Phys. Rev. B 79 245117\.
* (20) Şahin H, Ataca C, Ciraci S 2009 Appl. Phys. Lett. 95 222510\.
* (21) Jaguar, version 7.5. Schrödinger 2007. LLC: New York, NY.
* (22) Lee H, Son Y-W, Park N, Han S, Yu, J 2005 Phys. Rev. B 72 174431\.
* (23) Yazyev O V, Wang W L, Meng S, Kaxiras E 2008 Nano Lett. 8 766\.
* (24) Zhou J, Wang Q, Sun Q, Chen X S, Kawazoe Y, Jena P 2009 Nano Lett. 9 3867\.
|
arxiv-papers
| 2010-01-06T15:31:00 |
2024-09-04T02:49:07.545313
|
{
"license": "Public Domain",
"authors": "Julia Berashevich and Tapash Chakraborty",
"submitter": "Julia Berashevich",
"url": "https://arxiv.org/abs/1001.0919"
}
|
1001.0920
|
2010573-584Nancy, France 573
Claire Mathieu
Ocan Sankur
Warren Schudy
# Online Correlation Clustering
C. Mathieu Department of Computer Science, Brown University, 115 Waterman
Street, Providence, RI 02912 , O. Sankur Ecole Normale Supérieure, 45, rue
d’Ulm, 75005 Paris, France and W. Schudy
###### Abstract.
We study the online clustering problem where data items arrive in an online
fashion. The algorithm maintains a clustering of data items into similarity
classes. Upon arrival of v, the relation between v and previously arrived
items is revealed, so that for each u we are told whether v is similar to u.
The algorithm can create a new cluster for v and merge existing clusters.
When the objective is to minimize disagreements between the clustering and the
input, we prove that a natural greedy algorithm is O(n)-competitive, and this
is optimal.
When the objective is to maximize agreements between the clustering and the
input, we prove that the greedy algorithm is .5-competitive; that no online
algorithm can be better than .834-competitive; we prove that it is possible to
get better than 1/2, by exhibiting a randomized algorithm with competitive
ratio .5+c for a small positive fixed constant c.
###### Key words and phrases:
correlation clustering, online algorithms
###### 1991 Mathematics Subject Classification:
F.2.2 Nonnumerical Algorithms and Problems
Part of this work was funded by NSF grant CCF 0728816.
## 1\. Introduction
We study online correlation clustering. In correlation clustering [2, 15], the
input is a complete graph whose edges are labeled either positive, meaning
similar, or negative, meaning dissimilar. The goal is to produce a clustering
that agrees as much as possible with the edge labels. More precisely, the
output is a clustering that maximizes the number of agreements, i.e., the sum
of positive edges within clusters and the negative edges between clusters.
Equivalently, this clustering minimizes the disagreements. This has
applications in information retrieval, e.g. [8, 10].
In the online setting, vertices arrive one at a time and the total number of
vertices is unknown to the algorithm a priori. Upon the arrival of a vertex,
the labels of the edges that connect this new vertex to the previously
discovered vertices are revealed. The algorithm updates the clustering while
preserving the clusters already identified (it is not permitted to split any
pre-existing cluster). Motivated by information retrieval applications, this
online model was proposed by Charikar, Chekuri, Feder and Motwani [5] (for
another clustering problem). As in [5], our algorithms maintain Hierarchical
Agglomerative Clusterings at all times; this is well suited for the
applications of interest.
The problem of correlation clustering was introduced by Ben-Dor et al. [3] to
cluster gene expression patterns. Unfortunately, it was shown that even the
offline version of correlation clustering is NP-hard [15, 2]. The following
are the two approximation problems that have been studied [2, 7, 1]: Given a
complete graph whose edges are labeled positive or negative, find a clustering
that minimizes the number of disagreements, or maximizes the number of
agreements. We will call these problems MinDisAgree and MaxAgree respectively.
Bansal et al. [2] studied approximation algorithms both for minimization and
maximization problems, giving a constant factor algorithm for MinDisAgree, and
a Polynomial Time Approximation Scheme (PTAS) for MaxAgree. Charikar et al.
[7] proved that MinDisAgree is APX-hard and gave a factor $4$ approximation.
Ailon et al. [1] presented a randomized factor $2.5$ approximation for
MinDisAgree, which is currently the best known factor. The problem has
attracted significant attention, with further work on several variants [9, 6,
11, 13, 3, 12, 14].
In this paper, we study online algorithms for MinDisAgree and MaxAgree. We
prove that MinDisAgree is essentially hopeless in the online setting: the
natural greedy algorithm is $O(n)$-competitive, and this is optimal up to a
constant factor, even with randomization (Theorem 7). The situation is better
for MaxAgree: we prove that the greedy algorithm is a $.5$-competitive
(Theorem 1), but that no algorithm can be better than $0.803$ competitive
($0.834$ for randomized algorithms, see Theorem 2). What is the optimal
competitive ratio? We prove that it is better than $.5$ by exhibiting an
algorithm with competitive ratio $0.5+\epsilon_{0}$ where $\epsilon_{0}$ is a
small absolute constant (Theorem 2.2). Thus Greedy is not always the best
choice!
More formally, let $v_{1},\ldots,v_{n}$ denote the sequence of vertices of the
input graph, where $n$ is not known in advance. Between any two vertices,
$v_{i}$ and $v_{j}$ for $i\neq j$, there is an edge labeled positive or
negative. In MinDisAgree (resp. MaxAgree), the goal is to find a clustering
$\mathcal{C}$, i.e. a partition of the nodes, that minimizes the number of
disagreements $\text{cost}(\mathcal{C})$: the number of negative edges within
clusters plus the number of positive edges between clusters (resp. maximizes
the number of agreements $\text{profit}(\mathcal{C})$: the number of positive
edges within clusters plus the number of negative edges between clusters).
Although these problems are equivalent in terms of optimality, they differ
from the point of view of approximation. Let OPT denote the optimum solution
of MinDisAgree and of MaxAgree.
In the online setting, upon the arrival of a new vertex, the algorithm updates
the current clustering: it may either create a new singleton cluster or add
the new vertex to a pre-existing cluster, and may decide to merge some pre-
existing clusters. It is not allowed to split pre-existing clusters.
A $c$-competitive algorithm for MinDisAgree outputs, on any input $\sigma$, a
clustering $\mathcal{C}(\sigma)$ such that
$\text{cost}(\mathcal{C}(\sigma))\leq c\cdot\text{cost}(\text{OPT}(\sigma))$.
For MaxAgree, we must have $\text{profit}(\mathcal{C}(\sigma))\geq
c\cdot\text{profit}(\text{OPT}(\sigma))$. (When the algorithm is randomized,
this must hold in expectation).
## 2\. Maximizing Agreements Online
### 2.1. Competitiveness of Greedy
For subsets of vertices $S$ and $T$ we define $\Gamma(S,T)$ as the set of
edges between $S$ and $T$. We write $\Gamma^{+}(S,T)$ (resp.
$\Gamma^{-}(S,T)$) for the set of positive (resp. negative) edges of
$\Gamma(S,T)$. We define the gain of merging $S$ with $T$ as the change in the
profit when clusters $S$ and $T$ are merged:
$\text{gain}(S,T)={|\Gamma^{+}(S,T)|-|\Gamma^{-}(S,T)|}=2|\Gamma^{+}(S,T)|-|S||T|.$
We present the following greedy algorithm for online correlation clustering.
Algorithm 1 Algorithm Greedy
1: Upon the arrival of vertex $v$ do
2: Put $v$ in a new cluster consisting of $\\{v\\}$.
3: while there are two clusters $C$, $C^{\prime}$ such that
$\text{gain}(C,C^{\prime})>0$ do
4: Merge $C$ and $C^{\prime}$
5: end while
6: end for
###### Theorem 1.
Let OPT denote the offline optimum.
* •
For every instance,
$\text{profit}(\textsc{Greedy}{})\geq{0.5~{}\text{profit}(\text{OPT})}$.
* •
There are instances with
$\text{profit}(\textsc{Greedy}{})\leq{(0.5+o(1))\text{profit}(\text{OPT})}$.
### 2.2. Bounding the optimal competitive ratio
###### Theorem 2.
The competitive ratio of any randomized online algorithm for MaxAgree is at
most $0.834$. The competitive ratio of any deterministic online algorithm for
MaxAgree is at most $0.803$.
The proof uses Yao’s Min-Max Theorem [4] (maximization version).
###### Theorem 3 (Yao’s Min-Max Theorem).
Fix a distribution $D$ over a set of inputs $(I_{\sigma})_{\sigma}$. The
competitive ratio of any randomized online algorithm is at most
$\max\\{\frac{E_{I}[\text{profit}(\mathcal{A}(I))]}{E_{I}[\text{profit}(\text{OPT}(I))]}:\mathcal{A}\hbox{
deterministic online algorithm}\\},$
where the expectations are over a random input $I$ drawn from distribution
$D$.
To prove Theorem 2.2, we first define two generic inputs that we will use to
apply Theorem 2.3. The first input is a graph $G_{1}$ with $2m$ vertices and
all positive edges between them The second input is a graph with $6m$ vertices
defined as follows. The first $2m$ vertices have all positive edges between
them, the next $2m$ vertices have all positive edges between them, and the
last $2m$ vertices also have all positive edges between them. In each of these
three sets $G_{1},G_{2},G_{3}$ of $2m$ vertices, half are labelled “left side”
vertices and the other half are labelled ”right side” vertices. All edges
between left vertices are positive, but edges between a vertex $u$ on the left
side of $G_{i}$ and a vertex $v$ on the right side of $G_{j}$, $j\neq i$, are
all negative.
The online algorithm cannot distinguish between the two inputs until time
$2m+1$, so it must hedge against two very different possible optimal
structures.
### 2.3. Beating Greedy
#### 2.3.1. Designing the algorithm
Our algorithm is based on the observation that Algorithm Greedy always
satisfies at least half of the edges. Thus, if profit(OPT) is less than
$(1-\alpha/2)|E|$ for some constant $\alpha$, then the profit of Greedy is
better than half of optimal. We design an algorithm called Dense,
parameterized by constants $\alpha$ and $\tau$, such that if profit(OPT) is
greater than $(1-\alpha/2)|E|$, then the approximation factor is at least
$0.5+\eta$ for some positive constant $\eta$. We use both algorithms Greedy
and Dense to define Algorithm 2.
###### Theorem 4.
Let $\alpha\in(0,1)$, $\tau>1$ and $\eta\in(0,\frac{1}{2})$ be such that
$\eta\leq
1.5-\tau^{2}-((2\sqrt{3}+9/2)\alpha^{1/4}+\frac{\alpha^{1/4}}{1-\alpha^{1/4}}+\alpha/2)2\frac{2\tau-1}{(\tau-1)}.$
(2.1)
Then, for every instance such that $\text{OPT}\geq(1-\alpha/2)E$, Algorithm
$\textsc{Dense}_{\alpha,\tau}$ has profit at least $(1/2+\eta)\text{OPT}$.
Using Theorem 4 we can bound the competitive ratio of Algorithm 2.
###### Corollary 2.1.
Let $\alpha,\tau$ and $\eta$ be as above, and let
$p={\alpha}/({2+2\eta(2-\alpha)})$. Then Algorithm 2 has competitive ratio at
least $\frac{1}{2}+\frac{\alpha\eta/2}{1+2\eta(1-\alpha/2)}$.
###### Corollary 2.2.
For $\alpha=10^{-12}$, $\tau=1.0946$, $\eta=0.0555$ and $p=4,5\cdot 10^{-13}$,
Algorithm 2 is $\frac{1}{2}+2\cdot 10^{-14}$-competitive.
Algorithm 2 A $\frac{1}{2}+\epsilon_{0}$-competitive algorithm
Given $p$, $\alpha$, $\tau$,
With probability $1-p$, run Greedy,
With probability $p$, run $\text{{Dense}{}}_{\alpha,\tau}$.
Algorithm 3 Algorithm $\textsc{Dense}{}_{\alpha,\tau}$
1: Let $\mathcal{C}=\widehat{\text{OPT}}_{1}$ and for every cluster
$D\in\mathcal{C}$, let $\text{repr}_{1}(D):=D\in\widehat{\text{OPT}}_{1}$ .
2: Upon the arrival of a vertex $v$ at time $t$ do
3: Put $v$ in a new cluster $\\{v\\}$.
4: if $t=t_{i}$ for some $i$ then
5: for every cluster $D$ in $\widehat{\text{OPT}_{i}}$ do
6: Define a cluster $D^{\prime\prime}$ obtained by merging the restriction of
$D$ to $\\{t_{i-1},\ldots,t_{i}\\}$ with every cluster $C\in\mathcal{C}$ in
$\\{1,\ldots,t_{i-1}\\}$ such that $\text{repr}_{i-1}(C)$ is defined and is
half-contained in $D$.
7: If $D^{\prime\prime}$ is not empty, set
$\text{repr}_{i}(D^{\prime\prime}):=D\in\widehat{\text{OPT}_{i}}$.
8: end for
9: end if
10: end for
How do we define algorithm Dense? Using the PTAS of [2], one can compute
offline a factor $(1-\alpha/2)$ approximative solution $\text{OPT}^{\prime}$
of any instance of MaxAgree in polynomial time. We will design algorithm Dense
so that it guarantees an approximation factor of $0.5+\eta$ whenever
$\text{profit}(\text{$\text{OPT}^{\prime}$})\geq(1-\alpha)|E|$. Since
$\text{profit}(\text{OPT})\geq(1-\alpha/2)|E|$ implies that
$\text{profit}(\text{$\text{OPT}^{\prime}$})\geq(1-\alpha)|E|$, Theorem 4 will
follow.
We say that $\text{OPT}_{t}^{\prime}$ is large if
$\text{profit}(\text{$\text{OPT}_{t}^{\prime}$})\geq(1-\alpha)|E|$. We define
a sequence $(t_{i})_{i}$ of update times inductively as follows: By convention
$t_{0}=0$. Time $t_{1}$ is the earliest time $t\geq 100$ such that
$\text{OPT}_{t}^{\prime}$ is large. Assume $t_{i}$ is already defined, and let
$j$ be such that $\tau^{j-1}\leq t_{i}<\tau^{j}$. If
$\text{OPT}_{\tau^{j}}^{\prime}$ is large, then $t_{i+1}=\tau^{j}$, else
$t_{i+1}$ is the earliest time $t\geq\tau^{j}$ such that
$\text{OPT}_{t}^{\prime}$ is large. Let $t_{1},t_{2},\ldots,t_{K}$ be the
resulting sequence. We will note, with an abuse of notation,
$\text{OPT}_{i}^{\prime}$ instead of $\text{OPT}_{t_{i}}^{\prime}$ for $1\leq
i\leq K$.
We say that a cluster $A$ is half-contained in $B$ if $|A\cap B|>|A|/2$. Let
$\epsilon=\alpha^{1/4}$. For each $t_{i}$, we inductively define a near
optimal clustering of the nodes $[1,t_{i}]$. For the base case, let
$\widehat{\text{OPT}}_{1}$ be the clustering obtained from
$\text{OPT}_{1}^{\prime}$ by keeping the $1/\epsilon^{2}$ largest clusters and
splitting the other clusters into singletons. For the general case, to define
$\widehat{\text{OPT}_{i}}$ given $\widehat{\text{OPT}_{i-1}}$, mark the
clusters of $\text{OPT}_{i}^{\prime}$ as follows. For any $D$ in
$\text{OPT}_{i}^{\prime}$, mark $D$ if either one of the
$1/\epsilon^{2}-1/\epsilon$ largest clusters of $\widehat{\text{OPT}_{i-1}}$
is half-contained in $D$, or $D$ is one of the $1/\epsilon$ largest clusters
$\text{OPT}_{i}^{\prime}$. Then $\widehat{\text{OPT}_{i}}$ contains all the
marked clusters of $\text{OPT}_{i}^{\prime}$ and the rest of the vertices in
$[1,t_{i}]$ as singleton clusters. (Note that, by definition, any
$\widehat{\text{OPT}_{i}}$ contains at most $1/\epsilon^{2}$ non-singleton
clusters; this will be useful in the analysis.)
Note that Dense only depends on parameters $\alpha$ and $\tau$ indirectly via
the definition of update times and of $\widehat{\text{OPT}}$.
#### 2.3.2. Analysis: Proof of Theorem 4
The analysis is by induction on $i$, assuming that we start from clustering
$\widehat{\text{OPT}_{i}}$ at time $t_{i}$, then apply the above algorithm
from time $t_{i}$ to the final time $t$. If $i=1$ this is exactly our
algorithm, and if $i=K$ then this is simply $\widehat{\text{OPT}_{K}}$; in
general it is a mixture of the two constructions.
More formally, define a forest ${\mathcal{F}}$ (at time $t$) with one node for
each $t_{i}\leq t$ and cluster of $\widehat{\text{OPT}_{i}}$. The node
associated to a cluster $A$ of $\widehat{\text{OPT}_{i-1}}$ is a child of the
node associated to a cluster $B$ of $\widehat{\text{OPT}_{i}}$ if and only if
$A$ is half-contained in $B$. With a slight abuse of notation, we define the
following clustering $\mathcal{F}$ associated to the forest. There is one
cluster $T$ for each tree of the forest: for each node $A$ of the tree, if $i$
is such that $A\in\widehat{\text{OPT}_{i}}$, then cluster $T$ contains
$A\cap(t_{i-1},t_{i}]$. This defines $T$.
One interpretation of Dense is that at all times $t$, there is an associated
forest and clustering $\mathcal{F}$; and our algorithm Dense simply maintains
it. See Figure 1 for an example.
###### Lemma 2.3.
Algorithm 3 is an online algorithm that outputs clustering $\mathcal{F}$ at
time $t$.
Figure 1. An example of a forest $\mathcal{F}$ given in left, and the
corresponding clustering given in right. Here, we have
$\text{$\widehat{\text{OPT}_{i}}$}=\\{B_{1},B_{2}\\}$ and
$\text{$\widehat{\text{OPT}_{i-1}}$}=\\{A_{1},\ldots,A_{5}\\}$.
Let ${\mathcal{F}}_{i}$ be the forest obtained from ${\mathcal{F}}$ by erasing
every node associated to clusters of $\widehat{\text{OPT}_{j}}$ for every
$j<i$. With a slight abuse of notation, we define the following clustering
${\mathcal{F}}_{i}$ associated to that forest: there is one cluster $C$ for
each tree of the forest defined as follows. For each node $A$ of the tree, let
$k\geq i$ be such that $A\in\widehat{\text{OPT}_{k}}$: then $C$ contains
$A\cap(t_{k-1},t_{k}]$ if $k>i$, and $C$ contains $A$ if $k=i$. This defines a
sequence of clusterings such that ${\mathcal{F}}_{1}={\mathcal{F}}$ is the
output of the algorithm, and ${\mathcal{F}}_{K}=\widehat{\text{OPT}_{K}}$.
###### Lemma 2.4 (Main lemma).
For any $2\leq i\leq K$,
$\text{cost}({\mathcal{F}}_{i-1})-\text{cost}({\mathcal{F}}_{i})\leq\left((4+2\sqrt{3})\epsilon+\frac{\epsilon}{1-\epsilon}\right)t_{i}t_{K}.$
We defer the proof of Lemma 2.4 to next section. Assuming Lemma 2.4, we upper-
bound the cost of clustering $\mathcal{F}$.
###### Lemma 2.5 (Lemma 14, [2]).
For any $0<c<1$ and clustering $\mathcal{C}$, let $\mathcal{C}^{\prime}$ be
the clustering obtained from $\mathcal{C}$ by splitting all clusters of
$\mathcal{C}$ of size less than $cn$, where $n$ is the number of vertices.
Then $\text{cost}(\mathcal{C}^{\prime})\leq\text{cost}(\mathcal{C})+cn^{2}/2$.
###### Lemma 2.6.
$\text{cost}(\mathcal{F})\leq((2\sqrt{3}+9/2)\epsilon+\frac{\epsilon}{1-\epsilon}+\epsilon^{4}/2)\frac{2\tau-1}{\tau-1}t_{K}^{2}$.
###### Proof 2.7.
We write:
$\text{cost}(\mathcal{F})=\text{cost}(\widehat{\text{OPT}_{K}})+\sum_{i=2}^{K}(\text{cost}(\mathcal{F}_{i-1})-\text{cost}({\mathcal{F}}_{i}))$.By
definition, $\widehat{\text{OPT}_{K}}$ contains the $1/\epsilon$ largest
clusters of $\text{OPT}_{K}^{\prime}$. Then the remaining clusters of
$\text{OPT}_{K}^{\prime}$ are of size at most $\epsilon t_{K}$. By Lemma 2.5,
the cost of $\widehat{\text{OPT}_{K}}$ is at most
$\text{cost}(\text{$\text{OPT}_{K}^{\prime}$})+\epsilon
t_{K}^{2}/2\leq{(\alpha+\epsilon)t_{K}^{2}/2}$. Applying Lemma 2.4, and
summing over $2\leq i\leq K$, we get
$\text{cost}(\mathcal{F})\leq{(\alpha+\epsilon)t_{K}^{2}/2}+\left((4+2\sqrt{3})\epsilon+\frac{\epsilon}{1-\epsilon}\right)\sum_{i}{t_{i}}t_{K}.$
By definition of the update times $(t_{i})_{i}$, for any $j>0$ there exists at
most one $t_{i}$ such that $\tau^{j}\leq t_{i}<\tau^{j+1}$. Let $L$ be such
that $\tau^{L}\leq t_{K}<\tau^{L+1}$. Then
$\sum_{1\leq i\leq K}{t_{i}}\leq\sum_{1\leq i\leq
K-1}{t_{i}}+t_{K}\leq\sum_{1\leq j\leq
L}{\tau^{j}}+t_{K}\leq\frac{\tau^{L+1}}{\tau-1}+t_{K}\leq\frac{2\tau-1}{\tau-1}t_{K}.$
Hence the desired bound on $\text{cost}(\mathcal{F})$.
###### Proof 2.8 (Proof of Theorem 4).
Fix an input graph of size $n$, such that
$\text{profit}(\text{OPT})\geq(1-\alpha/2){n\choose 2}$. By Lemma 2.6, at time
$t_{K}$, Algorithm 3 has clustering $\mathcal{F}$ with
$\text{cost}(\mathcal{F})\leq O(\epsilon)\frac{2\tau-1}{\tau-1}t_{K}^{2}.$
By definition of the update times, $n<\tau t_{K}$. To guarantee a competitive
ratio of $0.5+\eta$, for some $\eta$, the cost must not exceed
$(0.5-\eta){n\choose 2}$ at time $n$, when all vertices $t_{K}+1,\ldots,n$ are
added as singleton clusters. The number of new edges added to the graph
between times $t_{K}$ and $n$ is ${n-t_{K}\choose 2}+t_{K}(n-t_{K})$. We must
have
$\frac{2\tau-1}{\tau-1}O(\epsilon)t_{K}^{2}+{n-t_{K}\choose
2}+t_{K}(n-t_{K})\leq(0.5-\eta){n\choose 2},$ (2.2)
for some $0<\eta<0.5$. Using the fact that $n-t_{K}\leq(\tau-1)t_{K}$ and
$t_{K}\leq n-1$, to satisfy (2.2), it suffices to have
$\frac{2\tau-1}{\tau-1}O(\epsilon)t_{K}^{2}+t_{K}^{2}(\tau-1)^{2}/2+(\tau-1)t_{K}^{2}\leq(0.5-\eta)t_{K}^{2}/2,$
which is equivalent to (2.1). Moreover we have the following natural
constraints on constants $\eta$, $\epsilon$ and $\tau$: $0<\eta<0.5$,
$0<\epsilon<1$, and $\tau>1$. Then, for any set of values of constants $\eta$,
$\epsilon$, $\tau$ verifying those constraints, Algorithm Dense is
$0.5+\eta$-competitive.
#### 2.3.3. The core of the analysis: proof of Lemma 2.4
###### Lemma 2.9.
Let $\mathcal{S}^{i}$ be the set of vertices of the non-singleton clusters
that are not among the $1/\epsilon^{2}-1/\epsilon$ largest clusters of
$\widehat{\text{OPT}_{i-1}}$. Then
$|\mathcal{S}^{i}|\leq\frac{\epsilon}{1-\epsilon}t_{i-1}.$
###### Proof 2.10.
Let $C$ be a cluster of $\widehat{\text{OPT}_{i-1}}$, such that
$C\subseteq\mathcal{S}^{i}$. Then
${|C|\leq(1/\epsilon^{2}-1/\epsilon)^{-1}t_{i-1}}$. Since there are at most
$1/\epsilon$ such clusters, the number of vertices of these are at most
$1/\epsilon(1/\epsilon^{2}-1/\epsilon)^{-1}t_{i-1}$.
###### Notation 5.
For any $i\neq j$, and a cluster $B$ of $\text{OPT}_{i}^{\prime}$, we denote
by $\gamma^{i,j}_{B}$ the square root of the number of edges of
$[1,t_{\min(i,j)}]\times[1,t_{\min(i,j)}]$, adjacent to at least one node of
$B$, and which are classified differently in $\text{OPT}_{i}^{\prime}$ and in
$\text{OPT}_{j}^{\prime}$.
We refer to non singleton clusters as large clusters.
###### Lemma 2.11.
Let $\mathcal{T}^{i}$ be the set of vertices of those
$1/\epsilon^{2}-1/\epsilon$ largest clusters of $\widehat{\text{OPT}_{i-1}}$
that are not half-contained in any cluster of $\text{OPT}_{i}^{\prime}$. Then
$|\mathcal{T}^{i}|\leq\sqrt{6}\sum_{\text{large
}C\in\text{$\widehat{\text{OPT}_{i-1}}$}}{\gamma^{i,i-1}_{C}}$.
Let $B$ be a cluster of $\widehat{\text{OPT}_{i}}$. For any $j\leq i$, we
define $\mathcal{C}_{j}(B)$ as the cluster associated with the tree of
$\mathcal{F}_{j}$ that contains $B$. For any
$B\in\text{$\widehat{\text{OPT}_{i}}$}$, we call $\mathcal{C}_{i-1}(B)$ the
extension of $\mathcal{C}_{i}(B)$ to $\mathcal{F}_{i-1}$. By definition of
$\mathcal{F}_{i}$, the following lemma is easy.
###### Lemma 2.12.
For any $B\in\text{$\widehat{\text{OPT}_{i}}$}$, the restriction of
$\mathcal{C}_{i-1}(B)$ to $(t_{i-1},t_{K}]$ is equal to the restriction of
$\mathcal{C}_{i}(B)$ to $(t_{i-1},t_{K}]$.
Let $(A_{j})_{j}$ denote the clusters of $\widehat{\text{OPT}_{i-1}}$ that are
half-contained in $B$. We define $\delta^{i}(B)$ as the symmetric difference
of the restriction of $B$ to $[1,t_{i-1}]$ and $\cup_{j}{A_{j}}$:
$\delta^{i}(B)=(B\cap[1,t_{i-1}])\Delta\cup_{j}{A_{j}}.$
###### Lemma 2.13.
For any cluster $C_{i}$ of $\mathcal{F}_{i}$, let $C_{i}^{\prime}$ denote the
extension of $C_{i}$ to $\mathcal{F}_{i-1}$. Then
$\bigcup_{C_{i}\in\mathcal{F}_{i}}{{C}_{i}\setminus{C}_{i}^{\prime}}\subseteq\mathcal{S}^{i}\cup\mathcal{T}^{i}\cup\bigcup_{\text{large
}B\in\text{$\widehat{\text{OPT}_{i}}$}}{\delta^{i}(B)}$
###### Proof 2.14.
By Lemma 2.12, the partition of the vertices $(t_{i-1},t_{K}]$ is the same for
$C_{i}$ as for $C_{i}^{\prime}$. So $C_{i}$ and $C_{i}^{\prime}$ only differ
in the vertices of $[1,t_{i-1})$:
$\bigcup_{C_{i}\in\mathcal{F}_{i}}{{C}_{i}\setminus{C}_{i}^{\prime}}\subseteq\bigcup_{B\in\text{$\widehat{\text{OPT}_{i}}$}}{\delta^{i}(B)}.$
We will show that for a singleton cluster $B$ of $\widehat{\text{OPT}_{i}}$,
$\delta^{i}(B)$ is included in
$\mathcal{S}^{i}\cup\mathcal{T}^{i}\bigcup_{\text{large
}B\in\text{$\widehat{\text{OPT}_{i}}$}}{\delta^{i}(B)}$, which yields the
lemma.
Let $B=\\{v\\}$ be a singleton cluster of $\widehat{\text{OPT}_{i}}$ such that
$\delta^{i}(B)\neq\\{\\}$. A non-singleton cluster cannot be half-contained in
a singleton cluster so we conclude no clusters are half-contained in $B$ and
hence $\delta^{i}(B)=\\{v\\}$. By definition of $\delta^{i}(B)$,
$v\in[1,t_{i-1}]$. So there exists a cluster $A$ of
$\widehat{\text{OPT}_{i-1}}$ that contains $v$. Clearly $A$ is not a singleton
since otherwise $\delta^{i}(B)$ would be $\\{\\}$. There are two cases.
First, if $A$ is half-contained in a cluster $B^{\prime}\neq B$ of
$\widehat{\text{OPT}_{i}}$ then cluster $B^{\prime}$ is necessarily large
since it contains more than one vertex of $A$. Then we have
$v\in\delta^{i}(B^{\prime})$.
Second, if $A$ is not half-contained in any cluster of
$\widehat{\text{OPT}_{i}}$ then
$A\subseteq\mathcal{S}^{i}\cup\mathcal{T}^{i}$. In fact, if $A$ is half-
contained in a cluster of $\text{OPT}_{i}^{\prime}$ which is split into
singletons in $\widehat{\text{OPT}_{i}}$, then $A$ is not one of the
$1/\epsilon^{2}-1/\epsilon$ largest clusters of $\widehat{\text{OPT}_{i-1}}$,
and $A\subseteq\mathcal{S}^{i}$. If $A$ is not half-contained in any cluster
of $\text{OPT}_{i}^{\prime}$, then $A\subseteq\mathcal{T}^{i}$ if $A$ is one
of the $1/\epsilon^{2}-1/\epsilon$ largest clusters of
$\widehat{\text{OPT}_{i-1}}$ and $A\subseteq\mathcal{S}^{i}$ otherwise.
###### Lemma 2.15.
For any large cluster $B$ of $\widehat{\text{OPT}_{i}}$, $|\delta^{i}(B)|\leq
2\sqrt{2}\gamma^{i,i-1}_{B}$.
###### Proof 2.16.
Let $B^{\prime}$ denote the restriction of $B$ to $[1,t_{i-1}]$. We first show
that
$1/2(|\cup_{j}{A_{j}}\setminus B^{\prime}|)^{2}\leq(\gamma^{i,i-1}_{B})^{2}.$
Observe that $(\gamma^{i,i-1}_{B})^{2}$ includes all edges $uv$ such that one
of the following two cases occurs.
First, if $u\in A_{j}\setminus B$ and $v\in A_{j}\cap B$: such edges are
internal in the clustering $\text{OPT}_{i-1}^{\prime}$ but external in the
clustering $\text{OPT}_{i}^{\prime}$. The number of edges of this type is
$\sum_{j}|A_{j}\setminus B|\cdot|A_{j}\cap B|$. Since $A_{j}$ is half-
contained in $B$, this is at least $\sum_{j}|A_{j}\setminus B|^{2}$.
Second, if $u\in A_{j}\cap B$ and $v\in A_{k}\cap B$ with $j\neq k$: such
edges are external in the clustering $\text{OPT}_{i-1}^{\prime}$ but internal
in the clustering $\text{OPT}_{i}^{\prime}$. The number of edges of this type
is $\sum_{j<k}|A_{j}\cap B|\cdot|A_{k}\cap B|\geq\sum_{j<k}|A_{j}\setminus
B|\cdot|A_{k}\setminus B|$.
Summing, it is easy to infer that
$(\gamma^{i,i-1}_{B})^{2}\geq(1/2)\left(\sum_{j}|A_{j}\setminus
B|\right)^{2}=(1/2)|\cup_{j}A_{j}\setminus B^{\prime}|^{2}$. Let
$(A^{\prime}_{j})_{j}$ denote the clusters of $\widehat{\text{OPT}_{i-1}}$
that are not half-contained in $B$, but have non-empty intersections with $B$.
We now show that
$1/2(|B^{\prime}\setminus\cup_{j}{A^{\prime}_{j}}|)^{2}\leq(\gamma^{i,i-1}_{B})^{2}.$
We have $B^{\prime}\setminus\cup_{j}{A_{j}}=\cup_{j}{(A^{\prime}_{j}\cap B)}$.
Observe that any $A^{\prime}_{j}$ is a large cluster of
$\widehat{\text{OPT}_{i-1}}$, thus a cluster of $\text{OPT}_{i-1}^{\prime}$.
Then $(\gamma^{i,i-1}_{B})^{2}$ includes all edges $uv$ such that one of the
following two cases occurs
First, if $u\in A^{\prime}_{j}\setminus B$ and $v\in A^{\prime}_{j}\cap B$:
such edges are internal in the clustering $\text{OPT}_{i-1}^{\prime}$ but
external in the clustering $\text{OPT}_{i}^{\prime}$. The number of edges of
this type is $\sum_{j}|A^{\prime}_{j}\setminus B|\cdot|A^{\prime}_{j}\cap B|$.
Since $A^{\prime}_{j}$ is not half-contained in $B$, this is at least
$\sum_{j}|A^{\prime}_{j}\cap B|^{2}$.
Second, if $u\in A^{\prime}_{j}\cap B$ and $v\in A^{\prime}_{k}\cap B$ with
$j\neq k$: such edges are external in the clustering
$\text{OPT}_{i-1}^{\prime}$ but internal in the clustering
$\text{OPT}_{i}^{\prime}$. The number of edges of this type is
$\sum_{j<k}|A^{\prime}_{j}\cap B|\cdot|A^{\prime}_{k}\cap B|$.
Summing, we get
$(\gamma^{i,i-1}_{B})^{2}\geq(1/2)\left(\sum_{j}|A^{\prime}_{j}\cap
B|\right)^{2}=(1/2)|B^{\prime}\setminus\cup_{j}A^{\prime}_{j}|^{2}.$
###### Lemma 2.17.
For any $i\geq 1$, $\widehat{\text{OPT}_{i}}$ has at most $1/\epsilon^{2}$ non
singleton clusters, all of which are clusters of $\text{OPT}_{i}^{\prime}$
###### Proof 2.18.
By definition, $\widehat{\text{OPT}_{1}}$ has at most $1/\epsilon^{2}$ non
singleton clusters. For any $i>1$, a cluster of $\widehat{\text{OPT}_{i-1}}$
can only be half-contained in one cluster of $\text{OPT}_{i}^{\prime}$.
Therefore given $\widehat{\text{OPT}_{i-1}}$, at most $1/\epsilon^{2}$
clusters of $\text{OPT}_{i}^{\prime}$ are marked. Thus
$\widehat{\text{OPT}_{i}}$ has at most $1/\epsilon^{2}$ clusters.
We can now prove Lemma 2.4.
###### Proof 2.19 (Proof of Lemma 2.4).
By Lemma 2.12, clusterings $\mathcal{F}_{i}$ and $\mathcal{F}_{i-1}$ only
differ in their partition of $[1,t_{i-1}]$. Then the set of the vertices that
are classified differently in $\mathcal{F}_{i}$ and $\mathcal{F}_{i-1}$ is
$\cup_{i}C_{i}\setminus C_{i-1}$. Each of these vertices creates at most
$t_{K}$ disagreements:
$\begin{split}\text{cost}({\mathcal{F}}_{i-1})-\text{cost}({\mathcal{F}}_{i})&\leq\sum_{C_{i}\in\mathcal{F}_{i}}{|C_{i}\setminus
C_{i-1}|}t_{K}\\\ \end{split}$ (2.3)
By Lemmas 2.13 and 2.15,
$\sum_{C_{i}\in\mathcal{F}_{i}}{|C_{i}\setminus
C_{i-1}|}t_{K}\leq\left(2\sqrt{2}\Bigg{(}\sum_{\text{large
}B\in\widehat{\text{OPT}_{i}}}\gamma^{i,i-1}_{B}\Bigg{)}+|\mathcal{S}^{i}|+|\mathcal{T}^{i}|\right)t_{K}.$
(2.4)
By Lemmas 2.9 and 2.11,
$|\mathcal{S}^{i}|\leq\frac{\epsilon}{1-\epsilon}t_{i-1}\hbox{ and
}|\mathcal{T}^{i}|\leq\sqrt{6}\sum_{\text{large
}B\in\text{$\widehat{\text{OPT}_{i-1}}$}}{\gamma^{i-1,i}_{B}}$ (2.5)
The term $\sum_{\text{large
}B\in\widehat{\text{OPT}_{i-1}}}\gamma^{i-1,i}_{B}$ can be seen as the
$\ell_{1}$ norm of the vector $(\gamma^{i-1,i}_{B})_{\text{large }B}$. Since
$\widehat{\text{OPT}_{i-1}}$ has at most $1/\epsilon^{2}$ large clusters by
Lemma 2.17, we can use Hölder’s inequality:
$\begin{split}\sum_{\text{large
}B\in\widehat{\text{OPT}_{i-1}}}\gamma^{i-1,i}_{B}&=\|(\gamma^{i-1,i}_{B})_{\text{large
}B}\|_{1}\leq 1/\epsilon\|(\gamma^{i-1,i}_{B})_{\text{large
}B}\|_{2}.\end{split}$
By definition we have $\|(\gamma^{i-1,i}_{B})_{\text{large
B}}\|_{2}\leq\sqrt{2(\text{cost}(\text{$\text{OPT}_{i-1}^{\prime}$})+\text{cost}(\text{$\text{OPT}_{i}^{\prime}$}))}$.
Thus
$\sum_{\text{large }B\in\widehat{\text{OPT}_{i-1}}}\gamma^{i-1,i}_{B}\leq
1/\epsilon\sqrt{2(\alpha t_{i-1}^{2}/2+\alpha
t_{i}^{2}/2)}\leq\frac{\sqrt{2\alpha}}{\epsilon}t_{i}.$ (2.6)
Similarly, we have
$\sum_{\text{large
}B\in\text{$\widehat{\text{OPT}_{i}}$}}{\gamma^{i,i-1}_{B}}\leq\frac{\sqrt{2\alpha}}{\epsilon}t_{i}.$
(2.7)
Combining equations (2.3) through (2.7) and $\alpha=\epsilon^{4}$ yields
$\text{cost}({\mathcal{F}}_{i-1})-\text{cost}({\mathcal{F}}_{i})\leq\left((4+2\sqrt{3})\epsilon+\frac{\epsilon}{1-\epsilon}\right)t_{i}t_{K}$
## 3\. Minimizing Disagreements Online
###### Theorem 6.
Algorithm Greedy is $(2n+1)$-competitive for MinDisAgree.
To prove Theorem 6, we need to compare the cost of the optimal clustering to
the cost of the clustering constructed by the algorithm. The following lemma
reduces this to, roughly, analyzing the number of vertices classified
differently.
###### Lemma 3.1.
Let $\mathcal{W}$ and $\mathcal{W}^{\prime}$ be two clusterings such that
there is an injection $W^{\prime}_{i}\in\mathcal{W}^{\prime}\rightarrow
W_{i}\in\mathcal{W}$. Then
$\text{cost}(\mathcal{W}^{\prime})-\text{cost}(\mathcal{W})\leq
n\sum_{i}|W_{i}^{\prime}\setminus W_{i}|$.
For subsets of vertices $S_{1},\ldots,S_{m}$, we will write, with a slight
abuse of notation, $\Gamma^{+}(S_{1},\ldots,S_{m})$ for the set of edges in
$\Gamma^{+}(S_{i},S_{j})$ for any $i\neq j$:
$\Gamma^{+}(S_{1},\ldots,S_{m})=\cup_{i\neq j}{\Gamma^{+}(S_{i},S_{j})}$.
###### Lemma 3.2.
Let $C$ be a cluster created by Greedy, and
$\mathcal{W}=\\{W_{1},\ldots,W_{K}\\}$ denote the clusters of OPT. Then
$|C|\leq\max_{i}|C\cap W_{i}|+2|\Gamma^{+}(C\cap W_{1},\ldots,C\cap W_{K})|.$.
We call $\displaystyle i_{0}=\arg\max_{i}|C\cap W_{i}|$ the leader of $C$.
###### Proof 3.3 (Proof of Theorem 6).
Let $\mathcal{C}$ denote the clustering given by Greedy. For every cluster
$W_{i}$ of OPT, merge all the clusters of $\mathcal{C}$ that have $i$ as their
leaders. Let $\mathcal{C}^{\prime}=(W^{\prime}_{i})$ be this new clustering.
By definition of the greedy algorithm, this operation can only increase the
cost since every pair of clusters have a negative-majority cut at the end of
the algorithm:$\text{cost}(\mathcal{C})\leq\text{cost}(\mathcal{C}^{\prime}).$
We apply Lemma 3.1 to $\mathcal{W}=$OPT and
$\mathcal{W^{\prime}}=\mathcal{C}^{\prime}$, and obtain:
$\text{cost}(\mathcal{C}^{\prime})\leq\text{cost}(\text{OPT})+n\sum_{i}|W_{i}^{\prime}\setminus
W_{i}|$. By definition of $\mathcal{C}^{\prime}$ we have
$|W_{i}^{\prime}\setminus
W_{i}|=\sum_{\begin{subarray}{c}C\in\mathcal{C}:\text{leader}(C)=i\end{subarray}}\sum_{j\neq
i}{|C\cap W_{j}|},$ hence
$\sum_{i}|W_{i}^{\prime}\setminus
W_{i}|=\sum_{C\in\mathcal{C}}\sum_{j\neq\text{leader}(C)}{|C\cap W_{j}|}.$
By Lemma 3.2, $\sum_{j\neq\text{leader}(C)}{|C\cap W_{j}|}\leq
2|\Gamma^{+}(C\cap W_{1},\ldots,C\cap W_{K})|$. Finally, to bound OPT from
below, we observe that, for any two clusterings $\mathcal{C}$ and
$\mathcal{W}$, it holds that the sum over $C\in\mathcal{C}$ of
${|\Gamma^{+}(C\cap W_{1},\ldots,C\cap W_{K})|}$ is less than
$\text{cost}(\mathcal{W})$. Combining these inequalities yields the theorem.
###### Theorem 7.
Let ALG be a randomized algorithm for MinDisAgree. Then there exists an
instance on which ALG has cost at least $n-1-\text{cost}(\text{OPT})$ where
OPT is the offline optimum. If OPT is constant then
$\text{cost}(\text{ALG})=\Omega(n)\text{cost}(\text{OPT})$.
###### Proof 3.4.
Consider two cliques $A$ and $B$, each of size $m$, where all the internal
edges of $A$ and $B$ are positive. Choose a vertex $a$ in $A$, and a set of
vertices $b_{1},\ldots,b_{k}$ in $B$. Define the edge labels of $ab_{i}$ as
positive, for all $1\leq i\leq k$ and the rest of the edges between $A$ and
$B$ as negative. Define an input sequence starting with
$a,b_{1},\ldots,b_{k}$, followed by the rest of the vertices in any order.
## References
* [1] Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: ranking and clustering. In STOC ’05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 684–693, New York, NY, USA, 2005. ACM Press.
* [2] Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Mach. Learn., 56(1-3):89–113, 2004.
* [3] Amir Ben-Dor, Ron Shamir, and Zohar Yakhini. Clustering gene expression patterns. Journal of Computational Biology, 6(3-4):281–297, 1999.
* [4] Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. Cambridge University Press, New York, NY, USA, 1998.
* [5] Moses Charikar, Chandra Chekuri, Tomas Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. SIAM J. Comput., 33(6):1417–1440, 2004.
* [6] Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. Clustering with qualitative information. In focs, volume 00, page 524, Los Alamitos, CA, USA, 2003. IEEE Computer Society.
* [7] Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. Clustering with qualitative information. J. Comput. Syst. Sci., 71(3):360–383, 2005.
* [8] William W. Cohen and Jacob Richman. Learning to match and cluster large high-dimensional data sets for data integration. In KDD ’02: Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 475–480, New York, NY, USA, 2002. ACM.
* [9] Erik D. Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica. Correlation clustering in general weighted graphs. Theor. Comput. Sci., 361(2):172–187, 2006.
* [10] Jenny Rose Finkel and Christopher D. Manning. Enforcing transitivity in coreference resolution. In Proceedings of ACL-08: HLT, Short Papers, pages 45–48, Columbus, Ohio, June 2008. Association for Computational Linguistics.
* [11] Ioannis Giotis and Venkatesan Guruswami. Correlation clustering with a fixed number of clusters. Theory of Computing, 2(1):249–266, 2006.
* [12] Thorsten Joachims and John Hopcroft. Error bounds for correlation clustering. In ICML ’05: Proceedings of the 22nd international conference on Machine learning, pages 385–392, New York, NY, USA, 2005. ACM.
* [13] Marek Karpinski and Warren Schudy. Linear time approximation schemes for the Gale-Berlekamp game and related minimization problems. In STOC ’09: Proceedings of the 41st annual ACM symposium on Theory of computing, pages 313–322, 2009.
* [14] Claire Mathieu and Warren Schudy. Correlation clustering with noisy input. In To appear in Procs. 21st SODA, preprint: http://www.cs.brown.edu/$\sim$ws/papers/cluster.pdf, 2010.
* [15] Ron Shamir, Roded Sharan, and Dekel Tsur. Cluster graph modification problems. Discrete Appl. Math., 144(1-2):173–182, 2004.
|
arxiv-papers
| 2010-01-06T15:54:38 |
2024-09-04T02:49:07.551655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Claire Mathieu, Ocan Sankur, Warren Schudy",
"submitter": "Ocan Sankur",
"url": "https://arxiv.org/abs/1001.0920"
}
|
1001.1041
|
# Dynamical control of two-level system’s decay and long time freezing
Wenxian Zhang The State Key Laboratory for Advanced Materials and Devices,
Department of Optical Science and Engineering, Fudan University, Shanghai
200433, China Ames Laboratory and Iowa State University, Ames, Iowa 50011,
USA Jun Zhuang The State Key Laboratory for Advanced Materials and Devices,
Department of Optical Science and Engineering, Fudan University, Shanghai
200433, China
###### Abstract
We investigate with exact numerical calculation coherent control of a two-
level quantum system’s decay by subjecting the two-level system to many
periodic ideal $2\pi$ phase modulation pulses. For three spectrum intensities
(Gaussian, Lorentzian, and exponential), we find both suppression and
acceleration of the decay of the two-level system, depending on difference
between the spectrum peak position and the eigen frequency of the two-level
system. Most interestingly, the decay of the two-level system freezes after
many control pulses if the pulse delay is short. The decay freezing value is
half of the decay in the first pulse delay.
###### pacs:
03.67.Pp, 03.65.Yz, 02.60.Cb
## I Introduction
It is of fundamental interests to study the quantum control of spontaneous
emission which inspired the development of the quantum electrodynamics during
the last century Milonni (1994); Scully and Zubairy (1997); Allen and Eberly
(1975). The unwanted spontaneous emission often sets the ultimate limit of
precise quantum measurement and many proposals have been made to suppress it
Frishman and Shapiro (2003); Evers and Keitel (2002); Frishman and Shapiro
(2001); Agarwal et al. (2001a). One way widely applied to control the
spontaneous emission of a small quantum system, such as a neutral atom, is to
place it either into or nearby a microcavity such that only a single mode or
several modes of the cavity are resonant to the eigenfrequency of the quantum
system Vogel and Welsch (2006). In this way the structure of the vacuum is
modified and the spontaneous emission could be controlled by the properties of
the cavity. Another way to control the spontaneous emission is dynamical
control of the quantum system such that the coupling between the quantum
system and the vacuum is effectively modified. For example, dynamical
decoupling of the quantum system from the vacuum via either phase modulation
or amplitude modulation could in principle extend the coherent time of the
system Itano et al. (1990, 1991). Analogous to the quantum Zeno effect (ZE)
which says frequent measurements of a quantum system would prevent the decay
of an unstable quantum system Misra and Sudarshan (1977); Itano et al. (1990,
1991), the extension of the coherent time through coherent modulations of the
quantum system are often called ZE as well. Under some unfavorable conditions,
it is also possible that frequent modulations lead to acceleration of the
spontaneous emission, the so-called quantum anti-Zeno effect (AZE) Kofman and
Kurizki (2000); Agarwal et al. (2001b); Fischer et al. (2001); Kofman and
Kurizki (2001).
In practice people employ both ways separately or combination of them to
realize the optimal control of a quantum system. G. S. Agarwal et al.’s work
Agarwal et al. (2001a, b) is among many of such works. In their paper, a two-
level model system with both structured vacuum and free space vacuum are
investigated. They found significant suppression of spontaneous emission rate
for structured vacua but either suppression or acceleration may appear in a
free space vacuum, depending on the frequency of the control pulses which is
$2\pi$ phase modulation pulses. They demonstrated for a free space vacuum that
ZE shows up for $\omega_{0}\tau=1$ while AZE appears for $\omega_{0}\tau=\pi$
where $\omega_{0}$ is the eigenfrequency of the two-level system and $\tau$ is
the delay time between control pulses. At the end they argued that AZE is
possible for $\tau>\omega_{0}^{-1}$ Agarwal et al. (2001a).
A puzzle arises if one accepts the AZE condition because neither ZE or AZE
appear if one leaves the quantum system alone in which case obviously
$\tau\gg\omega_{0}^{-1}$. In this paper we revisit this problem by adopting
the exact solution of a two-level quantum system which does not require the
weak coupling and short time approximations compared to Ref. Agarwal et al.
(2001a). By investigating several typical structured vacuum, we show the
conditions for quantum ZE and AZE and the boundary between them.
Another puzzle is the ZE effect of a large number of pulses. According to the
ZE, at a fixed evolution time $t$, the survival probability of the initially
excited state approaches 1 with infinite number of pulses $N\rightarrow\infty$
(the pulse delay $\tau=t/N$ approaches 0). It is unclear what the road map
looks like as $N$ increases. Two ways might be possible. One way is that the
rate of decay depends solely on the pulse delay $\tau$ (independent on $N$)
and approaches zero if $\tau\rightarrow 0$ Scully et al. (2003). Another way
is that the decay rate depends only on the number of pulses $N$ (independent
on $\tau$) and the decay rate becomes zero after initial several pulses. For
the latter one, we expect to see decay freezing at large number of pulses. We
will find which way is the correct one in this paper.
The paper is organized as follows. Section II reviews briefly Rabi oscillation
of a two-level system under a perturbation and Sec. III develops an exact
formula of Rabi oscillation under periodic pulses. In Sec. IV after
establishing a quasi-level picture of the pulsed two-level system with
constant spectrum intensity, we investigate the quantum Zeno and anti-Zeno
effect for three spectrum intensities, including the Gaussian, Lorentzian, and
exponential one. The boundary between quantum Zeno and anti-Zeno effect is
given. We discuss the decay freezing as pulse delay $\tau$ getting small in
Sec. V. Finally, conclusion and discussion are given in Sec. VI.
## II Rabi oscillations of a two-level system
Let us consider a simple model of a two-level quantum system with constant
coupling to its environment modes. Such a model serves as the base of further
discussions for more complicated systems with mode-dependent coupling. The
Hamiltonian of the system with detuning $\delta=\omega-\omega_{0}$ (we set
$\hbar=1$ for convenience) and coupling strength $\nu$ is described by Scully
and Zubairy (1997); dip
$\displaystyle H$ $\displaystyle=$ $\displaystyle{\delta\over
2}(|e\rangle\langle e|-|g\rangle\langle g|)+\nu(|e\rangle\langle
g|+|g\rangle\langle e|),$ (1)
where $\omega_{0}=E_{e}-E_{g}$ is the difference between eigenfrequencies of
the excited state $|e\rangle$ and the ground state $|g\rangle$, with
respectively eigenenergy $E_{e}$ and $E_{g}$, and $\omega$ is the frequency of
the external mode coupled to the two-level system picturenote . Utilizing
Pauli matrices, the Hamiltonian can be rewritten as
$\displaystyle H$ $\displaystyle=$
$\displaystyle{\delta^{\prime}}\sigma_{z}+\nu\sigma_{x}$ (2)
with $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$,
$\sigma_{x}=|e\rangle\langle g|+|g\rangle\langle e|$, and
$\delta^{\prime}=\delta/2$.
The evolution operator of the coupled two-level system is
$\displaystyle U$ $\displaystyle=$ $\displaystyle d_{1}|e\rangle\langle
e|+d_{1}^{*}|g\rangle\langle g|+d_{2}(|e\rangle\langle g|+|g\rangle\langle
e|)$ (3)
where $d_{1}=\cos\Omega t-i(\delta^{\prime}/\Omega)\sin\Omega t$ and
$d_{2}=-i(\nu/\Omega)\sin\Omega t$ with $\Omega^{2}=\delta^{\prime
2}+\nu^{2}$. The transition probability at time $t$ from $|e\rangle$ to
$|g\rangle$ is
$\displaystyle p_{eg}(t)$ $\displaystyle=$
$\displaystyle{\nu^{2}\over\Omega^{2}}\sin^{2}(\Omega t)$ (4)
providing the initial state is $\psi(0)=|e\rangle$. The system revives at
times such that $\Omega t=k\pi$ with $k$ being an integer.
## III Controlled Rabi oscillations of a two-level system
Figure 1: Diagram of a coupled two-level system subjected to ideal $2\pi$
pulses.
By applying an ideal $2\pi$ pulse (or parity kick, see Fig. 1), which is very
strong in amplitude and short in time but gives a $\pi$ phase shift solely to
the excited state $|e\rangle$ utilizing an auxiliary state $|a\rangle$ Agarwal
et al. (2001a), the state of the system changes according to
$\displaystyle x|e\rangle+y|g\rangle$
$\displaystyle\underrightarrow{\;\;2\pi\;{\rm pulse}\;\;}$
$\displaystyle-x|e\rangle+y|g\rangle$ (5)
where $|x|^{2}+|y|^{2}=1$. We denote such a pulse as $Z$ pulse hereafter
$\displaystyle Z(x|e\rangle+y|g\rangle)=-x|e\rangle+y|g\rangle.$ (6)
In fact, $Z=-\sigma_{z}=|g\rangle\langle g|-|e\rangle\langle e|$.
The evolution operator for the $2\pi$ pulse and the free evolution $\tau$ is
$\displaystyle U^{(1)}$ $\displaystyle=$ $\displaystyle
ZU=\left(\begin{array}[]{cc}d_{1}&d_{2}\\\
-d_{2}&-d_{1}^{*}\end{array}\right).$ (9)
For $N$ such operations, the evolution operator at time $t=N\tau$ becomes
$\displaystyle U^{(N)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}d_{1}&d_{2}\\\
-d_{2}&-d_{1}^{*}\end{array}\right)^{N}.$ (12)
Let $d_{1}=d_{1r}+id_{1i}$ and $d_{2}=id_{2i}$. After some straightforward
simplifications with the use of Pauli matrices, one easily obtains
$\displaystyle U^{(N)}$ $\displaystyle=$ $\displaystyle i^{N}(\cos
N\lambda-i\sigma_{\alpha}\sin N\lambda)$ (13)
where $\sin\lambda=\sqrt{1-d_{1i}^{2}}$ and
$\sigma_{\alpha}=(1/\sin\lambda)(d_{1r}\sigma_{z}-d_{2i}\sigma_{y})$. Note
that $\lambda$ depends on $\tau$ instead of $t=N\tau$.
The transition probability from the initial state $|e\rangle$ to the ground
state $|g\rangle$ is
$\displaystyle p_{eg}^{\prime}(t=N\tau)$ $\displaystyle=$
$\displaystyle|\langle
g|U^{(N)}|e\rangle|^{2}=p_{eg}(\tau){\sin^{2}N\lambda\over\sin^{2}\lambda}.$
(14)
The above result is exact for any coupling strength and pulse delay $\tau$.
For weak coupling $\nu\ll\delta$, $\Omega\approx\delta^{\prime}$ and
$\lambda\approx\delta^{\prime}\tau+\pi/2+2k\pi$ with $k$ an integer, then
$\displaystyle p_{eg}^{\prime}$ $\displaystyle\approx$
$\displaystyle{\nu^{2}\over\delta^{\prime
2}}\tan^{2}\delta^{\prime}\tau\sin^{2}N\delta^{\prime}\tau$ (15)
for even $N$, which is exactly Eq. (9) in Agarwal et al.’s paper Agarwal et
al. (2001a).
## IV Quantum Zeno and anti-Zeno effects
The above results are applicable only to single mode bath/environment which
couples to the central two-level system. In general the bath has multimodes
and the coupling may depend on the mode, e.g., the dipolar coupling between an
atom and an electric and magnetic field. For a many-mode bath, the transition
probability of the two-level system subjected to control pulses is in general
given by
$\displaystyle p_{eg}^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{n}{\nu^{2}\over\Omega_{n}^{2}}\sin^{2}\Omega_{n}\tau{\sin^{2}N\lambda_{n}\over\sin^{2}\lambda_{n}}$
(16)
with $n$ the bath mode index mmnote . Note that $\Omega_{n}$ and $\lambda_{n}$
become mode dependent in the many-mode bath case and $\nu$ is constant here
but will be considered as mode dependent in Sec. IV.4. Assuming the bath
spectrum is dense, we turn the summation over mode index $n$ into the
integration over mode frequency $\omega$,
$\displaystyle p_{eg}^{\prime}$ $\displaystyle=$ $\displaystyle\int
d\omega\rho(\omega){\nu^{2}\over\Omega^{2}}\sin^{2}\Omega\tau{\sin^{2}N\lambda\over\sin^{2}\lambda}$
(17)
where $\rho(\omega)$ is the density of states of the bath. For $N=1$ we get
the free decay results (no control pulse). Before considering the real
physical systems which usually have mode dependent coupling, we study several
toy models with constant coupling strength $\nu=\nu_{0}$ for all bath modes to
gain some ideas about the effect of control pulses.
### IV.1 Uniform spectrum intensity
Figure 2: Control pulse effect with fixed pulse delays. The parameters are
$\omega_{0}=1,\nu_{0}=0.001,\omega_{c}=100$, and $\tau=10,4,3,2,1$ from top to
bottom for solid lines. The dashed line denotes the free decay. Time is in
units of $1/\omega_{0}$ hereafter.
Taking $\rho(\omega)=\rho_{0}\equiv 1/\omega_{c}$ if $\omega\in[0,\omega_{c}]$
and $\rho(\omega)=0$ otherwise with $\omega_{c}$ denoting the cutoff frequency
of the bath, Fig. 2 shows typical free decay and controlled decay of the
excited state. Except at very short times, Fig. 2 shows that the transition
probability linearly depends on the total evolution time. Utilizing the linear
dependence, one defines decay rate (Einstein constant) at long time $t$ as
$\displaystyle A$ $\displaystyle\equiv$ $\displaystyle{\partial
p_{eg}(t)\over\partial t}=2\pi\rho_{0}\nu_{0}^{2}$ (18)
for free decay Agarwal et al. (2001a) and
$\displaystyle A^{\prime}$ $\displaystyle\equiv$
$\displaystyle{1\over\tau}{\partial p_{eg}^{\prime}\over\partial
N}={2\pi\rho_{0}\nu_{0}^{2}\over\tau^{2}}\sum_{k}{1\over\nu_{0}^{2}+(\omega_{k}-\omega_{0})^{2}/4}$
(19)
for controlled decay, where $\omega_{k}$ is the $k$th resonant mode frequency
Kofman and Kurizki (2001). For weak coupling $\nu_{0}\ll\pi/\tau$, the $k$th
resonant mode lies approximately at
$\omega_{k}\approx\omega_{0}+(2k+1)\pi/\tau$ where $\sin\lambda\ll 1$ and
$\sin
2N\lambda/\sin\lambda\approx\pi\delta(\lambda)\approx(2\pi/\tau)\delta(\omega-\omega_{k})$
for large $N$. Note that the two nearest neighbor peaks around $\omega_{0}$
with $k=0$ and $k=-1$ contribute equally about $40\%$ among all the peaks
Agarwal et al. (2001a); Kofman and Kurizki (2001).
If we assume the bath has only positive frequency and concentrate on the two
dominant peaks, we find
$\displaystyle{A^{\prime}\over A}$ $\displaystyle\simeq$
$\displaystyle\left\\{\begin{array}[]{cc}0,&\omega_{c}\tau<\pi\\\
{1/2},&\omega_{0}\tau<\pi<\omega_{c}\tau\\\ 1,&{\rm
otherwise}\end{array}\right.$ (23)
and $p_{eg}^{\prime}/p_{eg}$ also shows step-like behavior as depicted in Fig.
3. Moreover, Fig. 3 exhibits only quantum Zeno effect, i.e., suppression of
the decay by the control pulses. The decay are completely suppressed,
$A^{\prime}=0$, once the control pulse frequency is larger than the cutoff
frequency.
Figure 3: (Color online) Control pulse effect vs number of pulses $N$ (main
panel) and relative positions of resonant modes to $\omega_{0}$ and
$\omega_{c}$ (upper three panels A, B, and C corresponds to three regions in
the main panel, respectively). The parameters are
$N\tau=100,\omega_{0}=1,\nu_{0}=0.001$, and $\omega_{c}=100$. Two big steps
shows two major peaks out of the lower and upper cutoff frequency,
respectively. These small steps shows minor peaks going out of the cutoff
frequencies. Circles are for the frequency-dependent coupling case.
### IV.2 Gaussian spectrum intensity
A Gaussian spectrum intensity has a form as
$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2\pi}\;\Gamma}\;e^{-(\omega-\omega_{m})^{2}/2\Gamma^{2}},\;\;\omega\geq
0$ (24)
where $\Gamma$ and $\omega_{m}$ denotes the width and the position of the
maximal intensity, respectively.
#### IV.2.1 $\omega_{0}=\omega_{m}$
In this case we expect only quantum Zeno effect to appear, $A^{\prime}\leq A$,
because it is easy to check that
$\displaystyle{A^{\prime}\over A}$ $\displaystyle=$
$\displaystyle{4\over\pi^{2}}\sum_{k}{1\over(2k+1)^{2}}\;{\rho(\omega_{k})\over\rho(\omega_{0})}\leq
1$ (25)
where we have used the large $N$ and weak coupling assumptions. By considering
the two dominant peaks, we further obtain that
$\displaystyle{A^{\prime}\over A}$ $\displaystyle\approx$
$\displaystyle{8\over\pi^{2}}\;e^{-\pi^{2}/2\Gamma^{2}\tau^{2}}$ (26)
for small $\tau$ such that $\Gamma\tau\lesssim\pi$. Equation (26) shows that
the decay rate decreases rapidly with the control pulse frequency in a
Gaussian form. For exceedingly small $\tau$ which satisfies
$\Gamma\tau\ll\pi$, $A^{\prime}$ is essentially zero, which means the
transition is inhibited and the survival probability of the initial state
saturates. The red dashed line in Fig. 4 demonstrates the quantum Zeno effect
and the prohibition of the transition.
Figure 4: (Color online) Reduced transition probability at fixed time
($t=N\tau=100$) vs number of pulses $N$ for Gaussian spectrum intensity for
$\omega_{0}=\omega_{m}$ (red dashed line) and $\omega_{0}=\omega_{m}-2$ (blue
solid line). Circles and crosses are for the corresponding frequency-dependent
coupling case. The vertical dot-dashed line shows the boundary between quantum
Zeno and anti-Zeno effects for $\omega_{0}=\omega_{m}-2$. Other parameters are
$\nu_{0}=0.001,\omega_{c}=100,\omega_{m}=\omega_{c}/2$, $\Gamma=1$.
#### IV.2.2 $|\omega_{0}-\omega_{m}|\gtrsim\Gamma$
For $|\omega_{0}-\omega_{m}|<\Gamma$, the results are similar to
$\omega_{0}=\omega_{m}$ and exhibit only quantum Zeno effect. To observe
substantially enhancement of the transition, i.e., the quantum anti-Zeno
effect Kofman and Kurizki (2000), $|\omega_{0}-\omega_{m}|\gtrsim\Gamma$ is
required. More specifically, by taking the biggest peak around $\omega_{0}$,
one obtains the necessary condition for quantum anti-Zeno effect as
$\displaystyle|\omega_{0}-\omega_{m}|\geq\sqrt{4\ln{\pi\over
2}}\;\Gamma\approx 1.3\Gamma.$ (27)
The blue solid line in Fig. 4 shows the quantum Zeno and anti-Zeno effects for
$|\omega_{0}-\omega_{m}|=2$. Both quantum Zeno (large $N$ region) and anti-
Zeno (small $N$ region) effects are observed. At fixed time $t=N\tau$, large
$\tau$ (small N) gives anti-Zeno effect while small $\tau$ (large N) gives
Zeno effect. The boundary between quantum Zeno and anti-Zeno effect is about
$2|\omega_{m}-\omega_{0}|\tau\approx\pi$ if
$|\omega_{m}-\omega_{0}|\gg\Gamma$. Strongest anti-Zeno effect is obtained at
$|\omega_{0}-\omega_{m}|\tau=\pi$ where one of the two main resonant modes
lies near $\omega_{m}$.
### IV.3 Lorentzian and exponential spectrum intensity
The spectrum intensity for Lorentzian and exponential are, respectively,
$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{\Gamma/\pi\over(\omega-\omega_{m})^{2}+\Gamma^{2}},\;\omega\geq
0$ (28) $\displaystyle\rho(\omega)$ $\displaystyle=$ $\displaystyle{1\over
2\Gamma}e^{-|\omega-\omega_{m}|/\Gamma},\;\omega\geq 0.$ (29)
Similar to Gaussian spectral intensity, only the quantum Zeno effect is
observed if $\omega_{0}=\omega_{m}$ for the Lorentzian and exponential shape
(Fig. 5). The transition probability is essentially inhibited once
$\tau\ll\tau_{c}$ where $\tau_{c}\sim 1/\omega_{c}$. Once
$|\omega_{0}-\omega_{m}|\gtrsim\Gamma$, we find both quantum Zeno and anti-
Zeno effects and the boundary between Zeno and anti-Zeno regime is determined
approximately by $2|\omega_{m}-\omega_{0}|\tau\approx\pi$. The peak position
of the quantum anti-Zeno effect lies about at
$|\omega_{0}-\omega_{m}|\tau\approx\pi$. As shown in Fig. 5, the reduced
transition probability in the exponential case has a narrower peak width than
that in the Lorentzian case. Two small secondary peaks are noticeable in the
anti-Zeno region of the exponential spectrum case. These peaks are due to the
second and third quasi-level resonant to $\omega_{m}$. They satisfy the
condition, respectively, $|\omega_{0}-\omega_{m}|\tau\approx 3\pi,5\pi$.
Figure 5: (Color online) Same as Fig. 4 except that the spectrum intensity is
Lorentzian (top) and exponential (bottom).
### IV.4 Frequency dependent coupling
The widely adopted dipolar coupling in spontaneous emission of a two level
atom or molecule has a frequency dependence as
$\nu(\omega)=\nu_{0}\sqrt{\omega}$ where $\nu_{0}$ is taken as a constant,
which depends on the dipole matrix elements Scully and Zubairy (1997). We will
consider the same coupling spectrum intensity as before, i.e.,
$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\omega}{1\over\sqrt{2\pi}\;\Gamma}\;e^{-(\omega-\omega_{m})^{2}/2\Gamma^{2}},\;\omega\geq
0$ (30) $\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\omega}{\Gamma/\pi\over(\omega-\omega_{m})^{2}+\Gamma^{2}},\;\omega\geq
0$ (31) $\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{1\over\omega}{1\over
2\Gamma}e^{-|\omega-\omega_{m}|/\Gamma},\;\omega\geq 0$ (32)
for Gaussian, Lorentzian, and exponential density of state, respectively. As
shown in Figs. 4 and 5, the frequency dependent coupling has little effect on
the performance of the control pulses.
## V Decay freezing
Figure 6: Decay freezing (left column) and dependence of the differential
freezing value on pulse delay (right column) for the Gaussian (top row),
Lorentzian (middle row), and exponential (bottom row) spectrum. Only results
of odd number pulses are plotted for a better view. The dashed lines show
decay without control pulses. Crosses of dashed lines and solid lines in the
left column panels denote the specific pulse delay $\tau$.
By inspecting Eq. (17), the controlled transition probability freezes if
$N\lambda\gg 1$ and the freezing value is
$\displaystyle\lim_{N\rightarrow\infty}p_{eg}^{\prime}$ $\displaystyle=$
$\displaystyle{1\over 2}\int
d\omega\rho(\omega){\nu^{2}\over\Omega^{2}}\sin^{2}\Omega\tau{1\over\sin^{2}\lambda},$
(33)
where we have replaced the rapidly oscillating integrand $\sin^{2}(N\lambda)$
with its average $1/2$. The left column of Fig. 6 from numerical calculation
indeed shows that the transitions freeze at long times for three different
cases at small $\tau$. The smaller the $\tau$ is, the smaller the freezing
value is. Freezing of the transition probability is equivalent to freezing of
the survival probability of the excited state, $f=1-p^{\prime}_{eg}$.
Moreover, for small enough pulse delay $\tau$, we have $\sin^{2}\lambda\approx
1$ thus
$\displaystyle\lim_{N\rightarrow\infty}p_{eg}^{\prime}$ $\displaystyle\approx$
$\displaystyle{1\over 2}\int
d\omega\rho(\omega){\nu^{2}\over\Omega^{2}}\sin^{2}\Omega\tau={1\over
2}p_{eg}(\tau).$ (34)
The above relation indicates that the freezing value of the transition
probability at long times is one half of the free transition value at $t=\tau$
which is the pulse delay. Clearly, the road map to the quantum ZE is that the
decay rate becomes zero after initial pulses (decay freezing) and the freezing
value of the survival probability $f$ approaches 1 as $\tau$ decreases.
Define differential freezing value
$\varepsilon(\tau)=p_{eg}^{\prime}(N\tau=1000)-{1\over 2}p_{eg}(\tau)$, which
describes the difference of the transition probability after many pulses and
one half of the decay in the first pulse delay. The right column of Fig. 6
shows that $\varepsilon$ decreases with increasing of $1/\tau$ (decreasing of
$\tau$), confirming the analytical results of Eq. (34).
In fact, decay freezing exists not only in the model system we consider in the
paper, but also exists in many other pulse-controlled systems, such as gated
semiconductor quantum dot Zhang et al. (2007a, b, 2008); Lee et al. (2008);
Liu et al. (2007), spin-boson model Viola and Lloyd ; Faoro and Viola (2004),
and nuclear spins Haeberlen (1976). The basic idea behind the decay freezing
is that the control pulses creates an effective preferred direction along
which the decay is frozen. In terms of Pauli matrix (c.f. Eq. 2), the
preferred direction created by control pulses in the model we consider is $z$.
## VI Conclusion and discussion
By investigating three models of coupling spectrum intensity (Gaussian,
Lorentzian and exponential), we demonstrate that a two-level system subjected
to many ideal $2\pi$ pulses exhibits both quantum Zeno and anti-Zeno effect,
depending on the relative position of $\omega_{0}$ to the peak position
$\omega_{m}$ of the spectrum and the pulse delay $\tau$. Instead of decreasing
the decay rate, the pulsed two-level system shows decay freezing after many
pulses at small $\tau$ and the freezing value of the survival probability of
the initial excited state approaches 1 (no decay) with decreasing $\tau$.
In this paper, all the spectrums have single peak and we observe only single
quantum Zeno and/or anti-Zeno region. Under some special circumstances where a
multiple peaks spectrum exists, one would expect multiple quantum Zeno and
anti-Zeno regions. We have also assumed that the spectrum of the structured
vacuum is time independent where the back action exerted on the vacuum by the
two-level system has been neglected. A full quantum version of the coupling
between the two-level system and the vacuum could possibly change the picture
of the controlled decay at long times but the short time behavior would be
intact in the weak coupling regime, because the back action is weak and needs
a long time to manifest its effect on the two-level system dynamics.
We consider only periodic pulse sequence ($\tau$ is fixed) in this paper. In
principle, other pulse sequences with varying $\tau$ may also give similar
quantum Zeno and anti-Zeno effect. They may even have additional advantages
Uhrig (2007); Lee et al. (2008). In addition, the strength and the duration of
the $2\pi$ pulses are finite in practice. One could minimize the finite pulse
effect by employing the phase alternation techniques Slichter (1992) or the
Eulerian protocols Viola (2004).
## VII Acknowledgments
W. Z. is grateful for many helpful discussions with V. V. Dobrovitski, S. Y.
Zhu, and T. Yu. This work was partially carried out at the Ames Laboratory,
which is operated for the U. S. Department of Energy by Iowa State University
under Contract No. W-7405-82 and was supported by the Director of the Office
of Science, Office of Basic Energy Research of the U. S. Department of Energy.
Part of the calculations was performed at the National High Performance
Computing Center of Fudan University.
## References
* Milonni (1994) P. W. Milonni, _The Quantum Vacuum: An Introduction to Quantum Electrodynamics_ (Academic Press, New York, 1994).
* Scully and Zubairy (1997) M. O. Scully and M. S. Zubairy, _Quantum Optics_ (Cambridge University Press, Cambridge, 1997).
* Allen and Eberly (1975) L. Allen and J. H. Eberly, _Optical Resonance and Two-level Atoms_ (Dover Publication, New York, 1975).
* Frishman and Shapiro (2003) E. Frishman and M. Shapiro, Phys. Rev. A 68, 032717 (2003).
* Evers and Keitel (2002) J. Evers and C. H. Keitel, Phys. Rev. Lett. 89, 163601 (2002).
* Frishman and Shapiro (2001) E. Frishman and M. Shapiro, Phys. Rev. Lett. 87, 253001 (2001).
* Agarwal et al. (2001a) G. S. Agarwal, M. O. Scully, and H. Walther, Phys. Rev. Lett. 86, 4271 (2001a).
* Vogel and Welsch (2006) W. Vogel and D.-G. Welsch, _Quantum Optics_ (Wiley-VCH, Weinheim, 2006), 3rd ed.
* Itano et al. (1990) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 (1990).
* Itano et al. (1991) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 43, 5168 (1991).
* Misra and Sudarshan (1977) B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
* Kofman and Kurizki (2000) A. G. Kofman and G. Kurizki, Nature(London) 405, 546 (2000).
* Agarwal et al. (2001b) G. S. Agarwal, M. O. Scully, and H. Walther, Phys. Rev. A 63, 044101 (2001b).
* Fischer et al. (2001) M. C. Fischer, B. Gutiérrez-Medina, and M. G. Raizen, Phys. Rev. Lett. 87, 040402 (2001).
* Kofman and Kurizki (2001) A. G. Kofman and G. Kurizki, Phys. Rev. Lett. 87, 270405 (2001).
* Scully et al. (2003) M. O. Scully, S.-Y. Zhu, and M. S. Zubairy, Chaos, Solitons and Fractals 16, 403 (2003).
* (17) We have assumed the dipolar interaction has a symmetry such that $\langle e|\vec{d}\cdot\vec{E}|g\rangle=\langle g|\vec{d}\cdot\vec{E}|e\rangle$, where $\vec{d}$ and $\vec{E}$ are the dipole operator of the atom and the electric field of the light, respectively.
* (18) We adopt the Schrödinger picture instead of the usual interaction picture (e.g., Ref. Agarwal et al. (2001a)) for our convenience. Both pictures are equivalent in principle.
* (19) For weak couplings, which are the case in the decay problem we consider, high order nonlinear interaction between the two-level atom and multiple electric field modes is neglected.
* Zhang et al. (2007a) W. Zhang, V. V. Dobrovitski, L. F. Santos, L. Viola, and B. N. Harmon, Phys. Rev. B 75, 201302(R) (2007a).
* Zhang et al. (2007b) W. Zhang, N. P. Konstantinidis, K. A. Al-Hassanieh, and V. V. Dobrovitski, J. Phys.: Condens. Matter 19, 083202 (2007b).
* Zhang et al. (2008) W. Zhang, N. P. Konstantinidis, V. V. Dobrovitski, B. N. Harmon, L. F. Santos, and L. Viola, Phys. Rev. B 77, 125336 (2008).
* Lee et al. (2008) B. Lee, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett. 100, 160505 (2008).
* Liu et al. (2007) R.-B. Liu, W. Yao, and L. J. Sham, New J. Phys. 9, 226 (2007).
* (25) L. Viola and S. Lloyd, eprint quant-ph/9809058.
* Faoro and Viola (2004) L. Faoro and L. Viola, Phys. Rev. Lett. 92, 117905 (2004).
* Haeberlen (1976) U. Haeberlen, _High resolution NMR in solids: Selective averaging_ (Academic Press, New York, 1976).
* Uhrig (2007) G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007).
* Slichter (1992) C. P. Slichter, _Principles of Magnetic Resonance_ (Springer-Verlag, New York, 1992).
* Viola (2004) L. Viola, J. Mod. Opt. 51, 2357 (2004).
|
arxiv-papers
| 2010-01-07T09:43:16 |
2024-09-04T02:49:07.561248
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenxian Zhang and Jun Zhuang",
"submitter": "Wenxian Zhang",
"url": "https://arxiv.org/abs/1001.1041"
}
|
1001.1101
|
# NATURE OF W51e2: MASSIVE CORES AT DIFFERENT PHASES OF STAR FORMATION
Hui Shi11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Jia20 Datun Road, Chaoyang District, Beijing 100012, China. Email:
shihui, hjl @ nao.cas.cn , Jun-Hui Zhao22affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA. Email:
jzhao @ cfa.harvard.edu , J.L. Han11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Jia20 Datun Road, Chaoyang
District, Beijing 100012, China. Email: shihui, hjl @ nao.cas.cn
###### Abstract
We present high-resolution continuum images of the W51e2 complex processed
from archival data of the Submillimeter Array (SMA) at 0.85 and 1.3 mm and the
Very Large Array at 7 and 13 mm. We also made line images and profiles of
W51e2 for three hydrogen radio recombination lines (RRLs; H26$\alpha$,
H53$\alpha$, and H66$\alpha$) and absorption of two molecular lines of
HCN(4-3) and CO(2-1). At least four distinct continuum components have been
detected in the 3$\arcsec$ region of W51e2 from the SMA continuum images at
0.85 and 1.3 mm with resolutions of 0.3$\arcsec\times 0.2\arcsec$ and
1.4$\arcsec\times 0.7\arcsec$, respectively. The west component, W51e2-W,
coincides with the ultracompact HII region reported from previous radio
observations. The H26$\alpha$ line observation reveals an unresolved hyper-
compact ionized core ($<0.06\arcsec$ or $<310$ AU) with a high electron
temperature of $1.2\times 10^{4}$ K, with the corresponding emission measure
EM$>7\times 10^{10}{\rm pc~{}cm^{-6}}$ and the electron density $N_{e}>7\times
10^{6}$ cm-3. The inferred Lyman continuum flux implies that the HII region
W51e2-W requires a newly formed massive star, an O8 star or a cluster of
B-type stars, to maintain the ionization. W51e2-E, the brightest component at
0.85 mm, is located 0.9$\arcsec$ east from the hyper-compact ionized core. It
has a total mass of $\sim$140 M☉ according to our spectral energy distribution
analysis and a large infall rate of $>1.3\times 10^{-3}$ M☉yr-1 inferred from
the absorption of HCN. W51e2-E appears to be the accretion center in W51e2.
Given the fact that no free$-$free emission and no RRLs have been detected,
the massive core of W51e2-E appears to host one or more growing massive proto-
stars. Located $2\arcsec$ northwest from W51e2-E, W51e2-NW is detected in the
continuum emission at 0.85 and 1.3 mm. No continuum emission has been detected
at $\lambda\geq$ 7 mm. Along with the maser activities previously observed,
our analysis suggests that W51e2-NW is at an earlier phase of star formation.
W51e2-N is located 2$\arcsec$ north of W51e2-E and has only been detected at
1.3 mm with a lower angular resolution ($\sim 1\arcsec$), suggesting that it
is a primordial, massive gas clump in the W51e2 complex.
HII regions – ISM: individual objects (W51e2) – stars: formation
## 1 INTRODUCTION
Massive stars are formed in dense, massive molecular cores. Detailed physical
processes in star-forming regions have not been well studied until recent
high-resolution observations were available at submillimeter wavelengths. High
angular resolution observations are necessary to unveil the physical environs
and activities of the individual sub-cores as well as their impact on the
overall process of massive star formation.
W51 is a well-known complex of HII regions with about $1\arcdeg$ area in the
Galactic plane (Westerhout, 1958). W51A, also called G49.5-0.4, is the most
luminous region in W51 (Kundu & Velusamy, 1967). Several discrete components
have been found from W51A and referred alphabetically as a$-$i in the right
ascension (R.A.) order (Martin, 1972; Mehringer, 1994). W51e is one of the
brightest regions in W51A. High-resolution centimeter observations have
revealed that W51e consists of several ultracompact (UC) HII regions (Scott,
1978; Gaume et al., 1993), among which the UC HII region of W51e2 appears to
be the brightest. This region is severely obscured by the dust, and no IR
detection has been made at wavelengths shorter than 20 $\mu$m (Genzel et al.,
1982). The inferred distance to W51e2 is $\sim$5.1 kpc (Xu et al., 2009);
thus, 1$\arcsec$ corresponds to a linear scale of about 5100 AU.
Based on the radio continuum observations, spectral energy distribution (SED)
analyses of W51e2 suggest that the emission source from this region consists
of two components, namely, (1) thermal emission from a cold dust component and
(2) free$-$free emission from a hot HII region (Rudolph et al., 1990; Sollins
et al., 2004; Keto et al., 2008).
Observations of hydrogen radio recombination lines (RRLs) were carried out
toward W51e2 (Mehringer, 1994; Keto et al., 2008; Keto & Klaassen, 2008).
Mehringer (1994) failed to detect the H92$\alpha$ line in this region, which
might be attributed to the large optical depth and/or the large pressure
broadening of the UC HII region at 3.6 cm. Based on the observations of
H66$\alpha$, H53$\alpha$, and H30$\alpha$, Keto et al. (2008) argued that the
pressure broadening is responsible for the broad-line widths observed in the
RRLs.
Table 1: Observation parameters for W51e2 Parameters | SMA: 0.85 mm | SMA: 1.3 mm | VLA: 7 mm | VLA: 13 mm
---|---|---|---|---
Observation date | 2007 Jun. 18 | 2005 Sep. 1 | 2004 Feb. 14 | 2003 Sep. 29
Array configuration | Very extended (8 ants) | Extended (6 ants) | AB | BC
Pointing center R.A.(J2000) | 19:23:43.888 | 19:23:43.895 | 19:23:43.918 | 19:23:43.918
Pointing center decl.(J2000) | +14:30:34.798 | +14:30:34.798 | +14:30:28.164 | +14:30:28.164
Frequency(GHz) | 343, 353 | 221, 231 | 43 | 22
Bandwidth | 2.0+2.0 (GHz) | 2.0+2.0 (GHz) | 12.5 (MHz) | 12.5 (MHz)
On-source time (hr) | 2.10 | 4.75 | 5.90 | 3.93
System temperature(K) | 150$\sim$500 | 100$\sim$250 | |
Bandpass calibrators | J1229+020, J1751+096 | 3C 454.4 | 3C 84 | 3C 84
Phase calibrators | J1733$-$130, J1743$-$038 | J1751+096, J2025+337 | J1923+210 | J1923+210
| J2015+371 | | |
Flux calibrators | Callisto | Ceres | 3C 286 | 3C 286
Observations of molecular lines (e.g. Ho & Young, 1996; Zhang & Ho, 1997;
Zhang et al., 1998; Young et al., 1998; Sollins et al., 2004) showed evidence
for infall (or accretion) gas within 5$\arcsec$ ($<$ 0.2 pc) around W51e2. A
possible rotation was suggested by Zhang & Ho (1997) on the basis of fitting
the position$-$velocity (PV) diagram of the NH3(3,3) absorption line at 13 mm.
A spin-up rotation with an axis of position angle (P.A.) $\approx$20$\arcdeg$
was further suggested based on the velocity gradient of CH3CN(83$-$73) at 2 mm
(Zhang et al., 1998). However, based on the velocity gradient observed from
the H53$\alpha$ line, Keto & Klaassen (2008) interpreted their results as
evidence for rotational ionized accretion flow around the UC HII region and
derived a rotational axis of P.A.$\approx-30\arcdeg$. In addition, a possible
bipolar outflow along the northwest and southeast (NW$-$SE) directions was
suggested based on the observations of the CO(2-1) line emission (Keto &
Klaassen, 2008), which appears to be perpendicular to the axis of the
rotational ionized disk. However, the southwest (SW) elongation of the UC HII
region in Gaume et al.’s (1993) observations was explained as a collimated
ionized outflow. On the other hand, proper-motion measurements of H2O/OH
masers appeared to favor a hypothetical model with an outflow or expanding gas
bubble in W51e2 (Imai et al., 2002; Fish & Reid, 2007).
Thus, improved images and comprehensive analysis of high-resolution
observations at the wavelengths from radio to submillimeter are needed to
reconcile the differences in the interpretation of the results from the
previous observations of W51e2. In this paper, we show the high-resolution
continuum images with the Submillimeter Array (SMA) at 0.85 and 1.3 mm and
with the Very Large Array (VLA) at 7 and 13 mm, as well as line images and
profiles of the hydrogen recombination lines (H26$\alpha$, H53$\alpha$, and
H66$\alpha$) and the molecular lines of HCN(4-3) and CO(2-1). In Section 2, we
describe the details of the data processing and results. In Section 3, we
model the individual components on the basis of the high-resolution
observations of W51e2 with the SMA and VLA. The nature and astrophysical
processes of star formation cores in W51e2 are discussed. A summary and
conclusions are given in Section 4.
## 2 DATA REDUCTION AND RESULTS
We processed the archival data from the SMA and VLA observations of W51e2 and
constructed both the continuum and spectral line images. Details in
calibration and imaging are discussed below along with the relevant parameters
that are summarized in Table 1.
### 2.1 The SMA continuum images at 0.85 and 1.3 mm
The observations of W51e2 at 0.85 mm were carried out with the SMA in the very
extended array (eight antennas) and at 1.3 mm in the extended array (six
antennas). The data were calibrated in Miriad (Sault et al., 1995) following
the reduction instructions for SMA data
111http://www.cfa.harvard.edu/sma/miriad. The system temperature corrections
were applied to the visibility data. The bandpass calibration was made with
strong calibrators for each of the two data sets (Table 1). The flux density
scale was determined using Callisto at 0.85 mm and Ceres at 1.3 mm.
Corrections to the complex gains at 1.3 mm were made by applying the solutions
interpolated from the two nearby QSOs J1751+096 and J2025+337. For the high-
resolution data at 0.85 mm, the visibility data are initially calibrated with
a model of W51e2 derived from observations in a low angular resolution at 0.87
mm. The residual phase errors were further corrected using the self-
calibration technique. The final continuum images were constructed by
combining all the line-free channels in both upper sideband and lower sideband
data.
Figure 1: High-resolution 0.85 mm continuum image of W51e2 observed with the
SMA. Contours are $\pm 5\sigma\times 2^{n}$ ($n=0$, 1, 2, 3, … and
$\sigma=7.11$ mJy beam-1). The FWHM beam ($0.31\arcsec\times 0.22\arcsec$,
P.A.=$60.6\arcdeg$) is shown at bottom right. Three continuum emission
components e2-NW, e2-E, and e2-W are marked with “+”. The $\lambda 3.6$ cm
position of the UC HII region in W51e2 Gaume et al. (1993) is marked with
“$\times$”.
Figure 2: (a): 1.3 mm continuum image of W51e2 observed with the SMA.
Contours are $\pm 5\sigma\times 2^{n}$ ($n=0$, 1, 2, 3, … and $\sigma=14.0$
mJy beam-1). The FWHM beam ($1.35\arcsec\times 0.69\arcsec$,
P.A.=$-87.4\arcdeg$) is shown at bottom right. The three components detected
at 0.85 mm together with the additional component e2-N detected at 1.3 mm are
marked with “+”. The position “+” of the submillimeter components and the
peaks of the 3.6 cm continuum (“$\times$”) and H26$\alpha$ (“$\bigstar$”) are
also marked in the rest of the images. (b): 7 mm continuum image of W51e2
observed with the VLA. Contours are $\pm 5\sigma\times 2^{n/2}$ ($n=0$, 1, 2,
3, … and $\sigma=1.12$ mJy beam-1). The FWHM beam ($0.40\arcsec\times
0.15\arcsec$, P.A.=$80.3\arcdeg$) is shown at bottom-right. (c): 13 mm
continuum image of W51e2 observed with VLA. Contours are $\pm 5\sigma\times
2^{n}$ ($n=0$, 1, 2, 3, …, and $\sigma=0.50$ mJy beam-1). The FWHM beam
($0.23\arcsec\times 0.10\arcsec$, P.A.=$82.6\arcdeg$) is shown at bottom
right. (d): Image of the H26$\alpha$ line emission integrated from 25 km s-1
to 84 km s-1 from W51e2. Contours are $\pm 5\sigma\times 2^{n}$ ($n=0$, 1, 2,
3, … and $\sigma=0.146$ Jy beam-1 km s-1). The FWHM beam ($0.25\arcsec$) is
shown at bottom right. All the RRL images are convolved to a common circular
beam ($0.25\arcsec$). The star denotes the peak position of the H26$\alpha$
line and is marked in the rest of the RRL images. (e): Image of the
H53$\alpha$ line emission integrated from 22 km s-1 to 101 km s-1. Contours
are $\pm 5\sigma\times 2^{n}$ ($n=0$, 1, 2, 3, … and $\sigma=9.68\times
10^{-3}$ Jy beam-1 km s-1). (f): Image of the H66$\alpha$ line emission
integrated from 24 km s-1 to 120 km s-1. Contours are $\pm 5\sigma\times
2^{n/2}$ ($n=0$, 1, 2, 3, … and $\sigma=6.14\times 10^{-3}$ Jy beam-1 km s-1).
Figure 1 shows the high-resolution (0.3$\arcsec\times 0.2\arcsec$) continuum
image of W51e2 at 0.85 mm. The complex of W51e2 has been resolved into at
least three bright, compact components (see Figure 1, the symbol “+” marks the
positions of the components). The brightest source, W51e2-E, is located $\sim
0.9\arcsec$ east of the UC HII region which is marked with “$\times$” for the
peak position at 3.6 cm from Gaume et al. (1993). The secondary bright source,
W51e2-W, coincides with the 3.6 cm peak of the UC HII region. The third
component, W51e2-NW, is located about 2$\arcsec$ northwest of W51e2-E. In
addition, these compact emission components appear to be surrounded by an
amorphous halo in the continuum emission at 0.85 mm. The three bright
continuum sources were also detected in the low-resolution continuum image at
1.3 mm (Figure 2(a)). Moreover, a weak continuum component, W51e2-N, is
present $\sim 2\arcsec$ north of W51e2-E.
### 2.2 The VLA continuum images at 7 and 13 mm
The observations of W51e2 at 7 and 13 mm were carried out with the VLA (see
the summarized observing parameters in Table 1). We made the calibrations by
following the standard data-reduction procedure for VLA data with
AIPS222http://www.aips.nrao.edu. Then, the calibrated visibilities were loaded
into the Miriad environment. The imaging and further analysis were carried out
with Miriad.
The dirty images were made using INVERT with robust weighting (robust=0) and
were cleaned with the hybrid deconvolution algorithm. Figures 2(b) and 2(c)
show the continuum images at 7 and 13 mm, respectively. Strong continuum
emission is detected from W51e2-W at both 7 and 13 mm in good agreement with
previous observations at centimeters (Gaume et al., 1993). In addition, a weak
but significant ($>$10$\sigma$) emission feature at W51e2-E has been detected
at only 7 mm. As shown in Figure 2(b), the weak 7 mm continuum source at
W51e2-E has been clearly separated from the UC HII region, W51e2-W.
### 2.3 Hydrogen recombination lines
For the line cubes, the continuum emission was subtracted using the linear
interpolation from the line-free channels with UVLIN. The H26$\alpha$ line
(353.623 GHz) was included in the line cubes made from the SMA data at 0.85
mm. We also made a line image cube to cover the H30$\alpha$ line (231.901 GHz)
from the SMA data at 1.3 mm. Both the H26$\alpha$ and H30$\alpha$ line data
were resampled to 1 km s-1. The rms noises of 49 mJy beam-1 and 81 mJy beam-1
in each of the channel images are inferred for the H26$\alpha$ and H30$\alpha$
line image cubes, respectively. Figure 2(d) shows the integrated line
intensity image of the H26$\alpha$ line made with a 3$\sigma$ cutoff in the
velocity range from 25 to 84 km s-1. The H26$\alpha$ line emission from the UC
HII region (W51e2-W) appears to be very compact and has not been resolved with
the beam of 0.25$\arcsec$, suggesting that the intrinsic source size of the
hyper-compact HII core is $<0.06\arcsec$. The peak position of the integrated
H26$\alpha$ line emission (stars in Figure 2) appears to have a significant
offset of $\sim$0.15$\arcsec$ from the continuum peak at 13 mm (also see the
Gaume’s position in Figure 2(d)). Unlike the other RRLs and continuum emission
at centimeter wavelengths, the line emission at the frequency of the
H30$\alpha$ line (the image is not shown in this paper) shows an extended
distribution covering the entire 3$\arcsec$ region, suggesting that the
H30$\alpha$ line might have been severely contaminated by one or more
molecular lines in W51e2 (e.g., C3H7CN with a rest frequency of 231.9009 GHz).
The line of H66$\alpha$ is broad and weak, which is barely covered by the VLA
correlator band with which we had difficulty determining the line-free
channels. Instead, the continuum levels were determined by averaging all
channels excluding a few relatively strong line emission channels and were
subtracted from the visibility data. The uncertainty is $<5$% in the continuum
levels due to the line contamination. The image cubes of the H53$\alpha$ and
H66$\alpha$ lines were cleaned and convolved to a circular beam of
0.25$\arcsec$, giving the rms noises of 2.4 and 1.5 mJy beam-1 per channel,
respectively. The integrated intensities of H53$\alpha$ (with a 3$\sigma$
cutoff in each channel) and H66$\alpha$ (with a 2$\sigma$ cutoff in each
channel) are shown in Figures 2(e) and 2(f), respectively. Both H53$\alpha$,
and H66$\alpha$ lines are detected from only W51e2-W, the UC HII region. For
the H26$\alpha$, H$53\alpha$, and H66$\alpha$ lines from W51e2-W, we
integrated the line emission region with a size of $0.5\arcsec$ around the
H26$\alpha$ peak position. The profiles of the H26$\alpha$, H$53\alpha$ and
H66$\alpha$ lines are multiplied by the frequency ratios of
$\nu_{H26\alpha}/\nu_{H26\alpha}$, $\nu_{H26\alpha}/\nu_{H53\alpha}$ and
$\nu_{H26\alpha}/\nu_{H66\alpha}$, respectively, as shown in Figure 3. The
measurements of the peak intensity ($I_{peak}$), radial velocity ($V_{LSR}$),
and FWHM line width ($\Delta V$) for the H26$\alpha$, H$53\alpha$, and
H66$\alpha$ lines are given in Table 3.
Figure 3: Profiles of hydrogen recombination lines, H26$\alpha$, H53$\alpha$,
and H66$\alpha$, integrated from W51e2-W in an area of
0.5$\arcsec\times$0.5$\arcsec$ centered at the H26$\alpha$ line peak. All the
line images have been convolved with the same circular beam of 0.25$\arcsec$.
The spectra are multiplied with the ratios of the rest frequency of the
H26$\alpha$ line to the RRL rest frequencies ($\nu_{H26\alpha}/\nu_{RRL}$).
The vertical dashed line denotes the systematic velocity of 53.9$\pm$1.1 km
s-1 which we determined from the previous measurements of seven different hot
molecular lines (see Section 2.4).
Figure 4: (a): Image of the HCN(4-3) absorption line integrated from 44 to 62
km s-1. Dashed contours are $\pm 5\sigma\times 2^{n}$ ($n=0$, 1, 2, 3, … and
$\sigma=0.113$ Jy beam-1 km s-1), overlaid on the gray-scaled continuum image
at 0.85 mm. The FWHM beam $0.33\arcsec\times 0.24\arcsec$ (sup=0) is shown at
bottom right. (b): Image of the CO(2-1) absorption line integrated from 43 to
71 km s-1. White dashed contours are $\pm 5\sigma\times 2^{n}$ ($n=0$, 1, 2,
3, … and $\sigma=0.771$ Jy beam-1 km s-1), overlaid on the gray-scaled
continuum image at 0.85 mm. The FWHM beam $1.35\arcsec\times 0.73\arcsec$
(sup=0) is shown at bottom right. (c and d): The spectral profiles of the
HCN(4-3) absorption line toward W51e2-E and W51e2-W, respectively. The
vertical dashed line denotes the systematic velocity of 53.9$\pm$1.1 km s-1.
(e): The spectral profile of the CO(2-1) absorption line toward W51e2-E, with
a vertical dashed line for the systematic velocity. Table 2: Core fluxes of
W51e2 at Centimeter to Submillimeter bands
parameters | W51e2-W | W51e2-E | W51e2-NW | W51e2-N | References and Beamsize $\Theta$
---|---|---|---|---|---
R.A.(J2000) | 19:23:43.90 | 19:23:43.96 | 19:23:43.87 | 19:23:43.97 | This paper
Decl.(J2000) | +14:30:34.62 | +14:30:34.56 | +14:30:35.97 | +14:30:36.57 | This paper
Deconvolved size ($\arcsec$)∗ | 0.45 | 0.71 | 0.38 | 1.24 | This paper
S0.85mm(Jy) | 0.35$\pm$0.07 | 3.30$\pm$0.20 | 0.81$\pm$0.34 | | This paper: $\Theta=0.4\arcsec$
S1.3mm(Jy) | | 2.15$\pm$0.12 | 0.62$\pm$0.12 | 0.73$\pm$0.08 | This paper: $\Theta=1.1\arcsec$
S7mm(Jy) | 0.62$\pm$0.09 | 0.04$\pm$0.01 | $<$ 0.0052 | $<$ 0.0052 | This paper: $\Theta=0.4\arcsec$
S1.3cm(Jy) | 0.43$\pm$0.03 | $<$ 0.004 | $<$ 0.004 | $<$ 0.004 | This paper: $\Theta=0.4\arcsec$; Gaume et al. (1993): upper limit
S2cm(Jy) | 0.21$\pm$0.05 | | | | Rudolph et al. (1990): $\Theta=0.4\arcsec$
S3.6cm(Jy) | 0.067$\pm$0.004 | $<$ 0.001 | $<$ 0.001 | $<$ 0.001 | Gaume et al. (1993): $\Theta=0.21\arcsec$; Gaume et al. (1993): upper limit
S6cm(Jy) | 0.025$\pm$0.003 | – | – | – | Rudolph et al. (1990): $\Theta=0.6\arcsec$
∗The sizes of e2-W, e2-E, and e2-NW are derived from the 0.85 mm image, and
e2-N is from the 1.3 mm image.
---
### 2.4 Absorption of HCN(4-3) and CO(2-1)
The systematic velocity of W51e2 varies when it is measured using different
hot molecular lines (Sollins et al., 2004; Remijan et al., 2004; Zhang et al.,
1998; Rudolph et al., 1990). In order to minimize the possible effects of the
different distributions with different molecular lines, we averaged these
measurements of molecular lines H13CO+, SO2, SiO, CH3CN, and HCO+ and obtained
a mean value of 53.9$\pm$1.1 km s-1. We adopt this value as the systematic
velocity for W51e2-E hereafter.
The lines of HCN(4-3) at $\nu_{0}=$ 354.505 GHz and CO(2-1) at $\nu_{0}=$
230.538 GHz were covered in the SMA observations at 0.85 mm and 1.3 mm,
respectively. The dirty images were made with natural weighting and
deconvolved with the Hogbom Clark Steer hybrid algorithm. The final rms noises
of $\sigma=$ 65 and 94 mJy beam-1 per channel (1 km s-1 in the channel width)
were achieved for HCN(4-3) and CO(2-1), respectively. Figure 4 shows the
spectra of HCN(4-3) toward W51e2-E and W51e2-W and the spectrum of CO(2-1)
near the peak position of W51e2-E. A significant absorption of the HCN(4-3)
line is detected against the submillimeter core, W51e2-E, and most of which is
redshifted with respect to the systematic velocity of 53.9$\pm$1.1 km s-1 as
indicated by the vertical dashed line in Figure 4(c). Note that the redshifted
absorption of HCN observed with the high angular resolution against the
compact continuum core corresponds to the cold gas in front of the dust core
moving toward it, i.e., the infall. The CO(2-1) spectrum shows a broader
redshifted feature in absorption than that of HCN. Due to the poor angular
resolution in the CO(2-1) observations, the observed broad profile in
absorption is caused not only by the infall but also by the outflow. In
particular, the high redshifted absorption arises very likely from the outflow
gas. In comparison with that of W51e2-E, the spectrum toward W51e2-W only
shows a weak spectral feature (Figure 4(d)) which can be characterized as an
inverse P Cygni profile, suggesting that only little molecular gas (if
present) falls onto the UC HII region.
To show the spatial distribution of the absorption gas, we carried out a
moment analysis. The calculation of the zeroth moment corresponds to the
integrated intensity over the velocity. In order to avoid cancellation between
emission and absorption spectral features in the same beam area, we separate
the absorption and emission in the moment calculation. We found that both the
blueshifted and redshifted emission of the HCN(4-3) line (the images are not
shown in this paper) is extended from northwest to southeast across W51e2-E,
which is, in general, consistent with the molecular outflow direction as
suggested based on the CO(2-1) line observation (Keto & Klaassen, 2008). The
HCN(4-3) absorption, integrated from 44 to 62 km s-1 with a 5$\sigma$ cutoff
in each channel (Figure 4(a)), is found to be mainly concentrated on W51e2-E.
From the lower-resolution observation, the distribution of the CO(2-1)
absorption line integrated from 43 to 71 km s-1 (a 5$\sigma$ cutoff in each
channel, see Figure 4(b)), also shows the absorption peaks at W51e2-E instead
of W51e2-W. We also note that the CO(2-1) absorption shows a relatively
extended feature north of W51e2-E, further indicating that the absorption of
CO(2-1) might indeed be significantly contaminated by the outflow gas.
Along with the absorption spectra, the distribution of the absorbing gas in
W51e2 evidently demonstrates that the submillimeter (dust) core W51e2-E,
instead of W51e2-W (the UC HII region), is the center of accretion for the
majority of the high-density gas from this molecular core.
### 2.5 Spectral Energy Distributions
In lack of high angular resolution observations at millimeter/submillimeter
wavelengths, the flux densities from the entire region of W51e2 were
considered in the previous analyses of the SED (Rudolph et al., 1990; Sollins
et al., 2004; Keto et al., 2008). With the high-resolution observations at the
wavelengths from radio to submillimeter, we are now able to spatially separate
individual emission components in our SED analysis.
Adding the data at 3.6 cm (Gaume et al., 1993) and at 2 and 6 cm (Rudolph et
al., 1990) along with the continuum data at the four wavelengths (13, 7, 1.3,
and 0.8 mm) discussed in this paper, we have seven measurements in flux
densities covering nearly 2 order of magnitude in frequency. For 13, 7, and
0.8 mm data, the flux density measurements were made by fitting a Gaussian to
the individual emission components in the images with the same size of the
convolved FWHM beam (0.4$\arcsec$). For the 1.3 mm data, we determined the
flux density by fitting multiple Gaussian components to the emission features
in a relatively lower resolution (1.1$\arcsec$), which results in relatively
large uncertainties. For the emission cores W51e2-W, W51e2-E, W51e2-NW, and
W51e2-N, the flux densities determined are given in Table 2. The relevant
continuum spectra for the four emission components are shown in Figure 5.
## 3 Nature of the W51e2 complex
Four distinct components have been identified in the 3$\arcsec$ region of
W51e2 from the high-resolution images at wavelengths from centimeter to
submillimeter (Figure 1 and 2). The nature of these components is discussed as
follows.
Figure 5: The SED for W51e2 cores. (a): SED fitting for W51e2-W with a
free$-$free emission model. Black points denote the measurements from this
work, and the asterisks are from Rudolph et al. (1990) and Gaume et al.
(1993). All flux values are listed in Table 2. (b): The SED fitting for the
W51e2-E core with a thermal dust emission model. The arrows denote the upper
limits from Rudolph et al. (1990) and Gaume et al. (1993). (c): The SED
fitting for the W51e2-NW core. The upper limit of the flux density at 43 GHz
(7 mm) is from our data (3$\sigma$ of the 7 mm image). (d): The SED fitting
for the W51e2-N core.
### 3.1 W51e2-W: UC HII region
W51e2-W is the only core detected at centimeter wavelengths, and its continuum
emission is dominated by the free$-$free emission from the ionized core and
traces the thermal HII region around the central star. Figure 2(b)$-$(f) show
the UC HII region, W51e2-W, including an unresolved core (a possible ionized
disk) and a northeast and southwest (NE$-$SW) extension as a possible outflow.
The NE and SW extensions are best to be seen at 7 and 13 mm in Figure 2(b) and
(c), which is in good agreement with the previous observations at 3.6 and 1.3
cm by Gaume et al. (1993). Gaume et al. (1993) found the spectral indices
$\alpha$ of about 2 (optically thick) and 0.4 for the NE and SW extensions of
W51e2-W, respectively, and suggested that there is a one-side (SW) collimated
ionized outflow from the core.
The H26$\alpha$ line is an excellent tracer for hyper-compact ionized cores
(star in Figure 2(d)), providing a good diagnosis for the presence of an
ionized disk. On the basis of the SMA observations with an angular resolution
of 0.25$\arcsec$, the hyper-compact ionized core which peaked at 59.1 km s-1
in W51e2-W has not been resolved, giving a limit on the intrinsic size of
$\theta_{s}<0.06\arcsec$ and linear size $<310$ AU from a Gaussian fitting.
The elongation shown in the H53$\alpha$ image (Figure 2(e)) agrees with the
radio continuum images observed at 1.3 cm (Figure 2(c) and Gaume et al.,
1993), 3.6 cm (Gaume et al., 1993), and 7 mm (Figure 2(b)), suggesting that
the NE$-$SW extension observed in the H53$\alpha$ line corresponds to the
expansion of ionized gas (or outflow). The H66$\alpha$ line emission also has
a similar elongation (Figure 2(f)), with peak intensity near Gaume’s position
in the NE and an extension structure in the SW. We also found a significant
velocity gradient of the H53$\alpha$ line in the W51e2-W region, in agreement
with what was observed by Keto & Klaassen (2008), redshifted in the SW and
blueshifted in the NE. The broad wings of the recombination lines shown in
Figure 3 (also in Table 3) corresponding to extended, optically thin emission
indicate that it is hard to explain them as an ionized disk. The velocity
gradient observed in the H53$\alpha$ line is more likely produced by an
ionized outflow rather than an ionized accretion disk.
We fitted the SED of W51e2-W with a free$-$free emission model
$\displaystyle S_{\nu}$ $\displaystyle=$
$\displaystyle\Omega_{s}B_{\nu}(T_{e})(1-e^{-\tau_{c}})~{}~{}~{}~{}~{}({\rm
Jy}),$ (1)
where $S_{\nu}$ is the flux density at frequency $\nu$, $\Omega_{s}$ is the
solid angle of the source, $B_{\nu}(T_{e})$ is the Planck function, and
$T_{e}$ is the electron temperature. The continuum optical depth is expressed
by (Mezger & Henderson, 1967) as
$\displaystyle\tau_{c}$ $\displaystyle=$ $\displaystyle
0.0824\left(\frac{T_{e}}{{\rm K}}\right)^{-1.35}\left(\frac{\nu}{{\rm
GHz}}\right)^{-2.1}\left(\frac{\rm EM}{{\rm
pc~{}cm^{-6}}}\right)\alpha(\nu,T_{e}).$ (2)
The emission measure (EM) can be written as
$\displaystyle{\rm EM}=N_{e}^{2}Lf_{V},$ (3)
where $N_{e}$ is the mean electron density, $L$ is the path length, and
$f_{V}$ is the volume filling factor. The dimensionless factor
$\alpha(\nu,T_{e})$ is the order of unity (Mezger & Henderson, 1967). For a
homogeneous ionized core, the model can be expressed with two free parameters,
namely, $T_{e}$ and EM, for the first approximation described above.
The solid curve in Figure 5(a) shows the best fit to the data, suggesting a
turnover frequency of $\nu_{0}\approx 27$ GHz. The derived mean values for the
physical parameters for the overall HII region, $T_{e}$=4900$\pm$320 K and
EM=(1.3$\pm$0.2)$\times 10^{9}$ pc cm-6, are similar to the previous result in
Zhang et al. (1998). Thus, the mean electron density of
$N_{e}\sim$3$\times$105 cm-3 is inferred on the assumption of
$Lf_{V}\sim$0.5$\arcsec\times$5.1 kpc.
Both the Lyman continuum flux ($N_{L}$) and the excitation parameter ($U$) can
be derived (Rubin, 1968; Panagia, 1973; Matsakis et al., 1976) as follows:
$\displaystyle N_{L}$ $\displaystyle\gtrsim$ $\displaystyle 7.5\times
10^{46}{\rm s^{-1}}\left(\frac{S_{\nu}}{\rm Jy}\right)\left(\frac{D}{\rm
kpc}\right)^{2}\left(\frac{\nu}{\rm GHz}\right)^{0.1}\times$ (4)
$\displaystyle\left(\frac{T_{e}}{10^{4}{\rm K}}\right)^{-0.45},$
$\displaystyle U$ $\displaystyle=$ $\displaystyle 3.155\times 10^{-15}{\rm
pc~{}cm^{-2}}\left(\frac{N_{L}}{\rm
s^{-1}}\right)^{\frac{1}{3}}\left(\frac{T_{e}}{10^{4}{\rm
K}}\right)^{\frac{4}{15}},$ (5)
where $D$ is the distance to the source. We assumed $\alpha(\nu,T_{e})\sim 1$.
Based on our model fitting, the continuum emission from e2-W becomes optically
thin at a frequency $>$80 GHz ($\tau<$0.1), and we obtained
$N_{L}$=3.0$\times$1048 s-1 and $U$=37.6 pc cm-2. The inferred Lyman continuum
flux requires a massive star equivalent to a zero-age mean-sequence star of
type O8 located inside the UC HII region W51e2-W (Panagia, 1973), which is
consistent with the result of O7.5 in Rudolph et al. (1990). Alternatively, a
cluster of B-type stars can also be responsible for the ionization. The total
ionized mass within 0.5$\arcsec$ of W51e2-W is $\sim$0.02 M☉, which agrees
with the value derived by Gaume et al. (1993).
For the hyper-compact core ($\theta_{s}<0.06\arcsec$) as observed with the
H26$\alpha$ line, the electron temperature can be estimated from the
H26$\alpha$ line and the continuum flux density at 354 GHz (0.85 mm) on the
assumption of optically thin, LTE condition:
$\displaystyle T_{e}^{*}$ $\displaystyle=$
$\displaystyle\Bigg{[}\left(\frac{6985}{\alpha(\nu,T_{e})}\right)\left(\frac{\nu}{\rm
GHz}\right)^{1.1}\left(\frac{\Delta V_{\rm H26\alpha}}{\rm
km~{}s^{-1}}\right)^{-1}\left(\frac{S_{\rm 0.85\,mm}}{S_{\rm
H26\alpha}}\right)\times$ (6)
$\displaystyle\left(\frac{1}{1+\frac{N(He)}{N(H)}}\right)\Bigg{]}^{0.87}.$
Assuming $N(He)/N(H)=0.096$ (Mehringer, 1994) and $\alpha(\nu,T_{e})=1$, we
have $T_{e}^{*}=12,000\pm 2,000$ K which appears to be considerably higher
than the mean electron temperature of the overall HII region. The EM of the
hyper-compact HII core can be assessed as
$\displaystyle{\rm EM}$ $\displaystyle=$ $\displaystyle 7.1~{}{\rm
pc~{}cm^{-6}}\left(\frac{T_{e}}{\rm K}\right)^{3/2}\left(\frac{\theta_{s}}{\rm
arcsec}\right)^{-2}\left(\frac{\lambda}{\rm mm}\right)\left(\frac{S_{L}}{\rm
Jy}\right)\times$ (7) $\displaystyle\left(\frac{\Delta V}{\rm
km~{}s^{-1}}\right),$
where $T_{e}$ is the electron temperature, $\theta_{s}$ is the intrinsic size
of the HII region, $S_{L}$ is the peak line intensity in Jy, $\Delta V$ is the
FWHM line width in km s-1, and $\lambda$ is the observing wavelength in
millimeters. Based on the measurements of the H26$\alpha$ line from the hyper-
compact ionized core together with the assumption of $T_{e}\approx T_{e}^{*}$,
EM $>7\times 10^{10}$ pc cm-6 is found. The corresponding lower limit of the
volume electron density is $N_{e}>7\times 10^{6}$ cm-3 assuming
$Lf_{V}<0.06\arcsec\times 5.1$ kpc. The inferred high electron temperature and
high electron density suggest that the H26$\alpha$ line arises from a hot,
very compact region ($<0.06\arcsec$) which is probably close to the central
ionizing star(s). With the nature of optically thin and sensitive to the high-
density ionized gas, the H26$\alpha$ line is an excellent tracer of the hot,
high-density ionized region surrounding the ionizing source. In comparison
with the mean electron temperature ($T_{e}$=4900 K) inferred from the
free$-$free emission in a large area (0.5$\arcsec$), the high temperature of
$T_{e}$=12,000 K derived from the H26$\alpha$ in a compact area
($<0.06\arcsec$) suggests that a temperature gradient is present along the
radius of the UC HII region W51e2-W.
Observations of RRLs in a wide range at wavelengths from centimeters to
submillimeters also offer a means to further explore physical conditions of
W51e2-W. Table 3 summarizes the reliable measurements of three RRLs from
W51e2, namely, H26$\alpha$, H53$\alpha$, and H66$\alpha$. We plotted the
spectral profiles of the three RRLs together, and each of the profiles has
been multiplied by a factor of $\nu_{26\alpha}/\nu_{n\alpha}$, the ratio of
the H26$\alpha$ line frequency to that of RRLs (Figure 3). In the case of
optically thin and no-pressure broadening, the three line profiles should
match each other. Comparing H53$\alpha$ with H26$\alpha$, we find that the
peak intensities of the two lines agree with each other, which suggests that,
in general, both the H53$\alpha$ and H26$\alpha$ lines are under an optically
thin, LTE condition. Several velocity peaks in the H26$\alpha$ line indicate
multiple kinematic components in the hyper-compact HII core. The H53$\alpha$
line is characterized by a relatively smooth profile with large velocity
wings. The large velocity wings observed in the H53$\alpha$ line can be
explained by the large velocity gradient in the lower-density electron gas of
the ionized outflow (Gaume et al., 1993). However, the line intensity of
H66$\alpha$ appears to be significantly weaker than the other two high-
frequency RRLs. The peak of the modified line profile of H66$\alpha$ is a
factor of $\sim 5$ less than that of the other two. Considering the fact that
the frequency of H66$\alpha$ is below the turnover frequency of
$\nu_{0}\approx 27$ GHz, we found that the H66$\alpha$ line from the ionized
gas is obscured severely due to the self-absorption process in the hyper-
compact HII core. From Equation (2), a mean optical depth of $\tau_{c}=1.6$ at
$\nu_{H66\alpha}=22.36$ GHz is found. The exponential attenuation of
$exp(-1.6)\approx 0.2$ suggests that the H66$\alpha$ line is attenuated mainly
due to the self-absorption in the ionized core. In addition, the pressure
broadening effect which weakens the lower frequency lines needs to be
assessed.
Table 3: Parameters of RRLs derived from our data∗
| ${\rm H26\alpha}$ | ${\rm H53\alpha}$ | ${\rm H66\alpha}$
---|---|---|---
Rest Frequency (GHz) | 353.623 | 42.952 | 22.364
Measurements | | |
$I_{peak}$ (Jy) | 1.29$\pm$0.02 | 0.137$\pm$0.001 | 0.015$\pm$0.003
$V_{LSR}$ (km s-1) | 59.1$\pm$0.2 | 60.4$\pm$0.2 | 62.1$\pm$1.3
$\Delta V$ (km s-1) | 23.1$\pm$0.5 | 32.0$\pm$0.3 | 35.1$\pm$ 8.3
Derived line broadenings | | |
$\Delta V_{P}$ (km s-1) | 0.01 | 1.3 | 6.5
$\Delta V_{T}$ (km s-1) | 15.0 | 15.0 | 15.0
$\Delta V_{t}$ (km s-1) | 17.5 | 28.3 | 31.1
$\Delta V_{D}$ (km s-1) | 23.1 | 32.0 | 34.5
∗All line profiles are integrated from the 0.5$\arcsec\times$0.5$\arcsec$
region around the peak position of H26$\alpha$ from maps of the same beamsize
of 0.25$\arcsec$.
---
The total line width of RRLs ($\Delta V$) can be expressed by the Doppler
broadening ($\Delta V_{D}$) and pressure broadening ($\Delta V_{P}$).
According to Brocklehurst & Leeman (1971) and Gordon & Sorochenko (2002), we
have
$\displaystyle\Delta V_{D}$ $\displaystyle=$ $\displaystyle{\rm
km~{}s^{-1}}\sqrt{0.0458\times\left(\frac{T_{e}}{\rm
K}\right)+\left(\frac{\Delta V_{t}}{\rm km~{}s^{-1}}\right)^{2}},$ (8)
$\displaystyle\Delta V_{P}$ $\displaystyle=$ $\displaystyle 3.74\times
10^{-14}~{}{\rm km~{}s^{-1}}~{}n^{4.4}\left(\frac{\lambda}{\rm
mm}\right)\left(\frac{N_{e}}{\rm cm^{-3}}\right)\times$ (9)
$\displaystyle\left(\frac{T_{e}}{\rm K}\right)^{-0.1},$ $\displaystyle\Delta
V$ $\displaystyle=$ $\displaystyle\sqrt{\Delta V_{D}^{2}+\Delta V_{P}^{2}},$
(10)
where $\Delta V_{t}$ is the turbulent broadening of the gas including
broadening due to outflows and disk rotations and $n$ is the principal quantum
number of the transition. We calculated broadenings on the mean electron
temperature ($T_{e}\approx$4900 K) and the mean electron density
($N_{e}\approx$3$\times$105 cm-3) of the UC HII region. The thermal broadening
($\Delta V_{T}$) is 15 km s-1 for $T_{e}$=4900 K. The pressure broadening is a
strong function of the principal quantum number, i.e., $\Delta V_{P}$
increases drastically toward the low-frequency lines. For the high-frequency
line (H26$\alpha$), the pressure broadening ($\Delta V_{P}=0.01$ km s-1) can
be ignored, while for the H66$\alpha$ line, the pressure broadening of $\Delta
V_{P}$=7 km s-1 becomes significant. The pressure broadening is $\Delta
V_{P}$=76 km s-1 for the H92$\alpha$ line, which can actually wash out the
profile of the line emission. Thus, in addition to the opacity effect, the
pressure broadening may also make a considerable contribution to diminish the
H92$\alpha$ line which was not detected in the observations of Mehringer
(1994). The turbulent and/or the dynamical motions of the ionized gas
contribute a total of $\Delta V_{t}\approx$18 km s-1 in the line broadening
for the H26$\alpha$ line and $\Delta V_{t}\approx$28 and 31 km s-1 for the
H53$\alpha$ and H66$\alpha$ lines, respectively. The H26$\alpha$ line traces
the high-density ionized gas in the hyper-compact core or the ionized disk (if
it exists) and is less sensitive to the lower-density outflow which produces
the broad wings in the line profiles. The turbulent and/or dynamical
broadening becomes dominant in the H53$\alpha$ and H66$\alpha$ lines, and
these lower frequency RRLs (H53$\alpha$ and H66$\alpha$) better trace the
lower-density electron gas in the ionized outflow.
In short, W51e2-W consists of a hyper-compact HII core ($<310$ AU) with an
emission measure of EM $>7\times 10^{10}$ pc cm-6 and an ionized outflow. An
early-type star equivalent to an O8 star (or a cluster of B-type stars) is
postulated to have formed within the hyper-compact HII core. The broadening of
the line profiles is dominated by the Doppler broadening including both
thermal and turbulence/dynamical motions. Pressure broadening becomes
significant only for the H66$\alpha$ line and the lines with a larger
principal quantum number.
### 3.2 W51e2-E: A massive proto-stellar core
W51e2-E is located $\sim 0.9\arcsec$ east of the UC HII region W51e2-W. It is
the brightest source in the 0.85 mm continuum image (Figure 1). The flux
density drops drastically at 7 mm (Figure 2(b)). It cannot be detected at
longer wavelengths. The SED of the continuum emission from W51e2-E (Figure
5(b)) appears to arise from a dust core associated with proto-stars.
With the Rayleigh$-$Jeans approximation, the continuum flux density from a
homogeneous, isothermal dust core can be described as (Launhardt & Henning,
1997)
$\displaystyle S_{d}(\nu)$ $\displaystyle=$ $\displaystyle 6.41\times
10^{-6}{\rm Jy}\left(\frac{\kappa_{0}}{\rm
cm^{2}g^{-1}}\right)\left(\frac{\nu}{\nu_{0}}\right)^{\beta}\left(\frac{\nu}{\rm
GHz}\right)^{2}\times$ (11) $\displaystyle\left(\frac{M_{d}}{\rm
M_{\sun}}\right)\left(\frac{T_{d}}{\rm K}\right)\left(\frac{D}{\rm
kpc}\right)^{-2},$
where $\kappa_{0}$ is the dust opacity at frequency $\nu_{0}$, $\beta$ is the
power-law index of the dust emissivity, $M_{d}$ is the mass of dust, $T_{d}$
is the dust temperature in K, and $D$ is the distance to the source. We adopt
the dust opacity value of $\kappa(\lambda=1.3mm)=0.8~{}{\rm cm^{2}g^{-1}}$
(Ossenkopf & Henning, 1994; Launhardt & Henning, 1997) and have three free
parameters for the dust core ($M_{d}$, $T_{d}$, and $\beta$).
The solid line in Figure 5(b) shows the best least-squares fitting to the
observed data. The index of $\beta=0.26\pm 0.08$ derived from our fitting
indicates that emission from the e2-E core might be dominated by large grains
of dust (Miyake & Nakagawa, 1993; Chen et al., 1995; Andrews & Williams,
2007). In addition, because of a lack of flux density measurements at higher
frequencies or shorter wavelengths, the dust mass and temperature are
degenerate in our best fitting, i.e., $M_{d}T_{d}=142\pm 9$ M☉K. Remijan et
al. (2004) derived the kinetic temperature in W51e2 to be $153\pm 21$ K from
the high-resolution observations of CH3CN, which is close to the dust
temperature $T_{d}$=100 K used in Zhang et al. (1998). Adopting that
$T_{d}=$100 K and assuming that the ratio of H2 gas to dust is 100, we found
that the total mass of $\sim$140 M☉ is in the W51e2-E core. Emission from the
W51e2-E region has been slightly resolved, so that the dust core of W51e2-E
might host a number of proto-stellar cores which accrete the surrounding
molecular gas as indicated by both the HCN and CO absorption lines (see
Section 2.4). This proposed scenario for W51e2-E is also consistent with the
hourglass-like magnetic fields inferred from polarization measurements of Tang
et al. (2009) for the W51e2 region. The configuration center of $B$ vectors
coincides with the peak position of the W51e2-E dust core.
From a Gaussian fitting to the absorption spectrum of the HCN(4-3) toward the
center of W51e2-E (Figure 4(c)), we determined the peak line intensity $\Delta
I_{L}=-1.3\pm 0.1$ Jy beam-1, the line center velocity $V_{HCN}$ =$56.4\pm
0.2$ km s-1 and the line width of $\Delta V_{HCN}$=$9.5\pm 0.4$ km s-1. The
infall velocity of the gas is the offset between the center velocity and the
systematic velocity, i.e., $V_{in}=V_{HCN}-V_{sys}\approx 2.5$ km s-1. The
peak continuum intensity of W51e2-E at 0.85 mm is $I_{C}=1.2\pm 0.01$ Jy
beam-1 (corresponding to a brightness temperature of $144\pm 1$ K). The ratio
of $-\Delta I_{L}/I_{C}\sim 1$ suggests that the absorption line is saturated.
Thus, a lower limit on the optical depth (see Qin et al., 2008) is
$\tau_{HCN}>3$. Assuming that the excitation temperature of HCN(4-3) equals
the dust temperature, 100 K, we obtained a lower limit of HCN column density
of $7.9\times 10^{15}$ cm-2. Taking $[H_{2}]/[HCN]\sim 0.5\times 10^{8}$
(Irvine et al., 1987) and the size of the infall region as 0.75$\arcsec$
($\sim 4000$ AU, see Figure 4(a)), we found that the hydrogen volume density
$N_{H_{2}}\sim 6.9\times 10^{6}$ cm-3. The infall rate of the gas can be
estimated by
$\displaystyle{\rm d}M/{\rm d}t$ $\displaystyle=$ $\displaystyle 2.1\times
10^{-5}{\rm M_{\sun}yr^{-1}}\left(\frac{\theta_{in}}{\rm
arcsec}\right)^{2}\left(\frac{D}{\rm kpc}\right)^{2}\left(\frac{V_{in}}{\rm
km~{}s^{-1}}\right)\times$ (12) $\displaystyle\left(\frac{N_{H_{2}}}{\rm
10^{6}cm^{-3}}\right),$
where $2\theta_{in}=0.75\arcsec$ is the diameter of the infalling region, $D$
is the distance to the source, $V_{in}$ is the infall velocity, and the
molecular mass ratio $m/m_{H_{2}}=1.36$ is assumed. With the derived
parameters a lower limit of infall rate $1.3\times 10^{-3}$ M☉ yr-1 is
inferred, suggesting that W51e2-E is the accretion center of the W51e2
complex.
### 3.3 W51e2-NW and W51e2-N
W51e2-NW was clearly detected by Tang et al. (2009) in the continuum emission
at 0.87 mm. They also found a significant concentration of $B$ fields at the
e2-NW. Genzel et al. (1981) and Imai et al. (2002) detected bright and compact
H2O masers near this source. We also detected local enhanced continuum
emission at 0.85 and 1.3 mm. No significant continuum emission has been
detected at 7 and 13 mm (Figure1 and 2), and there are no significant
absorption lines of HCN(4-3) and CO(2-1) in this region. We fitted the SED of
continuum emission with a thermal dust model as shown in Figure 5(c). A lower
limit of $\beta\geq 0.35$ is derived from the fitting, which leads to an upper
limit of $M_{d}T_{d}\leq 39$ M☉K. Assuming the dust temperature in this region
to be $\sim$100 K, we inferred an upper limit of 40 M☉ in the total gas mass
if the ratio of H2 gas to dust is equal to 100\. W51e2-NW appears to be a
massive core at a very early phase of star formation.
Located $\sim 2\arcsec$ north from W51e2-E, W51e2-N is detected in continuum
emission from our lower-resolution (1.4$\arcsec\times 0.7\arcsec$) image at
1.3 mm (Figure 2(a)). This source appears to be resolved out with a high
angular resolution (0.3$\arcsec\times 0.2\arcsec$) observation at 0.85 mm. No
significant continuum emission was detected at the longer wavelengths. The SED
consisting of the flux density at 1.3 mm and the upper limits at 7, 13, and 36
mm is also fitted with a thermal dust model. From the fitted values of the
parameters $\beta\geq 0.75$ and $M_{d}T_{d}\leq 68$ M☉K, we found an upper
limit of $M_{d}\lesssim 70$ M☉ if $T_{d}\sim$100 K and the H2 gas to dust
ratio is 100\. No maser activities have been detected from W51e2-N, suggesting
that W51e2-N is probably a primordial molecular clump in W51e2.
### 3.4 A propagating scenario of star formation in W51e2
Apparently, in the massive molecular core W51e2, the gravitational collapse
occurred first at W51e2-W where an O8 star or a cluster of B-type stars were
formed. Based on the absorption spectrum of the HCN line, there appears to be
little molecular gas present in the immediate environs. The accretion appears
to be paused by the intensive radiation pressure from the central ionized
star(s). Most of the surrounding gas ($\sim$0.02 M⊙) has been ionized. The
thermal pressure in the UC HII region drives an ionized outflow from the
ionized core. The submillimeter observations of dust emission and molecular
line absorption show that the major accretion now has been re-directed to
W51e2-E, 0.9$\arcsec$ ($\sim$4600 AU) east from W51e2-W. W51e2-E becomes the
new dominant gravitational center accreting mass from the surroundings with a
rate of $10^{-3}$ M⊙ yr-1. Star formation activities take place in this
massive core, as shown by maser activities, outflow, and organized $B$-field
structure. Since no free$-$free emission and no RRLs have been detected, the
massive core W51e2-E likely hosts one or more massive proto-stars. The offset
of the radial velocities between the two cores, W51e2-E and W51e2-W, is
$\delta V=59.1-53.9=5.2$ km s-1. If W51e2-W and W51e2-E are gravitationally
bound and W51e2-W is circularly orbiting around W51e2-E, W51e2-E has a
dynamical mass of $>140$ M⊙, in good agreement with the mass derived from the
SED analysis.
A bipolar outflow from W51e2-E has been suggested from the observations of
molecular lines. The impact of the outflow on the medium in its path may
trigger the collapse of the sub-core W51e2-NW and induce further star
formation activities there. W51e2-N represents a gas clump with a considerable
amount of mass for the star formation. This speculative scenario appears to
reasonably explain what we have observed in the W51e2 complex.
## 4 Summary and conclusions
We have presented high-resolution images of the W51e2 region at 0.85, 1.3, 7,
and 13 mm for continuum and hydrogen recombination lines (H26$\alpha$,
H53$\alpha$, and H66$\alpha$) and the molecular lines (HCN(4-3) and CO(2-1)).
The W51e2 complex has been resolved into four distinct components, W51e2-W,
W51e2-E, W51e2-NW, and W51e2-N. We have carried out a comprehensive analysis
of the continuum SED, the RRLs, and the molecular absorption lines and found
that the four cores are at different phases of massive star formation.
1\. W51e2-W, associated with the UC HII region, is the only source from which
the H26$\alpha$, H53$\alpha$, and H66$\alpha$ lines have been detected. The
Lyman continuum flux inferred from the SED analysis suggests that a massive O8
star or a cluster of B-type stars have been formed within the HII region. The
unresolved H26$\alpha$ line emission region suggests the presence of a hot
($T_{e}^{*}=12,000\pm 2,000$ K), hyper-compact ionized core with a velocity of
$59.1\pm 0.2$ km s-1, a linear size of $<310$ AU, and a large emission measure
EM$>7\times 10^{10}$ pc cm-6 (or a high density $N_{e}>7\times 10^{6}$ cm-3).
The H53$\alpha$ and H66$\alpha$ line images show an SW elongation from the
hyper-compact ionized core, suggesting either an expansion or an outflow of
the ionized gas with relatively lower density. No significant detection of the
HCN molecular line in absorption against the compact HII region indicates that
the significant accretion onto this core has been stopped. The line ratios
between the H26$\alpha$, H53$\alpha$, and H66$\alpha$ show that both the
H26$\alpha$ and H53$\alpha$ lines from W51e2-W are in an optically thin, LTE
condition. The emission of the H66$\alpha$ line from the hyper-compact core
appears to be attenuated mainly due to the self-absorption in this HII region.
The line profile of the H26$\alpha$ appears to be dominated by the Doppler
broadening due to the thermal motions of the hot electrons while the line
broadening in both the H53$\alpha$ and H66$\alpha$ is dominated by the Doppler
effect due to the dynamical motions of the ionized outflow. The pressure
broadening in both the H26$\alpha$ and H53$\alpha$ lines is negligible and
becomes significant for the H66$\alpha$ line corresponding to $\Delta
V_{P}\sim$ 7 km s-1.
2\. W51e2-E is a massive ($\sim$140 M☉) dust core as suggested based on the
SED analysis. No hydrogen recombination lines and no radio continuum emission
($\lambda>1$ cm) have been detected from this region, which suggests that
W51e2-E is the major core hosting massive proto-stellar objects. Both the
absorptions of HCN(4-3) and CO(2-1) are redshifted with respect to the
systematic velocity, $53.9\pm 1.1$ km s-1, indicating that a large amount of
molecular gas moves toward the massive core. From the ratios of the absorption
line to continuum, a large infall rate of $>1.3\times 10^{-3}$ M☉yr-1 is
inferred from the HCN absorption line, corresponding to the accretion radius
of 2000 AU.
3\. W51e2-NW appears to be a massive ($\lesssim$ 40 M☉) core at an earlier
phase of star formation. Given its location in the projected path of the
molecular outflow from W51e2-E, the star formation activities of W51e2-NW are
likely to be triggered by the outflow impact on the medium in the W51e2
complex.
4\. W51e2-N appears to be a relatively less dense but massive clump
($\lesssim$ 70 M☉) from which no obvious evidence has been found for star
formation activities.
We thank the referee for helpful comments, Dr. S.-L. Qin for providing help
with the data reduction and analysis, and Professor Y.-F. Wu for helpful
discussions. H. Shi and J.-L. Han are supported by the National Natural
Science Foundation (NNSF) of China (10773016, 10821061, and 10833003) and the
National Key Basic Research Science Foundation of China (2007CB815403).
## References
* Andrews & Williams (2007) Andrews, S. M. & Williams, J. P. 2007, ApJ, 659, 705
* Brocklehurst & Leeman (1971) Brocklehurst, M. & Leeman, S. 1971, Astrophys. Lett., 9, 35
* Chen et al. (1995) Chen, H., Zhao, J.-H., & Ohashi, N. 1995, ApJ, 450, L71+
* Fish & Reid (2007) Fish, V. L. & Reid, M. J. 2007, ApJ, 670, 1159
* Gaume et al. (1993) Gaume, R. A., Johnston, K. J., & Wilson, T. L. 1993, ApJ, 417, 645
* Genzel et al. (1982) Genzel, R., Becklin, E. E., Moran, J. M., Reid, M. J., Jaffe, D. T., Downes, D., & Wynn-Williams, C. G. 1982, ApJ, 255, 527
* Genzel et al. (1981) Genzel, R., Downes, D., Schneps, M. H., Reid, M. J., Moran, J. M., Kogan, L. R., Kostenko, V. I., Matveenko, L. I., & Ronnang, B. 1981, ApJ, 247, 1039
* Gordon & Sorochenko (2002) Gordon, M. A. & Sorochenko, R. L., eds. 2002, Astrophysics and Space Science Library, Vol. 282, Radio Recombination Lines. Their Physics and Astronomical Applications
* Ho & Young (1996) Ho, P. T. P. & Young, L. M. 1996, ApJ, 472, 742
* Imai et al. (2002) Imai, H., Watanabe, T., Omodaka, T., Nishio, M., Kameya, O., Miyaji, T., & Nakajima, J. 2002, PASJ, 54, 741
* Irvine et al. (1987) Irvine, W. M., Goldsmith, P. F., & Hjalmarson, A. 1987, in Astrophysics and Space Science Library, Vol. 134, Interstellar Processes, ed. D. J. Hollenbach & H. A. Thronson, Jr., 561–609
* Keto & Klaassen (2008) Keto, E. & Klaassen, P. 2008, ApJ, 678, L109
* Keto et al. (2008) Keto, E., Zhang, Q., & Kurtz, S. 2008, ApJ, 672, 423
* Kundu & Velusamy (1967) Kundu, M. R. & Velusamy, T. 1967, Annales d’Astrophysique, 30, 59
* Launhardt & Henning (1997) Launhardt, R. & Henning, T. 1997, A&A, 326, 329
* Martin (1972) Martin, A. H. M. 1972, MNRAS, 157, 31
* Matsakis et al. (1976) Matsakis, D. N., Evans, II, N. J., Sato, T., & Zuckerman, B. 1976, AJ, 81, 172
* Mehringer (1994) Mehringer, D. M. 1994, ApJS, 91, 713
* Mezger & Henderson (1967) Mezger, P. G. & Henderson, A. P. 1967, ApJ, 147, 471
* Miyake & Nakagawa (1993) Miyake, K. & Nakagawa, Y. 1993, Icarus, 106, 20
* Ossenkopf & Henning (1994) Ossenkopf, V. & Henning, T. 1994, A&A, 291, 943
* Panagia (1973) Panagia, N. 1973, AJ, 78, 929
* Qin et al. (2008) Qin, S.-L., Zhao, J.-H., Moran, J. M., Marrone, D. P., Patel, N. A., Wang, J.-J., Liu, S.-Y., & Kuan, Y.-J. 2008, ApJ, 677, 353
* Remijan et al. (2004) Remijan, A., Sutton, E. C., Snyder, L. E., Friedel, D. N., Liu, S.-Y., & Pei, C.-C. 2004, ApJ, 606, 917
* Rubin (1968) Rubin, R. H. 1968, ApJ, 154, 391
* Rudolph et al. (1990) Rudolph, A., Welch, W. J., Palmer, P., & Dubrulle, B. 1990, ApJ, 363, 528
* Sault et al. (1995) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes, 433–+
* Scott (1978) Scott, P. F. 1978, MNRAS, 183, 435
* Sollins et al. (2004) Sollins, P. K., Zhang, Q., & Ho, P. T. P. 2004, ApJ, 606, 943
* Tang et al. (2009) Tang, Y., Ho, P. T. P., Koch, P. M., Girart, J. M., Lai, S., & Rao, R. 2009, ApJ, 700, 251
* Westerhout (1958) Westerhout, G. 1958, Bull. Astron. Inst. Netherlands, 14, 215
* Xu et al. (2009) Xu, Y., Reid, M. J., Menten, K. M., Brunthaler, A., Zheng, X. W., & Moscadelli, L. 2009, ApJ, 693, 413
* Young et al. (1998) Young, L. M., Keto, E., & Ho, P. T. P. 1998, ApJ, 507, 270
* Zhang & Ho (1997) Zhang, Q. & Ho, P. T. P. 1997, ApJ, 488, 241
* Zhang et al. (1998) Zhang, Q., Ho, P. T. P., & Ohashi, N. 1998, ApJ, 494, 636
|
arxiv-papers
| 2010-01-07T16:02:48 |
2024-09-04T02:49:07.569799
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui Shi, Jun-Hui Zhao, J.L. Han",
"submitter": "Hui Shi",
"url": "https://arxiv.org/abs/1001.1101"
}
|
1001.1132
|
# The parton bubble model compared to central Au Au collisions (0% to 5%) at
$\sqrt{s_{NN}}$=200 GeV.
R.S. Longacre Brookhaven National Laboratory, Upton, New York 11973
###### Abstract
In an earlier paper we developed a Parton Bubble Model (PBM) for RHIC, high-
energy heavy-ion collisions. PBM was based on a substructure of a ring of
localized bubbles (gluonic hot spots) which initially contain 3-4 partons
composed of almost entirely gluons. The bubble ring was perpendicular to the
collider beam direction, centered on the beam, at midrapidity, and located on
the expanding fireball surface of Au Au central collisions (0-10%) at
$\sqrt{s_{NN}}$=200 GeV. The bubbles emitted correlated particles at kinetic
freezeout, leading to a lumpy fireball surface. For a selection of charged
particles (0.8 GeV/c $<$ $p_{t}$ $<$ 4.0 GeV/c), the PBM reasonably
quantitatively (within a few percent) explained high precision RHIC
experimental correlation analyses in a manner which was consistent with the
small observed HBT source size in this transverse momentum range. We
demonstrated that surface emission from a distributed set of surface sources
(as in the PBM) was necessary to obtain this consistency. In this paper we
give a review of the above comparison to central Au Au collisions. The bubble
formation can be associated with gluonic objects predicted by a Glasma Flux
Tube Model (GFTM) that formed longitudinal flux tubes in the transverse plane
of two colliding sheets of Color Glass Condensate (CGC), which pass through
one another. These sheets create boost invariant flux tubes of longitudinal
color electric and magnetic fields. A blast wave gives the tubes near the
surface transverse flow in the same way it gave transverse flow to the bubbles
in the PBM. In this paper we also consider the equivalent characteristics of
the PBM and GFTM and connect the two models. In the GFTM the longitudinal
color electric and magnetic fields have a non-zero topological charge density
$F\widetilde{F}$. These fields cause a local strong CP violation which effects
charged particle production coming from quarks and anti-quarks created in the
tube or bubble.
###### pacs:
25.75.Nq, 11.30.Er, 25.75.Gz, 12.38.Mh
## I Parton bubble model development
Our interest in and the eventual development of the PBM goes back many years
to the early nineteen eighties. Van Hove’s workVanHove in the early eighties
considered the bubbles as small droplets of quark-gluon plasma (QGP) assumed
to be produced in ultra-relativistic collisions of hadrons and/or nuclei. His
work recognized that experimental detectors only see what is emitted in the
final state at kinetic freezeout. Therefore final state predictions of his
string model work on plasma bubbles or any other theory required specific
convincing predictions that are observable in the final state which can be
measured by experiment. His work predicted that the final state rapidity
distribution dn/dy of hadrons would exhibit isolated maxima of width $\Delta
y\sim 1$ (single bubble) or rapidity bumps of a few units (due to those cases
producing a few bubbles). These rapidity regions would also contain
traditional signals of QGP formation. Even this early work by Van Hove needed
his bubbles near the surface of the final state fireball in order to
experimentally measure the particles emitted in the final state at kinetic
freezeout. We and other searched for these Van Hove bubbles, but no one ever
found any significant evidence for them.
Our next paper2000 was our final attempt to make a theoretical treatment of
the single bubble case (similar to the Van Hove case) which could be
experimentally verified. It is possible that with enough statistics one in
principle could find single to a few bubble events. We developed a number of
event generators which could possibly provide evidence for striking signals
resulting from these bubbles. One should note that in that paper we had
included in Sec. 11 mathematical expressions describing how charged pions are
effected by strong color electric and color magnetic fields which are present
and parallel in a CP-odd bubble of metastable vacuum. This section was
motivated by the work of Kharzeev and PisarskiPisarski . For example eq. 3-4
of the section show the boosts of the $\pi^{+}$ and the $\pi^{-}$ we
estimated. Thus in 2000 we were already interested in and were investigating
and publishing predictions for this phenomenon.
When we considered the RHIC quantum interference data in 2000Adler it became
clear that the fireball surface was rapidly moving outward. This implied a
very large phase space region covered by the fireball, and also implied that
it would be very unlikely for there to be only one isolated bubble (small
source size) with a large amount of energy sitting on the surface. It seemed
more likely that there would be many bubbles or gluonic hot spots around the
expanding surface. This led to a paperthemodel which was our earlier version
of Ref.PBM . In that paper we concluded that the behavior of the Hanbury-Brown
and Twiss (HBT) measurementsAdler ; HBT should be interpreted as evidence of
a substructure of bubbles located on the surface of the final state fireball
in the central rapidity region at kinetic freezeout. The HBT radii were
decreasing almost linearly with transverse momentum and implying to us a
source size of $\sim$2 fm radii bubbles were on the surface and could be
selected for if we considered transverse momenta above 0.8 GeV/c. These
momenta would allow sufficient resolution to resolve individual bubbles of
$\sim$2 fm radii. We further concluded that these HBT quantum interference
observations were likely due to phase space focusing of the bubbles pushed by
the expanding fireball. Thus the HBT measurements of source size extrapolated
to transverse momenta above 0.8 GeV/c were images of the bubbles. The HBT
correlation has the property of focusing these images on top of each other for
a ring of bubbles transverse to and centered on the beam forming an average
HBT radius.
Thus our model was a ring of bubbles sitting on the freezeout surface with
average size $\sim$2 fm radius perpendicular to the beam at mid-rapidity. The
phase space focusing would also lead to angular correlations between particles
emitted by the bubbles and be observable. Fig. 1 shows the geometry of the
bubbles. For the background particles that account for particles in addition
to the bubble particles we used unquenched HIJINGhijing with its jets
removed. We assumed that most of the jets were eliminated from the central
events because of the strong jet quenching observed at RHICquench1 . We
investigated the feasibility of using charged-particle-pair correlations as a
function of angles which should be observable due to the phase space focusing
of the particles coming from individual bubbles. At that point we fully
expected that these correlations would be observable in the STAR detectorstar
at RHIC if one analyzed central Au Au collisions. Correlation analyses are
powerful tools in detecting substructures. Historically substructures have
played an important role in advancing our understanding of strong (non
perturbative) interactions. Our earlier paper explained the general
characteristics of the angular correlation data, and was also consistent with
HBT measurements in a qualitative manner. This motivated us to develop a
reasonably quantitative model, the parton bubble modelPBM which is discussed
in the following.
Figure 1: The bubble geometry is an 8 fm radius ring perpendicular to and
centered on the beam axis. It is composed of twelve adjacent 2 fm radius
spherical bubbles elongated along the beam direction by the Landau
longitudinal expansion. The upper left figure is a projection on a plane
section perpendicular to the beam axis. The lower left figure is a projection
of the bubble geometry on a plane containing the beam axis. The lower right
figure is a perspective view of the bubble geometry. Due to rotational
invariance about the beam axis the only direction that is meaningful to define
is the beam axis shown in the lower 2 figures.
## II Parton bubble modelPBM
In this publicationPBM we developed a QCD inspired parton bubble model (PBM)
for central (impact parameter near zero) high energy heavy ion collisions at
RHIC. The PBM is based on a substructure consisting of a single ring of a
dozen adjoining 2-fm-radius bubbles (gluonic hot spots) transverse to the
collider beam direction, centered on the beam, and located at or near mid-
rapidity. The at or near mid-rapidity refers to the boost invariant region
that exist in the RHIC heavy ion collisions which spans 2-4 units of rapidity.
Because of this spread in $\eta$, we will use the relative spread of particles
in $\eta$ ($\Delta\eta$) in order to capture the angular correlations along
the beam axis. The ring resides on the fireball blast wave surface (see Fig.
1). We assumed these bubbles are likely the final state result of quark-gluon-
plasma (QGP) formation since the energy densities produced experimentally are
greater than those estimated as necessary for formation of a quark-gluon-
plasma. Thus this is the geometry for the final state kinetic freezeout of the
QGP bubbles on the surface of the expanding fireball treated in a blast wave
model.
The average behavior of emitted final state particles coming from the surface
bubbles at kinetic freezeout is given by the ring of twelve bubbles formed by
energy density fluctuations near the surface of the expanding fireball of the
blast wave. One should note that the blast wave surface is moving at its
maximum velocity at freezeout (3c/4). For central events each of the twelve
bubbles have 3-4 partons per bubble each at a fixed $\phi$ for a given bubble.
This number of partons was determined from correlation data analyzed by the
STAR experimentPBM . The transverse momentum ($p_{t}$) distribution of the
charged particles is similar to pQCD but has a suppression at high $p_{t}$
like the data.
The bubble ring radius of our model was estimated by blast wave, HBT and other
general considerations to be approximately 8 fm. The bubbles emit correlated
charged particles at final-state kinetic freezeout where we select a $p_{t}$
range (0.8 GeV/c $<$ $p_{t}$ $<$ 4.0 GeV/c) in order to increase signal to
background. The 0.8 GeV/c $p_{t}$ cut increases the resolution to allow
resolving individual bubbles which have a radius of $\sim$2 fm. This space
momentum correlation of the blast wave provides us with a strong angular
correlation signal. PYTHIA fragmentation functionspythia were used for the
bubbles fragmentation that generate the final state particles emitted from the
bubbles. A single parton using PYTHIA forms a jet with the parton having a
fixed $\eta$ and $\phi$ (see Fig. 2). The 3-4 partons in the bubble which
shower using PYTHIA all have a different $\eta$ value but all have the same
$\phi$ (see Fig. 3). The PBM explained the high precision Au Au central
(0-10%) collisions at $\sqrt{s_{NN}}=$ 200 GeVcentralproduction (the highest
RHIC energy).
Figure 2: A jet parton shower.
Figure 3: Each bubble contains 3-4 partons as shown.
### II.1 The Correlation Function
We utilize a two particle correlation function in the two dimensional (2-D)
space of $\Delta\phi$ vs $\Delta\eta$. The azimuthal angle $\phi$ of a
particle is defined by the angle of the particle with respect to the vertical
axis which is perpendicular to the beam axis and is measured in a clock-wise
direction about the beam. $\Delta\phi$ is the difference, $\phi_{1}$ \-
$\phi_{2}$, of the $\phi$ angle of a pair of particles (1 and 2). The pseudo-
rapidity $\eta$ of a particle is measured along one of the beam directions.
$\Delta\eta$ is the difference, $\eta_{1}$ \- $\eta_{2}$, of the $\eta$ values
of a pair of particles (1 and 2).
The two dimensional (2-D) total correlation function is defined as:
$C(\Delta\phi,\Delta\eta)=S(\Delta\phi,\Delta\eta)/M(\Delta\phi,\Delta\eta).$
(1)
Where S($\Delta\phi,\Delta\eta$) is the number of pairs at the corresponding
values of $\Delta\phi,\Delta\eta$ coming from the same event, after we have
summed over all the events. M($\Delta\phi,\Delta\eta$) is the number of pairs
at the corresponding values of $\Delta\phi,\Delta\eta$ coming from the mixed
events, after we have summed over all our created mixed events. A mixed event
pair has each of the two particles chosen from a different event. We make on
the order of ten times the number of mixed events as real events. We rescale
the number of pairs in the mixed events to be equal to the number of pairs in
the real events. This procedure implies a binning in order to deal with finite
statistics. To enhance comparison we use the same binning in our simulations
as is used by the STAR high precision experimental analysescentralproduction .
The division by M($\Delta\phi,\Delta\eta$) for experimental data essentially
removes or drastically reduces acceptance and instrumental effects. If the
mixed pair distribution was the same as the real pair distribution
C($\Delta\phi,\Delta\eta$) would be one for all values of the binned
$\Delta\phi,\Delta\eta$. In the correlations used in this paper we select
particles independent of its charge. The correlation of this type is called a
Charge Independent (CI) Correlation. This difference correlation function has
the defined property that it only depends on the differences of the azimuthal
angle ($\Delta\phi$) and the beam angle ($\Delta\eta$) for the two particle
pair. Thus the two dimensional difference correlation distribution for each
bubble which is part of C($\Delta\phi,\Delta\eta$) is similar for each of our
12 bubbles and will image on top of each other. The PBM fit to the angular
correlation data was reasonably quantitative to within a few percent (see Fig.
4 and Fig. 5)).
Figure 4: The CI (sum of all charged-particle-pairs) correlation for the 0-10%
centrality bin with a charged particle $p_{t}$ selection (0.8 $<$ $p_{t}$ $<$
4.0 GeV/c), generated by the PBM. It is plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot.
Figure 5: In each of the five labeled $\Delta\eta$ bins we show the
$\Delta\phi$ total correlation for the CI as a function of $\Delta\phi$. The
STAR Au + Au central trigger analysis results from the formulae of
Ref.centralproduction are shown as a solid line. The parton bubble model
predictions are shown by the circular points(o) which are large enough to
include the statistical errors from a 2 million event sample. The vertical
correlation scale is not offset and is correct for the largest $\Delta\eta$
bin, 1.2 $<\Delta\eta<$ 1.5, and is the lowest bin on the figure. As one
proceeds upwards to the next bin $\Delta\eta$ the correlation is offset by
+0.01. This is added to the correlation of each subsequent $\Delta\eta$ bin.
The smallest $\Delta\eta$ and top of Figure 12 has a +0.04 offset. A solid
straight horizontal line shows the offset for each $\Delta\eta$ bin. Each
solid straight horizontal line is at 1.0 in correlation strength. The 0-10%
centrality in HIJING corresponds to impact parameter (b) range 0.0 to 4.0 fm.
The agreement is very good.
The correlation functions we employed (like the HBT correlation functions)
have the property that for the difference in angles (difference of momentum
for HBT) these correlations will image all 12 bubbles on top of each other.
This leads to average observed angles of approximately $30^{\circ}$ in
$\Delta\phi$, $70^{\circ}$ in $\Delta\eta$, originating from a source size of
about $\sim$2 fm radius which is consistent with the HBT correlation. Thus the
PBM generates $\Delta\phi$ vs $\Delta\eta$ charged-particle-pair correlations
for charged particles with $p_{t}$ in the range 0.8 Gev/c to 4.0 GeV/c as
displayed in Fig. 4 and Fig. 5. We show the CI correlation in Fig. 5 a
reasonably quantitative successful comparison with data. It should be noted
that the spread in $\Delta\eta$ of the bubble is determined by the
longitudinal momentum distribution of the 3-4 partons that make up the bubble.
This distribution which is quassian in nature was adjusted to the observed
$\Delta\eta$ width of correlation data analyzed by the STAR experimentPBM .
Furthermore the model results were consistent with the Hanbury-Brown and Twiss
(HBT) observationsHBT that the observed source radii determined by quantum
statistical interference were reducing by a considerable factor with
increasing transverse momentum ($p_{t}$). The HBT radii are interpreted to
reduce from $\sim$6 fm at $p_{t}$ $\sim$0.2 GeV/c to $\sim$2 fm at $p_{t}$
$\sim$1 GeV/c for our $p_{t}$ range. The generally accepted explanation for
this behavior is that as $p_{t}$ increases radial flow increasingly focused
the viewed region of the final state into smaller volumes. If just one small
HBT size bubble were emitting all the correlated particles, this phenomena
would lead to large spikes of particles emitted at one limited $\phi$ angular
region in individual events. This is definitely not observed in the Au Au or
other collision data at RHIC. Therefore, a distributed ring of small sources
around the beam as assumed in the PBM is necessary to explain both the HBT
resultsHBT and the correlation datacentralproduction . The particles emitted
from the same bubble are virtually uncorrelated to particles emitted from any
other bubble except for momentum conservation requirements. An away side peak
in the total correlation is built up from momentum conservation between the
bubbles.
## III The PBM for other centralities(PBMEPBME )
The PBM was also recently extended to PBMEPBME which is identical to the PBM
for central collisions (0-5%). For centralities running from 30-80% jet
quenching is not strong enough to make jets negligible therefore, a jet
component was added which was based on HIJING calculations. This jet component
accounts for more of the correlation as one moves toward peripheral bins, and
explains all of it for the most peripheral collisions. The PBME explained in a
reasonably quantitative manner (within a few percent) the behavior of the
recent quantitative experimental analysis of charge pair correlations as a
function of centralitycentralitydependence . This further strengthened the
substantial evidence for bubble substructure. The agreement of the PBM and the
PBME surface emission models with experimental analyses strongly implied that
at kinetic freezeout the fireball was dense and opaque in the central region
and most centralities (except the peripheral region) in the intermediate
transverse momentum region (0.8 $<$ $p_{t}$ $<$ 4.0 GeV/c). Thus we conclude
that the observed correlated particles are both formed and emitted from or
near the surface of the fireball. In the peripheral region the path to the
surface is always small.
The CI correlation used in Fig. 4 and the PBMPBM is also defined as the sum
of two charged-particle-pair correlations. They are the unlike-sign charge
pairs (US) and the like-sign charge pairs (LS). This Charge Independent
correlation (CI) is defined as the correlation made up of charged-particle-
pairs independent of what the sign is, which is the average of the US plus the
LS correlations. Thus CI = (US + LS)/2. Therefore the CI is the total charge
pair correlation observed in the experimental detection system within its
acceptance.
The CI was generated for $\sqrt{s_{NN}}$ = 200 GeV central (0-5%) Au Au
collisions in the $p_{t}$ range 0.8 GeV/c to 4.0 GeV/c at RHIC. When the CI
was compared to the corresponding experimental analysescentralproduction ;
centralitydependence a reasonably quantitative agreement within a few percent
of the observed CI was attained. See Ref.PBM Sec. 4.2, Ref.PBME Sec. V, and
Ref.centralitydependence Sec. VI B111The STAR collaboration preferred to use
the conventional definition of CI = US + LS which is larger by a factor of 2
than the CI definition used in this present paper that is directly physically
meaningful.. This analysis and our previous workPBM ; PBME was done without
using a jet trigger. In fact jet production in the central region was
negligible due to strong jet quenchingquench1 ; quench2 ; quench3 . As one can
see in Fig. 4 the central CI correlation of $\Delta\phi$ on the near side
($\Delta\phi$ $<$ $90^{\circ}$), is sharp and approximately jetlike at all
$\Delta\eta$. In contrast the $\Delta\eta$ dependence is relatively flat. In
the central production the average observed angles are approximately
$30^{\circ}$ in $\Delta\phi$ and $70^{\circ}$ in $\Delta\eta$. It is of
interest to note that the sharp jetlike collimation in $\Delta\phi$ persists
at all centralities (0-80%). The elongation in $\Delta\eta$ persists in the
0-30% centrality range and then decreases with decreasing centrality becoming
jetlike in the peripheral bins. These characteristics are in reasonable
quantitative agreement with the fits of the PBM and the PBME to experimental
data analysesPBM ; centralproduction ; PBME ; centralitydependence .
The difference of the US and LS correlations is defined as the Charge
Dependent (CD) correlation (CD = US - LS). The 2-D experimental CD correlation
for centralities (0-80%) has a jetlike shape which is consistent with PYTHIA
jets (vacuum fragmentation) for all centralities. This clearly implies that
both particle hadronization and emission occur from the fireball surface
region as explained in Ref.PBME Sec. II.
## IV Glasma flux tube model
A glasma flux tube model (GFTM)Dumitru that had been developed considers that
the wavefunctions of the incoming projectiles, form sheets of color glass
condensates (CGC)CGC that at high energies collide, interact, and evolve into
high intensity color electric and magnetic fields. This collection of
primordial fields is the GlasmaLappi ; Gelis , and initially it is composed of
only rapidity independent longitudinal color electric and magnetic fields. An
essential feature of the Glasma is that the fields are localized in the
transverse space of the collision zone with a size of 1/$Q_{s}$. $Q_{s}$ is
the saturation momentum of partons in the nuclear wavefunction. These
longitudinal color electric and magnetic fields generate topological Chern-
Simons chargeSimons which becomes a source for particle production.
The transverse space is filled with flux tubes of large longitudinal extent
but small transverse size $\sim$$Q^{-1}_{s}$. Particle production from a flux
tube is a Poisson process, since the flux tube is a coherent state. As the
partons emitted from these flux tubes locally equilibrate, transverse flow
builds due to the radial flow of the blast waveGavin . The flux tubes that are
near the surface of the fireball get the largest radial flow and are emitted
from the surface. As in the parton bubble model these partons shower and the
higher $p_{t}$ particles escape the surface and do not interact. These flux
tubes are in the present paper considered strongly connected to the bubbles of
the PBM. The method used to connect the PBM and the GFTM is described and
discussed in the next section V.
$Q_{s}$ is around 1 GeV/c thus the transverse size of the flux tube is about
1/4 fm. The flux tubes near the surface are initially at a radius $\sim$5 fm.
The $\phi$ angle wedge of the flux tube is $\sim$1/20 radians or
$\sim$$3^{\circ}$. Thus the flux tube initially has a narrow range in $\phi$.
The large width in the $\Delta\eta$ correlation which in the PBM depended on
the large spread in $\Delta\eta$ of the bubble partons results from the
independent longitudinal color electric and magnetic fields that created the
Glasma flux tubes. How much of these longitudinal color electric and magnetic
fields are still present in the surface flux tubes when they have been pushed
by the blast wave will be a speculation of this paper for measuring strong CP
violation?
It has been noted that significant features of the PBM that generated final
state correlations which fit the experimentally observed correlation dataPBM ;
centralproduction ; PBME ; centralitydependence are similar to those
predicted by the GFTMDumitru . Therefore in the immediately following section
we assume a direct connection of the PBM and the GTM, give reasons to justify
it, and then discuss its consequences, predictions and successes.
## V The connection of the PBM and the GFTM
The successes of the PBM have strongly implied that the final state surface
region bubbles of the PBM represent a significant substructure. In this
subsection we show that the characteristics of our PBM originally developed to
fit the precision STAR Au Au correlation data in a manner consistent with the
HBT data; implies that the bubble substructure we originally used to fit these
previous data is closely related to the GFTM.
A natural way to connect the PBM bubbles to the GFTM flux tubes is to assume
that the final state at kinetic freezeout of a flux tube is a PBM final state
bubble. Thus the initial transverse size of a flux tube $\sim$1/4 fm has
expanded to the size of $\sim$2 fm at kinetic freezeout. With this assumption
we find consistency with the theoretical expectations of the GFTM. We can
generate and explain the triggered ridge phenomenon and data (see Sec. VI),
thus implying the ridge is connected to the bubble substructure. We can
predict and obtain very strong evidence for the color electric field of the
glasma from comparing our multi-particle charged particle correlation
predictions, and existing experimental correlation publications (see Sec.
VII). We have also predicted correlations which can be used to search for
evidence for the glasma color magnetic field (Sec. VII).
Of course an obvious question that arises is that since a flux tube is an
isolated system does a PBM bubble, that we assume is the final state of a flux
tube at kinetic freezeout, also act as an isolated system when emitting the
final state particles? The near side correlations signals since the original
PBMPBM have always come virtually entirely from particles emitted from the
same bubble. Thus each bubble has always acted as an isolated system similar
to the behavior of a flux tube.
## VI The Ridge is formed by the bubbles when a jet trigger is added to the
PBM
In heavy ion collisions at RHIC there has been observed a phenomenon called
the ridge which has many different explanationsDumitru ; Armesto ; Romatschke
; Shuryak ; Nara ; Pantuev ; Mizukawa ; Wong ; Hwa . The ridge is a long range
charged particle correlation in $\Delta\eta$ (very flat), while the
$\Delta\phi$ correlation is approximately jet-like (a narrow Gaussian). There
also appears with the ridge a jet-like charged-particle-pair correlation which
is symmetric in $\Delta\eta$ and $\Delta\phi$ such that the peak on the jet-
like correlation is at $\Delta\eta$ = 0 and $\Delta\phi$ = 0. The $\Delta\phi$
correlation of the jet and the ridge are approximately the same and smoothly
blend into each other. The ridge correlation is generated when one triggers on
an intermediate $p_{t}$ range charged particle and then forms pairs between
that trigger particle and each of all other intermediate charged particles
with a smaller $p_{t}$ down to some lower limit. The first case we will study
in this paper is a trigger charged particle between 3.0 to 4.0 GeV/c
correlated with all other charged particles which have a $p_{t}$ between 1.1
GeV/c to 3.0 GeV/c.
In this paper we will investigate whether the PBM can account for the ridge
once we add a jet trigger to our PBM generatorPBM . However this trigger will
also select jets which previously could be neglected because there was such
strong quenchingquench1 ; quench2 ; quench3 of jets in central collisions. A
jet trigger had not been used in the PBM comparison to all previous data. We
use HIJINGhijing merely to determine the expected number of jets to add for
our added jet trigger. These jet particles were added to our PBM generator.
Thus our PBM generator now had HIJING generated background particles, bubbles
of the PBM and added jet particles from HIJING. We have already shown that our
final state particles come from hadrons at or near the fireball surface. We
reduce the number of jets by 80% which corresponds to the estimate that only
the parton interactions on or near the surface are not quenched away, and thus
at kinetic freezeout form and emit hadrons which enter the detector. This 80%
reduction is consistent with single $\pi^{0}$ suppression observed in
Ref.quench3 . We find for the reduced HIJING jets that 4% of the Au Au central
events (0-5%) centrality at $\sqrt{s_{NN}}=$ 200 have a charged particle with
a $p_{t}$ between 3.0 and 4.0 GeV/c with at least one other charged particle
with its $p_{t}$ greater than 1.1 GeV/c coming from the same jet. The addition
of the jets to the PBM generator provides the appropriate particles which are
picked up by the trigger in order to form a narrow $\Delta\eta$ correlation
signal at 0 which is also a narrow signal in $\Delta\phi$ at 0 (Fig. 14). This
narrow jet signal is present in the data and is what remains of jets after 80%
are quenched away.
We then form two-charged-particle correlations between one-charged-particle
with a $p_{t}$ between 3.0 to 4.0 GeV/c and another charged particle whose
$p_{t}$ is greater than 1.1 GeV/c. The results of these correlations are shown
in Fig. 6. Fig. 6 is the CI correlation for the 0-5% centrality bin with the
just above stated $p_{t}$ selections on charge pairs. Since we know in our
Monte Carlo which particles are emitted from bubbles and thus form the ridge,
we can predict the shape of the ridge for the above $p_{t}$ cut by plotting
only the correlation formed from pairs of particles that are emitted by the
same bubble (see Fig. 7). The charge pair correlations are virtually all
emitted from the same bubble. Those charge pair correlations formed from
particles originating from different bubbles are small contributors which
mainly have some effect on the away side correlation. Thus Fig. 7 is the ridge
signal which is the piece of the CI correlation for the 0-5% centrality of
Fig. 6, after removing all other pairs except the pairs emitted from the same
bubble.
Figure 6: The CI correlation for the 0-5% centrality bin that results from
requiring one trigger particle $p_{t}$ above 3 GeV/c and another particle
$p_{t}$ above 1.1 GeV/c. It is plotted as a two dimensional $\Delta\phi$ vs.
$\Delta\eta$ perspective plot.
Figure 7: The ridge signal is the piece of the CI correlation for the 0-5%
centrality of Fig. 6 after removing all other particle pairs except the pairs
that come from the same bubble. It is plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot. As explained in the text
charge pair correlations formed by particles emitted from different bubbles
are small contributors which mainly have some effect on the away side
correlation for $\Delta\phi$ greater than about $120^{\circ}$.
### VI.1 Parton $p_{t}$ correlated vs. random
A very important aspect of the GFTM is the boost that the flux tubes get from
the radial flow of the blast wave. This boost is the same all along the flux
tube and only depends on how far away from the center axis of the blast wave
the flux tube is. In the PBM the partons in the bubble received a boost in
$p_{t}$ from the radial flow field in the blast wave; that after final state
fragmentation of the bubbles gave the generated particles a consistent $p_{t}$
spectrum with the data. Since the boost from the blast wave depended on the
position of the bubble in the radial flow field, there should be a correlation
between partons $p_{t}$ within a bubble. If one redistributed the partons with
their $p_{t}$ boosts uniformly among the bubbles, the over all results for the
generated correlations without a trigger would be unchanged. One should note
none of our prior work (i.e.PBMPBM or PBMEPBME ) contained a trigger. We have
only added a trigger here to treat the “triggered ridge” which obviously
requires it. In our treatment of the GFTM (Sec. VII) we have removed the
trigger.
Once we require a trigger demanding higher $p_{t}$ particles, we start picking
out bubbles which have more radial flow (harder particles). Thus correlated
particles which pass our $p_{t}$ selection and trigger come almost entirely
from the harder bubbles with more radial flow, while the softer bubbles which
were subject to less radial flow mainly generate low enough $p_{t}$ particles
that become background particles or are lost to the correlation analysis due
to $p_{t}$ selection. Figure 6 shows the result of our trigger and $p_{t}$
selection. If one redistributed the partons with their $p_{t}$ radial flow
boost among all the bubbles, then all bubbles become equal and there is no
longer soft and hard bubbles. With this change we can generate bubble events.
We find that both the non-triggered correlation and the $\Delta\phi$
correlation are virtually unchanged in the region less than about
$120^{\circ}$ (see Fig. 8). It seems that we move from the situation where we
have a few bubbles with a lot of correlated particles above our $p_{t}$ cut to
a lot of bubbles that have few correlated particles above our cuts. However
there is a difference in the triggered correlation’s away side or $\Delta\phi$
near the $180^{\circ}$ peak.
In Ref.PBM we discuss this away side effect and attributed this peak to
momentum conservation. This is however not the whole story. The away side peak
is generated by the geometry of the bubble ring which requires that for every
bubble there is another bubble nearly $180^{\circ}$ away that is emitting
particles. These symmetry requirements on the geometry contribute more to the
away side peak especially for the case without a trigger or redistributed
random partons.
This geometry effect is what causes the away side correlation of elliptic flow
in a peripheral heavy ion collision. In such a collision there is a higher
energy density aligned with the reaction plane. On one side of the reaction
plane there are more particles produced because of increased energy density,
and because of the geometry of the situation. On the other side of the
reaction plane there are also more particles produced for the same reasons.
Thus there is a correlation between particles that are near each other on one
side of the reaction plane and a correlation between particles that are on
both sides of the reaction plane. Momentum conservation requires particle
communication by bouncing into each other, while geometry has no such
communication. The right hand does not need to know what the left hand is
doing. However, if physics symmetry makes them do the same thing then they can
show a correlation.
### VI.2 The away side peak
The away side peak depends on the fragmentation of the away side bubble. For
the triggered case where there is a hard bubble with a strong correlation of
parton $p_{t}$ inside the bubble (GFTM like), we will have more correlated
particles adding to the signal on the near side. On the other hand the away
side bubble will be on average a softer bubble which will have a lot less
particles passing the $p_{t}$ cuts and thus have a smaller signal. If we
consider the case of a random parton $p_{t}$ bubble the particles passing the
$p_{t}$ cuts should be very similar on the near and the away side. In Fig. 8
we compare the two correlations generated by the correlated and the random
cases. We compare the $\Delta\phi$ correlation for the two cases (solid is
correlated and open is random)for each of the five $\Delta\eta$ bins which
cover the entire $\Delta\eta$ range (0.0 to 1.5). The vertical correlation
scale is not offset and is correct for the largest $\Delta\eta$ bin, 1.2
$<\Delta\eta<$ 1.5, which is the lowest bin on the figure. As one proceeds
upwards to the next $\Delta\eta$ bin the correlation is offset by +0.05. This
is added to the correlation of each subsequent $\Delta\eta$ bin. The smallest
$\Delta\eta$ bin on top of Fig. 8 has a +0.2 offset. A solid straight
horizontal line shows the offset for each $\Delta\eta$ bin. Each solid
straight horizontal line is at 1.0 in correlation strength. We see that the
away side correlation in the $\Delta\phi$ region greater than about
$120^{\circ}$ is larger for the random case.
Figure 8: The $\Delta\phi$ CI correlation for the 0-5% centrality bin
requiring one trigger particle $p_{t}$ above 3 GeV/c and another particle
$p_{t}$ above 1.1 GeV/c. We compare the two correlations generated by the
correlated and the random cases. We compare the $\Delta\phi$ correlation for
the two cases (solid is correlated and open is random) for each of the five
$\Delta\eta$ bins which cover the entire $\Delta\eta$ range (0.0 to 1.5). The
vertical correlation scale is not offset and is correct for the largest
$\Delta\eta$ bin, 1.2 $<\Delta\eta<$ 1.5, which is the lowest bin on the
figure. As one proceeds upwards to the next $\Delta\eta$ bin the correlation
is offset by +0.05. This is added to the correlation of each subsequent
$\Delta\eta$ bin. The smallest $\Delta\eta$ bin on top of Fig. 8 has a +0.2
offset. A solid straight horizontal line shows the offset for each
$\Delta\eta$ bin. Each solid straight horizontal line is at 1.0 in correlation
strength. We see that the away side correlation for $\Delta\phi$ greater than
about $120^{\circ}$ is larger for the random case.
Figure 9: In this figure we display the $\phi$ and $\eta$ ranges considered
for producing ridge particles in the Au Au central collisions. A typical
trigger particle ($p_{t}$ between 3.0 to 4.0 GeV/c) implies two associated
regions in the considered $\phi$ and $\eta$ ranges where the ridge particles
lie. The ridge particles we consider are separated by 0.7 in $\eta$
($\Delta\eta$) so that one eliminates particles coming from jet production.
The bulk of the ridge particles lie within $20^{\circ}$ of the $\phi$ of the
trigger particle. The width of the $\phi$ spread is $\Delta\phi$ =
$40^{\circ}$.
Figure 10: On a trigger by trigger basis we accumulate particles from the
ridge region (see Fig. 9). We then form a ratio the of $p_{t}$ spectrum for
the ridge region particles divided by the $p_{t}$ spectrum for all particles
in central Au Au events. We normalize the two $p_{t}$ spectra to have the same
counts in the 2.0 GeV/c bin. This ratio or correlation for the bubbles which
have a correlated $p_{t}$ among the partons (GFTM like) are the solid points.
The random $p_{t}$ distribution among partons in the bubble are the open
points.
### VI.3 Predicted transverse momentum dependence of correlation
Particles which come from bubbles that satisfy the trigger (3.0 $<$ $p_{t}$
$<$ 4.0 GeV/c) are harder and have a different $p_{t}$ distribution compared
to the case without trigger requirements. In a given triggered event if one
could select particles that come from the bubble or ridge, one would find that
the $p_{t}$ spectrum is harder than the average $p_{t}$ spectrum. In Fig. 9 we
show how one can by trigger choose particles that are rich with ridge
particles. We consider a typical trigger particle and the associated regions
in the available experimental $\phi$ and $\eta$ ranges. The ridge particles we
consider are separated from the trigger particle by 0.7 in $\eta$. This is
done to eliminate particles coming from jet production that could also be
associated with the trigger particle. The bulk of the ridge particles lie
within $20^{\circ}$ of the $\phi$ of the trigger particle. The width of the
$\phi$ spread is $\Delta\phi$ = $40^{\circ}$. Therefore on a trigger by
trigger basis we accumulate particles from the ridge region (see Fig. 9) and
form a $p_{t}$ spectrum of charged particles. We then form a ratio of the
$p_{t}$ spectrum for the ridge region particles divided by the $p_{t}$
spectrum for all particles in central Au Au events. This ratio is highly
dependent on our ridge cut area. We can virtually remove this dependency by
normalizing the two $p_{t}$ spectra to have the same counts for the 2.0 GeV/c
bin. Figure 10 is this ratio which can be considered a correlation of
particles as a function of $p_{t}$ in the ridge region compared to particles
from the event in general. In Fig. 10 we form this ratio or correlation for
the bubbles which have a correlated $p_{t}$ among the partons (GFTM like). We
also show the correlation for bubbles which have a random $p_{t}$ distribution
among partons in the bubble. For the random case the $p_{t}$ spectrum does not
differ much from the average $p_{t}$ spectrum while there is a big difference
for the flux tube like case. This correlation could be easily measured in the
RHIC data.
### VI.4 Comparison to data
Triggered angular correlation data showing the ridge was presented at Quark
Matter 2006Putschke . Figure 11 shows the experimental $\Delta\phi$ vs.
$\Delta\eta$ CI correlation for the 0-10% central Au Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV; requiring one trigger particle $p_{t}$ between 3 and
4 GeV/c and an associated particle $p_{t}$ above 2.0 GeV/c. The yield is
corrected for the finite $\Delta\eta$ pair acceptance. For the PBM generator,
we then form a two-charged-particle correlation between one charged particle
with a $p_{t}$ between 3.0 to 4.0 GeV/c and another charged particle whose
$p_{t}$ is greater than 2.0 GeV/c. The results of this correlation is shown in
Fig. 12. Figure 11 shows the corrected pair yield determined in the central
data whereas Fig. 12 shows the correlation function generated by the PBM which
does not depend on the number of events analyzed. We can compare the two
figures, if we realize that the away side ridge has around 420,000 pairs in
Fig. 11 while in Fig. 12 the away side ridge has a correlation of around
0.995. If we multiply the correlation scale of Fig. 12 by 422,111 in order to
achieve the number of pairs seen in Fig. 11, the away side ridge would be at
420,000 and the peak would be at 465,000. This would make a good agreement
between the two figures.
The correlation formed by the ridge particles is generated almost entirely by
particles emitted by the same bubble. We have shown in all our publications
that the same side correlation signals are almost entirely formed by particles
coming from the same bubble. Thus we can predict the shape and the yield of
the ridge for the above $p_{t}$ trigger selection and lower cut, by plotting
only the correlation coming from pairs of particles that are emitted by the
same bubble (see Fig. 13).
In Ref.Putschke it was assumed that the ridge yield was flat across the
acceptance while in Fig. 13 we see that this is not the case. Therefore our
ridge yield is approximately 35% larger than estimated in Ref.Putschke .
Finally we can plot the jet yield that we had put into our Monte Carlo. We
used HIJINGhijing to determine the number of expected jets, and then reduced
the number of jets by 80%. This assumes that only the parton interactions on
or near the surface that form hadrons at kinetic freezeout are not quenched
away and thus enter the detector. This 80% reduction is consistent with single
$\pi^{0}$ suppression observed in Ref.quench3 . This jet yield is plotted in
Fig. 14 where we subtracted contributions from the bubbles and the background
particles from Fig. 11.
Figure 11: Raw $\Delta\phi$ vs. $\Delta\eta$ CI preliminary correlation
dataPutschke for the 0-10% centrality bin for Au Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV requiring one trigger particle $p_{t}$ between 3 to 4
GeV/c and an associated particle $p_{t}$ above 2.0 GeV/c.
Figure 12: The PBM generated CI correlation for the 0-5% centrality bin
requiring one trigger particle $p_{t}$ above 3 GeV/c and less than 4 GeV/c and
another particle $p_{t}$ above 2.0 GeV/c plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot. The trigger requirements on
this figure are the same as those on the experimental data in Fig. 11.
Figure 13: The ridge signal is the piece of the CI correlation for the 0-5%
centrality of Fig. 12 after removing all other particle pairs except the pairs
that come from the same bubble. It is plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot.
Figure 14: The jet signal is left in the CI correlation after the
contributions from the background and all the bubble particles are removed
from the 0-5% centrality (with trigger requirements) of Fig. 12. It is plotted
as a two dimensional $\Delta\phi$ vs. $\Delta\eta$ perspective plot.
## VII Strong CP violations or Chern-Simons topological charge
### VII.1 The Source $F\widetilde{F}$.
The strong CP problem remains one of the most outstanding puzzles of the
Standard Model. Even though several possible solutions have been put forward
it is not clear why CP invariance is respected by the strong interaction. It
was shown however through a theorem by Vafa-WittenWitten1 that the true
ground state of QCD cannot break CP. The part of the QCD Lagrangian that
breaks CP is related to the gluon-gluon interaction term $F\widetilde{F}$. The
part of $F\widetilde{F}$ related to CP violations can be separated into a
separate term which then can be varied by multiplying this term by a parameter
called $\theta$. For the true QCD ground state $\theta$ = 0 (Vafa-Witten
theorem). In the vicinity of the deconfined QCD vacuum metastable
domainsTytgat $\theta$ non-zero could exist and not contradict the Vafa-
Witten theorem since these are not the QCD ground state. The metastable
domains CP phenomenon would manifest itself in specific correlations of pion
momentaPisarski ; Tytgat .
### VII.2 Pionic measures of CP violation.
The glasma flux tube model (GFTM)Dumitru considers the wavefunctions of the
incoming projectiles, form sheets of CGCCGC at high energies that collide,
interact, and evolve into high intensity color electric and magnetic fields.
This collection of primordial fields is the GlasmaLappi ; Gelis . Initially
the Glasma is composed of only rapidity independent longitudinal (along the
beam axis) color electric and magnetic fields. These longitudinal color
electric and magnetic fields generate topological Chern-Simons chargeSimons
through the $F\widetilde{F}$ term and becomes a source of CP violation. How
much of these longitudinal color electric and magnetic fields are still
present in the surface flux tubes when they have been pushed by the blast wave
is a speculation of this paper for measuring strong CP violation? The color
electric field which points along the flux tube axis causes an up quark to be
accelerated in one direction along the beam axis, while the anti-up quark is
accelerated in the other direction. So when a pair of quarks and anti-quarks
are formed they separate along the beam axis leading to a separated $\pi^{+}$
$\pi^{-}$ pair along this axis. The color magnetic field which also points
along the flux tube axis (which is parallel to the beam axis) causes an up
quark to rotate around the flux tube axis in one direction, while the anti-up
quark is rotated in the other direction. So when a pair of quarks and anti-
quarks are formed they will pickup or lose transverse momentum after the
bubble is boosted radial outwards due to radial flow of the blast wave. These
changes in $p_{t}$ will be transmitted to the $\pi^{+}$ $\pi^{-}$ pairs.
It is important to note that the CP violating asymmetries in $\pi^{+}$ and
$\pi^{-}$ momenta arise through the Witten-Wess-ZuminoWitten2 ; Wess term.
The quarks and anti quarks which, later form the $\pi^{+}$ and $\pi^{-}$,
directly respond to the color electric and color magnetic fields and receive
their boosts at the quark and anti quark level before they are pions. These
boosts are transmitted to the pions at hadronization. Thus no external
magnetic field is required in the methodology followed in this paper. The
following references demonstrate this:2000 ; Pisarski ; Tytgat ; Finch .
To represent the color electric field effect we assume as the first step for
the results being shown in this paper; that we generate bubbles which have an
added boost of 100 MeV/c to the quarks in the longitudinal momentum which
represents the color electric effect. The $\pi^{+}$ and the $\pi^{-}$ which
form a pair are boosted in opposite directions along the beam axis. In a given
bubble all the boosts are the same but vary in direction from bubble to
bubble. For the color magnetic field effect, we give 100 MeV/c boosts to the
transverse momentum in an opposite way to the $\pi^{+}$ and the $\pi^{-}$
which form a pair. For each pair on one side of a bubble one of the charged
pions boost is increased and the other is decreased. While on the other side
of the bubble the pion for which the boost was increased is now decreased and
the pion which was decreased is now increased by the 100 MeV/c. All pairs for
a given bubble are treated in the same way however each bubble is random on
the sign of the pion which is chosen to be boosted on a given side. This
addition to our model is used in the simulations of the following subsections.
### VII.3 Color electric field pionic measure.
Above we saw that pairs of positive and negative pions should show a charge
separation along the beam axis due to a boost in longitudinal momentum caused
by the color electric field. A measure of this separation should be a
difference in the pseudorapidity ($\Delta\eta$) of the opposite sign pairs.
This $\Delta\eta$ measure has a well defined sign since we defined this
difference measuring from the $\pi^{-}$ to the $\pi^{+}$. In order to form a
correlation we must pick two pairs for comparison. The pairs have to come from
the same bubble (a final state of an expanded flux tube) since we have shown
by investigating the events generated by the model that pairs originating from
different bubbles will not show this correlation. Therefore we require the
$\pi^{+}$ and the $\pi^{-}$ differ by $20^{\circ}$ or less in $\phi$. Let us
call the first pair $\Delta\eta_{1}$. The next pair ($\Delta\eta_{2}$ having
the same $\phi$ requirement) has to also lie inside the same bubble to show
this correlation. Thus we require that there is only $10^{\circ}$ between the
average $\phi$ of the pairs. This implies that at most in $\phi$ no two pions
from a given bubble can differ by more than $30^{\circ}$. In Fig. 15 we show
two pairs which would fall into the above cuts. $\Delta\eta_{1}$ and
$\Delta\eta_{2}$ are positive in Fig. 15. However if we would interchange the
$\pi^{+}$ and $\pi^{-}$ on either pair the value of its $\Delta\eta$ would
change sign. Finally the mean value shown on Fig. 15 is the mid-point between
the $\pi^{+}$ and the $\pi^{-}$ where one really uses the vector sum of the
$\pi^{+}$ and the $\pi^{-}$ which moves this point toward the harder pion.
Considering the above cuts we defined a correlation function where we combine
pairs each having a $\Delta\eta$. Our variable is related to the sum of the
absolute values of the individual $\Delta\eta$’s
($|\Delta\eta_{1}|+|\Delta\eta_{2}|$). We assign a sign to this sum such that
if the sign of the individual $\Delta\eta$’s are the same it is a plus sign,
while if they are different it is a minus sign. For the flux tube the color
electric field extends over a large pseudorapidity range therefore let us
consider the separation of pairs $|\Delta\eta|$ greater than 0.9. For the
numerator of the correlation function we consider all combinations of unique
pairs (sign ($|\Delta\eta_{1}|+|\Delta\eta_{2}|$)) from a given central Au Au
event divided by a mixed event denominator created from pairs in different
events. We determine the rescale of the mixed event denominator by considering
the number of pairs of pairs for the case $|\Delta\phi|$ lying between
$50^{\circ}$ and $60^{\circ}$ for events and mixed events so that the overall
ratio of this sample numerator to denominator is 1. By picking $50^{\circ}$
$<$ $|\Delta\phi|$ $<$ $60^{\circ}$ we make sure we are not choosing pairs
from the same bubble. For a simpler notation let (sign
($|\Delta\eta_{1}|+|\Delta\eta_{2}|$)) = $\Delta\eta_{1}+\Delta\eta_{2}$ which
varies from -4 to +4 since we have an over all $\eta$ acceptance -1 to +1 (for
the STAR TPC detector for which we calculated). The value being near $\pm$ 4
can happen when one has a hard pion with $p_{t}$ of 4 GeV/c (upper cut) at
$\eta$ = 1 with a soft pion $p_{t}$ of 0.8 GeV/c (lower cut) at $\eta$ = -1
combined with another pair; a hard pion with $p_{t}$ of 4 GeV/c at $\eta$ = -1
with a soft pion with $p_{t}$ of 0.8 GeV/c at $\eta$ = 1.
Figure 15: Pairs of pairs selected for forming a correlation due to the boost
in longitudinal momentum caused by the color electric field. The largest
separation we allow in $\phi$ for plus minus pair 1 is $\Delta\phi_{1}$ =
$20^{0}$. This assures the pair is in the same bubble. The same is true for
pair 2 so that it will also be contained in this same bubble. The mid-point
for pair 1 and 2 represents the vector sum of pair 1 and 2 which moves toward
the harder particle when the momenta differ. These mid-points can not be
separated by more than $10^{0}$ in $\Delta\phi$ in order to keep all four
particles inside the same bubble since the correlation function is almost
entirely generated within the same bubble. The $\Delta\eta$ measure is the
angle between the vector sum 1 compared to the vector sum 2 along the beam
axis (for this case $|\Delta\eta|$ = 1.1). The positive sign for
$\Delta\eta_{1}$ comes from the fact that one moves in a positive $\eta$
direction from negative to positive. The same is true for $\Delta\eta_{2}$. If
we would interchange the charge of the particles of the pairs the sign would
change.
Figure 16: Correlation function of pairs formed to exhibit the effects of
longitudinal momentum boosts by the color electric field as defined in the
text. This correlation shows that there are more aligned pairs (correlation is
larger by $\sim$0.5% between 1.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ 2.0
compared to between -2.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ -1.0).
$\Delta\eta_{1}+\Delta\eta_{2}$ is equal to
$|\Delta\eta_{1}|+|\Delta\eta_{2}|$. As explained in the text this means there
are more pairs of pairs aligned in the same direction compared to pairs not
aligned as predicted by the color electric field. The signs and more detail
are also explained in the text.
In Fig. 16 we show the correlation function of opposite sign charge-particle-
pairs paired and binned by the variable $\Delta\eta_{1}+\Delta\eta_{2}$ with a
cut $|\Delta\eta|$ greater than 0.9 between the vector sums of the two pairs.
The events are generated by the PBMPBM and are charged particles of 0.8 $<$
$p_{t}$ $<$ 4.0 GeV/c, and $|\eta|$ $<$ 1, from Au Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. Since we select pairs of pairs which are near each
other in $\phi$ they all together pick up the bubble correlation and thus
these pairs of pairs over all show about a 4% correlation. In the 1.0 $<$
$\Delta\eta_{1}+\Delta\eta_{2}$ $<$ 2.0 region the correlation is 0.5% larger
than the -2.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ -1.0 region. This means
there are more pairs of pairs aligned in the same direction compared to pairs
of pairs not aligned. This alignment is what is predicted by the color
electric field effect presented above. In fact if one has plus minus pairs all
aligned in the same direction and spread across a pseudorapidity range,
locally at any place in the pseudorapidity range one would observe an increase
of unlike sign charge pairs compared to like sign charge pairs. At small
$\Delta\eta$ unlike sign charge pairs are much larger than like sign charge
pairs in both the PBM and the data which agree. See Figs. 10, 11, and 14 of
Ref.PBM . Figure 11 of Ref.PBM compares the total correlation for unlike-sign
charge pairs and like-sign charge pairs in the precision STAR central
production experiment for Au Au central collisions (0-10% centrality) at
$\sqrt{s_{NN}}$=200 GeV, in the transverse momentum range 0.8 $<$ $p_{t}$ $<$
2.0 GeV/ccentralproduction . The unlike-sign charge pairs are clearly larger
in the region near $\Delta\phi$ = $\Delta\eta$ = 0.0. The increased
correlation of the unlike-sign pairs is 0.8% $\pm$ 0.002%. Figure 10 of
Ref.PBM shows that the PBM fit to these data gives the same results. Figure
14 of Ref.PBM shows that the CD = unlike-sign charge pairs minus like-sign
charge pairs is positive for the experimental analysis. Therefore the unlike-
sign charge pairs are considerably larger than the like-sign charge pairs. In
fact this effect is so large and the alignment is so great that when one adds
the unlike and like sign charge pairs correlations together there is still a
dip at small $\Delta\eta$ and $\Delta\phi$ see Fig. 4 and Fig. 7 of the
present paper. The statistical significance of this dip in the two high
precision experiments done independently from different data sets gathered 2
years apartcentralproduction ; centralitydependence is huge. It would require
a fluctuation of $14\sigma$ to remove the dip. This dip is also not due to any
systematic error since both of the just cited precision experiments carefully
investigated that possibility; and found no evidence to challenge the reality
of this dip. This highly significant dip ($\sim$$14\sigma$) means that like-
sign pairs are removed as one approaches the region $\Delta\eta$ =
$\Delta\phi$ = 0.0 at a rate much larger than regular jet fragmentation. Jet
fragmentation does not have a dip in its CI correlation. Thus this is very
strong evidence for the predicted effect of the color electric field.
### VII.4 Color magnetic pionic measure.
After considering the color electric effect we turn to the color magnetic
effect which causes up quarks to rotate around the flux tube axis in one
direction, while the anti-up quarks rotate in the other direction. So when a
pair of quarks and anti-quarks are formed they will pickup or lose transverse
momentum. These changes in $p_{t}$ will be transmitted to the $\pi^{+}$ and
$\pi^{-}$ pairs. It was previously shown in Sec. VII A that the CP violating
asymmetries in the $\pi^{+}$ and the $\pi^{-}$ momenta arise through the
Witten-Wess-Zumino term. The quarks and anti quarks which, later form the
$\pi^{+}$ and the $\pi^{-}$, directly respond to the color electric and color
magnetic fields and receive their boosts at the quark and anti quark level
before they hadronize into pions. These boosts are transmitted to the
$\pi^{+}$ and the $\pi^{-}$.
In order to observe these differential $p_{t}$ changes one must select pairs
on one side of the bubble in $\phi$ and compare to other pairs on the other
side of the same bubble which would lie around $40^{\circ}$ to $48^{\circ}$
away in $\phi$. We defined a pair as a plus particle and minus particle with
an opening angle ($\theta$) of $16^{\circ}$ or less. We are also interested in
pairs that are directly on the other side of the bubble. We require they are
near in pseudorapidity ($\Delta\eta$ $<$ 0.2). The above requirements
constrain the four charged particles comprising both pairs to be contained in
the same bubble and be close to being directly on opposite sides of the bubble
(a final state expanded flux tube). The difference in $p_{t}$ changes due to
and predicted by the color magnetic effect should give the $\pi^{+}$ on one
side of the bubble an increased $p_{t}$ and a decreased $p_{t}$ on the other
side, while for the $\pi^{-}$ it will be the other way around. This will lead
to an anti-alignment between pairs. In Fig. 17 we show two pairs which would
fall into the above cuts. Both pairs are at the limit of the opening angle cut
$\theta_{1}$ and $\theta_{2}$ equal $16^{\circ}$. The $p_{t}$ of the plus
particle for pair number 1 is 1.14 GeV/c, while the minus particle is 1.39
GeV/c. Thus $\Delta P_{t1}$ is equal to -0.25 GeV/c. The $p_{t}$ of the plus
particle for pair number 2 is 1.31 GeV/c, the minus particle is 0.91 GeV/c and
$\Delta P_{t2}$ is equal to 0.40 GeV/c. Finally the mean value shown on Fig.
17 is the mid-point between the $\pi^{+}$ and the $\pi^{-}$ where one really
uses the vector sum of the $\pi^{+}$ and the $\pi^{-}$ which moves this point
toward the harder pion.
Figure 17: Pairs of pairs selected for forming a correlation exhibiting the
effect of changes in $p_{t}$ due to the color magnetic field. The largest
opening angle $\theta$ for plus minus pairs is $16^{\circ}$ or less. This
opening angle assures that each pair has a high probability that it arises
from a quark anti-quark pair. In this figure we have picked two pairs at this
limit ($\theta_{1}$ = $16^{\circ}$ and $\theta_{2}$ = $16^{\circ}$). The mid-
point for pair 1 and 2 represents the vector sum of pair 1 and 2 which moves
toward the harder particle when the momenta differ. These mid-points are
chosen to have $40^{\circ}$ $<$ $|\Delta\phi|$ $<$ $48^{\circ}$ in order for
the pairs to be on opposite sides of the bubble. Since we are interested in
pairs directly across the bubble we make the $\Delta\eta$ separation be no
more than 0.2. The difference in $p_{t}$ for pair 1 is $\Delta P_{t1}$ = -0.25
GeV/c, while the difference in $p_{t}$ for pair 2 is $\Delta P_{t2}$ = 0.40
GeV/c. The minus sign for 1 follows from the fact that the plus particle has
1.14 GeV/c and the minus particle has 1.39 GeV/c. The plus sign for 2 follows
from the fact that the plus particle has 1.31 GeV/c and the minus particle has
0.91 GeV/c.
Considering the above cuts we defined a correlation function where we combine
pairs each having a $\Delta P_{t}$. Our variable is related to the sum of the
absolute values of the individual $\Delta P_{t}$’s ($|\Delta P_{t1}|+|\Delta
P_{t2}|$). We assign a sign to this sum such that if the sign of the
individual $\Delta P_{t}$’s are the same it is a plus sign, while if they are
different it is a minus sign. For the flux tube the color magnetic field
extends over a large pseudorapidity range where quarks and anti-quarks rotate
around the flux tube axis, therefore we want to sample pairs at the different
sides of the tube making a separation in $\phi$ ($\Delta\phi$) between
$40^{\circ}$ to $48^{\circ}$. We are interested in sampling the pairs on the
other side so we require the separation in $\eta$ ($\Delta\eta$) be 0.2 or
less. For the numerator of the correlation function we consider all
combinations of unique pairs (sign ($|\Delta P_{t1}|+|\Delta P_{t2}|$)) from a
given central Au Au event divided by a mixed event denominator created from
pairs in different events. We determine the rescale of the mixed event
denominator by considering the number of pairs of pairs for the case
$|\Delta\eta|$ lying between 1.2 and 1.5 plus any value of $|\Delta\phi|$ for
events and mixed events so that the overall ratio of this sample numerator to
denominator is 1. By picking this $\Delta\eta$ bin for all $|\Delta\phi|$ we
have around the same pair count as the signal cut with the $\Delta\phi$
correlation of the bubbles being washed out. For a simpler notation let (sign
($|\Delta P_{t1}|+|\Delta P_{t2}|$)) = $\Delta P_{t1}+\Delta P_{t2}$ which we
plot in the range from -4 to +4 since we have an over all $p_{t}$ range 0.8 to
4.0 GeV/c. Thus the maximum magnitude of $\Delta P_{t}$’s is 3.2 GeV/c which
makes $\Delta P_{t1}+\Delta P_{t2}$ have a range of $\pm$6.4. However the
larger values near these range limits occur very rarely.
Figure 18: Correlation function of pairs of pairs formed for exhibiting the
effect of changes in $p_{t}$ due to the color magnetic field as defined in the
text. This correlation shows that there are more anti-aligned pairs. The
correlation is larger by $\sim$1.2% for $\Delta P_{t1}+\Delta P_{t2}$ = -4.0
compared to $\Delta P_{t1}+\Delta P_{t2}$ = 4.0. $\Delta P_{t1}+\Delta P_{t2}$
is equal to $|\Delta P_{t1}|+|\Delta P_{t2}|$ where the sign is also explained
in the text.
In Fig. 18 we show the correlation function of opposite sign charged-particle-
pairs paired and binned by the variable $\Delta P_{t1}+\Delta P_{t2}$ with a
cut $|\Delta\eta|$ less than 0.2 between the vector sums of the two pairs, and
with $40^{\circ}$ $<$ $|\Delta\phi|$ $<$ $48^{\circ}$. The events are
generated by the PBMPBM and are charged particles of 0.8 $<$ $p_{t}$ $<$ 4.0
GeV/c, with $|\eta|$ $<$ 1, from Au Au collisions at $\sqrt{s_{NN}}=$ 200 GeV.
Since we select pairs of pairs which are near each other in $\phi$
($40^{\circ}$ to $48^{\circ}$) they all together pick up the bubble
correlation and thus these pairs of pairs over all show about a 0.4%
correlation. In the -4.0 $<$ $\Delta P_{t1}+\Delta P_{t2}$ $<$ -1.0 region the
correlation increases from 0.4% to 1%, while in the 1.0 $<$ $\Delta
P_{t1}+\Delta P_{t2}$ $<$ 4.0 region the correlation decreases from 0.4% to
-0.2%. This means the pairs of pairs are anti-aligned at a higher rate than
aligned. This anti-alignment is what is predicted by the color magnetic field
effect presented above. In fact the anti-alignment increases with $p_{t}$ as
the ratio bubble particle to background increases. However these predicted
color magnetic effects have not been searched for yet and therefore there is
no experimental evidence for them. Experimental investigations of STAR TPC
detector charged particle correlations provide a method of obtaining
experimental confirmation of the anti-alignment caused by the color magnetic
field.
## VIII Summary and Discussion
In this article we summarize the assumptions made and the reasoning that led
to the development and construction of our parton bubble model (PBM)PBM ,
which successfully explained the charge-particle-pair correlations in the
central (0-10% centrality) $\sqrt{s_{NN}}=$ 200 GeV Au Au
datacentralproduction . The PBM is also consistent with the Au Au central
collision HBT results. This is presented and discussed in Sec. 4 of Ref.PBM
and Sec. I and II of this paper. In Sec. III we discussed extending our model,
which was a central collision model, to be able to treat the geomery of bubble
production for 0-80% collision centralities (PBMEPBME ) such as measured and
analyzed in RHIC datacentralitydependence .
The bubbles represent a significant substructure of gluonic hot spots formed
on the surface of a dense opaque fireball at kinetic freezeout. The origins of
these hot spots come from a direct connection between our Parton Bubble Model
(PBMPBM ) and the Glasma Flux Tube Model (GFTM)Dumitru ). In the GFTM a flux
tube is formed right after the initial collision of the Au Au system. This
flux tube extends over many units of pseudorapidity ($\Delta\eta$). A blast
wave gives the tubes near the surface transverse flow in the same way it gave
flow to the bubbles in the PBM. This means the transverse momentum ($p_{t}$)
distribution of the flux tube is directly translated to the $p_{t}$ spectrum.
In the PBM this flux tube is approximated by a sum of partons which are
distributed over this same large $\Delta\eta$ region. Initially the transverse
space is filled with flux tubes of large longitudinal extent but small
transverse size $\sim$$Q^{-1}_{s}$. The flux tubes that are near the surface
of the fireball get the largest radial flow and are emitted from the surface.
As in the parton bubble model these partons shower and the higher $p_{t}$
particles escape the surface and do not interact. $Q_{s}$ is around 1 GeV/c
thus the size of the flux tube is about 1/4 fm initially. The flux tubes near
the surface are initially at a radius $\sim$5 fm. The $\phi$ angle wedge of
the flux tube $\sim$ 1/20 radians or $\sim$$3^{\circ}$. In Sec. V (also see
Sec. IV) we connect the GFTM to the PBM: by assuming the bubbles are the final
state of a flux tube at kinetic freezeout, and discussing evidence for this
connection which is further developed in Sec. VI and Sec. VII.
With the connection of the PBM to the GFTM two new predictions become possible
for our PBM. The first is related to the fact that the blast wave radial flow
given to the flux tube depends on where the tube is initially in the
transverse plane of the colliding Au Au system. The tube gets the same radial
boost all along its longitudinal length. This means that there is correlated
$p_{t}$ among the partons of the bubble. In this paper we consider predictions
we can make in regard to interesting topics and comparisons with relevant data
which exist by utilizing the parton bubble model (PBMPBM ), and its features
related to the glasma flux tube model (GFTM)Dumitru ).
Topic 1: The ridge is treated in Sec. VI.
Topic 2: Strong CP violation (Chern-Simons topological charge) is treated in
Sec. VII.
We show in Sec. VI that if we trigger on particles with 3 to 4 GeV/c and
correlate this trigger particle with an other charged particle of greater than
1.1 GeV/c, the PBM can produce a phenomenon very similar to the ridgeDumitru ;
Armesto ; Romatschke ; Shuryak ; Nara ; Pantuev ; Mizukawa ; Wong ; Hwa . (See
Figs. 6-12). We then selected charged particles inside the ridge and predicted
the correlation that one should observe when compared to the average charged
particles of the central Au Au collisions at $\sqrt{s_{NN}}$ = 200
GeVcentralproduction ; centralitydependence .
In Sec. VI D (Comparison to data): Triggered experimental angular correlations
showing the ridge were presented at Quark Matter 2006Putschke . Figure 7 shows
the experimental $\Delta\phi$ vs. $\Delta\eta$ CI correlation for 0-10%
central Au Au collisions at $\sqrt{s_{NN}}$ = 200; requiring one trigger
particle $p_{t}$ between 3 to 4 GeV/c and an associated particle $p_{t}$ above
2.0 GeV/c. The yield is corrected for the finite $\Delta\eta$ pair acceptance.
For the PBM generator, we then form a two charged particle correlation between
one charged particle with a $p_{t}$ between 3.0 to 4.0 GeV/c and another
charged particle whose $p_{t}$ is greater than 2.0 GeV/c. These are the same
trigger conditions as in Ref.Putschke which is shown in Fig. 11, that shows
the corrected pair yield in the central data. Fig. 12 shows the correlation
function generated by the PBM which does not depend on the number of events
analyzed. The two figures were shown to be in reasonable agreement when
compared as explained previously in Sec. VI D. In Fig. 13 we show the ridge
signal predicted by the PBM for very similar data but with 0-5% centrality.
Figure 14 shows the extraction of the jet signal. Explanations are given in
the text.
The second prediction is a development of a predictive pionic measure of the
strong CP Violation. The GFTM flux tubes are made up of longitudinal color
electric and magnetic fields which generate topological Chern-Simons
chargeSimons through the $F\widetilde{F}$ term that becomes a source of CP
violation. The color electric field which points along the flux tube axis
causes an up quark to be accelerated in one direction along the beam axis,
while the anti-up quark is accelerated in the other direction. So when a pair
of quarks and anti-quarks are formed they separate along the beam axis leading
to a separated $\pi^{+}$ and $\pi^{-}$ pair along this axis. The color
magnetic field which also points along the flux tube axis (which is parallel
to the beam axis) causes an up quark to rotate around the flux tube axis in
one direction, while the anti-up quark rotates in the other direction. So when
a pair of quarks and anti-quarks are formed they will pickup or lose
transverse momentum. These changes in $p_{t}$ will be transmitted to the
$\pi^{+}$ and $\pi^{-}$ pairs.
The above pionic measures of strong CP violation are used to form correlation
functions based on four particles composed of two pairs which are opposite
sign charge-particle-pairs that are paired and binned. These four particle
correlations accumulate from bubble to bubble by particles that are pushed or
pulled (by the color electric field) and rotated (by the color magnetic field)
in a right or left handed direction. The longitudinal color electric field
predicts aligned pairs in a pseudorapidity or $\Delta\eta$ measure. The
longitudinal color magnetic field predicts anti-aligned pairs in a transverse
momentum or $\Delta P_{t}$ measure. The observations of these correlations
would be a strong confirmation of this theory. The much larger unlike-sign
pairs than like-sign pairs in the PBM and the data; and the strong dip of the
CI correlation at small $\Delta\eta$ (see Fig. 4 and Sec. VII C for full
details) shows very strong evidence supporting the color electric alignment
prediction in the $\sqrt{s_{NN}}$ = 200 GeV central Au Au collision data
analyses at RHICPBM ; centralproduction . This highly significant dip
($\sim$$14\sigma$) means that like-sign pairs are removed as one approaches
the region $\Delta\eta$ = $\Delta\phi$ = 0.0. Thus this is very strong
evidence for the predicted effect of the color electric field. The color
magnetic anti-aligned pairs in the transverse momentum prediction, treated in
Sec. VII D as of now has not been observed or looked for. However our
predicted specific four charged particle correlations can be used to search
for experimental evidence for the color magnetic fields.
Our success in demonstrating strong experimental evidence for the expected
color electric field effects from previously published data suggests that the
unique detailed correlations we have presented for searching for evidence for
the predicted color magnetic field effects should be urgently investigated. If
we are lucky and the predicted color magnetic effects can be confirmed
experimentally we would have strong evidence for the following:
1) CP is violated in the strong interaction in isolated local space time
regions where topological chargeSimons is generated.
2) The glasma flux tube model (GFTM) which was evolved from the color glass
condensate (CGC) would be found to be consistent with a very significant
experimental check.
3) The parton bubble model event generator (PBM) is clearly closely connected
to the GFTM. The bubble substructure strongly supported by the PBM is likely
due to the final state of the flux tube at kinetic freezeout.
## IX Acknowledgments
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886. The author thank William Love for valuable discussion
and assistance in production of figures. Finally this manuscript is in memory
of my fellow collaborator Prof. Sam Lindenbaum which help form many of the
ideas of bubbles in heavy ion collisions.
## References
* (1) L. Van Hove, Z. Phys. C 27, 135 (1985).
* (2) S.J. Lindenbaum, R.S. Longacre, J. Phys. G 26, 937 (2000).
* (3) D. Kharzeev, R.D. Pisarski, Phys. Rev. D 61, 111901 (2000).
* (4) C. Adler et al., Phys. Rev. Lett. 87, 082301 (2001).
* (5) S.J. Lindenbaum, R.S. Longacre M. Kramer, Eur. Phys. J. C. 30, 241 (2003).
* (6) S.J. Lindenbaum, R.S. Longacre, Eur. Phys. J. C. 49, 767-782 (2007).
* (7) J. Adams et al., Phys. Rev. C. 71, 044906 (2005), S.S. Adler et al., Phys Rev. Lett. 93, 152302 (2004).
* (8) X.N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501 (1991).
* (9) K. Adcox et al., Phys. Rev. Lett. 88, 022301 (2001).
* (10) Nucl. Instrum. Meth. A 499,(2003), C. Adler et al., 433-436, K.H. Ackermann et al.(STAR Collaboration), 624-632, M. Anderson et al., 659-678, F.S. Bieser et al., 766-777.
* (11) T. Sjostrand, M. van Zijil, Phys. Rev. D 36, 2019 (1987).
* (12) J. Adams et al., Phys. Rev. C 75, 034901 (2007).
* (13) S.J. Lindenbaum and R.S. Longacre, Phys. Rev. C 78, 054904 (2008).
* (14) B.I. Abelev et al., arXiv:0806.0513[nucl-ex].
* (15) J. Adams et al., Phys. Rev. Lett. 91, 172302 (2003).
* (16) A. Adare et al., Rev. Lett. 101, 232301 (2008).
* (17) A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan, Nucl. Phys. A 810, 91 (2008).
* (18) L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994); Phys. Rev. D 49, 3352 (1994); Phys. Rev. D 50, 2225 (1994).
* (19) T. Lappi and L. McLerran, Nucl. Phys. A 772, 200 (2006).
* (20) F. Gelis and R. Venugopalan, Acta Phys. Polon. B 37, 3253 (2006).
* (21) D. Kharzeev, A. Krasnitz and R. Venugopalan, Phys. Lett. B 545, 298 (2002).
* (22) S. Gavin, L. McLerran and G. Moschelli, Eur. Phys. J. C. 62, 277-280 (2009).
* (23) N. Armesto, C. Salgado, U.A. Wiedemann, Phys Rev. Lett. 93, 242301 (2004).
* (24) P. Romatschke, Phys. Rev. C. 75, 014901 (2007).
* (25) E. Shuryak, Phys. Rev. C. 76, 047901 (2007).
* (26) A. Dumitru, Y. Nara, B. Schenke, M. Strickland, Phys. Rev. C 78, 024909 (2008).
* (27) V.S. Pantuev, arXiv:0710.1882[hep-ph].
* (28) R. Mizukawa, T. Hirano, M. Isse, Y. Nara, A. Ohnishi, J. Phys. G 35, 104083 (2008).
* (29) C.Y. Wong, Phys. Rev. C. 78, 064905 (2008)..
* (30) R.C. Hwa, arXiv:0708.1508[nucl-th].
* (31) J. Putschke, J. Phys. G Phys. 34, S679 (2007).
* (32) C. Vafa, E. Witten, Phys. Rev. Lett. 53, 535 (1984), Nucl. Phys. B 234, 173 (1984).
* (33) D. Kharzeev, R.D. Pisarski, M.H.G. Tytgat, Phys. Rev. Lett. 81, 512 (1998).
* (34) C. Vafa, E. Witten, Nucl. Phys. B 233, 422 (1983).
* (35) J. Wess, and B. Zumino, Phys. Lett. 37B, 95 (1971)
* (36) L.E. Finch et al., Phys. Rev. C 65, 014908 (2001).
|
arxiv-papers
| 2010-01-07T18:46:48 |
2024-09-04T02:49:07.581898
|
{
"license": "Public Domain",
"authors": "R. S. Longacre",
"submitter": "Ron S. Longacre",
"url": "https://arxiv.org/abs/1001.1132"
}
|
1001.1186
|
# A Bivariate Preprocessing Paradigm for Buchberger-Möller Algorithm
Xiaoying Wang, Shugong Zhang Tian Dong dongtian@jlu.edu.cn School of
Mathemathics, Key Lab. of Symbolic Computation and Knowledge Engineering
(Ministry of Education), Jilin University, Changchun 130012, PR China
###### Abstract
For the last almost three decades, since the famous Buchberger-Möller(BM)
algorithm emerged, there has been wide interest in vanishing ideals of points
and associated interpolation polynomials. Our paradigm is based on the theory
of bivariate polynomial interpolation on cartesian point sets that gives us
related degree reducing interpolation monomial and Newton bases directly.
Since the bases are involved in the computation process as well as contained
in the final output of BM algorithm, our paradigm obviously simplifies the
computation and accelerates the BM process. The experiments show that the
paradigm is best suited for the computation over finite prime fields that have
many applications.
###### keywords:
Buchberger-Möller algorithm , Bivariate Lagrange interpolation , Degree
reducing interpolation space , Cartesian set
###### MSC:
13P10 , 65D05 , 12Y05
††journal: Journal of Computational and Applied
Mathematicsfundfundfootnotetext: This work was supported in part by the
National Grand Fundamental Research 973 Program of China (No. 2004CB318000).
## 1 Introduction
For an arbitrary field $\mathbb{F}$, we let $\mathbb{F}_{q}$ a finite prime
field of size $q$ and $\Pi^{d}:=\mathbb{F}[x_{1},\ldots,x_{d}]$ the
$d$-variate polynomial ring over $\mathbb{F}$. Given a preassigned set of
distinct affine points $\Xi\subset\mathbb{F}^{d}$, it is well-known that the
set of all polynomials in $\Pi^{d}$ vanishing at $\Xi$ constitutes a radical
zero-dimensional ideal, denoted by $\mathcal{I}(\Xi)$, which is called the
_vanishing ideal_ of $\Xi$.
Recent years, there has been considerable interest in vanishing ideals of
points in many branches of mathematics such as algebraic geometry[1],
multivariate interpolation[2, 3], coding theory[4, 5], statistics[6], and even
computational molecular biology[7, 8]. As is well known, the most significant
milestone of the computation of vanishing ideals is the algorithm presented in
[9] by Hans Michael Möller and Bruno Buchberger known as Buchberger-Möller
algorithm(BM algorithm for short). For any point set
$\Xi\subset\mathbb{F}^{d}$ and fixed term order $\prec$, BM algorithm yields
the reduced Gröbner basis for $\mathcal{I}(\Xi)$ w.r.t. $\prec$ and a
$\prec$-degree reducing interpolation Newton basis for $d$-variate Lagrange
interpolation on $\Xi$. The algorithm also produces the Gröbner éscalier of
$\mathcal{I}(\Xi)$ w.r.t. $\prec$ as a byproduct. Afterwards, in 1993, BM
algorithm was applied in [10] in order to solve the renowned FGLM-problem. In
the same year, [11] merged BM and FGLM algorithms into four variations that
can solve more general zero-dimensional ideals therefore related ideal
interpolation problems [3]. The algorithms are referred as MMM algorithms.
Although very important, BM algorithm (and MMM algorithms) has a very poor
complexity that limits its applications. In this decade, many authors proposed
new algorithms that can reduce the complexity but mostly suitable for special
cases. [12] presented a modular version of BM algorithm that is best suited to
the computation over $\mathbb{Q}$. [13, 14, 15] presented algorithms for
obtaining, with relatively little effort, the Gröbner éscalier of a vanishing
ideal w.r.t. the (inverse) lexicographic order that can lead to an
interpolation Newton basis or the reduced Gröbner basis for the vanishing
ideal after solving a linear system.
For a fixed point set $\Xi$ in $\mathbb{F}^{d}$ and a term order $\prec$, it
is well known that there are _two_ factors that determine the Gröbner éscalier
of $\mathcal{I}(\Xi)$ w.r.t. $\prec$ thereby the reduced Gröbner basis for
$\mathcal{I}(\Xi)$ and related degree reducing interpolation Newton bases (up
to coefficients). One is apparently the cardinal of $\Xi$. It is the unique
determinate factor in univariate cases. Another one is the geometry (the
distribution of the points) of $\Xi$ that is dominating in multivariate cases
but not taken into consideration by BM and MMM algorithms. Recent years, [16,
17, 18] studied multivariate Lagrange interpolation on a special kind of point
sets, cartesian point sets (aka lower point sets), and constructed the
associated Gröbner éscalier and degree reducing interpolation Newton bases
theoretically. We know from [9, 11] that, for a cartesian subset of $\Xi$ (it
always exists!), certain associated degree reducing interpolation Newton basis
forms part of the output of BM algorithm w.r.t. some reordering of $\Xi$.
Therefore, finding a large enough cartesian subset of $\Xi$ with little enough
effort will reduce the complexity of BM algorithm.
Following this idea, the paper proposes a preprocessing paradigm for BM
algorithm with the organization as follows. The next section is devoted as a
preparation for the paper. And then, the main results of us are presented in
two sections. Section 3 will pursue the paradigm for two special term orders
while Section 4 will set forth our solution for other more general cases. In
the last section, Section 5, some implementation issues and experimental
results will be illustrated.
## 2 Preliminary
In this section, we will introduce some notation and recall some basic facts
for the reader’s convenience. For more details, we refer the reader to [19,
20].
We let $\mathbb{N}_{0}$ denote the monoid of nonnegative integers. A
_polynomial_ $f\in\Pi^{2}$ is of the form
$f=\sum_{\bm{\alpha}\in\mathbb{N}_{0}^{2}}f_{\bm{\alpha}}X^{\bm{\alpha}},\hskip
22.76228pt\\#\\{\bm{\alpha}\in\mathbb{N}_{0}^{2}:0\neq
f_{\bm{\alpha}}\in\mathbb{F}\\}<\infty,$
where _monomial_ $X^{\bm{\alpha}}=x^{\alpha_{1}}y^{\alpha_{2}}$ with
$\bm{\alpha}=(\alpha_{1},\alpha_{2})$. The set of bivariate monomials in
$\Pi^{2}$ is denoted by $\mathbb{T}^{2}$.
Fix a term order $\prec$ on $\Pi^{2}$ that may be lexicographical order
$\prec_{\text{lex}}$, inverse lexicographical order $\prec_{\text{inlex}}$, or
total degree inverse lexicographical order $\prec_{\text{tdinlex}}$ etc. For
all $f\in\Pi^{2}$, with $f\neq 0$, we may write
$f=f_{\bm{\gamma_{1}}}X^{\bm{\gamma_{1}}}+f_{\bm{\gamma_{2}}}X^{\bm{\gamma_{2}}}+\cdots+f_{\bm{\gamma_{r}}}X^{\bm{\gamma_{r}}},$
where $0\neq
f_{\bm{\gamma_{i}}}\in\mathbb{F},\bm{\gamma_{i}}\in\mathbb{N}_{0}^{2},i=1,\ldots,r$,
and $X^{\bm{\gamma_{1}}}\succ X^{\bm{\gamma_{2}}}\succ\cdots\succ
X^{\bm{\gamma_{r}}}$. We shall call
$\mathrm{LT}(f):=f_{\bm{\gamma_{1}}}X^{\bm{\gamma_{1}}}$ the _leading term_
and $\mathrm{LM}(f):=X^{\bm{\gamma_{1}}}$ the _leading monomial_ of $f$.
Furthermore, for a non-empty subset $F\subset\Pi^{2}$, put
$\mathrm{LT}(F):=\\{\mathrm{LT}(f):f\in F\\}.$
As in [21], we define the $\prec\mspace{-8.0mu}-$_degree_ of a polynomial
$f\in\Pi^{2}$ to be the leading bidegree w.r.t. $\prec$
$\delta(f):=\bm{\gamma},\quad X^{\bm{\gamma}}=\mathrm{LM}(f),$
with $\delta(0)$ undefined. Further, for any finite dimensional subset
$F\subset\Pi^{2}$, define
$\delta(F):=\max_{f\in F}\delta(f).$
Finally, for any $f,g\in\Pi^{2}$, if $\delta(f)\prec\delta(g)$ then we say
that $f$ is of _lower degree_ than $g$ and use the abbreviation
$f\prec g:=\delta(f)\prec\delta(g).$
In addition, $f\preceq g$ is interpreted as the degree of $f$ is lower than or
equal to that of $g$.
Let $\mathcal{A}$ be a finite subset of $\mathbb{N}_{0}^{2}$. $\mathcal{A}$ is
called a _lower_ set if, for any
$\bm{\alpha}=(\alpha_{1},\alpha_{2})\in\mathcal{A}$, we always have
$\mathrm{R}(\bm{\alpha}):=\\{(\alpha^{\prime}_{1},\alpha^{\prime}_{2})\in\mathbb{N}_{0}^{2}:0\leq\alpha^{\prime}_{i}\leq\alpha_{i},i=1,2\\}\subset\mathcal{A}.$
Especially, $\bm{0}\in\mathcal{A}$. Moreover, we set
$m_{j}=\max_{(h,j)\in\mathcal{A}}h,0\leq j\leq\nu$, with
$\nu=\max_{(0,k)\in\mathcal{A}}k$. Clearly, $\mathcal{A}$ can be determined
uniquely by the ordered $(\nu+1)$-tuple $(m_{0},m_{1},\ldots,m_{\nu})$ hence
represented as $\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$. Swapping the
roles of $x$ and $y$, we can also represent $\mathcal{A}$ as
$\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{m_{0}})$ with
$n_{i}=\max_{(i,k)\in\mathcal{A}}k,0\leq i\leq m_{0}$. It should be noticed
that $\nu=n_{0}$.
Given a set $\Xi=\\{\xi^{(1)},\ldots,\xi^{(\mu)}\\}\subset\mathbb{F}^{2}$ of
$\mu$ distinct points. For prescribed values
$f_{i}\in\mathbb{F},i=1,\ldots,\mu$, find all polynomials $p\in\Pi^{2}$
satisfying
$p(\xi^{(i)})=f_{i},\quad i=1,\ldots,\mu.$ (1)
We call it the problem of _bivariate Lagrange interpolation_. Note that in
most cases, especially from a numerical point of view, we are not interested
in all such $p$’s but a “degree reducing” one, as in the univariate cases.
###### Definition 1.
[2] Fix term order $\prec$. We call a subspace $\mathcal{P}\subset\Pi^{2}$ a
_degree reducing interpolation space_ w.r.t. $\prec$ for the bivariate
Lagrange interpolation (1) if
DR1.
$\mathcal{P}$ is an _interpolation space_ , i.e., for any
$f_{i}\in\mathbb{F},i=1,\ldots,\mu$, there is a unique $p\in\mathcal{P}$ such
that $p$ satisfies (1). In other words, the interpolation problem is _regular_
w.r.t. $\mathcal{P}$.
DR2.
$\mathcal{P}$ is $\prec\mspace{-8.0mu}-$_reducing_ , i.e., when
$L_{\mathcal{P}}$ denotes the Lagrange projector with range $\mathcal{P}$,
then the interpolation polynomial
$L_{\mathcal{P}}q\preceq q,\hskip 14.22636pt\forall q\in\Pi^{2}.$
For interpolation problem (1), a given interpolation space
$\mathcal{P}\subset\Pi^{2}$ will give rise to an _interpolation scheme_ that
is referred as $(\Xi,\mathcal{P})$, cf. [20]. Since (1) is regular w.r.t.
$\mathcal{P}$, we can also say that $(\Xi,\mathcal{P})$ is regular. Moreover,
if $\mathcal{P}$ is degree reducing w.r.t. $\prec$, a basis
$\\{p_{1},\ldots,p_{\mu}\\}$ for $\mathcal{P}$ will be called a _degree
reducing interpolation basis_ w.r.t. $\prec$ for (1). Assume that $p_{1}\prec
p_{2}\prec\cdots\prec p_{\mu}$. If
$p_{j}(\xi^{(i)})=\delta_{ij},\quad 1\leq i\leq j\leq\mu,$
for some suitable reordering of $\Xi$, then we call
$\\{p_{1},\ldots,p_{\mu}\\}$ a _degree reducing interpolation Newton
basis_(DRINB) w.r.t. $\prec$ for (1).
Let $G_{\prec}$ be the reduced Gröbner basis for the vanishing ideal
$\mathcal{I}(\Xi)$ w.r.t.$\prec$. The set
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi)):=\\{x^{\bm{\alpha}}\in\mathbb{T}^{2}:\mathrm{LT}(g)\nmid
x^{\bm{\alpha}},\forall g\in G_{\prec}\\}$
is called the _Gröbner éscalier_ of $\mathcal{I}(\Xi)$ w.r.t. $\prec$. From
[2, 21], the interpolation space spanned by
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$, denoted by $\mathcal{P}_{\prec}(\Xi)$,
is canonical since it is the unique degree reducing interpolation space
spanned by monomials w.r.t. $\prec$ for (1). Hence, we call
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$ the _degree reducing interpolation
monomial basis_(DRIMB) w.r.t. $\prec$ for (1), with
$\\#\mathrm{N}_{\prec}(\mathcal{I}(\Xi))=\mu$. Let
$\mathrm{N}_{\prec}(\Xi):=\\{\bm{\alpha}:x^{\bm{\alpha}}\in\mathrm{N}_{\prec}(\mathcal{I}(\Xi))\\}\subset\mathbb{N}_{0}^{2}.$
We can deduce easily that $\mathrm{N}_{\prec}(\Xi)$ is a lower set and
obviously has a one-to-one correspondence with
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$. Therefore, interpolation scheme
$(\Xi,\mathcal{P}_{\prec}(\Xi))$ can be equivalently represented as
$(\Xi,\mathrm{N}_{\prec}(\Xi))$.
According to [17], we can construct two particular lower sets from $\Xi$,
denoted by $S_{x}(\Xi),S_{y}(\Xi)$, which reflect the geometry of $\Xi$ in
certain sense.
Specifically, we cover the points in $\Xi$ by lines
$l_{0}^{x},l_{1}^{x},\ldots,l_{\nu}^{x}$ parallel to the $x$-axis and assume
that, without loss of generality, there are $m_{j}+1$ points, say
$u_{0j}^{x},u_{1j}^{x},\ldots,u_{m_{j},j}^{x}$, on $l_{j}^{x}$ with $m_{0}\geq
m_{1}\geq\cdots\geq m_{\nu}\geq 0$ hence the ordinates of $u_{ij}^{x}$ and
$u_{i^{\prime}j}^{x},i\neq i^{\prime}$, same. Now, we set
$S_{x}(\Xi):=\\{(i,j):0\leq i\leq m_{j},\ 0\leq j\leq\nu\\},$
which apparently equals to $\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$. We
can also cover the points by lines
$l_{0}^{y},l_{1}^{y},\ldots,l_{\lambda}^{y}$ parallel to the $y$-axis and
denote the points on line $l_{i}^{y}$ by
$u_{i0}^{y},u_{i1}^{y},\ldots,u_{i,n_{i}}^{y}$ with $n_{0}\geq
n_{1}\geq\cdots\geq n_{\lambda}\geq 0$ hence the abscissae of $u_{ij}^{y}$ and
$u_{ij^{\prime}}^{y},j\neq j^{\prime}$, same. Similarly, we put
$S_{y}(\Xi):=\\{(i,j):0\leq i\leq\lambda,\ 0\leq j\leq
n_{i}\\}=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda}).$
In addition, we can also define the sets of abscissae and ordinates
$\displaystyle H_{j}(\Xi):=$ $\displaystyle\\{\bar{x}:(\bar{x},\bar{y})\in
l_{j}^{x}\cap\Xi\\},\quad 0\leq j\leq\nu,$ (2) $\displaystyle V_{i}(\Xi):=$
$\displaystyle\\{\bar{y}:(\bar{x},\bar{y})\in l_{i}^{y}\cap\Xi\\},\quad 0\leq
i\leq\lambda.$
###### Definition 2.
[17] We say that a set $\Xi$ of distinct points in $\mathbb{F}^{2}$ is
_cartesian_ if there exists a lower set $\mathcal{A}$ such that $\Xi$ can be
written as
$\Xi=\\{(x_{i},y_{j}):(i,j)\in\mathcal{A}\\},$
where the $x_{i}$’s are distinct numbers, and similarly the $y_{j}$’s. We also
say that $\Xi$ is $\mathcal{A}$-cartesian.
To the best of our knowledge, there are two criteria for determining whether a
2-dimensional point set is cartesian.
###### Theorem 1.
[17] A set of distinct points $\Xi\subset\mathbb{F}^{2}$ is cartesian if and
only if $S_{x}(\Xi)=S_{y}(\Xi)$.
###### Theorem 2.
[18] A set of distinct points $\Xi\subset\mathbb{F}^{2}$ is cartesian if and
only if
$H_{0}(\Xi)\supseteq H_{1}(\Xi)\supseteq\cdots\supseteq H_{\nu}(\Xi),\quad
V_{0}(\Xi)\supseteq V_{1}(\Xi)\supseteq\cdots\supseteq V_{\lambda}(\Xi).$
About the bivariate Lagrange interpolation on a cartesian set, [17] proved the
succeeding theorem.
###### Theorem 3.
[17] Given a cartesian set $\Xi\subset\mathbb{F}^{2}$, there exists a unique
lower set $\mathcal{A}\in\mathbb{N}_{0}^{2}$ such that $\Xi$ is
$\mathcal{A}$-cartesian and the Lagrange interpolation scheme
$(\Xi,\mathcal{A})$ is regular.
Finally, we will redescribe the classical BM algorithm with the notation
established above.
###### Algorithm 1.
(BM Algorithm)
Input: A set of distinct points
$\Xi=\\{\xi^{(i)}:i=1,\ldots,\mu\\}\subset\mathbb{F}^{d}$ and a fixed term
order $\prec$.
Output: The 3-tuple $(G,N,Q)$, where $G$ is the reduced Gröbner basis for
$\mathcal{I}(\Xi)$ w.r.t. $\prec$, $N$ is the Gröbner éscalier of
$\mathcal{I}(\Xi)$ (the DRIMB for (1) also) w.r.t. $\prec$, and $Q$ is a DRINB
w.r.t. $\prec$ for (1).
BM1. Start with lists $G=[\ ],N=[\ ],Q=[\ ],L=[1]$, and a matrix $B=(b_{ij})$
over $\mathbb{F}$ with $\mu$ columns and zero rows initially.
BM2. If $L=[\ ]$, return $(G,N,Q)$ and stop. Otherwise, choose the monomial
$t=\mbox{min}_{\prec}L$, and delete $t$ from $L$.
BM3. Compute the evaluation vector $(t(\xi^{(1)}),\ldots,t(\xi^{(\mu)}))$, and
reduce it against the rows of $B$ to obtain
$(v_{1},\ldots,v_{\mu})=(t(\xi^{(1)}),\ldots,t(\xi^{(\mu)}))-\sum_{i}a_{i}(b_{i1},\ldots,b_{i\mu}),\quad
a_{i}\in\mathbb{F}.$
BM4.. If $(v_{1},\ldots,v_{\mu})=(0,\ldots,0)$, then append the polynomial
$t-\sum_{i}a_{i}q_{i}$ to the list $G$, where $q_{i}$ is the $i$th element of
$Q$. Remove from $L$ all the multiples of $t$. Continue with BM2.
BM5. Otherwise $(v_{1},\ldots,v_{\mu})\neq(0,\ldots,0)$, add
$(v_{1},\ldots,v_{\mu})$ as a new row to $B$ and $t-\sum_{i}a_{i}q_{i}$ as a
new element to $Q$. Append the monomial $t$ to $N$, and add to $L$ those
elements of $\\{x_{1}t,\ldots,x_{d}t\\}$ that are neither multiples of an
element of $L$ nor of $\mathrm{LT}(G)$. Continue with BM2.
## 3 Special cases
In this section, we will focus on $\prec_{\mathrm{lex}}$ and
$\prec_{\mathrm{inlex}}$ that may be the most talked about term orders. For
these special cases, our preprocessing paradigm will first provide exact $N,Q$
of the 3-tuple output $(G,N,Q)$ to BM algorithm directly and effortlessly. And
then, $G$ can be obtained by BM algorithm easily. Note that we will continue
with all the notation that we established for $S_{x}(\Xi)$ and $S_{y}(\Xi)$ in
the previous section.
###### Proposition 4.
Let $\Xi$ be a set of $\mu$ distinct points
$u_{mn}^{x}=(x_{mn},y_{mn})\in\mathbb{F}^{2},(m,n)\in S_{x}(\Xi)$. The points
give rise to polynomials
$\phi_{ij}^{x}=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y-y_{0t})\prod_{s=0}^{i-1}(x-x_{sj}),\quad(i,j)\in
S_{x}(\Xi),$ (3)
where
$\varphi_{ij}^{x}=1/\prod_{t=0}^{j-1}(y_{0j}-y_{0t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj})\in\mathbb{F}$,
and the empty products are taken as 1. Then we have
$\phi_{ij}^{x}(u_{mn}^{x})=\delta_{(i,j),(m,n)},\quad(i,j)\succeq_{\mathrm{inlex}}(m,n).$
###### Proof 1.
Fix $(i,j)\in S_{x}(\Xi)$. Recalling the definition of $u_{ij}^{x}$, we have
$y_{0j}=y_{ij}$. If $(i,j)=(m,n)$, by $y_{00}\neq y_{01}\neq\cdots\neq y_{0j}$
and $x_{0j}\neq x_{1j}\neq\cdots\neq x_{ij}$, we have
$\phi_{ij}^{x}(u_{ij}^{x})=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{ij}-y_{0t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj})=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{0j}-y_{0t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj}),$
which implies $\phi_{ij}^{x}(u_{ij}^{x})=1$.
Otherwise, if $(i,j)\succ_{\mathrm{inlex}}(m,n)$, we have $j>n$, or $j=n,i>m$.
When $j>n$, we have
$\displaystyle\phi_{ij}^{x}(u_{mn}^{x})$
$\displaystyle=\varphi_{ij}^{x}(y_{mn}-y_{00})\cdots(y_{mn}-y_{0n})\cdots(y_{mn}-y_{0,j-1})\prod_{s=0}^{i-1}(x_{mn}-x_{sj})$
$\displaystyle=\varphi_{ij}^{x}(y_{0n}-y_{00})\cdots(y_{0n}-y_{0n})\cdots(y_{0n}-y_{0,j-1})\prod_{s=0}^{i-1}(x_{mn}-x_{sj})$
$\displaystyle=0,$
and when $j=n,i>m$,
$\displaystyle\phi_{ij}^{x}(u_{mn}^{x})$
$\displaystyle=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{mn}-y_{0t})(x_{mn}-x_{0j})\cdots(x_{mn}-x_{mj})\cdots(x_{mn}-x_{i-1,j})$
$\displaystyle=\varphi_{ij}^{x}\prod_{t=0}^{n-1}(y_{mn}-y_{0t})(x_{mn}-x_{0n})\cdots(x_{mn}-x_{mn})\cdots(x_{mn}-x_{i-1,n})$
$\displaystyle=0,$
which leads to
$\phi_{ij}^{x}(u_{mn}^{x})=0,\quad(i,j)\succ_{\mathrm{inlex}}(m,n).$
∎
Similarly, we can prove the following proposition:
###### Proposition 5.
Let $\Xi$ be a set of $\mu$ distinct points
$u_{mn}^{y}=(x_{mn},y_{mn})\in\mathbb{F}^{2},(m,n)\in S_{y}(\Xi)$. We define
the polynomials
$\phi_{ij}^{y}=\varphi_{ij}^{y}\prod_{s=0}^{i-1}(x-x_{s0})\prod_{t=0}^{j-1}(y-y_{it}),\quad(i,j)\in
S_{y}(\Xi),$ (4)
where
$\varphi_{ij}^{y}=1/\prod_{s=0}^{i-1}(x_{i0}-x_{s0})\prod_{t=0}^{j-1}(y_{ij}-y_{it})\in\mathbb{F}$.
The empty products are taken as 1. Then,
$\phi_{ij}^{y}(u_{mn}^{y})=\delta_{(i,j),(m,n)},\quad(i,j)\succeq_{\mathrm{lex}}(m,n).$
In 2004, [17] proved that the Lagrange interpolation schemes
$(\Xi,S_{x}(\Xi))$ and $(\Xi,S_{y}(\Xi))$ are both regular. Here we reprove
the regularities in another way for the purpose of presenting the degree
reducing interpolation bases theoretically .
###### Theorem 6.
Resume the notation in _Proposition 4_ and _5_. Then the Lagrange
interpolation schemes $(\Xi,S_{x}(\Xi))$ and $(\Xi,S_{y}(\Xi))$ are regular.
Furthermore,
(i) the set $N_{x}:=\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi)\\}$ is the _DRIMB_ as
well as $Q_{x}:=\\{\phi_{ij}^{x}:(i,j)\in S_{x}(\Xi)\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{lex}}$ for the interpolation problem (1).
(ii) the set $N_{y}:=\\{x^{i}y^{j}:(i,j)\in S_{y}(\Xi)\\}$ is the _DRIMB_ as
well as $Q_{y}:=\\{\phi_{ij}^{y}:(i,j)\in S_{y}(\Xi)\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{inlex}}$ for (1).
###### Proof 2.
We only give the proof for $S_{x}(\Xi)$. The statements about $S_{y}(\Xi)$ can
be proved likewise.
First, we will show the regularity of the interpolation scheme
$(\Xi,S_{x}(\Xi))$. Let
$\mathcal{P}_{x}:=\mathrm{Span}_{\mathbb{F}}N_{x}\subset\Pi^{2}$ with
$\dim\mathcal{P}_{x}=\\#\Xi=\mu$. Obviously, $N_{x}$ is the monomial basis for
it. By (3), we can check easily that
$\mathrm{Span}_{\mathbb{F}}Q_{x}\subseteq\mathcal{P}_{x}.$
Construct a square matrix $B_{\mu\times\mu}$ whose $(h,k)$ entry is
$\phi_{h}^{x}(u_{k}^{x})$ where $\phi_{h}^{x},u_{k}^{x}$ are $h$th and $k$th
elements of $Q_{x}$ and $\Xi=\\{u_{mn}^{x}:(m,n)\in S_{x}(\Xi)\\}$ w.r.t. the
increasing $\prec_{\mathrm{inlex}}$ on $(i,j)$ and $(m,n)$ respectively. From
Proposition 4, $B_{\mu\times\mu}$ is upper unitriangular which implies that
$\mathrm{Span}_{\mathbb{F}}Q_{x}=\mathcal{P}_{x}$ and $Q_{x}$ forms a Newton
basis for $\mathcal{P}_{x}$. It follows that $\mathcal{P}_{x}$ is an
interpolation space for Lagrange interpolation (1) therefore the scheme
$(\Xi,\mathcal{P}_{x})$ is regular. Since
$(\Xi,S_{x}(\Xi))=(\Xi,\mathcal{P}_{x})$, according to Section 2,
$(\Xi,S_{x}(\Xi))$ is regular.
Next, we shall verify that the statements in (i), which is equivalent to the
statement that $\mathcal{P}_{x}$ is a degree reducing interpolation space
w.r.t. $\prec_{\mathrm{lex}}$ for (1) that coincides with
$\mathcal{P}_{\prec_{\mathrm{lex}}}(\Xi)$. Since the arguments above have
proved that $\mathcal{P}_{x}$ satisfies the DR1 condition in Definition 1,
what is left for us is to check the DR2 condition. From [21], we only need to
check it for monomials.
Take a monomial $x^{i_{0}}y^{j_{0}}\in\mathbb{T}^{2}$. We shall prove that
$L_{\mathcal{P}_{x}}x^{i_{0}}y^{j_{0}}\preceq_{\mathrm{lex}}x^{i_{0}}y^{j_{0}}.$
(5)
Since $\mathcal{P}_{x}$ satisfies DR1, $L_{\mathcal{P}_{x}}x^{i_{0}}y^{j_{0}}$
is the unique polynomial in $\mathcal{P}_{x}$ that matches
$x^{i_{0}}y^{j_{0}}$ on $\Xi$. Therefore, when $x^{i_{0}}y^{j_{0}}\in N_{x}$,
we have $L_{\mathcal{P}_{x}}x^{i_{0}}y^{j_{0}}=x^{i_{0}}y^{j_{0}}$ , namely
(5) is true for this case. Assume that
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},\ldots,m_{n_{0}})=\mathrm{L}_{y}(n_{0},\ldots,n_{m_{0}}).$
It is easy to see that $\delta(\mathcal{P}_{x})=(m_{0},n_{m_{0}})$. If
$x^{m_{0}}y^{n_{m_{0}}}\prec_{\mathrm{lex}}x^{i_{0}}y^{j_{0}}$ then
$\delta(L_{\mathcal{P}_{x}}x^{i_{0}}y^{j_{0}})\preceq_{\mathrm{lex}}\delta(\mathcal{P}_{x})=(m_{0},n_{m_{0}})\prec_{\mathrm{lex}}(i_{0},j_{0})=\delta(x^{i_{0}}y^{j_{0}})$
that leads to (5) for the case.
Thus, what remains for us is to check (5) for $x^{i_{0}}y^{j_{0}}\notin N_{x}$
with $(i_{0},j_{0})\prec_{\mathrm{lex}}(m_{0},n_{m_{0}})$ that implies $0\leq
i_{0}<m_{0},j_{0}>n_{i_{0}}$. For this, we only need to verify that
$L_{\mathcal{P}_{x}}x^{i_{0}}y^{j_{0}}\in\mathrm{Span}_{\mathbb{F}}\\{x^{i}y^{j}:(i,j)\in
F_{i_{0}}\\},$ (6)
where $F_{i_{0}}=\\{(i,j)\in
S_{x}(\Xi):(i,j)\prec_{\mathrm{lex}}(i_{0},j_{0})\\}\subset S_{x}(\Xi).$ If
$x^{i_{0}}y^{j_{0}}\in\mathcal{I}(\Xi)$, then
$L_{\mathcal{P}_{x}}x^{i_{0}}y^{j_{0}}=0\prec_{\mathrm{lex}}x^{i_{0}}y^{j_{0}}$.
The statement (6) becomes trivial in this case. Otherwise, if we can find a
polynomial $p\in\Pi^{2}$ such that
$\displaystyle p=x^{i_{0}}y^{j_{0}}-\sum_{(i,j)\in
F_{i_{0}}}a_{ij}x^{i}y^{j}\in\mathcal{I}(\Xi),$ (7)
where $a_{ij}\in\mathbb{F}$ are not all zero, then (6) follows.
According to Section 2, our point set
$\Xi=\\{u_{ij}^{x}=(x_{ij},y_{ij}):(i,j)\in S_{x}(\Xi)\\}$. Let
$\Xi^{\prime}=\\{u_{mn}^{x}\in\Xi:(m,n)\in F_{i_{0}}\\}\subset\Xi$. Now, we
claim that there exists a unique polynomial $p$ of the form (7) such that
$p\in\mathcal{I}(\Xi^{\prime})$, which is equivalent to the statement that the
linear system
$\displaystyle\sum_{(i,j)\in
F_{i_{0}}}a_{ij}x_{mn}^{i}y_{mn}^{j}=x_{mn}^{i_{0}}y_{mn}^{j_{0}},\quad
u_{mn}^{x}\in\Xi^{\prime},$ (8)
has a unique solution.
Note that $\mathrm{Span}_{\mathbb{F}}\\{x^{i}y^{j}:(i,j)\in
F_{i_{0}}\\}=\mathrm{Span}_{\mathbb{F}}\\{\phi^{x}_{ij}:(i,j)\in
F_{i_{0}}\\}$. We can conclude that the rank of the coefficient matrix of (8)
is equal of that of the matrix $B^{\prime}_{\\#F_{i_{0}}\times\\#F_{i_{0}}}$,
which is a submatrix of $B$ whose $(h,k)$ entry is $\phi_{h}^{x}(u_{k}^{x})$
where $\phi_{h}^{x},u_{k}^{x}$ are $h$th and $k$th elements of
$\\{\phi_{ij}^{x}:(i,j)\in F_{i_{0}}\\}$ and $\Xi^{\prime}=\\{u_{mn}^{x}\\}$
w.r.t. the increasing $\prec_{\mathrm{inlex}}$ on $(i,j)$ and $(m,n)$
respectively. By (3), we see easily that $B^{\prime}$ is upper unitriangular
which implies that the coefficient matrix of (8) is of full rank. Accordingly,
there is a unique polynomial $p\in\mathcal{I}(\Xi^{\prime})$ that has the form
(7).
Now we shall verify that
$p(u_{ij}^{x})=0,u_{ij}^{x}\in\Xi\setminus\Xi^{\prime}$. By the definition of
$\Xi^{\prime}$, we know that $i>i_{0}$ here. Let
$q(x):=p(x,y_{ij})=\sum_{s=0}^{i_{0}}b_{s}x^{s}\in\Pi^{1},\quad
b_{s}\in\mathbb{F}.$
Since $y_{0j}=y_{1j}=\cdots=y_{i_{0},j}=y_{ij}$ and
$u_{0j}^{x},u_{1j}^{x},\ldots,u_{i_{0},j}^{x}\in\Xi^{\prime}$, it follows that
$q(x_{sj})=p(x_{sj},y_{ij})=p(x_{sj},y_{sj})=p(u_{sj}^{x})=0,\quad
s=0,\ldots,i_{0},$
namely $q(x)$ has $i_{0}+1$ zero points that clearly implies $q(x)\equiv 0$.
Since $p(u_{ij}^{x})=q(x_{ij})=0$, we have $p\in\mathcal{I}(\Xi)$. By (6), (5)
is true in this case. As a result, for any $f\in\Pi^{2}$, we have
$L_{\mathcal{P}_{x}}f\preceq_{\mathrm{lex}}f,$
that is to say $\mathcal{P}_{x}$ satisfies DR2.
Consequently, by Definition 1, $\mathcal{P}_{x}$ is a degree reducing
interpolation space w.r.t. $\prec_{\mathrm{lex}}$ for Lagrange interpolation
(1). Hence $N_{x}$ is the DRIMB and $Q_{x}$ is a Newton basis w.r.t.
$\prec_{\mathrm{lex}}$ for (1).∎
Note that $\mathcal{P}_{\prec_{\mathrm{lex}}}(\Xi)$ is the unique degree
reducing interpolation space spanned by monomials w.r.t.
$\prec_{\mathrm{lex}}$, thus we have
$\mathcal{P}_{x}=\mathcal{P}_{\prec_{\mathrm{lex}}}(\Xi)$. Therefore,
$N_{x}=N_{\prec_{\mathrm{lex}}}(\mathcal{I}(\Xi))$ holds, which means that
$N_{x}$ is also the Gröbner éscalier of $\mathcal{I}(\Xi)$ w.r.t.
$\prec_{\mathrm{lex}}$.
###### Corollary 7.
If $\Xi\subset\mathbb{F}^{2}$ is an $\mathcal{A}$-cartesian set, then
$\mathcal{A}=S_{x}(\Xi)=S_{y}(\Xi)$.
###### Proof 3.
Since $\Xi$ is cartesian, by Theorem 1 and 6, we have $S_{x}(\Xi)=S_{y}(\Xi)$
hence $(\Xi,S_{x}(\Xi))=(\Xi,S_{y}(\Xi))$ are both regular. But from Theorem
3, only $\mathcal{A}$ can make $(\Xi,\mathcal{A})$ regular, therefore
$\mathcal{A}=S_{x}(\Xi)=S_{y}(\Xi)$.∎
From Algorithm 1 we know that $G,N,Q$ are essential elements of BM algorithm
and compose its output. For $\prec_{\mathrm{lex}}$ and
$\prec_{\mathrm{inlex}}$ cases, Theorem 6 presents us $N$ and $Q$
theoretically hence we can obtain them with little effort. According to [11],
the leading terms of $G$ are contained in the border set of $N$. Therefore, we
can get $G$ faster than compute $G$ directly with BM algorithm. Now is our
algorithm.
###### Algorithm 2.
(SPBM)
Input: A set of distinct affine points $\Xi\subset\mathbb{F}^{2}$ and fixed
$\prec_{\mathrm{lex}}$ or $\prec_{\mathrm{inlex}}$.
Output: The 3-tuple $(G,N,Q)$, where $G$ is the reduced Gröbner basis of
$\mathcal{I}(\Xi)$, $N$ is the Gröbner éscalier
$\mathrm{N}(\mathcal{I}(\Xi))$, and $Q$ is a DRINB for the Lagrange
interpolation on $\Xi$.
SPBM1. Construct lower set $S_{x}(\Xi)$ or $S_{y}(\Xi)$ according to Section
2.
SPBM2. Compute the sets $N$ and $Q$ by Theorem 6.
SPBM3. Construct the border set $L:=\\{x\cdot t:t\in N\\}\bigcup\\{y\cdot
t:t\in N\\}\setminus N$ and the matrix $B$ that is same to the
$B_{\mu\times\mu}$ in the proof of Theorem 6.
SPBM4. Goto BM2 of BM algorithm for the reduced Gröbner basis $G$.
###### Example 1.
Let
$\Xi=\\{(0,1),(0,3),(1,0),(1,2),(1,3),(1,4),(2,1),(2,2),(3,1)\\}\subset\mathbb{Q}^{2}.$
First, we choose lines $x=1,x=0,x=2,x=3$ as
$l_{0}^{y},l_{1}^{y},l_{2}^{y},l_{3}^{y}$ respectively (Shown in (a) of Figure
1), therefore we have
$S_{y}=\\{(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(2,0),(2,1),(3,0)\\},$
which is illutrated in (b) of Figure 1.
(a) $\Xi$ (b) $S_{y}$
Figure 1: The point set and related $S_{y}$ of Example 1.
Thus, by Theorem 6, we have
$\displaystyle N=\\{$ $\displaystyle
1,y,y^{2},y^{3},x,xy,x^{2},x^{2}y,x^{3}\\};$ $\displaystyle Q=\\{$
$\displaystyle
1,\frac{1}{2}y,\frac{1}{3}y^{2}-\frac{2}{3}y,\frac{1}{8}y^{3}-\frac{5}{8}y^{2}+\frac{3}{4}y,-x+1,-\frac{1}{2}xy+\frac{1}{2}y+\frac{1}{2}x-\frac{1}{2},$
$\displaystyle\frac{1}{2}x^{2}-\frac{1}{2}x,\frac{1}{2}x^{2}y-\frac{1}{2}xy-\frac{1}{2}x^{2}+\frac{1}{2}x,\frac{1}{6}x^{3}-\frac{1}{2}x^{2}+\frac{1}{3}x\\}.$
Next, from SPBM3, the border set
$L=\\{y^{4},xy^{2},xy^{3},x^{2}y^{2},x^{3}y,x^{4}\\}$ and the matrix
$B=\left(\begin{array}[]{cccc}1&1&1&\cdots\\\ 0&1&3/2&\cdots\\\
0&0&1&\cdots\\\ \vdots&\vdots&\vdots&\ddots\\\ \end{array}\right).$
Finally, turn to BM2 with these $N,Q,L,B$, we can get the reduced Gröbner
basis
$\displaystyle G=\\{$ $\displaystyle
x^{4}-6x^{3}+11x^{2}-6x,x^{3}y-3x^{2}y+2xy-x^{3}+3x^{2}-2x,$ $\displaystyle
xy^{2}-y^{2}+\frac{1}{2}x^{2}y-\frac{9}{2}xy+4y-\frac{1}{2}x^{2}+\frac{7}{2}x-3,$
$\displaystyle
y^{4}-9y^{3}+26y^{2}-\frac{9}{2}x^{2}y+\frac{15}{2}xy-27y-3x^{3}+\frac{39}{2}x^{2}-\frac{51}{2}x+9\\}.$
for $\mathcal{I}(\Xi)$ w.r.t. $\prec_{\mathrm{inlex}}$.
###### Example 2.
Given a bivariate point set
$\Xi=\\{(0,0),(0,2),(0,3),(1,1),(\frac{5}{2},0),(\frac{5}{2},1),(\frac{5}{2},2),(4,0),(4,2)\\}\subset\mathbb{Q}^{2}.$
We choose lines $y=0,y=2,y=1,y=3$ as $l_{0}^{x},l_{1}^{x},l_{2}^{x},l_{3}^{x}$
respectively (Illustrated in (a) of Figure 2), which follows that
$S_{x}=\\{(0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(0,3)\\}.$
(a) $\Xi$ (b) $S_{x}$
Figure 2: Illustrations for Example 2.
Thus, with SPBM algorithm, we have
$\displaystyle N=\\{$ $\displaystyle
1,x,x^{2},y,xy,x^{2}y,y^{2},xy^{2},y^{3}\\},$ $\displaystyle Q=\\{$
$\displaystyle
1,\frac{1}{4}x,-\frac{4}{15}x^{2}+\frac{16}{15}x,\frac{1}{2}y,\frac{1}{8}xy,-\frac{2}{15}x^{2}y+\frac{8}{15}xy,-y^{2}+2y,$
$\displaystyle-\frac{2}{3}xy^{2}+\frac{2}{3}y^{2}+\frac{4}{3}xy-\frac{4}{3}y,\frac{1}{6}y^{3}-\frac{1}{2}y^{2}+\frac{1}{3}y\\},$
$\displaystyle G=\\{$ $\displaystyle
y^{4}-6y^{3}+11y^{2}-6y,xy^{3}-3xy^{2}+2xy,x^{2}y^{2}-2x^{2}y-\frac{7}{2}xy^{2}$
$\displaystyle+7xy-\frac{5}{4}y^{3}+\frac{25}{4}y^{2}-\frac{15}{2}y,x^{3}-\frac{13}{2}x^{2}-3xy^{2}+6xy$
$\displaystyle+10x-\frac{15}{4}y^{3}+\frac{75}{4}y^{2}-\frac{45}{2}y\\}.$
## 4 General cases
Next, we will discuss how to accelerate BM algorithm with respect to term
orders other than $\prec_{\mathrm{lex}}$ or $\prec_{\mathrm{inlex}}$. In [17],
the author proposed that if the set of points $\Xi$ is cartesian, then we can
obtain the interpolation basis without any difficulty, see Theorem 3. But in
general $\Xi$ may not be cartesian. However, we have the following
proposition.
###### Proposition 8.
There must exist at least one cartesian subset for any non-empty set of points
in $\mathbb{F}^{2}$.
###### Proof 4.
Let $\Xi$ be a non-empty set of points. Hence, there exists at least one point
$\xi\in\Xi$. But $\xi$ itself can construct a cartesian subset
$\\{\xi\\}\subset\Xi$.∎
###### Definition 3.
Let $\Xi$ be a set of points in $\mathbb{F}^{2}$ and $\Xi^{\prime}$ be a
cartesian subset of $\Xi$. We say that $\Xi^{\prime}$ is a _maximal cartesian
subset_ of $\Xi$ if any cartesian proper subset $\Xi^{\prime\prime}$ of $\Xi$
containing $\Xi^{\prime}$ is such that $\Xi^{\prime\prime}=\Xi^{\prime}$. In
addition, a _maximal row subset_ of $\Xi$ is a non-empty subset that equals
the intersection of $\Xi$ and a horizontal line.
From Proposition 8 we know that, for a set of given points, we can surely find
a maximal cartesian subset of it. Is it unique? Unfortunately, the answer is
often false.
###### Example 3.
Recall Example 2, let
$\displaystyle\Xi^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\\{(0,0),(0,2),(\frac{5}{2},0),(\frac{5}{2},1),(\frac{5}{2},2),(4,0),(4,2)\\},$
$\displaystyle\Xi^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\\{(0,0),(0,2),(0,3),(\frac{5}{2},0),(\frac{5}{2},2),(4,0),(4,2)\\},$
$\displaystyle\Xi^{\prime}_{3}$ $\displaystyle=$
$\displaystyle\\{(1,1),(\frac{5}{2},0),(\frac{5}{2},1),(\frac{5}{2},2)\\}.$
We can check easily that $\Xi_{1}^{\prime},\Xi_{2}^{\prime},\Xi_{3}^{\prime}$
are all maximal cartesian subsets of $\Xi$ (Illustrated in Figure 3).
(a) $\Xi^{\prime}_{1}$ (b) $\Xi^{\prime}_{2}$
(c) $\Xi^{\prime}_{3}$
Figure 3: Maximal cartesian subsets of $\Xi$, where $\bullet$ denotes the
points in $\Xi^{\prime}_{i},i=1,2,3$, while $\circ$ denotes the points in
$\Xi\backslash\Xi^{\prime}_{i}$.
###### Lemma 9.
Let $\Xi$ be a set of distinct points in $\mathbb{F}^{2}$ and $\prec$ a fixed
term order. If $\Xi^{\prime}$ is an $\mathcal{A}^{\prime}$-cartesian subset of
$\Xi$, then
$\mathcal{A}^{\prime}=\mathrm{N}_{\prec}(\Xi^{\prime})\subset\mathrm{N}_{\prec}(\Xi),$
or equivalently,
$\\{x^{i}y^{j}:(i,j)\in\mathcal{A}^{\prime}\\}=\mathrm{N}_{\prec}(\mathcal{I}(\Xi^{\prime}))\subset\mathrm{N}_{\prec}(\mathcal{I}(\Xi)).$
###### Proof 5.
From Section 2, the Gröbner éscalier
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi^{\prime}))$ is the DRIMB w.r.t. $\prec$
for the bivariate Lagrange interpolation on $\Xi^{\prime}$ hence the
interpolation scheme $(\Xi^{\prime},\mathrm{N}_{\prec}(\Xi^{\prime}))$ is
regular. Since $\mathcal{A}^{\prime}\subset\mathbb{N}_{0}^{2}$ is lower and
$\Xi^{\prime}$ is $\mathcal{A}^{\prime}$-cartesian, according to Theorem 3,
$\mathcal{A}^{\prime}$ is the unique lower set making the bivariate Lagrange
interpolation on $\Xi^{\prime}$ regular. This gives
$\mathcal{A^{\prime}}=\mathrm{N}_{\prec}(\Xi^{\prime}).$
Since $\Xi^{\prime}\subset\Xi$, from [19], we know that the vanishing ideals
satisfy $\mathcal{I}(\Xi^{\prime})\supset\mathcal{I}(\Xi)$. Denote by
$G^{\prime},G$ the reduced Gröbner bases for $\mathcal{I}(\Xi^{\prime})$ and
$\mathcal{I}(\Xi)$ w.r.t. $\prec$ respectively. We will prove
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi^{\prime}))\subset\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$
by contradiction. For any
$x^{i}y^{j}\in\mathrm{N}_{\prec}(\mathcal{I}(\Xi^{\prime}))$, we suppose there
were some $g\in G$ such that $\mathrm{LT}(g)|x^{i}y^{j}$. By [19],
$\langle\mathrm{LT}(G^{\prime})\rangle=\langle\mathrm{LT}(\mathcal{I}(\Xi^{\prime})\rangle\supset\mathrm{LT}(\mathcal{I}(\Xi))\supset\mathrm{LT}(G).$
Therefore,
$\mathrm{LT}(g)\in\mathrm{LT}(G)\subset\langle\mathrm{LT}(G^{\prime})\rangle$
implies that there exists some $g^{\prime}\in G^{\prime}$ such that
$\mathrm{LT}(g^{\prime})|\mathrm{LT}(g)$. Since $\mathrm{LT}(g)|x^{i}y^{j}$,
we have $\mathrm{LT}(g^{\prime})|x^{i}y^{j}$ that contradicts our assumption
on $x^{i}y^{j}$, which proves that
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi^{\prime}))\subset\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$
due to the definition of $\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$. Finally,
$\mathrm{N}_{\prec}(\Xi^{\prime})\cong\mathrm{N}_{\prec}(\mathcal{I}(\Xi^{\prime}))$
and $\mathrm{N}_{\prec}(\Xi)\cong\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$
complete the proof. ∎
###### Remark 1.
For any $\mathcal{A}$-cartesian set $\Xi$, by Corollary 7, we have
$\mathcal{A}=S_{x}(\Xi)=S_{y}(\Xi)$ that obviously leads to
$\mathcal{A}=S_{x}(\Xi)=S_{y}(\Xi)=\mathrm{N}_{\prec}(\Xi)$, according to the
Lemma above, where term order $\prec$ is arbitrary.
Now comes an algorithm for constructing a maximal cartesian subset of a given
point set in $\mathbb{F}^{2}$.
###### Algorithm 3.
(Maximal Cartesian Subset Construction Algorithm) Input: A set of distinct
points $\Xi=\\{\xi^{(i)}:i=1,\ldots,\mu\\}\subset\mathbb{F}^{2}$.
Output: A maximal cartesian subset $\Xi^{\prime}$ of $\Xi$.
MCS1. Start with an empty list $\Xi^{\prime}=[\ ]$.
MCS2. If $\Xi=[\ ]$, return the set $\Xi^{\prime}$ and stop. Otherwise,
compute lower sets $S_{x}(\Xi)$ and $S_{y}(\Xi)$.
MCS3. If $S_{x}(\Xi)=S_{y}(\Xi)$, then replace $\Xi^{\prime}$ by
$\Xi^{\prime}\cup\Xi$, return the set $\Xi^{\prime}$ and stop.
MCS4. Otherwise, we first choose a maximal row subset of $\Xi$ with maximal
cardinal number, denoted by $A$. Next, delete from $\Xi$ the points either in
$A$ or have different abscissae from the points in $A$. Finally, replace
$\Xi^{\prime}$ by $\Xi^{\prime}\cup A$ and continue with MCS2.
The following theorem ensure that this algorithm will terminate in finite
steps with a maximal cartesian subset as its output.
###### Theorem 10.
The algorithm described above will stop in a finite number of loops.
Furthermore, the set $\Xi^{\prime}$ returned by the algorithm is a maximal
cartesian subset.
###### Proof 6.
As input data of Algorithm 3, point set $\Xi$ is finite. Observing that
$\\#\Xi$ decreases actually in every loop, the algorithm will terminate in a
finite number, say $M$, of loops for sure. We assume that $M>1$ since $M=1$ is
trivial.
$\Xi^{\prime}_{\mathrm{in}}$ and $\Xi^{\prime}_{\mathrm{out}}$ signify the
input and output $\Xi^{\prime}$ of MCS4 in some loop respectively. Next, we
will prove by induction on $1\leq r\leq M-1$ that in the $r$th loop
$\Xi^{\prime}_{\mathrm{out}}$ is a cartesian set. The case $r=1$ is obvious
since $\Xi^{\prime}_{\mathrm{in}}=[\ ]$ and $\Xi^{\prime}_{\mathrm{out}}$ is
clearly cartesian as a maximal row subset of $\Xi$. Assume the statement is
true for $r=l<M-1$. When $r=l+1$, by the induction hypothesis,
$\Xi^{\prime}_{\mathrm{in}}$ is cartesian. Therefore, by Corollary 7, we
assume that
$\Xi^{\prime}_{\mathrm{in}}=\\{(x_{i},y_{j}):(i,j)\in
S_{x}(\Xi^{\prime}_{\mathrm{in}})\\},$
where
$S_{x}(\Xi^{\prime}_{\mathrm{in}})=\mathrm{L}_{x}(m_{0},\ldots,m_{n_{0}})=\mathrm{L}_{y}(n_{0},\ldots,n_{m_{0}})$.
Observing the construction process of $\Xi^{\prime}$ in the algorithm, we see
easily that $n_{0}=n_{1}=\cdots=n_{m_{n_{0}}}$. Let the maximal row subset of
$\Xi$ we choose at this moment be
$A=\\{(\overline{x}^{(0)},\overline{y}),(\overline{x}^{(1)},\overline{y}),\ldots,(\overline{x}^{(k)},\overline{y})\\}$.
Due to the nature of $A$, we have $k\leq m_{n_{0}}$ and $\overline{y}\neq
y_{j},j=0,\ldots,n_{0}$.
We claim that the set $\Xi^{\prime}_{\mathrm{in}}\cup A$ is cartesian. In
fact, we will focus on the horizontal parallel lines
$l_{j}^{x}:y=y_{j},j=0,\ldots,n_{0},$ and $l_{n_{0}+1}^{x}:y=\overline{y}$.
Resume the notation in (2). $H_{j}(\Xi^{\prime}_{\mathrm{in}}\cup
A)=H_{j}(\Xi^{\prime}_{\mathrm{in}})=\\{x_{i}:0\leq i\leq
m_{j}\\},j=0,\ldots,n_{0}$, and $H_{n_{0}+1}(\Xi^{\prime}_{\mathrm{in}}\cup
A)=\\{\overline{x}^{(i)}:0\leq i\leq k\\}$. Since $\Xi^{\prime}_{\mathrm{in}}$
is $S_{x}(\Xi^{\prime}_{\mathrm{in}})$-cartesian, by Theorem 2, the relation
$H_{0}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq
H_{1}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq\cdots\supseteq
H_{n_{0}}(\Xi^{\prime}_{\mathrm{in}}\cup A)$ holds. From the description of
MCS4, we can deduce that $H_{n_{0}}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq
H_{n_{0}+1}(\Xi^{\prime}_{\mathrm{in}}\cup A)$, which leads to
$\displaystyle H_{0}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq
H_{1}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq\cdots\supseteq
H_{n_{0}+1}(\Xi^{\prime}_{\mathrm{in}}\cup A).$ (9)
Note that for any $\overline{x}^{(i)},0\leq i\leq k$, there exists
$h_{i}\in\\{0,1,\ldots,m_{n_{0}}\\}$ such that $\overline{x}^{(i)}=x_{h_{i}}$.
Therefore, we could find a permutation $\sigma$ of $\\{0,1,\ldots,m_{0}\\}$
satisfying $\sigma(i)=h_{i},i=0,\ldots,k,$ and
$\sigma(i)=i,i=m_{n_{0}}+1,\ldots,m_{0}$. Choose lines
$l_{i}^{y}:x=x_{\sigma(i)},i=0,\ldots,m_{0}$, that give rise to
$V_{i}(\Xi^{\prime}_{\mathrm{in}})=\\{y_{j}:0\leq j\leq
n_{\sigma(i)}\\},i=0,\ldots,m_{0}$. Since $n_{0}=n_{1}=\cdots=n_{m_{n_{0}}}$,
the relation
$V_{0}(\Xi^{\prime}_{\mathrm{in}})=V_{1}(\Xi^{\prime}_{\mathrm{in}})=\cdots=V_{m_{n_{0}}}(\Xi^{\prime}_{\mathrm{in}})\supseteq
V_{m_{n_{0}}+1}(\Xi^{\prime}_{\mathrm{in}})\supseteq\cdots\supseteq
V_{m_{0}}(\Xi^{\prime}_{\mathrm{in}})$ holds. Observing that
$V_{i}(\Xi^{\prime}_{\mathrm{in}}\cup
A)=V_{i}(\Xi^{\prime}_{\mathrm{in}})\cup\\{\overline{y}\\},i=0,\ldots,k$, and
$V_{i}(\Xi^{\prime}_{\mathrm{in}}\cup
A)=V_{i}(\Xi^{\prime}_{\mathrm{in}}),i=k+1,\ldots,m_{0}$, it is easy to get
$\displaystyle V_{0}(\Xi^{\prime}_{\mathrm{in}}\cup
A)=\cdots=V_{k}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq
V_{k+1}(\Xi^{\prime}_{\mathrm{in}}\cup A)\supseteq\cdots\supseteq
V_{m_{0}}(\Xi^{\prime}_{\mathrm{in}}\cup A).$
Thus together with (9),
$\Xi^{\prime}_{\mathrm{out}}=\Xi^{\prime}_{\mathrm{in}}\cup A$ is cartesian
due to Theorem 2, hence our statement is true.
For the $M$th loop, if $\Xi=[\ ]$, then $\Xi^{\prime}$ here equals to the
$\Xi^{\prime}_{\mathrm{out}}$ of the MCS4 of the $(M-1)$th loop that is
cartesian due to the statement above. Otherwise, since the algorithm stops in
MCS3 of this loop, $\Xi$ is a non-empty cartesian set. Similar to the
arguments above, we can prove that
$\Xi^{\prime}=\Xi^{\prime}_{\mathrm{out}}\cup\Xi$ is also cartesian.
Finally, we should verify that the output $\Xi^{\prime}$ of the algorithm is
maximal. Otherwise, there must exist a maximal
$S_{x}(\Xi^{\prime\prime})$-cartesian subset $\Xi^{\prime\prime}$ of $\Xi$
satisfying $\Xi^{\prime\prime}\supsetneqq\Xi^{\prime}$. Take a point
$\xi_{0}=(x_{i_{0}},y_{j_{0}})$ with
$(i_{0},j_{0})=\min_{\prec_{\mathrm{inlex}}}\\{(i,j)\in
S_{x}(\Xi^{\prime\prime}):(x_{i},y_{j})\in\Xi^{\prime\prime}\setminus\Xi^{\prime}\\}$.
Suppose there exists a point in $\Xi^{\prime}$ sharing the ordinate with
$\xi_{0}$. If it is chosen as a point in the maximal row subset in MCS4 of
some loop, by the definition of $\xi_{0}$, we know that $\xi_{0}$ is surely
contained in the set $\Xi$ of that step, which contradicts the definition of
the maximal row subset. Otherwise, it must appear in the cartesian set $\Xi$
in MCS3 in the final loop. Then, by the definition of $\xi_{0}$, it should be
contained in $\Xi$ hence the output set $\Xi^{\prime}$, which introduces a
contradiction. If there does not exist a point in $\Xi^{\prime}$ sharing the
ordinate with $\xi_{0}$, since $\Xi^{\prime\prime}$ is also cartesian, by
Theorem 2, it is easily to see that $\xi_{0}$ must remain in $\Xi$ in every
loop, which contradicts the termination condition. As a result, the output of
the Algorithm 3 is a maximal cartesian subset. ∎
Let us continue with the setup and notation in Algorithm 3, and assume that
the final output of it is $\Xi^{\prime}$ who is
$S_{x}(\Xi^{\prime})$-cartesian . We now discuss how to preprocess the BM
algorithm with the help of $\Xi^{\prime}$.
Define an order $\prec_{\Xi}$ on the set $\Xi$. Let
$\xi^{(1)},\xi^{(2)}\in\Xi$. We say that $\xi^{(1)}\prec_{\Xi}\xi^{(2)}$ if
one of the following conditions holds:
* (1)
$\xi^{(1)}\in\Xi^{\prime}$, and $\xi^{(2)}\in\Xi\backslash\Xi^{\prime}$.
* (2)
$\xi^{(1)}=(x_{i_{1}},y_{j_{1}}),\xi^{(2)}=(x_{i_{2}},y_{j_{2}})\in\Xi^{\prime}$
and $(i_{1},j_{1})\prec_{\mathrm{inlex}}(i_{2},j_{2})$ with $(i_{k},j_{k})\in
S_{x}(\Xi^{\prime}),k=1,2$.
It should be noticed that the order is not total. For the points in
$\Xi\backslash\Xi^{\prime}$, any order of them can be interpreted as
increasing. Hereafter, we will suppose that the points in
$\Xi=\\{\xi^{(1)},\ldots,\xi^{(\\#\Xi)}\\}$ have been ordered increasingly
w.r.t. $\prec_{\Xi}$, namely $\xi^{(i)}\prec_{\Xi}\xi^{(j)},0\leq
i<j\leq\\#\Xi$. By the definition of $\prec_{\Xi}$, we have
$\Xi^{\prime}=\\{\xi^{(1)},\ldots,\xi^{(\\#\Xi^{\prime})}\\}$.
According to Lemma 9, $N^{\prime}=\\{x^{i}y^{j}:(i,j)\in
S_{x}(\Xi^{\prime})\\}\subset N$, with $N$ as a member of the 3-tuple output
of BM algorithm. Thus the other monomials of $N$ are obviously contained in
$\mathbb{T}^{2}\backslash N^{\prime}$. Notice that the generators of
$\mathbb{T}^{2}\backslash N^{\prime}$ are located in the border of
$N^{\prime}$, denoted by $L$, we can continue to spot the elements in $L$ by
BM algorithm to complete $N$.
Next, we will pay attention to the computation of the Newton basis. Since
$\Xi^{\prime}$ is cartesian, recalling Proposition 4, we can construct the
polynomials $\phi_{ij}^{x}$ w.r.t. $S_{x}(\Xi^{\prime})$. Order
$\phi_{ij}^{x},(i,j)\in S_{x}(\Xi^{\prime})$, increasingly w.r.t. $(i,j)$
under $\prec_{\mathrm{inlex}}$, and denote them as
$q_{1},q_{2},\ldots,q_{\\#\Xi^{\prime}}$. Set the matrix
$B=\left(\begin{array}[]{cccc}q_{1}(\xi^{(1)})&q_{1}(\xi^{(2)})&\cdots&q_{1}(\xi^{(\\#\Xi^{\prime})})\\\
q_{2}(\xi^{(1)})&q_{2}(\xi^{(2)})&\cdots&q_{2}(\xi^{(\\#\Xi^{\prime})})\\\
\vdots&\vdots&&\vdots\\\
q_{\\#\Xi^{\prime}}(\xi^{(1)})&q_{\\#\Xi^{\prime}}(\xi^{(2)})&\cdots&q_{\\#\Xi^{\prime}}(\xi^{(\\#\Xi^{\prime})})\\\
\end{array}\right).$ (10)
By Proposition 4, $B$ is obviously upper unitriangular which implies that the
polynomials $q_{1},q_{2},\ldots,q_{\\#\Xi^{\prime}}$ constitute a Newton basis
for $\mathcal{P}_{\prec}(\Xi^{\prime})=\mathrm{Span}_{\mathbb{F}}N^{\prime}$.
All in all, with the notation above, we get our preprocessing procedure for BM
algorithm.
###### Algorithm 4.
(GPBM)
Input: A set of distinct points $\Xi\subset\mathbb{F}^{2}$ and a term order
$\prec$.
Output: The 3-tuple $(G,N,Q)$.
GPBM1: Get a maximal cartesian subset $\Xi^{\prime}$ of $\Xi$ by Algorithm 3;
GPBM2: Compute the lower set $S_{x}(\Xi^{\prime})$ w.r.t. $\Xi^{\prime}$, the
set $N:=\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi^{\prime})\\}$, and the set
$Q:=\\{q_{1},q_{2},\ldots,q_{\\#\Xi^{\prime}}\\}$ where the $q_{i}$’s are as
in (10).
GPBM3: Construct $L:=\\{x\cdot t:t\in N\\}\bigcup\\{y\cdot t:t\in
N\\}\setminus N$ and the matrix $B$ that is same to (10).
GPBM4: Goto BM2 of the BM algorithm to complete the computation and get the
whole output.
## 5 Implementation and Timings
From the above section, we can see easily that our preprocessing paradigm is
more suitable to the cases where the constructed maximal cartesian subset
$\Xi^{\prime}$ forms a relatively large proposition in $\Xi$. Especially, when
the field $\mathbb{F}$ is finite, our preprocessing will play a more important
role in consideration of the nature of finite fields. In this section, we will
present some experimental results to compare the effectiveness of our paradigm
with the classical BM. First see an example with point set of small size.
###### Example 4.
We choose the field $\mathbb{F}_{7}$, and let
$\displaystyle\Xi=\\{$ $\displaystyle(0,0),(0,1),(0,4),(0,5),(1,0),$
$\displaystyle(1,1),(1,4),(1,6),(2,1),(2,2),$
$\displaystyle(2,6),(3,2),(4,2),(4,5),(4,6),$
$\displaystyle(5,1),(5,5),(5,6),(6,0),(6,2)\\}.$
By Algorithm 3, we can construct the maximal cartesian subset
$\Xi^{\prime}=\\{(0,1),(1,1),(2,1),(5,1),(1,6),(2,6),(5,6),(1,0),(1,4)\\}$
hence get
$\displaystyle N=$ $\displaystyle\\{1,x,x^{2},x^{3},y,xy,x^{2}y,y^{2}\\},$
$\displaystyle Q=$
$\displaystyle\\{1,x,4x^{2}+3x,2x^{3}+x^{2}+4x,3y+4,3xy+4x+4y+3,$
$\displaystyle\phantom{\\{}2x^{2}y+5x^{2}+xy+6x+4y+3,6y^{2}+1,2y^{3}+5y\\},$
$\displaystyle L=$
$\displaystyle\\{y^{4},xy^{2},xy^{3},x^{2}y^{2},x^{3}y,x^{4}\\},$
$\displaystyle B=$ $\displaystyle\left(\begin{array}[]{cccc}1&1&1&\cdots\\\
0&1&2&\cdots\\\ 0&0&1&\cdots\\\ \vdots&\vdots&\vdots&\ddots\\\
\end{array}\right).$
Put these $N,Q,L,B$ into BM algorithm, we can get the final output
$\displaystyle N=$
$\displaystyle\\{1,x,x^{2},x^{3},y,xy,x^{2}y,y^{2},y^{3},xy^{2},y^{4},xy^{3},x^{2}y^{2},x^{3}y,x^{4},y^{5},xy^{4},x^{2}y^{3},$
$\displaystyle\phantom{\\{}x^{3}y^{2},x^{4}y\\},$ $\displaystyle Q=$
$\displaystyle\\{1,x,4x^{2}+3x,2x^{3}+x^{2}+4x,3y+4,3xy+4x+4y+3,$
$\displaystyle\phantom{\\{}2x^{2}y+5x^{2}+xy+6x+4y+3,6y^{2}+1,2y^{3}+5y,xy^{2}+6y^{2}+6x+1,$
$\displaystyle\phantom{\\{}y^{4}+3y^{3}+6y^{2}+4y,5xy^{3}+5y^{4}+3y^{3}+2xy+2y^{2}+4y,$
$\displaystyle\phantom{\\{}6x^{2}y^{2}+xy^{2}+x^{2}+6x,\ldots\\},$
$\displaystyle G=$ $\displaystyle\\{y^{6}+3y^{5}+2y^{4}+6y^{3}+4y^{2}+5y,$
$\displaystyle\phantom{\\{}xy^{5}+x^{4}y+6x^{3}y^{2}+x^{2}y^{3}+5xy^{4}+6y^{5}+6x^{4}+2x^{3}y+6x^{2}y^{2}+3xy^{3}+$
$\displaystyle\phantom{\\{}3y^{4}+6x^{3}+6x^{2}y+2xy^{2}+6y^{3}+x^{2}+2xy+6y^{2}+x,$
$\displaystyle\phantom{\\{}x^{2}y^{4}+x^{4}y+3x^{2}y^{3}+3xy^{4}+5y^{5}+x^{4}+6x^{3}y+3x^{2}y^{2}+2xy^{3}+4y^{4}+$
$\displaystyle\phantom{\\{}6x^{3}+4y^{3}+6x^{2}+2xy+3y^{2}+x+5y,\ldots\\}.$
In the following, several tables show the timings for the computations of BM-
problems on sets of distinct random points w.r.t. the term order
$\prec_{\mathrm{lex}}$ or $\prec_{\mathrm{tdinlex}}$. The algorithms presented
in the paper were implemented on Maple 12 installed on a laptop with 2 Gb RAM
and 1.8 GHz CPU.
Take the field $\mathbb{F}_{23}$, we have
$\\#\Xi$ | 200 | 300 | 400 | 500
---|---|---|---|---
BM | 4.968 s | 15.359 s | 34.609 s | 61.172 s
SPBM | 1.438 s | 3.766 s | 7.141 s | 7.969 s
For $\mathbb{F}_{37}$, we have
$\\#\Xi$ | 300 | 600 | 900 | 1200
---|---|---|---|---
BM | 16.265 s | 121.766 s | 420.219 s | 1060.203 s
SPBM | 4.172 s | 25.125 s | 82.000 s | 132.719 s
For $\mathbb{F}_{17}$, we have
$\\#\Xi$ | 100 | 150 | 200 | 250
---|---|---|---|---
BM | 0.875 s | 2.421 s | 4.953 s | 8.188 s
GPBM | 0.797 s | 2.125 s | 4.250 s | 5.641 s
Preprocessing | 0.015 s | 0.094 s | 0.172 s | 0.391 s
$\\#\Xi^{\prime}/\\#\Xi$ | 0.310 | 0.393 | 0.430 | 0.616
Take the field $\mathbb{F}_{29}$, we have
$\\#\Xi$ | 200 | 400 | 600 | 800
---|---|---|---|---
BM | 5.672 s | 38.063 s | 112.156 s | 235.813 s
GPBM | 5.562 s | 36.906 s | 105.828 s | 135.609 s
Preprocessing | 0.046 s | 0.313 s | 1.671 s | 8.125 s
$\\#\Xi^{\prime}/\\#\Xi$ | 0.125 | 0.178 | 0.328 | 0.711
## References
* [1] D. A. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, 2nd Edition, Vol. 185 of Graduate Texts in Mathematics, Springer, New York, 2005.
* [2] T. Sauer, Polynomial interpolation in several variables: Lattices, differences, and ideals, in: K. Jetter, M. Buhmann, W. Haussmann, R. Schaback, J. Stöckler (Eds.), Topics in Multivariate Approximation and Interpolation, Vol. 12 of Studies in Computational Mathematics, Elsevier, Amsterdam, 2006, pp. 191–230.
* [3] C. de Boor, Ideal interpolation, in: C. K. Chui, M. Neamtu, L. L. Schumaker (Eds.), Approximation Theory XI: Gatlinburg 2004, Nashboro Press, Brentwood TN, 2005, pp. 59–91.
* [4] S. Sakata, Gröbner bases and coding theory, in: B. Buchberger, F. Winkler (Eds.), Gröbner Bases and Applications, Vol. 251 of London Mathematical Society Lecture Notes Series, Cambridge University Press, New York, 1998, pp. 470–485.
* [5] M. Sala, Gröbner bases, coding, and cryptography: a guide to the state-of-art, in: M. Sala, T. Mora, L. Perret, S. Sakata, C. Traverso (Eds.), Gröbner Bases, Coding, and Cryptography, Springer, Berlin, 2009, pp. 1–8.
* [6] L. Robbiano, Gröbner bases and statistics, in: B. Buchberger, F. Winkler (Eds.), Gröbner Bases and Applications, Vol. 251 of London Mathematical Society Lecture Notes Series, Cambridge University Press, New York, 1998, pp. 179–204.
* [7] R. Laubenbacher, B. Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, J. Theoret. Biol. 229 (4) (2004) 523–537.
* [8] W. Just, B. Stigler, Computing Gröbner bases of ideals of few points in high dimensions, ACM Commun. Comput. Algebra 40 (3-4) (2006) 67–78.
* [9] H. Möller, B. Buchberger, The construction of multivariate polynomials with preassigned zeros, in: J. Calmet (Ed.), Computer Algebra: EUROCAM ’82, Vol. 144 of Lecture Notes in Computer Science, Springer, Berlin, 1982, pp. 24–31.
* [10] J. C. Faugère, P. Gianni, D. Lazard, T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symbolic Comput. 16 (4) (1993) 329–344.
* [11] M. G. Marinari, H. M. Möller, T. Mora, Gröbner bases of ideals defined by functionals with an application to ideals of projective points, Appl. Algebra Engrg. Comm. Comput. 4 (2) (1993) 103–145.
* [12] J.Abbott, A.Bigatti, M.Kreuzer, L.Robbiano, Computing ideals of points, J. Symbolic Comput. 30 (2000) 341–356.
* [13] L. Cerlienco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1-3) (1995) 73–87.
* [14] S. Gao, V. M. Rodrigues, J. Stroomer, Gröbner basis structure of finite sets of points (Preprint).
* [15] B. Felszeghy, B. Ráth, L. Rónyai, The lex game and some applications, J. Symbolic Comput. 41 (6) (2006) 663–681.
* [16] T. Sauer, Lagrange interpolation on subgrids of tensor product grids, Math. Comp. 73 (245) (2004) 181–190.
* [17] N. Crainic, Multivariate Birkhoff-Lagrange interpolation schemes and cartesian sets of nodes, Acta Math. Univ. Comenian.(N.S.) LXXIII (2) (2004) 217–221.
* [18] T. Chen, T. Dong, S. Zhang, The Newton interpolation bases on lower sets, J. Inf. Comput. Sci. 3 (3) (2006) 385–394.
* [19] T. Becker, V. Weispfenning, Gröbner Bases, Vol. 141 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1993.
* [20] R. Lorentz, Multivariate Birkhoff Interpolation, Vol. 1516 of Lecture Notes in Mathematics, Springer, Heidelberg, 1992.
* [21] C. de Boor, Interpolation from spaces spanned by monomials, Adv. Comput. Math. 26 (1) (2007) 63–70.
|
arxiv-papers
| 2010-01-08T03:03:01 |
2024-09-04T02:49:07.595817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoying Wang, Shugong Zhang and Tian Dong",
"submitter": "Tian Dong",
"url": "https://arxiv.org/abs/1001.1186"
}
|
1001.1196
|
# Bivariate Lagrange Interpolation on Tower Interpolation Sites 111Dedicated
to Professor Wen-Tsun Wu on the occasion of his ninetieth birthday.
Tian Dong dongtian@jlu.edu.cn Tao Chen chentao@cueb.edu.cn Shugong Zhang
sgzh@jlu.edu.cn Xiaoying Wang wang.xiao.ying@163.com School of Mathemathics,
Key Lab. of Symbolic Computation and Knowledge Engineering (Ministry of
Education), Jilin University, Changchun 130012, China School of Statistics,
Capital University of Economics and Business, Beijing 100070, China
###### Abstract
As is well known, the geometry of the interpolation site of a multivariate
polynomial interpolation problem constitutes a dominant factor for the
structures of the interpolation polynomials. Solving interpolation problems on
interpolation sites with special geometries in theory may be a key step to the
development of general multivariate interpolation theory. In this paper, we
introduce a new type of 2-dimensional interpolation sites, tower interpolation
sites, whose associated degree reducing Lagrange interpolation monomial and
Newton bases w.r.t. fixed standard term orders such as lexicographical order,
total degree lexicographical order, etc. can be figured out theoretically.
Inputting these interpolation bases into Buchberger-Möller(BM) algorithm, we
can also construct the reduced Gröbner bases for related vanishing ideals.
Experimental results show that in this way we can get the bases much faster
than inputting tower sites directly into BM algorithm.
###### keywords:
Bivariate Lagrange interpolation , Degree reducing interpolation space , Tower
interpolation site , Gröbner basis
††thanks: This research was partially supported by the National Grand
Fundamental Research 973 Program of China(No. 2004CB318000) and the National
Natural Science Foundation of China (Nos. 10601020).
## 1 Introduction
Let $\mathbb{F}$ be an arbitrary field and $\mathbb{F}_{q}$ a finite prime
field of size $q$. $\Pi^{d}:=\mathbb{F}[x_{1},x_{2},\ldots,x_{d}]$ stands for
the $d$-variate polynomial ring over $\mathbb{F}$. Given a set
$\Xi=\\{\xi^{(1)},\ldots,\xi^{(\mu)}\\}\subset\mathbb{F}^{d}$ of $\mu$
distinct points. For prescribed values $f_{i}\in\mathbb{F},i=1,\ldots,\mu$,
find all polynomials $p\in\Pi^{d}$ such that
$p(\xi^{(i)})=f_{i},\quad i=1,\ldots,\mu.$ (1)
The problem is called _$d$ -variate Lagrange interpolation_, and the points
are named _interpolation nodes_ , $\Xi$ the _interpolation site_ , and $p$ an
_interpolation polynomial_. Generally, we only search interpolation
polynomials in a subspace $\mathcal{P}\subset\Pi^{d}$. If there is a unique
$p\in\mathcal{P}$ satisfying (1) for any fixed $f_{i}$’s, we call
$\mathcal{P}$ an _interpolation space_ that gives rise to a _regular_(or
_poised_) _interpolation scheme_ $(\Xi,\mathcal{P})$(see Lorentz (1992)).
As a basic subject in Approximation Theory and Numerical Analysis,
multivariate Lagrange interpolation has many applications in various fields of
mathematics and computer science. Nevertheless, unlike univariate Lagrange
interpolation, it is a relatively new topic with many basic problems unsolved.
Since the set of all polynomials in $\Pi^{d}$ vanishing at $\Xi$ constitutes
the vanishing ideal $\mathcal{I}(\Xi)$ of $\Xi$, multivariate Lagrange
interpolation becomes a particular example of _ideal interpolation_ introduced
in Birkhoff (1979) and surveyed by de Boor (2005). Recent years, there has
been considerable interest in the topic, for example, cf. Sauer and Xu (1995);
Bos et al. (2007); Shekhtman (2009) etc.
In most cases, especially from a numerical point of view, we are not
interested in all polynomials satisfying (1) but the one of “minimal degree”,
as in the univariate cases. In Sauer (2006), the notion of _degree reducing
interpolation spaces_ w.r.t. a fixed term order $\prec$ for interpolation
problem (1) was introduced. Let $\mathcal{P}_{\prec}(\Xi)\subset\Pi^{d}$ be
the subspace spanned by the Gröbner éscalier
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$ of $\mathcal{I}(\Xi)$ w.r.t. $\prec$.
de Boor (2007) pointed out that $\mathcal{P}_{\prec}(\Xi)$ is a canonical
$\prec$-degree reducing interpolation space as the unique one spanned by
monomials. When $d=1$, since
$\mathcal{I}(\Xi)=\langle(x-\xi^{(1)})(x-\xi^{(2)})\cdots(x-\xi^{(\mu)})\rangle$,
$\mathcal{P}_{\prec}(\Xi)=\mathrm{Span}_{\mathbb{F}}\\{x^{k}:0\leq
k\leq\mu-1\\}$ is a classical one. But when $d>1$, we can not obtain
$\mathcal{P}_{\prec}(\Xi)$ theoretically for any $\Xi$ any more since the
geometry of $\Xi$, as a dominant factor for
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$, becomes very complex. In these cases,
$\mathcal{P}_{\prec}(\Xi)$ can only be acquired by Buchberger-Möller(BM for
short) algorithm or variations thereof.
As a milestone algorithm for the computation of vanishing ideals, BM algorithm
was presented in Möller and Buchberger (1982). For any interpolation site
$\Xi$ and fixed term order $\prec$, BM algorithm yields the reduced Gröbner
basis for $\mathcal{I}(\Xi)$ w.r.t. $\prec$ and the monomial basis(i.e.
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$) as well as a Newton basis for
$\mathcal{P}_{\prec}(\Xi)$. However, as we all known, BM algorithm(and the
variations) has a poor complexity that limits its applications. One reason for
this may be that the geometry of $\Xi$ is not concerned by these algorithms.
Therefore, if we can solve the interpolation problems on some special
distributed nodes in theory, general interpolation theory will certainly
benefit from it and the algorithms will also be improved for some cases.
In this decade, Sauer (2004); Crainic (2004); Chen et al. (2006) studied
multivariate Lagrange interpolation on a special type of interpolation sites,
lower interpolation sites, and constructed the associated degree reducing
interpolation monomial and Newton bases in theory. Enlightened by them, this
paper introduces a new type of bivariate interpolation sites, tower
interpolation sites, whose geometry is more complex than lower sites
nonetheless the related degree reducing interpolation bases can still be
obtained theoretically.
In this paper, we will study bivariate Lagrange interpolation on tower
interpolation sites. After introducing tower sites and comparing them with
lower sites in Section 3, we will present the degree reducing interpolation
bases w.r.t. several standard term orders in Section 4 as main results. The
last section, Section 5, will give our improved BM algorithm for tower sites
and illustrate some experimental results.
Next comes Section 2 that serves as a preparation for the paper.
## 2 Preliminaries
For the reader’s convenience, this section will introduce some notation and
recall some basic facts.
Let $\mathbb{N}_{0}$ stand for the monoid of nonnegative integers. A
_polynomial_ $f\in\Pi^{2}$ is of the form
$f=\sum_{(i,j)\in\mathbb{N}_{0}^{2}}f_{ij}x^{i}y^{j},\hskip
22.76228pt\\#\\{(i,j)\in\mathbb{N}_{0}^{2}:0\neq
f_{ij}\in\mathbb{F}\\}<\infty.$
The set of bivariate monomials in $\Pi^{2}$ is denoted by $\mathbb{T}^{2}$.
Fix a term order $\prec$ on $\Pi^{2}$ that may be lexicographical order
$\prec_{\text{lex}}$, inverse lexicographical order $\prec_{\text{inlex}}$,
total degree lexicographical order $\prec_{\text{tdlex}}$, or total degree
inverse lexicographical order $\prec_{\text{tdinlex}}$ etc. (see Becker and
Weispfenning (1993)). For all nonzero $f\in\Pi^{2}$, we let $\mathrm{LT}(f)$
signify the _leading term_ ,$\mathrm{LM}(f)$ the _leading monomial_ , and
$\mathrm{LC}(f)$ the _leading coefficient_ of $f$. Furthermore, for a non-
empty subset $F\subset\Pi^{2}$, put
$\mathrm{LT}(F):=\\{\mathrm{LT}(f):f\in F\\}.$
Let $\mathcal{A}$ be a finite subset of $\mathbb{N}_{0}^{2}$. $\mathcal{A}$ is
called a _lower_ set if, for any $(i,j)\in\mathcal{A}$, we always have
$\\{(i^{\prime},j^{\prime})\in\mathbb{N}_{0}^{2}:0\leq i^{\prime}\leq i,0\leq
j^{\prime}\leq j\\}\subset\mathcal{A}.$ (2)
Especially, $(0,0)\in\mathcal{A}$. Let
$m_{j}=\max_{(h,j)\in\mathcal{A}}h,0\leq
j\leq\nu=\max_{(0,k)\in\mathcal{A}}k$. Obviously, $\mathcal{A}$ can be
determined uniquely by the ordered $(\nu+1)$-tuple
$(m_{0},m_{1},\ldots,m_{\nu})$ hence represented as
$\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$. In like manner, we can also
represent $\mathcal{A}$ as $\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{m_{0}})$ with
$n_{i}=\max_{(i,k)\in\mathcal{A}}k,0\leq i\leq m_{0}$. It is easy to see that
$\nu=n_{0}$.
###### Definition 1 (Crainic (2004)).
We say that an interpolation site $\Xi$ in $\mathbb{F}^{2}$ is _lower_ if
there exists a lower set $\mathcal{A}\subset\mathbb{N}_{0}^{2}$ such that
$\Xi$ can be written as
$\Xi=\\{(x_{i},y_{j}):(i,j)\in\mathcal{A}\\},$ (3)
where the $x_{i}$’s are distinct numbers, and similarly the $y_{j}$’s. We also
say that $\Xi$ is $\mathcal{A}$-lower.
In de Boor (2007), the $\prec\mspace{-8.0mu}-$_degree_ of a polynomial
$f\in\Pi^{2}$ was defined to be the leading bidegree w.r.t. $\prec$
$\delta(f):=(i,j),\quad x^{i}y^{j}=\mathrm{LM}(f),$
with $\delta(0)$ undefined. For any $f,g\in\Pi^{2}$, if
$\delta(f)\prec\delta(g)$ then we say that $f$ is of _lower degree_ than $g$
and use the abbreviation
$f\prec g:=\delta(f)\prec\delta(g).$
In addition, $f\preceq g$ is interpreted as the degree of $f$ is lower than or
equal to that of $g$.
###### Definition 2 (Sauer (2006)).
Fix term order $\prec$. We call a subspace $\mathcal{P}\subset\Pi^{2}$ a
_degree reducing interpolation space_ w.r.t. $\prec$ for the bivariate
Lagrange interpolation (1) if
1. (i)
$\mathcal{P}$ is an _interpolation space_.
2. (ii)
$\mathcal{P}$ is $\prec\mspace{-8.0mu}-$_reducing_ , i.e., when
$L_{\mathcal{P}}$ denotes the Lagrange projector with range $\mathcal{P}$,
then the interpolation polynomial
$L_{\mathcal{P}}q\preceq q,\hskip 14.22636pt\forall q\in\Pi^{2}.$
If an interpolation space $\mathcal{P}$ for (1) is degree reducing w.r.t.
$\prec$, a basis $\\{p_{1},\ldots,p_{\mu}\\}$ for $\mathcal{P}$ will be called
a _degree reducing interpolation basis_ w.r.t. $\prec$ for (1). Assume that
$p_{1}\prec p_{2}\prec\cdots\prec p_{\mu}$. If
$p_{j}(\xi^{(i)})=\delta_{ij},\quad 1\leq i\leq j\leq\mu,$
for some suitable reorder of $\Xi$, then we call the basis a _degree reducing
interpolation Newton basis_(DRINB) w.r.t. $\prec$ for (1). Let $G_{\prec}$ be
the reduced Gröbner basis for the vanishing ideal $\mathcal{I}(\Xi)$
w.r.t.$\prec$. The set
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi)):=\\{x^{i}y^{j}\in\mathbb{T}^{2}:\mathrm{LT}(g)\nmid
x^{i}y^{j},\forall g\in G_{\prec}\\}$ (4)
is the _Gröbner éscalier_ of $\mathcal{I}(\Xi)$ w.r.t. $\prec$. Recall Section
1. As the monomial basis for $\mathcal{P}_{\prec}(\Xi)$,
$\mathrm{N}_{\prec}(\mathcal{I}(\Xi))$ is named the _degree reducing
interpolation monomial basis_(DRIMB) w.r.t. $\prec$ for (1).
Finally, we will redescribe BM algorithm with the notation established above.
###### Algorithm 1.
(BM Algorithm)
Input: An interpolation site
$\Xi=\\{\xi^{(i)}:i=1,\ldots,\mu\\}\subset\mathbb{F}^{d}$ and a fixed term
order $\prec$.
Output: The 3-tuple $(G,N,Q)$, where $G$ is the reduced Gröbner basis for
$\mathcal{I}(\Xi)$ w.r.t. $\prec$, $N$ is the Gröbner éscalier of
$\mathcal{I}(\Xi)$ (the DRIMB for (1) also) w.r.t. $\prec$, and $Q$ is a DRINB
w.r.t. $\prec$ for (1).
BM1. Start with lists $G=[\ ],N=[\ ],Q=[\ ],L=[1]$, and a matrix $B=(b_{ij})$
over $\mathbb{F}$ with $\mu$ columns and zero rows initially.
BM2. If $L=[\ ]$, return $(G,N,Q)$ and stop. Otherwise, choose the monomial
$t=\mbox{min}_{\prec}L$, and delete $t$ from $L$.
BM3. Compute the evaluation vector $(t(\xi^{(1)}),\ldots,t(\xi^{(\mu)}))$, and
reduce it against the rows of $B$ to obtain
$(v_{1},\ldots,v_{\mu})=(t(\xi^{(1)}),\ldots,t(\xi^{(\mu)}))-\sum_{i}a_{i}(b_{i1},\ldots,b_{i\mu}),\quad
a_{i}\in\mathbb{F}.$
BM4.. If $(v_{1},\ldots,v_{\mu})=(0,\ldots,0)$, then append the polynomial
$t-\sum_{i}a_{i}q_{i}$ to the list $G$, where $q_{i}$ is the $i$th element of
$Q$. Remove from $L$ all the multiples of $t$. Continue with BM2.
BM5. Otherwise $(v_{1},\ldots,v_{\mu})\neq(0,\ldots,0)$, add
$(v_{1},\ldots,v_{\mu})$ as a new row to $B$ and $t-\sum_{i}a_{i}q_{i}$ as a
new element to $Q$. Append the monomial $t$ to $N$, and add to $L$ those
elements of $\\{x_{1}t,\ldots,x_{d}t\\}$ that are neither multiples of an
element of $L$ nor of $\mathrm{LT}(G)$. Continue with BM2.
## 3 Tower interpolation sites
Given an interpolation site $\Xi\subset\mathbb{F}^{2}$. In Crainic (2004), two
particular lower sets are constructed from $\Xi$, denoted by
$S_{x}(\Xi),S_{y}(\Xi)$, which reflect the geometry of $\Xi$ in certain sense.
Concretely, we cover the nodes in $\Xi$ by lines
$l_{0}^{x},l_{1}^{x},\ldots,l_{\nu}^{x}$ parallel to the $x$-axis and assume
that, without loss of generality, there are $m_{j}+1$ nodes, say
$u_{0j}^{x},u_{1j}^{x},\ldots,u_{m_{j},j}^{x}$, on $l_{j}^{x}$ with $m_{0}\geq
m_{1}\geq\cdots\geq m_{\nu}\geq 0$ hence the ordinates of $u_{ij}^{x}$ and
$u_{i^{\prime}j}^{x},i\neq i^{\prime}$, same. Now, we define
$S_{x}(\Xi):=\\{(i,j):0\leq i\leq m_{j},\ 0\leq j\leq\nu\\},$ (5)
which clearly equals to $\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})$.
Similarly, we can also cover the nodes by lines
$l_{0}^{y},l_{1}^{y},\ldots,l_{\lambda}^{y}$ parallel to the $y$-axis and
denote the nodes on line $l_{i}^{y}$ by
$u_{i0}^{y},u_{i1}^{y},\ldots,u_{i,n_{i}}^{y}$ with $n_{0}\geq
n_{1}\geq\cdots\geq n_{\lambda}\geq 0$ hence the abscissae of $u_{ij}^{y}$ and
$u_{ij^{\prime}}^{y},j\neq j^{\prime}$, same. Here, we put
$S_{y}(\Xi):=\\{(i,j):0\leq i\leq\lambda,\ 0\leq j\leq
n_{i}\\}=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda}).$
In addition, we can also define the sets of abscissae and ordinates
$\displaystyle H_{j}(\Xi):=$ $\displaystyle\\{\bar{x}:(\bar{x},\bar{y})\in
l_{j}^{x}\cap\Xi\\},\quad 0\leq j\leq\nu,$ (6) $\displaystyle V_{i}(\Xi):=$
$\displaystyle\\{\bar{y}:(\bar{x},\bar{y})\in l_{i}^{y}\cap\Xi\\},\quad 0\leq
i\leq\lambda.$
Now, we will introduce our key notion: tower interpolation site.
###### Definition 3.
We say that an interpolation site $\Xi$ in $\mathbb{F}^{2}$ is $x$-_tower_ if
lower set
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})\subset\mathbb{N}_{0}^{2}$
with $m_{0}>m_{1}>\cdots>m_{\nu}\geq 0$ such that
$\Xi:=\\{(x_{ij},y_{j}):(i,j)\in S_{x}(\Xi)\\},$ (7)
where $x_{ij}\in H_{0}(\Xi),(i,j)\in S_{x}(\Xi)$, are distinct for fixed $j$.
We also call $\Xi$ a $S_{x}(\Xi)$-$x$-_tower_ interpolation site. In like
fashion, if lower set
$S_{y}(\Xi)=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda})\subset\mathbb{N}_{0}^{2}$,
$n_{0}>n_{1}>\cdots>n_{\lambda}\geq 0$, such that
$\Xi:=\\{(x_{i},y_{ij}):(i,j)\in S_{y}(\Xi)\\},$ (8)
where $y_{ij}\in V_{0}(\Xi),(i,j)\in S_{y}(\Xi)$, are distinct for fixed $i$,
we will call $\Xi$ a $y$-_tower_ interpolation site.
We have mentioned in Section 1, Definition 3 is enlightened by the notion of
lower sites. Next, we will compare them in detail. Recall Definition 1. We
have the following criterion.
###### Theorem 4 (Crainic (2004)).
An interpolation site $\Xi\subset\mathbb{F}^{2}$ is lower if and only if
$S_{x}(\Xi)=S_{y}(\Xi)$.
Let $\Xi$ be an $\mathcal{A}$-lower site. By Definition 1 and Theorem 4, we
can deduce that $\mathcal{A}=S_{x}(\Xi)=S_{y}(\Xi)$. Hence, (3) can be
redescribed as
$\Xi:=\\{(x_{i},y_{j}):(i,j)\in S_{x}(\Xi)\\}\phantom{.}$ (9)
or
$\Xi:=\\{(x_{i},y_{j}):(i,j)\in S_{y}(\Xi)\\}.$ (10)
Observing (7), (9) and (8), (10), we find that they are similar in form each
to each nevertheless it seems that tower sites make the nodes on a line much
freer than lower sites, which is supported by the following lemma as an
alternative criterion for lower sites.
###### Lemma 5.
Resume the notation established above. An interpolation site
$\Xi\subset\mathbb{F}^{2}$ is lower if and only if
$H_{0}(\Xi)\supseteq H_{1}(\Xi)\supseteq\cdots\supseteq H_{\nu}(\Xi)$ (11)
or
$V_{0}(\Xi)\supseteq V_{1}(\Xi)\supseteq\cdots\supseteq V_{\lambda}(\Xi).$
(12)
Proof. Assume that interpolation site $\Xi$ is lower. By (5), (6), and (9), we
have $H_{j}(\Xi)=\\{x_{0},x_{1},\ldots,x_{m_{j}}\\},j=0,\ldots,\nu$. Since
$m_{0}\geq m_{1}\geq\cdots\geq m_{\nu}\geq 0$, (11) follows. (12) can be
proved in like manner.
Conversely, we assume that (11) holds. Suppose
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu})=\mathrm{L}_{y}(n^{\prime}_{0},n^{\prime}_{1},\ldots,n^{\prime}_{\lambda^{\prime}})$
and $S_{y}(\Xi)=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{\lambda})$. Since
$S_{x}(\Xi)$ is lower, we have $\lambda^{\prime}=m_{0}$ and
$n^{\prime}_{0}=\nu$. By (6), $\\#H_{j}(\Xi)=m_{j}+1,j=0,\ldots,\nu$.
Next, we shall verify that $\lambda=m_{0}=\lambda^{\prime}$, namely we can
cover $\Xi$ by exactly $m_{0}+1$ vertical lines $l_{i}^{y},i=0,\ldots,m_{0}$,
which pass the $m_{0}+1$ nodes on line $l_{0}^{x}$ respectively. Prove this by
contradiction. Since $\lambda\geq m_{0}$ apparently, equality can fail only
when $\lambda>m_{0}$, i.e., there exists at least one node
$(\bar{x},\bar{y})\in\Xi$ such that $(\bar{x},\bar{y})\notin
l_{i}^{y},i=0,\ldots,m_{0}$. Because $(\bar{x},\bar{y})\in\Xi$, from (6),
there must exist some $H_{j^{*}}(\Xi),0\leq j^{*}\leq\nu$, such that
$\bar{x}\in H_{j^{*}}(\Xi)$. (11) implies that $\bar{x}\in H_{0}(\Xi)$, which
contradicts $(\bar{x},\bar{y})\notin l_{i}^{y},i=0,\ldots,m_{0}$.
Finally, we will prove
$n^{\prime}_{h}=n_{h},\quad h=0,\ldots,m_{0},$ (13)
which implies $S_{x}(\Xi)=S_{y}(\Xi)$ immediately. We prove it by induction on
$h$. When $h=0$, for any node $(x^{(\nu)},y^{(\nu)})\in l_{\nu}^{x}$,
$x^{(\nu)}\in H_{\nu}(\Xi)$ hence $x^{(\nu)}\in H_{j}(\Xi),0\leq j\leq\nu$,
due to (11). So we have found $\nu+1$ nodes on line $x=x^{(\nu)}$ that
obviously equals to some $l_{i^{*}}^{y},0\leq i^{*}\leq m_{0}$. From the
definition of $\nu$ and $S_{y}(\Xi)$, we have $n_{0}=n^{\prime}_{0}=\nu$,
namely (13) is true for $h=0$. Now, we assume (13) for $0\leq h\leq k<m_{0}$.
When $h=k+1$, by the induction hypothesis, we can find $k+1$ lines, without
loss of generality, $l_{0}^{y},\ldots,l_{k}^{y}$ that contain exactly
$n_{h}+1$ nodes, $0\leq h\leq k$, of $\Xi$ respectively. Since
$S_{x}(\Xi)=\mathrm{L}_{y}(n^{\prime}_{0},n^{\prime}_{1},\ldots,n^{\prime}_{m_{0}})$,
we have $(k+1,n^{\prime}_{k+1})\in S_{x}(\Xi)$ which implies that there must
exist at least one node $(x^{(n^{\prime}_{k+1})},y^{(n^{\prime}_{k+1})})\in
l_{n^{\prime}_{k+1}}^{x}$ which is not on $l_{0}^{y},\ldots,l_{k}^{y}$. By
(11), we can find $n^{\prime}_{k+1}+1$ nodes on line
$x=x^{(n^{\prime}_{k+1})}$ that is a member of
$\\{l_{k+1}^{y},\ldots,l_{m_{0}}^{y}\\}$ hence $n_{k+1}\geq n^{\prime}_{k+1}$.
From the definition of $S_{y}(\Xi)$, there exists one line in
$\\{l_{k+1}^{y},\ldots,l_{m_{0}}^{y}\\}$ that have exactly $n_{k+1}+1$ nodes
distributed among some lines of $\\{l_{0}^{x},\ldots,l_{n_{k+1}}^{x}\\}$ due
to (11). Therefore, we have $n^{\prime}_{k+1}\geq n_{k+1}$ that leads to
$n_{k+1}=n^{\prime}_{k+1}$, namely (13) holds for $h=k+1$. Consequently,
$S_{x}(\Xi)=\mathrm{L}_{y}(n^{\prime}_{0},n^{\prime}_{1},\ldots,n^{\prime}_{m_{0}})=\mathrm{L}_{y}(n_{0},n_{1},\ldots,n_{m_{0}})=S_{y}(\Xi)$
which implies that $\Xi$ is lower due to Theorem 4.
Swapping the roles of $x$ and $y$, we can also prove the other statement
similarly therefore complete the proof. $\Box$
Note that it is easy for Lemma 5 to extend to higher dimensions but difficult
for Theorem 4.
Now, we go back to tower sites. For an $x$-tower site $\Xi$, although
$H_{j}(\Xi)\varsubsetneq H_{0}(\Xi),j=1,\ldots,\nu$, it may have nothing to do
with any other $H_{i}(\Xi),i=1,\ldots,j-1,j+1,\ldots,\nu$. Comparing with
(11), we find that, on one line, lower sites allow free amounts of nodes while
tower sites allow free positions of nodes. The following proposition reveals
the relation between them.
###### Proposition 6.
Resume the notation established above. If $\Xi$ satisfies
$H_{0}(\Xi)\varsupsetneq H_{1}(\Xi)\varsupsetneq\cdots\varsupsetneq
H_{\nu}(\Xi)$, then $\Xi$ is lower if and only if it is $x$-tower. Similarly,
if $\Xi$ satisfies $V_{0}(\Xi)\varsupsetneq
V_{1}(\Xi)\varsupsetneq\cdots\varsupsetneq V_{\lambda}(\Xi)$, then $\Xi$ is
lower iff it is $y$-tower.
Proof. Assume that $\Xi$ is lower with
$S_{x}(\Xi)=\mathrm{L}_{x}(m_{0},m_{1},\ldots,m_{\nu}).$ Since
$H_{0}(\Xi)\varsupsetneq H_{1}(\Xi)\varsupsetneq\cdots\varsupsetneq
H_{\nu}(\Xi)$, by (6), we have $m_{0}>m_{1}>\cdots>m_{\nu}\geq 0$ which
implies that $\Xi$ is a $S_{x}(\Xi)$-$x$-tower site according to (7).
Conversely, if $\Xi$ is $x$-tower, since $H_{0}(\Xi)\varsupsetneq
H_{1}(\Xi)\varsupsetneq\cdots\varsupsetneq H_{\nu}(\Xi)$, by Lemma 5, $\Xi$ is
lower follows.
In very like fashion, we can prove the other statement. $\Box$
Proposition 6 shows that the notions of lower sites and tower sites are not
mutually exclusive as there exist lower sites that are tower sites, but there
are also tower sites are not lower and vice versa. The following examples show
them with illustrations.
###### Example 7.
Observe (a), (b), (c) of Fig. 1. $\Xi_{1}$ is an $x$-tower site that is not
lower while $\Xi_{2}$ is a lower site that is not $x$-tower or $y$-tower.
$\Xi_{3}$ is a $y$-tower site that is lower too. (d) of Fig. 1 illustrates
$S_{y}(\Xi_{3})$ that is equal to $S_{x}(\Xi_{3})$.
(a) $\Xi_{1}$ (b) $\Xi_{2}$
(c) $\Xi_{3}$ (d) $S_{y}(\Xi_{3})$
Figure 1: Illustrations for Example 7
## 4 Main Results
In this section, we will pursue the degree reducing interpolation bases w.r.t.
a fixed term order for bivariate Lagrange interpolation on a tower site $\Xi$.
Since an interpolation polynomial of minimal total degree is prefered in most
situations, the $\prec_{\mathrm{tdlex}}$ and $\prec_{\mathrm{tdinlex}}$ cases
will be discussed at the beginning. Note that we will continue with the
notation established in the previous sections.
###### Lemma 8 (Dong et al. (2005)).
Let
$\Xi=\\{(x_{0},y_{0}),(x_{1},y_{0}),\ldots,(x_{m},y_{0})\\}\subset\mathbb{F}^{2}$
be a set of distinct nodes on line $y=y_{0}$. Then
$\mathcal{I}(\Xi)=\langle(x-x_{0})(x-x_{1})\cdots(x-x_{m}),y-y_{0}\rangle.$
As a main result of the paper, the following theorem gives the DRIMB w.r.t.
$\prec_{\mathrm{tdlex}}$ or $\prec_{\mathrm{tdinlex}}$ for bivariate Lagrange
interpolation on an $x$-tower or $y$-tower site respectively in theory.
###### Theorem 9.
Given an $x$-tower site $\Xi\subset\mathbb{F}^{2}$. The DRIMB w.r.t.
$\prec_{\mathrm{tdlex}}$ for (1) is
$N_{x}=\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi)\\}.$
If $\Xi$ is $y$-tower, then the DRIMB w.r.t. $\prec_{\mathrm{tdinlex}}$ for
(1) is
$N_{y}=\\{x^{i}y^{j}:(i,j)\in S_{y}(\Xi)\\}.$
Proof. We will only prove the first statement. The second one can be verified
in very like fashion.
For fixed $j$, by Lemma 8, the ideal
$\mathcal{I}_{j}:=\mathcal{I}(\\{(x_{0j},y_{j}),(x_{1j},y_{j}),\ldots,(x_{m_{j},j},y_{j})\\})$,
$j=0,\ldots,\nu$, equals
$\langle(x-x_{0j})(x-x_{1j})\cdots(x-x_{m_{j},j}),y-y_{j}\rangle.$
Obviously, $\mathcal{I}_{j}$’s are pairwise comaximal hence the ideal
$\mathcal{I}(\Xi)$ is equal to
$\bigcap_{j=0}^{\nu}\mathcal{I}_{j}=\prod_{j=0}^{\nu}\mathcal{I}_{j}=\prod_{j=0}^{\nu}\langle(x-x_{0j})(x-x_{1j})\cdots(x-x_{m_{j},j}),y-y_{j}\rangle.$
Let $G$ be the reduced Gröbner basis for $\mathcal{I}(\Xi)$ and $G_{k}$ the
reduced Gröbner basis for $\prod_{j=0}^{k}\mathcal{I}_{j}$ w.r.t.
$\prec_{\mathrm{tdlex}}$, $0\leq k\leq\nu$. We will use induction on $\nu$ to
prove
$\mathrm{LT}(G)=\\{x^{m_{0}+1},x^{m_{1}+1}y,x^{m_{2}+1}y^{2},\ldots,x^{m_{\nu}+1}y^{\nu},y^{\nu+1}\\}.$
(14)
First of all, since
$\mathcal{I}_{0}=\langle(x-x_{0,0})(x-x_{1,0})\cdots(x-x_{m_{0},0}),y-y_{0}\rangle,$
(14) is true for $\nu=0$.
Now, let $\nu=1$. $\Xi$ is $x$-tower implies that $m_{0}>m_{1}$ and
$H_{1}(\Xi)\subsetneq H_{0}(\Xi)$. Therefore,
$\displaystyle\mathcal{I}_{0}\cdot\mathcal{I}_{1}=$
$\displaystyle\langle(x-x_{0,0})\cdots(x-x_{m_{0},0}),y-y_{0}\rangle\cdot\langle(x-x_{0,1})\cdots(x-x_{m_{1},1}),y-y_{1}\rangle$
$\displaystyle=$
$\displaystyle\langle(x-x_{0,0})\cdots(x-x_{m_{0},0}),(x-x_{0,0})\cdots(x-x_{m_{0},0})(y-y_{1}),$
$\displaystyle(x-x_{0,1})\cdots(x-x_{m_{1},1})(y-y_{0}),(y-y_{0})(y-y_{1})\rangle$
$\displaystyle=$
$\displaystyle\langle(x-x_{0,0})\cdots(x-x_{m_{0},0}),(x-x_{0,1})\cdots(x-x_{m_{1},1})(y-y_{0}),(y-y_{0})(y-y_{1})\rangle.$
It is easy to check that
$\mathrm{LT}(G_{1})=\\{x^{m_{0}+1},x^{m_{1}+1}y,y^{2}\\}$ which means that
(14) holds for $\nu=1$.
When $\nu=2$, since $m_{1}>m_{2}$ and $H_{2}(\Xi)\subsetneq H_{0}(\Xi)$, we
get, after some easy computations,
$\displaystyle\mathcal{I}_{0}\cdot\mathcal{I}_{1}\cdot\mathcal{I}_{2}=$
$\displaystyle\langle(x-x_{0,0})\cdots(x-x_{m_{0},0}),(y-y_{0})(y-y_{1})(y-y_{2}),$
$\displaystyle(x-x_{0,1})\cdots(x-x_{m_{1},1})(x-x_{0,2})\cdots(x-x_{m_{2},2})(y-y_{0}),$
$\displaystyle(x-x_{0,1})\cdots(x-x_{m_{1},1})(y-y_{0})(y-y_{2}),$
$\displaystyle(x-x_{0,2})\cdots(x-x_{m_{2},2})(y-y_{0})(y-y_{1})\rangle$
$\displaystyle=:$ $\displaystyle\langle
g_{0}^{(2)},g_{1}^{(2)},g_{2}^{(2)},g_{3}^{(2)},g_{4}^{(2)}\rangle.$
Noticing that $m_{1}>m_{2}$, we let $\hat{q},\hat{r}\in\Pi^{1}$ be the
quotient and remainder respectively of the division of
$(x-x_{0,1})\cdots(x-x_{m_{1},1})$ by $(x-x_{0,2})\cdots(x-x_{m_{2},2})$,
namely
$(x-x_{0,1})\cdots(x-x_{m_{1},1})=\hat{q}(x-x_{0,2})\cdots(x-x_{m_{2},2})+\hat{r}.$
Denote the remainder of $g_{3}^{(2)}$ w.r.t. $g_{4}^{(2)}$ by
$\bar{g}_{3}^{(2)}$. One can check readily that
$\bar{g}_{3}^{(2)}=g_{3}^{(2)}-\hat{q}g_{4}^{(2)}.$
On the other hand, since
$g_{2}^{(2)}=\frac{1}{y_{1}-y_{2}}\left[(x-x_{0,2})\cdots(x-x_{m_{2},2})g_{3}^{(2)}-(x-x_{0,1})\cdots(x-x_{m_{1},1})g_{4}^{(2)}\right],$
we have $g_{2}^{(2)}\xrightarrow{\\{g_{3}^{(2)},g_{4}^{(2)}\\}}_{+}0$. It
follows that
$\displaystyle\mathcal{I}_{0}\cdot\mathcal{I}_{1}\cdot\mathcal{I}_{2}=$
$\displaystyle\langle\
g_{0}^{(2)},g_{1}^{(2)},\bar{g}_{3}^{(2)},g_{4}^{(2)}\rangle.$
We claim that
$G_{2}=G^{\prime}_{2}:=\\{g_{0}^{(2)},g_{1}^{(2)},(y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)},g_{4}^{(2)}\\},$
where $y_{1}-y_{2}=\mathrm{LC}(\bar{g}_{3}^{(2)})$, i.e.,
$(y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)}$ is monic.
In fact, if $S(f,g)$ stands for the S-polynomial of arbitrary polynomials
$f,g\in\Pi^{2}$, $S(g_{0}^{(2)},g_{1}^{(2)})\xrightarrow{G^{\prime}_{2}}_{+}0$
follows immediately because $\mathrm{LM}(g_{0}^{(2)})$ and
$\mathrm{LM}(g_{1}^{(2)})$ are relatively prime. Notice that
$(x-x_{0,2})\cdots(x-x_{m_{2},2})$ is a factor of
$(x-x_{0,0})\cdots(x-x_{m_{0},0})$, we get
$\displaystyle S(g_{0}^{(2)},g_{4}^{(2)})=$ $\displaystyle
y^{2}g_{0}^{(2)}-x^{m_{0}-m_{2}}g_{4}^{(2)}$ $\displaystyle=$ $\displaystyle
y^{2}g_{0}^{(2)}-x^{m_{0}-m_{2}}g_{4}^{(2)}-(y-y_{0})(y-y_{1})g_{0}^{(2)}+(y-y_{0})(y-y_{1})g_{0}^{(2)}$
$\displaystyle=$ $\displaystyle
y^{2}g_{0}^{(2)}-x^{m_{0}-m_{2}}g_{4}^{(2)}-(y-y_{0})(y-y_{1})g_{0}^{(2)}+\frac{(x-x_{0,0})\cdots(x-x_{m_{0},0})}{(x-x_{0,2})\cdots(x-x_{m_{2},2})}g_{4}^{(2)}$
$\displaystyle=$
$\displaystyle((y_{0}+y_{1})y-y_{0}y_{1})g_{0}^{(2)}+\left(\frac{(x-x_{0,0})\cdots(x-x_{m_{0},0})}{(x-x_{0,2})\cdots(x-x_{m_{2},2})}-x^{m_{0}-m_{2}}\right)g_{4}^{(2)}.$
It is easy to check that
$\mathrm{LM}(S(g_{0}^{(2)},g_{4}^{(2)}))=x^{m_{0}+1}y$. Since
$\displaystyle\mathrm{LM}(((y_{0}+y_{1})y-y_{0}y_{1})g_{0}^{(2)})$
$\displaystyle=x^{m_{0}+1}y,$
$\displaystyle\mathrm{LM}\left(\left(\frac{(x-x_{0,0})\cdots(x-x_{m_{0},0})}{(x-x_{0,2})\cdots(x-x_{m_{2},2})}-x^{m_{0}-m_{2}}\right)g_{4}^{(2)}\right)$
$\displaystyle=x^{m_{0}}y^{2},$
we have
$\displaystyle\mathrm{LM}(S(g_{0}^{(2)},g_{4}^{(2)}))=$
$\displaystyle\max_{\prec_{\mathrm{tdlex}}}\Bigg{(}\mathrm{LM}(((y_{0}+y_{1})y-y_{0}y_{1})g_{0}^{(2)}),$
$\displaystyle\mathrm{LM}\left(\left(\frac{(x-x_{0,0})\cdots(x-x_{m_{0},0})}{(x-x_{0,2})\cdots(x-x_{m_{2},2})}-x^{m_{0}-m_{2}}\right)g_{4}^{(2)}\right)\Bigg{)}$
$\displaystyle=$ $\displaystyle x^{m_{0}+1}y,$
which implies that
$S(g_{0}^{(2)},g_{4}^{(2)})\xrightarrow{G_{2}^{\prime}}_{+}0$. Similarly, by
$S(g_{1}^{(2)},g_{4}^{(2)})=(x^{m_{2}+1}-(x-x_{0,2})\cdots(x-x_{m_{2},2}))g_{1}^{(2)}-y_{2}g_{4}^{(2)},$
and $\mathrm{LM}(S(g_{1}^{(2)},g_{4}^{(2)}))=x^{m_{2}+1}y^{2}$,
$\displaystyle\max_{\prec_{\mathrm{tdlex}}}\big{(}\mathrm{LM}((x^{m_{2}+1}-(x-x_{0,2})\cdots(x-x_{m_{2},2}))g_{1}^{(2)}),\mathrm{LM}(-y_{2}g_{4}^{(2)})\big{)}$
$\displaystyle=$
$\displaystyle\max_{\prec_{\mathrm{tdlex}}}(x^{m_{2}}y^{3},x^{m_{2}+1}y^{2})=x^{m_{2}+1}y^{2}=\mathrm{LM}(S(g_{1}^{(2)},g_{4}^{(2)}))$
follows, namely $S(g_{1}^{(2)},g_{4}^{(2)})\xrightarrow{G_{2}^{\prime}}_{+}0$.
In like manner, we can also prove that
$S(g_{0}^{(2)},(y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)})\xrightarrow{G_{2}^{\prime}}_{+}0,\quad
S(g_{1}^{(2)},(y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)})\xrightarrow{G_{2}^{\prime}}_{+}0.$
Hence, there only remains
$S((y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)},g_{4}^{(2)})\xrightarrow{G_{2}^{\prime}}_{+}0$
to be checked. Actually, it can be deduced that
$S((y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)},g_{4}^{(2)})$ equals
$\frac{\hat{r}}{y_{1}-y_{2}}g_{1}^{(2)}+y_{1}\cdot(y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)}+\left(\hat{q}-x^{m_{1}-m_{2}}\right)g_{4}^{(2)}.$
Observing the structure of
$S((y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)},g_{4}^{(2)})$, we know that its leading
monomial is $x^{m_{1}+1}y$, therefore, from
$\displaystyle\mathrm{LM}\left(\frac{\hat{r}}{y_{1}-y_{2}}g_{1}^{(2)}\right)$
$\displaystyle=x^{m_{2}}y^{3},$
$\displaystyle\mathrm{LM}(y_{1}\cdot(y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)})$
$\displaystyle=x^{m_{1}+1}y,$
$\displaystyle\mathrm{LM}\left(\left(\hat{q}-x^{m_{1}-m_{2}}\right)g_{4}^{(2)}\right)$
$\displaystyle=x^{m_{1}}y^{2},$
$S((y_{1}-y_{2})^{-1}\bar{g}_{3}^{(2)},g_{4}^{(2)})\xrightarrow{G_{2}^{\prime}}_{+}0$.
The above arguments lead to the conclusion: S-polynomial
$S(g,g^{\prime})\xrightarrow{G^{\prime}_{2}}_{+}0$ for all $g,g^{\prime}\in
G^{\prime}_{2},g\neq g^{\prime}$. Consequently, from Buchberger’s S-pair
criterion, $G_{2}^{\prime}$ is a Gröbner basis for
$\mathcal{I}_{0}\cdot\mathcal{I}_{1}\cdot\mathcal{I}_{2}$ w.r.t.
$\prec_{\mathrm{tdlex}}$.
Moreover, for any polynomial $g\in G_{2}^{\prime}$, it is evident that
1. 1.
$\mathrm{LC}(g)=1$,
2. 2.
No monomial of $g$ lies in
$\langle\mathrm{LT}(G_{2}^{\prime}-\\{g\\})\rangle$,
which implies that $G_{2}^{\prime}$ is the reduced Gröbner basis for
$\mathcal{I}_{0}\cdot\mathcal{I}_{1}\cdot\mathcal{I}_{2}$ w.r.t.
$\prec_{\mathrm{tdlex}}$. Hence (14) holds for $\nu=2$.
Now, for $\nu=k$, assume (14). That is to say
$\mathrm{LT}(G_{k})=\\{x^{m_{0}+1},x^{m_{1}+1}y,x^{m_{2}+1}y^{2},\ldots,x^{m_{k}+1}y^{k},y^{k+1}\\}.$
Let $G_{k}=\\{g_{0}^{(k)},\ldots,g_{k+1}^{(k)}\\}$. Without loss of
generality, we may assume that
$\displaystyle\mathrm{LT}(g_{i}^{(k)})=$ $\displaystyle x^{m_{i}+1}y^{i},\quad
i=0,\ldots,k,$ $\displaystyle\mathrm{LT}(g_{k+1}^{(k)})=$ $\displaystyle
y^{k+1},$
which imply that
$\displaystyle g_{0}^{(k)}$ $\displaystyle=(x-x_{0,0})\cdots(x-x_{m_{0},0}),$
$\displaystyle g_{k+1}^{(k)}$
$\displaystyle=(y-y_{0})(y-y_{1})\cdots(y-y_{k}).$
When $\nu=k+1$, since $\Xi$ is $x$-tower, we obtain
$\displaystyle\prod_{i=0}^{k+1}\mathcal{I}_{i}=$
$\displaystyle(\prod_{i=0}^{k}\mathcal{I}_{i})\mathcal{I}_{k+1}$
$\displaystyle=\langle$ $\displaystyle\
g_{0}^{(k)},\ldots,g_{k+1}^{(k)}\rangle\cdot\langle(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1}),y-y_{k+1}\rangle$
$\displaystyle=\langle$ $\displaystyle\ g_{0}^{(k+1)},g_{0}^{(k)}(y-y_{k+1}),$
$\displaystyle\
g_{1}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1}),g_{1}^{(k)}(y-y_{k+1}),$
$\displaystyle\cdots$ $\displaystyle\
g_{k-1}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1}),g_{k-1}^{(k)}(y-y_{k+1}),$
$\displaystyle\
g_{k}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1}),g_{k}^{(k)}(y-y_{k+1}),$
$\displaystyle\
g_{k+1}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1}),g_{k+1}^{(k)}(y-y_{k+1})\rangle,$
where $g_{0}^{(k+1)}=g_{0}^{(k)}$. Thus, $g_{0}^{(k)}(y-y_{k+1})$ can be
removed from the ideal basis above. By the induction hypothesis, we have
$g_{k+2}^{(k+1)}:=g_{k+1}^{(k)}(y-y_{k+1})=(y-y_{0})(y-y_{1})\cdots(y-y_{k+1}).$
We denote polynomial $g_{k+1}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1})$ by
$g_{k+1}^{(k+1)}$. For $E_{1}:=\\{g_{k+1}^{(k+1)}\\}$, suppose
$g_{k}^{(k)}(y-y_{k+1})\xrightarrow{E_{1}}_{+}g_{k}^{(k+1)}$. Since
$m_{k}>m_{k+1}$, we have
$\mathrm{LM}(g_{k}^{(k+1)})=x^{m_{k}+1}y^{k}.$
Furthermore, recalling the case $\nu=2$, we can check easily that
$g_{k}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1})\xrightarrow{F_{1}}_{+}0,$
where $F_{1}:=\\{g_{k+1}^{(k+1)},g_{k}^{(k+1)}\\}$, which implies that the
polynomial can be removed from the basis.
Let $E_{2}:=F_{1}$ and suppose
$g_{k-1}^{(k)}(y-y_{k+1})\xrightarrow{E_{2}}_{+}g_{k-1}^{(k+1)}$. We can also
deduce that $\mathrm{LM}(g_{k-1}^{(k+1)})=x^{m_{k-1}+1}y^{k-1}$ and
$g_{k-1}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1})\xrightarrow{F_{2}}_{+}0,$
where $F_{2}:=\\{g_{k+1}^{(k+1)},g_{k-1}^{(k+1)}\\}$.
In this way we can construct two sequences $E_{1},E_{2},\ldots,E_{k}$ and
$F_{1},F_{2},\ldots,F_{k}$ such that
$\displaystyle g_{i}^{(k)}(y-y_{k+1})\xrightarrow{E_{k+1-i}}_{+}$
$\displaystyle g_{i}^{(k+1)},$ $\displaystyle
g_{i}^{(k)}(x-x_{0,k+1})\cdots(x-x_{m_{k+1},k+1})\xrightarrow{F_{k+1-i}}_{+}$
$\displaystyle 0,$
where, for $i=1,\ldots,k$,
$\displaystyle E_{i}$
$\displaystyle=\\{g_{k+2-i}^{(k+1)},\ldots,g_{k+1}^{(k+1)}\\},$ $\displaystyle
F_{i}$ $\displaystyle=\\{g_{k+1}^{(k+1)},g_{k+1-i}^{(k+1)}\\}.$
Let $\bar{g}_{i}^{(k+1)}=\mathrm{LC}(g_{i}^{(k+1)})^{-1}g_{i}^{(k+1)}$. With
quite similar methods that we used in the case $\nu=2$, we can prove finally
that
$\\{\bar{g}_{0}^{(k+1)},\bar{g}_{1}^{(k+1)},\ldots,\bar{g}_{k+1}^{(k+1)}\\}$
is the reduced Gröbner basis for $\prod_{i=0}^{k+1}\mathcal{I}_{i}$ w.r.t.
$\prec_{\mathrm{tdlex}}$, i.e., (14) holds when $\nu=k+1$. By (4) and the
definition of $S_{x}(\Xi)$, we have
$\mathrm{N}_{\prec_{\mathrm{tdlex}}}(\mathcal{I}(\Xi))=N_{x}=\\{x^{i}y^{j}:(i,j)\in
S_{x}(\Xi)\\},$
which complete the proof. $\Box$
The next two theorems present degree reducing interpolation Newton bases
w.r.t. $\prec_{\mathrm{tdlex}}$ or $\prec_{\mathrm{tdinlex}}$ for bivariate
Lagrange interpolation on an $x$-tower or $y$-tower site respectively.
###### Theorem 10.
Let $\Xi\subset\mathbb{F}^{2}$ be an $x$-tower site of nodes
$u_{mn}^{x}=(x_{mn},y_{n}),(m,n)\in S_{x}(\Xi)$, which gives rise to
polynomials
$\phi_{ij}^{x}=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y-y_{t})\prod_{s=0}^{i-1}(x-x_{sj}),\quad(i,j)\in
S_{x}(\Xi),$
where
$\varphi_{ij}^{x}=1/\prod_{t=0}^{j-1}(y_{j}-y_{t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj})\in\mathbb{F}$,
and the empty products are taken as 1. Then
$Q_{x}=\\{\phi_{ij}^{x}:(i,j)\in S_{x}(\Xi)\\}$ (15)
is a _DRINB_ w.r.t. $\prec_{\mathrm{tdlex}}$ for (1) satisfying
$\phi_{ij}^{x}(u_{mn}^{x})=\delta_{(i,j),(m,n)},\quad(i,j)\succeq_{\mathrm{inlex}}(m,n).$
Proof. Fix $(i,j)\in S_{x}(\Xi)$. If $(i,j)=(m,n)$, by $y_{0}\neq
y_{1}\neq\cdots\neq y_{j}$ and $x_{0j}\neq x_{1j}\neq\cdots\neq x_{ij}$, we
have
$\phi_{ij}^{x}(u_{ij}^{x})=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{j}-y_{t})\prod_{s=0}^{i-1}(x_{ij}-x_{sj})\\\
=\varphi_{ij}^{x}/\varphi_{ij}^{x}=1.$
Otherwise, if $(i,j)\succ_{\mathrm{inlex}}(m,n)$, we have $j>n$, or $j=n,i>m$.
When $j>n$, we have
$\phi_{ij}^{x}(u_{mn}^{x})=\varphi_{ij}^{x}(y_{n}-y_{0})\cdots(y_{n}-y_{n})\cdots(y_{n}-y_{j-1})\prod_{s=0}^{i-1}(x_{mn}-x_{sj})=0,$
and when $j=n,i>m$,
$\displaystyle\phi_{ij}^{x}(u_{mn}^{x})$
$\displaystyle=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y_{n}-y_{t})(x_{mn}-x_{0j})\cdots(x_{mn}-x_{mj})\cdots(x_{mn}-x_{i-1,j})$
$\displaystyle=\varphi_{ij}^{x}\prod_{t=0}^{n-1}(y_{n}-y_{t})(x_{mn}-x_{0n})\cdots(x_{mn}-x_{mn})\cdots(x_{mn}-x_{i-1,n})$
$\displaystyle=0,$
which leads to
$\phi_{ij}^{x}(u_{mn}^{x})=0,\quad(i,j)\succ_{\mathrm{inlex}}(m,n),$
namely $Q_{x}$ is a Newton basis for $\mathrm{Span}_{\mathbb{F}}Q_{x}$. By
Theorem 9, it is easy to check that
$\mathrm{Span}_{\mathbb{F}}Q_{x}=\mathrm{Span}_{\mathbb{F}}N_{x}=\mathcal{P}_{\prec_{\mathrm{tdlex}}}(\Xi)$.
Therefore, $Q_{x}$ is a DRINB w.r.t. $\prec_{\mathrm{tdlex}}$ for (1). $\Box$
Similarly, we can prove the following theorem:
###### Theorem 11.
Let $\Xi\subset\mathbb{F}^{2}$ be a $y$-tower site of nodes
$u_{mn}^{y}=(x_{m},y_{mn}),(m,n)\in S_{y}(\Xi)$. We define the polynomials
$\phi_{ij}^{y}=\varphi_{ij}^{y}\prod_{s=0}^{i-1}(x-x_{s})\prod_{t=0}^{j-1}(y-y_{it}),\quad(i,j)\in
S_{y}(\Xi),$
where
$\varphi_{ij}^{y}=1/\prod_{s=0}^{i-1}(x_{i}-x_{s})\prod_{t=0}^{j-1}(y_{ij}-y_{it})\in\mathbb{F}$.
The empty products are taken as 1. Then,
$Q_{y}=\\{\phi_{ij}^{x}:(i,j)\in S_{y}(\Xi)\\}$ (16)
is a _DRINB_ w.r.t. $\prec_{\mathrm{tdinlex}}$ for (1) satisfying
$\phi_{ij}^{y}(u_{mn}^{y})=\delta_{(i,j),(m,n)},\quad(i,j)\succeq_{\mathrm{lex}}(m,n).$
Now, we will turn to $\prec_{\mathrm{lex}}$ and $\prec_{\mathrm{inlex}}$
cases. For any interpolation site $\Xi\subset\mathbb{F}^{2}$, the following
theorem presents the degree reducing interpolation monomial and Newton bases
w.r.t. $\prec_{\mathrm{lex}}$ or $\prec_{\mathrm{inlex}}$ for (1)
theoretically.
###### Theorem 12 (Wang et al. (2009)).
Let
$\Xi=\\{u_{mn}^{x}=(x_{mn}^{x},y_{mn}^{x}):(m,n)\in
S_{x}(\Xi)\\}=\\{u_{mn}^{y}=(x_{mn}^{y},y_{mn}^{y}):(m,n)\in S_{y}(\Xi)\\}$
be an interpolation site in $\mathbb{F}^{2}$. Then
(i) the set $N_{x}:=\\{x^{i}y^{j}:(i,j)\in S_{x}(\Xi)\\}$ is the _DRIMB_ as
well as $Q_{x}:=\\{\phi_{ij}^{x}:(i,j)\in S_{x}(\Xi)\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{lex}}$ for (1), where
$\phi_{ij}^{x}=\varphi_{ij}^{x}\prod_{t=0}^{j-1}(y-y_{0t}^{x})\prod_{s=0}^{i-1}(x-x_{sj}^{x}),\quad(i,j)\in
S_{x}(\Xi),$
with
$\varphi_{ij}^{x}=1/\prod_{t=0}^{j-1}(y_{0j}^{x}-y_{0t}^{x})\prod_{s=0}^{i-1}(x_{ij}^{x}-x_{sj}^{x})\in\mathbb{F}$
and the empty products taken as 1;
(ii) the set $N_{y}:=\\{x^{i}y^{j}:(i,j)\in S_{y}(\Xi)\\}$ is the _DRIMB_ as
well as $Q_{y}:=\\{\phi_{ij}^{y}:(i,j)\in S_{y}(\Xi)\\}$ is a _DRINB_ w.r.t.
$\prec_{\mathrm{inlex}}$ for (1), where
$\phi_{ij}^{y}=\varphi_{ij}^{y}\prod_{s=0}^{i-1}(x-x_{s0}^{y})\prod_{t=0}^{j-1}(y-y_{it}^{y}),\quad(i,j)\in
S_{y}(\Xi),$
with
$\varphi_{ij}^{y}=1/\prod_{s=0}^{i-1}(x_{i0}^{y}-x_{s0}^{y})\prod_{t=0}^{j-1}(y_{ij}^{y}-y_{it}^{y})\in\mathbb{F}$,
and the empty products are taken as 1.
Since Theorem 12 holds for any interpolation site in $\mathbb{F}^{2}$, it
obviously holds for tower sites. It is easy to find that the $N_{x}$ in the
theorem is same to the one in Theorem 9, namely the DRIMB w.r.t.
$\prec_{\mathrm{lex}}$ is same to the one w.r.t. $\prec_{\mathrm{tdlex}}$ for
interpolation on an $x$-tower site. In addition, the $Q_{x}$ here is same to
(15) when $\Xi$ is $x$-tower. For $N_{y}$ and $Q_{y}$, similar statements are
also true for a $y$-tower site.
In a word, for a preassigned $x$-tower site, we can obtain the DRIMB w.r.t.
$\prec_{\mathrm{tdlex}}$ or $\prec_{\mathrm{lex}}$ by Theorem 9, and a DRINB
w.r.t. $\prec_{\mathrm{tdlex}}$ or $\prec_{\mathrm{lex}}$ by Theorem 10. The
$y$-tower cases can also be solved by Theorem 9 and 11.
## 5 Algorithm and Experimental results
Let $\Xi$ be a tower site in $\mathbb{F}^{2}$. Suppose $(G,N,Q)$ be the
3-tuple output of BM algorithm. Theorem 9-11 present us $N$ and $Q$
theoretically hence we can obtain them with little effort. According to
Marinari et al. (1993), $\mathrm{LT}(G)$ is contained in the border set of
$N$. Therefore, inputting $N,Q$ into BM algorithm can make us obtain $G$ much
faster than compute $G$ directly with BM algorithm.
###### Algorithm 2.
(TBM algorithm)
Input: A $x$-tower($y$-tower) interpolation site $\Xi\subset\mathbb{F}^{2}$ of
$\mu$ nodes and fixed term order
$\prec_{\mathrm{tdlex}}$($\prec_{\mathrm{tdinlex}}$) or
$\prec_{\mathrm{lex}}$($\prec_{\mathrm{inlex}}$).
Output: The 3-tuple $(G,N,Q)$, where $G$ is the reduced Gröbner basis for
$\mathcal{I}(\Xi)$, $N$ is the Gröbner éscalier
$\mathrm{N}(\mathcal{I}(\Xi))$, and $Q$ is a DRINB for (1).
TBM1. Construct lower set $S_{x}(\Xi)$($S_{y}(\Xi)$) following the process
introduced in Section 3.
TBM2. Compute the sets $N$ and $Q$ according to Theorem 9-11.
TBM3. Compute the border set $L:=\\{x\cdot t:t\in N\\}\bigcup\\{y\cdot t:t\in
N\\}\setminus N$.
TBM4. Construct $\mu\times\mu$ matrix $B$ whose $(h,k)$ entry is
$\phi_{h}^{x}(u_{k}^{x})$($\phi_{h}^{y}(u_{k}^{y})$) where
$\phi_{h}^{x}$($\phi_{h}^{y}$), $u_{k}^{x}$($u_{k}^{y}$) are $h$th and $k$th
elements of $Q=\\{\phi_{ij}^{x}(\phi_{ij}^{y}):(i,j)\in
S_{x}(\Xi)(S_{y}(\Xi))\\}$ and $\Xi=\\{u_{mn}^{x}(u_{mn}^{y}):(m,n)\in
S_{x}(\Xi)(S_{y}(\Xi))\\}$ w.r.t. the increasing
$\prec_{\mathrm{inlex}}$($\prec_{\mathrm{lex}}$) on $(i,j)$ and $(m,n)$
respectively.
TBM5. Goto BM2 with $N,Q,L,B$ for the reduced Gröbner basis $G$.
In the following, we will show the timings for the computations of BM-problems
on tower sites in finite prime fields w.r.t. term order
$\prec_{\mathrm{inlex}}$ or $\prec_{\mathrm{tdlex}}$. BM and TBM algorithms
were implemented on Maple 12 installed on a laptop with 2 Gb RAM and 1.8 GHz
CPU.
For field $\mathbb{F}_{23}$ and $\prec_{\mathrm{tdlex}}$,
Algorithms $\\#\Xi$ | 100 | 300 | 500 | 900 | 1300
---|---|---|---|---|---
TBM | 0.125 s | 1.297 s | 4.953 s | 20.609 s | 56.906 s
BM | 0.531 s | 9.437 s | 53.625 s | 224.812 s | 861.531 s
For field $\mathbb{F}_{43}$ and $\prec_{\mathrm{inlex}}$,
Algorithms $\\#\Xi$ | 600 | 900 | 1200 | 1500 | 1800
---|---|---|---|---|---
TBM | 7.781 s | 21.094 s | 45.282 s | 87.734 s | 134.265 s
BM | 121.094 s | 362.000 s | 1002.578 s | 2077.907 | 2729.750 s
## References
* Becker and Weispfenning (1993) Becker, T., Weispfenning, V., 1993. Gröbner Bases. Vol. 141 of Graduate Texts in Mathematics. Springer-Verlag, New York.
* Birkhoff (1979) Birkhoff, G., 1979. The algebra of multivariate interpolation. In: Coffman, C. V., Fix, G. J. (Eds.), Constructive Approaches to Mathematical Models. Academic Press, New York, pp. 345–363.
* Bos et al. (2007) Bos, L., De Marchi, S., Vianello, M., Xu, Y., 2007. Bivariate Lagrange interpolation at the Padua points: The ideal theory approach. Numer. Math. 108 (1), 43–57.
* Chen et al. (2006) Chen, T., Dong, T., Zhang, S., 2006. The Newton interpolation bases on lower sets. J. Inf. Comput. Sci. 3 (3), 385–394.
* Crainic (2004) Crainic, N., 2004. Multivariate Birkhoff-Lagrange interpolation schemes and cartesian sets of nodes. Acta Math. Univ. Comenian.(N.S.) LXXIII (2), 217–221.
* de Boor (2005) de Boor, C., 2005. Ideal interpolation. In: Chui, C. K., Neamtu, M., Schumaker, L. L. (Eds.), Approximation Theory XI: Gatlinburg 2004. Nashboro Press, Brentwood TN, pp. 59–91.
* de Boor (2007) de Boor, C., 2007. Interpolation from spaces spanned by monomials. Adv. Comput. Math. 26 (1), 63–70.
* Dong et al. (2005) Dong, T., Zhang, S., Lei, N., 2005. Interpolation basis for nonuniform rectangular grid. J. Inf. Comput. Sci. 2 (4), 671–680.
* Lorentz (1992) Lorentz, R., 1992. Multivariate Birkhoff Interpolation. Vol. 1516 of Lecture Notes in Mathematics. Springer, Heidelberg.
* Marinari et al. (1993) Marinari, M. G., Möller, H. M., Mora, T., 1993. Gröbner bases of ideals defined by functionals with an application to ideals of projective points. Appl. Algebra Engrg. Comm. Comput. 4 (2), 103–145.
* Möller and Buchberger (1982) Möller, H., Buchberger, B., 1982. The construction of multivariate polynomials with preassigned zeros. In: Calmet, J. (Ed.), Computer Algebra: EUROCAM ’82. Vol. 144 of Lecture Notes in Computer Science. Springer, Berlin, pp. 24–31.
* Sauer (2004) Sauer, T., 2004. Lagrange interpolation on subgrids of tensor product grids. Math. Comp. 73 (245), 181–190.
* Sauer (2006) Sauer, T., 2006. Polynomial interpolation in several variables: Lattices, differences, and ideals. In: Jetter, K., Buhmann, M., Haussmann, W., Schaback, R., Stöckler, J. (Eds.), Topics in Multivariate Approximation and Interpolation. Vol. 12 of Studies in Computational Mathematics. Elsevier, Amsterdam, pp. 191–230.
* Sauer and Xu (1995) Sauer, T., Xu, Y., 1995. On multivariate Lagrange interpolation. Math. Comp. 64 (211), 1147–1170.
* Shekhtman (2009) Shekhtman, B., 2009. On the limits of Lagrange projectors. Constr. Approx. 29 (3), 293–301.
* Wang et al. (2009) Wang, X., Zhang, S., Dong, T., 2009. A bivariate preprocessing paradigm for Buchberger-Möller algorithm, summited for Publication.
|
arxiv-papers
| 2010-01-08T03:46:46 |
2024-09-04T02:49:07.605592
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tian Dong",
"submitter": "Tian Dong",
"url": "https://arxiv.org/abs/1001.1196"
}
|
1001.1402
|
Bilinear approach to the quasi-periodic wave solutions of supersymmetric
equations in superspace $\mathbb{R}_{\Lambda}^{2,1}$
Engui Fan 111 E-mail address: faneg@fudan.edu.cn
School of Mathematical Sciences and Key Laboratory of Mathematics for
Nonlinear Science,
Fudan University, Shanghai, 200433, P.R. China
Abstract
We devise a lucid and straightforward way for explicitly constructing quasi-
periodic wave solutions (also called multi-periodic wave solutions) of
supersymmetric equations in superspace $\mathbb{R}_{\Lambda}^{2,1}$ over two-
dimensional Grassmann algebra $G_{1}(\sigma)$. Once a nonlinear equation is
written in a bilinear form, its quasi-periodic wave solutions can be directly
obtained by using a formula. Moreover, properties of these solutions are
investigated in detail by analyzing their structures, plots and asymptotic
behaviors. The relations between the quasi-periodic wave solutions and soliton
solutions are rigorously established. It is shown that the soliton solutions
can be obtained only as limiting cases of the quasi-periodic wave solutions
under small amplitude limits in superspace $\mathbb{R}_{\Lambda}^{2,1}$. We
find that, in contrast to the purely bosonic case, there is an interesting
influencing band occurred among the quasi-periodic waves under the presence of
the Grassmann variable. The quasi-periodic waves are symmetric about the band
but collapse along with the band. Furthermore, the amplitudes of the quasi-
periodic waves increase as the waves move away from the band. The efficiency
of our proposed method can be demonstrated on a class variety of
supersymmetric equations such as those considered in this paper,
$\mathcal{N}=1$ supersymmetric KdV, Sawada-Kotera-Ramani and Ito’s equations,
as well as $\mathcal{N}=2$ supersymmetric KdV equation.
Keywords: supersymmetric equations; super-Hirota’s bilinear form; Riemann
theta function; quasi-periodic wave solutions; soliton solutions.
PACS numbers: 11.30.Pb; 05.45.Yv; 02.30.Gp; 45.10.-b.
1\. Introduction
The algebro-geometric solutions or finite gap solutions of nonlinear equations
were originally obtained on the KdV equation based on inverse spectral theory
and algebro-geometric method developed by pioneers such as Novikov, Dubrovin,
Mckean, Lax, Its and Matveev et al. [2]-[6] in the late 1970s. In fact, such a
solution is an expression written in terms of the Riemann theta functions.
Hence it is also called a quasi-periodic solution due to the quasi-periodicity
of the theta functions. By now this theory has been extended to a large class
of nonlinear integrable equations including sine-Gordon equation, Camassa-Holm
equation, Thirring model equation, Kadomtsev-Petviashvili equation, Ablowitz-
Ladik lattice and Toda lattice [7]-[17].
The quasi-periodic solutions have important applications in physics. For
instance, they can describe the nonlinear interaction of several modes. All
the main physical characteristics of the quasi-periodic solutions (wave
numbers, phase velocities, amplitudes of the interacting modes) are defined by
a compact Riemann surface. There are numerous applications of the finite-gap
integration theory in condensed matter physics, state physics and fluid
mechanics. For example, in peierls state, phonon produce a finite-gap
potential for electrons, and the peierls state is a lattice of solutions at
low densities of electrons [7]. A most famous mechanical system, the
Kowalewski top, was the focus of interest in the 19th century. The equation of
motion of the top can be solved through finite-gap theory [7]. A problem of
fundamental interest in fluid mechanics is to provide an accurate description
of waves on a water surface. The Kadomtsev-Petviashvili equation is known to
describe the evolution of waves in shallow water and admits a large family of
quasi-periodic solutions. Each solution has $N$ independent phases.
Experiments demonstrate the existence of genuinely two-dimensional shallow
water waves that are full periodic in two spatial directions and time. The
comparisons with experiments showed that the two-periodic wave solutions of
the KP equation describe shallow water waves with much accuracy [18, 19].
The algebro-geometric theory, however, needs Lax pairs and involves
complicated calculus on the Riemann surfaces. It is rather difficult to
directly determine the characteristic parameters of waves such as frequencies
and phase shifts for a function of given wave-numbers and amplitudes. On the
other hand, the bilinear derivative method developed by Hirota is a powerful
approach for constructing exact solution of nonlinear equations. Once a
nonlinear equation is written in bilinear forms by a dependent variable
transformation, then multi-soliton solutions are usually obtained [20]–[26].
It was based on Hirota forms that Nakamura proposed a convenient way to
construct a kind of quasi-periodic solutions of nonlinear equations [27, 28,
29], where the periodic wave solutions of the KdV equation and the Boussinesq
equation were obtained. Such a method indeed exhibits some advantages over
algebro-gometric methods. For example, it does not need any Lax pairs and
Riemann surface for the considered equation, allows the explicit construction
of multi-periodic wave solutions, only relies on the existence of the Hirota’s
bilinear form, as well as all parameters appearing in Riemann matrix are
arbitrary. Recently, further development was made to investigate the discrete
Toda lattice, (2+1)-dimensional Kadomtsev-Petviashvili equation and
Bogoyavlenskii’s breaking soliton equation [30]-[35]. Indeed there are some
differences between quasi-periodic solutions and algebro-geometric solutions.
A quasi-periodic solution needs not be an algebro-geometric one. Sometimes a
quasi-periodic solution may not correspond to any Riemann surface and is
generically associated with infinite bands, not just finitely-many, for
instance with a Riemann surface of infinite genus.
The concept of supersymmetry was originally introduced and developed for
applications in elementary particle physics thirty years ago [36]–[38]. It is
found that supersymmetry can be applied to a variety of problems such as
relativistic, non-relativistic physics and nuclear physics. In recent years,
supersymmetry has been a subject of considerable interest both in physics and
mathematics. The mathematical formulation of the supersymmetry is based on the
introduction of Grassmann variables along with the standard ones [39]. In a
such way, a number of well known mathematical physical equations have been
generalized into the supersymmetric analogues, such as supersymmetric versions
of sine-Gordon, KdV, KP hierarchy, Boussinesq, MKdV etc. [40]–[50]. It has
been shown that these supersymmetric integrable systems possess bi-Hamiltonian
structure, Painleve property, infinite many symmetries, Darboux
transformation, Backlund transformation, bilinear form and multi-soliton
solutions. The systematic bilinear transcription of supersymmetric equations
was introduced by Carstea [43, 44]. This required an extension of the Hirota’s
bilinear operator to supersymmetric case. Despite this bilinearization of
supersymmetric equations, the standard construction did not lead to malti-
soliton solutions. In recent years, Carsta, Liu, Ghosh et al. have done much
on the construction of soliton solutions of supersymmetric equations [43]–[50]
. However, the quasi-periodic solutions of the supersymmetric systems, which
can be considered as a generalization of the soliton solutions, are still not
available (both by algebro-geometric method and by bilinear methods or others)
to the knowledge of the author.
The motivation of this paper is to show how the quasi-periodic wave solutions
of nonlinear supersymmetric equations can be constructed with Hirota’s
bilinear method in superspace. To achieve this aim, we devise a Riemannn theta
function formula, which actually provides us a lucid and straightforward way
for applying in a class of nonlinear supersymmetric equations. Once a
nonlinear equation is written in bilinear forms, then the quasi-periodic wave
solutions of the nonlinear equation can be obtained directly by using the
formula. This method considerably improves the key steps of the existing
methods, where repetitive recursion and computation must be preformed for each
equation [30]-[35]. As illustrative example, we shall construct quasi-periodic
wave solutions to the $\mathcal{N}=1$ supersymmetric Sawada-Kotera-Ramani
equation and $\mathcal{N}=2$ supersymmetric KdV equation.
The organization of this paper is as follows. In section 2, we briefly give
some properties on superspace and super-Hirota bilinear operators. In section
3, we introduce a super Riemann theta function and discuss its quasi-
periodicity. In particular, we provide a key formula for constructing periodic
wave solutions of supersymmetric equations. As applications of our method, in
section 4 and section 5, we construct one- and two-periodic wave solutions to
the $\mathcal{N}=1$ supersymmetric Sawada-Kotera-Ramani equation and
$\mathcal{N}=2$ supersymmetric KdV equation, respectively. The propagation of
the quasi-periodic waves are displayed with help of software Mathematica. In
addition, we further present a simple and effective limiting procedure to
analyze asymptotic behavior of the periodic wave solutions. It is rigorously
shown that the quasi-periodic wave solutions tend to the soliton solutions
under small amplitude limits. At last, we briefly discuss the conditions on
the construction of multi-periodic wave solutions of supersymmetric equations
in section 6.
2\. Super space and super-Hirota bilinear form
To fix the notations and make our presentation self-contained, we briefly
recall some properties about superanalysis and super-Hirota bilinear
operators. The details about superanalysis refer, for instance, to
Vladimirov’s work [51, 52].
A linear space $\Lambda$ is called $Z_{2}$-graded if it represented as a
direct sum of two subspaces
$\Lambda=\Lambda_{0}\oplus\Lambda,$
where elements of the spaces $\Lambda_{0}$ and $\Lambda_{1}$ are homogeneous.
We assume that $\Lambda_{0}$ is a subspace consisting of even elements and
$\Lambda_{1}$ is a subspace consisting of odd elements. For the element
$f\in\Lambda$ we denote by $f_{0}$ and $f_{1}$ its even and odd components. A
parity function is introduced on the $\Lambda$, namely,
$|f|=\left\\{\begin{matrix}0,\ \ {\rm if}\ \ f\in\Lambda_{0},\\\ \ 1,\ \ {\rm
if}\ \ f\in\Lambda_{1}.\end{matrix}\right.$
We introduce an annihilator of the set of odd elements by setting
${}^{\perp}\Lambda_{1}=\\{\lambda\in\Lambda:\lambda\Lambda_{1}=0\\}.$
A superalgebra is a $Z_{2}$-graded space $\Lambda=\Lambda_{0}\oplus\Lambda$ in
which, besides usual operations of addition and multiplication by numbers, a
product of elements is defined with the usual distribution law:
$a(\alpha b+\beta c)=\alpha ab+\beta ac,\ \ (\alpha b+\beta c)a=\alpha
ba+\beta ca,$
where $a,b,c\in\Lambda$ and $\alpha,\beta\in\mathbb{C}.$ Moreover, a structure
on $\Lambda$ is introduced of an associative algebra with a unite $e$ and even
multiplication i.e., the product of two even and two odd elements is an even
element and the product of an even element by an odd one is an odd element:
$|ab|=|a|+|b|$ mod (2).
A commutative superalgebra with unit $e=1$ is called a finite-dimensional
Grassmann algebra if it contains a system of anticommuting generators
$\sigma_{j},j=1,\cdots,n$ with the property:
$\sigma_{j}\sigma_{k}+\sigma_{k}\sigma_{j}=0,\ j,k=1,2,\cdots,n$, in
particular, $\sigma_{j}^{2}=0$. The Grassmann algebra will be denote by
$G_{n}=G_{n}(\sigma_{1},\cdots,\sigma_{n})$.
The monomials $\\{e_{0},e_{i}=\sigma_{j_{1}}\cdots\sigma_{j_{n}}\\}$,
$j=(j_{1}<\cdots<j_{n})$ form a basis in the Grassmann algebra $G_{n}$, $\dim
G_{n}=2^{n}$. Then it follows that any element of $G_{n}$ is a linear
combination of monomials $\sigma_{j_{1}}\cdots\sigma_{j_{k}},\
j_{1}<\cdots<j_{k}$, that is,
$f=f_{0}+\sum_{k\geq 0}\sum_{j_{1}<\cdots<j_{k}}f_{j_{1}\cdots
j_{k}}\sigma_{j_{1}}\cdots\sigma_{j_{k}},$
where the coefficients $f_{j_{1}\cdots j_{k}}\in\mathbb{C}$.
Definition 1. Let $\Lambda=\Lambda_{0}\oplus\Lambda$ be a commutative Banach
superalgebra, then the Banach space
$\mathbb{R}_{\Lambda}^{m,n}=\Lambda_{0}^{m}\times\Lambda_{1}^{n}$
is called a superspace of dimension $(m,n)$ over $\Lambda$. In particular, if
$\Lambda_{0}=\mathbb{C}$ and $\Lambda_{1}=0$, then
$\mathbb{R}_{\Lambda}^{m,n}=\mathbb{C}^{m}.$
A function $f(\boldsymbol{x}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda$ is
said to be superdifferentiable at the point $x\in\mathbb{R}_{\Lambda}^{m,n}$,
if there exist elements $F_{j}(\boldsymbol{x})$ in $\Lambda,\ j=1,\cdots,m+n$,
such that
$f(\boldsymbol{x}+\boldsymbol{h})=f(\boldsymbol{x})+\sum_{j=1}^{m+n}\langle
F_{j}(\boldsymbol{x}),h_{j}\rangle+o(\boldsymbol{x},\boldsymbol{h}),$
where $\boldsymbol{x}=(x_{1},\cdots,x_{m},x_{m+1},\cdots,x_{n})$ with
components $x_{j},j=1,\cdots,m$ being even variable and
$x_{m+j}=\theta_{j},j=1,\cdots,n$ being Grassmann odd ones. The vector
$\boldsymbol{h}=(h_{1},\cdots,h_{m}$, $h_{m+1},\cdots,h_{m+n})$ with
$(h_{1},\cdots,h_{m})\in\Lambda_{0}^{m}$ and
$(h_{m+1},\cdots,h_{m+n})\in\Lambda_{1}^{n}$. Moreover,
$\lim_{\parallel\boldsymbol{h}\parallel\rightarrow 0}\frac{\parallel
o(\boldsymbol{x},\boldsymbol{h})\parallel}{\parallel\boldsymbol{h}\parallel}\longrightarrow
0.$
The $F_{j}(\boldsymbol{x})$ are called the super partial derivative of $f$
with respect to $x_{j}$ at the point $\boldsymbol{x}$ and are denoted,
respectively, by
$\frac{\partial f(\boldsymbol{x})}{\partial x_{j}}=F_{j}(\boldsymbol{x}),\
j=1,\cdots,m+n.$
The derivatives $\frac{\partial f(\boldsymbol{x})}{\partial x_{j}}$ with
respect to even variables $x_{j},\ j=1,2,\cdots n$ are uniquely defined. While
the derivatives $\frac{\partial f(\boldsymbol{x})}{\partial\theta_{j}}$ to odd
variables $\theta_{j}=x_{j+n},\ j=1,2,\cdots m$ are not uniquely defined, but
with an accuracy to within an addition constant
$c\sigma_{1}\cdots\sigma_{n},c\in\mathbb{C}$ from an annihilator
${}^{\perp}G_{n}$ of finite-dimensional Grassmann algebra $G_{n}$.
The super derivative also satisfies Leibniz formula
$None$ $\frac{\partial(f(\boldsymbol{x})g(\boldsymbol{x}))}{\partial
x_{j}}=\frac{\partial f(\boldsymbol{x})}{\partial
x_{j}}g(\boldsymbol{x})+(-1)^{|x_{j}||f|}f(\boldsymbol{x})\frac{\partial
g(\boldsymbol{x})}{\partial x_{j}},\ j=1,\cdots,m+n.$
Denote by $\mathcal{P}(\Lambda_{1}^{n},\Lambda)$ the set of polynomials
defined on $\Lambda_{1}^{n}$ with value in $\Lambda$. We say that a super
integral is a map $I:\mathcal{P}(\Lambda_{1}^{n},\Lambda)\rightarrow\Lambda$
satisfying the following condition is an super integral about Grassmann
variable
(1) A linearity: $I(\mu f+\nu g)=\mu I(f)+\nu I(g),\ \mu,\nu\in\Lambda,\
f,g\in\mathcal{P}(\Lambda_{1}^{n},\Lambda);$
(2) translation invariance: $I(f_{\xi})=I(f)$, where
$f_{\xi}=f(\boldsymbol{\theta}+\boldsymbol{\xi})$ for all
$\boldsymbol{\xi}\in\Lambda_{1}^{n}$,
$f\in\mathcal{P}(\Lambda_{1}^{n},\Lambda).$
We denote $I(\theta^{\varepsilon})=I_{\varepsilon}$, where $\varepsilon$
belongs to the set of multiindices
$N_{n}=\\{\boldsymbol{\epsilon}=(\varepsilon_{1},\cdots,\varepsilon_{n}),\varepsilon_{j}=0,1,\boldsymbol{\theta}^{\varepsilon}=\theta_{1}^{\varepsilon_{1}}\cdots\theta_{n}^{\varepsilon_{n}}\not\equiv
0\\}$. In the case when $I_{\varepsilon}=0,\varepsilon\in
N_{n},|\varepsilon|\leq n=n-1$, such kind of integral has the form
$I(f)=J(f)I(1,\cdots,1),$
where
$J(f)=\frac{\partial^{n}f(0)}{\partial\theta_{1}\cdots\partial\theta_{n}}.$
Since the derivative is defined with an accurcy to with an additive constant
form the annihilator ${}^{\perp}L_{n}$,
$L_{n}=\\{\theta_{1}\cdots\theta_{n},\boldsymbol{\theta}\in\Lambda_{1}^{n}\\}$,
it follows that $J:\mathcal{P}\rightarrow\Lambda/^{\perp}L_{n}$ is single-
valued mapping. This mapping also satisfies the conditions 1 and 2, and
therefore we shall call it an integral and denote
$J(f)=\int
f(\boldsymbol{\theta})d\boldsymbol{\theta}=\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots
d\theta_{n},$
which has properties:
$None$ $\displaystyle\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots
d\theta_{n}=1,$ $\displaystyle\int\frac{\partial
f}{\partial\theta_{j}}d\theta_{1}\cdots d\theta_{n}=0,\ j=1,\cdots,n.$
$\displaystyle\int f(\boldsymbol{\theta})\frac{\partial
g(\boldsymbol{\theta})}{\partial\theta_{j}}d\boldsymbol{\theta}=(-1)^{1+|g|}\int\frac{\partial
f(\boldsymbol{\theta})}{\partial\theta_{j}}g(\boldsymbol{\theta})d\boldsymbol{\theta}.$
In this paper, we consider functions with two ordinary even variables $x,t$
and a Grassmann odd variable $\theta$. The associated space
$\mathbb{R}_{\Lambda}^{2,1}=\Lambda_{0}^{2}\times\Lambda_{1}$ (we may take
$\Lambda_{0}=\mathbb{R}$ or $\mathbb{C}$) is a superspace over Grassmann
algebra $G_{1}(\sigma)=G_{1,0}\oplus G_{1,1}$, whose elements have the form
$f=f_{0}+f_{1}\sigma.$
where $e=1$ is a unit, $\sigma$ is anticommuting generator. The monomials
$\\{1,\sigma\\}$ form a basis of the $G_{1}(\sigma)$, dim$G_{1}(\sigma)=2$.
Therefore, any $\mu\in G_{1,1}$ have the form $\mu=\beta\sigma,\
\beta\in\mathbb{C}$. Under traveling wave frame in space
$\mathbb{R}_{\Lambda}^{2,1}$, the phase variable should have the form
$\xi=\alpha x+\omega t+\beta\theta\sigma.$
For the functions
$f(x,t,\theta),g(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda$,
the Hirota bilinear differential operators $D_{x}$ and $D_{t}$ about ordinary
variables $x,t$ are defined by
$\displaystyle D_{x}^{m}D_{t}^{n}f(x,t,\theta)\cdot
g(x,t,\theta)=(\partial_{x}-\partial_{x^{\prime}})^{m}(\partial_{t}-\partial_{t^{\prime}})^{n}f(x,t,\theta)g(x^{\prime},t^{\prime},\theta)|_{x^{\prime}=x,t^{\prime}=t}.$
The super-Hirota bilinear operator is defined as [43]
$S_{x}^{N}f(x,t,\theta)\cdot
g(x,t,\theta)=\sum_{j=0}^{N}(-1)^{j|f|+\frac{1}{2}j(j+1)}\left[\begin{matrix}N\\\
j\end{matrix}\right]\mathfrak{D}^{N-j}f(x,t,\theta)\mathfrak{D}^{j}g(x,t,\theta),$
where the differential operator
$\mathfrak{D}=\partial_{\theta}+\theta\partial_{x}$ is the super derivative,
and the super binomial coefficients are defined by
$\left[\begin{matrix}N\\\
j\end{matrix}\right]=\left\\{\begin{matrix}\left(\begin{matrix}[N/2]\cr[j/2]\end{matrix}\right),{\rm
if}\ \ (N,j)\not=(0,1)\ \ {\rm mod}\ \ 2,\\\ \\\ 0,\ \ \ \ \ \ \ {\rm
otherwise}\ \ \ \ \ \ \ \ \ \ \ \ \ \ .\end{matrix}\right.$
$[k]$ is the integer part of the real number $k$ ($[k]\leq k\leq[k]+1$).
We point out here that throughout this paper the natural number $N$ (which
will denote powers, the number of phase variables, number of terms etc.) is
different form $\mathcal{N}$ which is related to supersymmetry or superspace.
Proposition 1. Suppose that functions
$f(x,t,\theta),g(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda$,
then Hirota bilinear operators $D_{x},D_{t}$ and super-Hirota bilinear
operator $S_{x}$ have properties [43]
$\displaystyle S_{x}^{2N}f\cdot g=D_{x}^{N}f\cdot g,$ $\displaystyle
D_{x}^{m}D_{t}^{n}e^{\xi_{1}}\cdot
e^{\xi_{2}}=(\alpha_{1}-\alpha_{2})^{m}(\omega_{1}-\omega_{2})^{n}e^{\xi_{1}+\xi_{2}},$
$\displaystyle S_{x}e^{\xi_{1}}\cdot
e^{\xi_{2}}=[\sigma(\beta_{1}-\beta_{2})+\theta(\alpha_{1}-\alpha_{2})]e^{\xi_{1}+\xi_{2}},$
where $\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j}$, $\
\alpha_{j},\omega_{j},\sigma_{j},\delta_{j}\in\Lambda_{0}$ are parameters,
$j=1,2$. In fact, the third formula above is defined with an accuracy to
within an addition constant of the $c\sigma\in^{\perp}\Lambda_{1}$. More
generally, we have
$None$ $\displaystyle F(S_{x},D_{x},D_{t})e^{\xi_{1}}\cdot
e^{\xi_{2}}=F(\sigma(\beta_{1}-\beta_{2})+\theta(\alpha_{1}-\alpha_{2}),\alpha_{1}-\alpha_{2},\omega_{1}-\omega_{2})e^{\xi_{1}+\xi_{2}},$
where $F(S_{t},D_{x},D_{t})$ is a polynomial about operators $S_{t},D_{x}$ and
$D_{t}$. This properties are useful in deriving Hirota’s bilinear form and
constructing the quasi-periodic wave solutions of the supersymmetric
equations.
3\. Super Riemann theta function and addition formulae
In the following, we introduce a multi-dimensional super Riemann theta
function on superspace $\mathbb{R}_{\Lambda}^{2,1}$ and discuss its quasi-
periodicity, which plays a central role in the construction of quasi-periodic
solutions of supersymmetric equations. The multi-dimensional Riemann theta
function reads
$None$
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{s}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{{Z}}^{N}}\exp\\{2\pi
i\langle\boldsymbol{\xi}+\boldsymbol{\varepsilon},\boldsymbol{n}+\boldsymbol{s}\rangle-\pi\langle\boldsymbol{\tau}(\boldsymbol{n}+\boldsymbol{s}),\boldsymbol{n}+\boldsymbol{s}\rangle\\}.$
Here the integer value vector
$\boldsymbol{n}=(n_{1},\cdots,n_{N})^{T}\in\mathbb{Z}^{N}$, complex parameter
vectors
$\boldsymbol{s}=(s_{1},\cdots,s_{N})^{T},\boldsymbol{\varepsilon}=(\varepsilon_{1},\cdots,\varepsilon_{N})^{T}\in\mathbb{{C}}^{N}$.
The complex phase variables $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T},\
\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j}$, $\
\alpha_{j},\omega_{j},\beta_{j},\delta_{j}\in\Lambda_{0}$, $j=1,2,\cdots,N$,
where $x,t$ are ordinary variables and $\theta$ is Grassmann variable.
Moreover, for two vectors $\boldsymbol{f}=(f_{1},\cdots,f_{N})^{T}$ and
$\boldsymbol{g}=(g_{1},\cdots,g_{N})^{T}$, their inner product is defined by
$\langle\boldsymbol{f},\boldsymbol{g}\rangle=f_{1}g_{1}+f_{2}g_{2}+\cdots+f_{N}g_{N}.$
The $\boldsymbol{\tau}=(\tau_{ij})$ is a positive definite and real-valued
symmetric $N\times N$ matrix, which is independent of $\theta$ and $\sigma$ in
superspace $\mathbb{R}_{\Lambda}^{2,1}$. The entries $\tau_{ij}$ of the period
matrix $\boldsymbol{\tau}$ can be considered as free parameters of the theta
function (3.1).
In this paper, we take the $\tau$ to be pure imaginary matrix to make the
theta function (3.1) real-valued. In the definition of the theta function
(3.1), for the case $\boldsymbol{s}=\boldsymbol{\varepsilon}=\boldsymbol{0}$,
hereafter we use
$\vartheta(\boldsymbol{\xi},{\boldsymbol{\tau}})=\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})$
for simplicity. Moreover, we have
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=\vartheta(\boldsymbol{\xi}+\boldsymbol{\varepsilon},\boldsymbol{\tau})$.
It is obvious that the Riemann theta function (3.1) converges absolutely and
superdifferentiable on superspace $\mathbb{R}_{\Lambda}^{2,1}$.
Remark 1. The period matrix $\boldsymbol{\tau}$ here is different form
algebro-geometric theory discussed in [2]-[34], where it is usually
constructed via a compact Riemann surface $\Gamma$ of genus $N\in\mathbb{N}$.
We take two sets of regular cycle paths: $a_{1},a_{2},\cdots,a_{N}$;
$b_{1},b_{2},\cdots,b_{N}$ on $\Gamma$ in such a way that the intersection
numbers of cycles satisfies
$a_{k}\circ a_{j}=b_{k}\circ b_{j}=0,a_{k}\circ b_{j}=\delta_{kj},\ \
k,j=1,\cdots,N.$
We choose the normalized holomorphic differentials $\omega_{j},j=1,\cdots,N$
on $\Gamma$ and let
$a_{jk}=\int_{a_{k}}{{\omega}}_{j},\ \ b_{jk}=\int_{b_{k}}{{\omega}}_{j},$
then $N\times N$ matrices $\boldsymbol{A}=(a_{jk})$ and
$\boldsymbol{B}=(b_{jk})$ are invertible. Define matrices $\boldsymbol{C}$ and
$\boldsymbol{\tau}$ by
$\boldsymbol{C}=(c_{jk})=\boldsymbol{A}^{-1},\ \
\boldsymbol{\tau}=(\tau_{jk})=\boldsymbol{A}^{-1}\boldsymbol{B}.$
It is can be shown that the matrix $\boldsymbol{\tau}$ is symmetric and has
positive definite imaginary part. However, we see that the entries in such a
matrix $\boldsymbol{\tau}$ are not free and difficult to be explicitly given.
$\square$
Definition 2. A function $g(\boldsymbol{x},t)$ on
$\mathbb{C}^{N}\times\mathbb{C}$ is said to be quasi-periodic in $t$ with
fundamental periods $T_{1},\cdots,T_{k}\in\mathbb{C}$ if $T_{1},\cdots,T_{k}$
are linearly dependent over $\mathbb{Z}$ and there exists a function
$G(\boldsymbol{x},t)\in\mathbb{C}^{N}\times\mathbb{C}^{k}$ such that
$G(\boldsymbol{x},y_{1},\cdots,y_{j}+T_{j},\cdots,y_{k})=G(\boldsymbol{x},y_{1},\cdots,y_{j},\cdots,y_{k}),\
\ {\rm for\ all}\ y_{j}\in\mathbb{C},\ j=1,\cdots,k.$
$G(\boldsymbol{x},t,\cdots,t,\cdots,t)=g(x,t).$
In particular, $g(\boldsymbol{x},t)$ becomes periodic with $T$ if and only if
$T_{j}=m_{j}T$. $\square$
Let’s first see the periodicity of the theta function
$\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$.
Proposition 2. [53] Let $\boldsymbol{e_{j}}$ be the $j-$th column of $N\times
N$ identity matrix $I_{N}$; ${\tau_{j}}$ be the $j-$th column of
$\boldsymbol{\tau}$, and $\tau_{jj}$ the $(j,j)$-entry of $\boldsymbol{\tau}$.
Then the theta function $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ has
the periodic properties
$None$
$\displaystyle\vartheta(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})=\exp(-2\pi
i\xi_{j}+\pi\tau_{jj})\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}).$
The theta function $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ which
satisfies the condition (4.4) is called a multiplicative function. We regard
the vectors $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau_{j}},\ \ j=1,\cdots,N\\}$ as periods of the theta
function $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ with multipliers $1$
and $\exp({-2\pi i\xi_{j}+\pi\tau_{jj}})$, respectively. Here, only the first
$N$ vectors are actually periods of the theta function
$\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$, but the last $N$ vectors are
the periods of the functions
$\partial^{2}_{\xi_{k},\xi_{l}}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$
and
$\partial_{\xi_{k}}\ln[\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})/\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})],\
k,l=1,\cdots,N$.
Proposition 3. Let $\boldsymbol{e_{j}}$ and $\boldsymbol{\tau_{j}}$ be defined
as above proposition 2. The meromorphic functions $f(\boldsymbol{\xi})$ on
$\mathbb{R}_{\Lambda}^{2,1}$ are as follow
$\displaystyle(i)\ \ \ \ \
f(\boldsymbol{\xi})=\partial_{\xi_{k}\xi_{l}}^{2}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),\
\ \boldsymbol{\xi}\in C^{N},\ \ \ k,l=1,\cdots,N,$ $\displaystyle(ii)\ \ \ \
f(\boldsymbol{\xi})=\partial_{\xi_{k}}\ln\frac{\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})},\
\ \boldsymbol{\xi},\ \boldsymbol{e},\ \boldsymbol{h}\in C^{N},\ \
j=1,\cdots,N.$
then in all two cases (i) and (ii), it holds that
$None$ $\displaystyle
f(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}})=f(\boldsymbol{\xi}),\
\ \ \boldsymbol{\xi}\in C^{N},\ \ \ j=1,\cdots,N.$
Proof. By using (3.2), it is easy to see that
$\displaystyle\frac{\vartheta^{\prime}_{\xi_{k}}(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})}=-2\pi
i\delta_{jk}+\frac{\vartheta^{\prime}_{\xi_{k}}(\boldsymbol{\xi},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})},$
or equivalently
$None$
$\displaystyle\partial_{\xi_{k}}\ln\vartheta(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})=-2\pi
i\delta_{jk}+\partial_{\xi_{k}}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}).$
Differentiating (3.4) with respective to $\xi_{l}$ again immediately proves
the formula (3.3) for the case (i). The formula (3.4) can be proved for the
case (ii) in a similar manner. $\square$
Theorem 1. Suppose that
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})$
and
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
are two Riemann theta functions on $\mathbb{R}_{\Lambda}^{2,1}$, in which
$\boldsymbol{\varepsilon}=(\varepsilon_{1},\dots,\varepsilon_{N})$,
$\boldsymbol{\varepsilon^{\prime}}=(\varepsilon_{1}^{\prime},\dots,\varepsilon_{N}^{\prime})$,
and $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})$,
$\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j},\ \
j=1,2,\cdots,N$. Then Hirota bilinear operators $D_{x},D_{t}$ and super-Hirota
bilinear operator $S_{x}$ exhibit the following perfect properties when they
act on a pair of theta functions
$None$ $\displaystyle
D_{x}\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
$\displaystyle=\left[\sum_{\boldsymbol{\mu}}\partial_{x}\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon},-\boldsymbol{\mu}/2|2\boldsymbol{\tau})|_{\boldsymbol{\xi}=\boldsymbol{0}}\right]\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$
$None$ $\displaystyle
S_{x}\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
$\displaystyle=\left[\sum_{\boldsymbol{\mu}}\mathfrak{D}_{x}\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon},-\boldsymbol{\mu}/2|2\boldsymbol{\tau})|_{\boldsymbol{\xi}=\boldsymbol{0}}\right]\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$
where $\boldsymbol{\mu}=(\mu_{1},\cdots,\mu_{N})$, and the notation
$\sum_{\boldsymbol{\mu}}$ represents $2^{N}$ different transformations
corresponding to all possible combinations $\mu_{1}=0,1;\cdots;\mu_{N}=0,1$.
In general, for a polynomial operator $F(S_{x},D_{x},D_{t})$ with respect to
$S_{x},D_{x}$ and $D_{t}$, we have the following useful formula
$None$ $\displaystyle
F(S_{x},D_{x},D_{t})\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=\left[\sum_{\boldsymbol{\mu}}C(\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\varepsilon},\boldsymbol{\mu})\right]\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$
in which, explicitly
$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}F(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right].$
where we denote $\boldsymbol{\mathcal{M}}=(4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\alpha}\rangle,\ 4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\omega}\rangle,\ 4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\sigma}+\theta\boldsymbol{\alpha}\rangle).$
Proof. For simplicity we prove the formula (3.6) for one-dimensional case. The
proof for $N$-dimensional case can be performed simply by replacing one-
dimensional vectors by $N$-dimensional ones.
Making use of Proposition 1, we obtain the relation
$\displaystyle\Delta\equiv
S_{x}\vartheta(\xi,\varepsilon^{\prime},0|{\tau})\cdot\vartheta(\xi,\varepsilon,0|{\tau})$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}\mathfrak{D}_{x}\exp\\{2\pi
im^{\prime}(\xi+\varepsilon^{\prime})-\pi m^{\prime
2}{\tau}\\}\cdot\exp\\{2\pi im(\xi+\varepsilon)-\pi m^{2}{\tau}\\},$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}2\pi
i(\beta\sigma+\theta\alpha)(m^{\prime}-m)\exp\left\\{2\pi
i(m^{\prime}+m)\xi-2\pi
i(m^{\prime}\varepsilon^{\prime}+m\varepsilon)-\pi{\tau}[m^{\prime
2}+m^{2}]\right\\}$
By shifting sum index as $m=l^{\prime}-m^{\prime}$, then
$\displaystyle\Delta=\sum_{l^{\prime},m^{\prime}\in\mathbb{Z}}2\pi
i(\sigma+\theta\alpha)(2m^{\prime}-l^{\prime})\exp\left\\{2\pi
il^{\prime}\xi-2\pi
i[m^{\prime}\varepsilon^{\prime}+(l^{\prime}-m^{\prime})\varepsilon]-\pi{\tau}[m^{\prime
2}+(l^{\prime}-m^{\prime})^{2}]\right\\}$
$\displaystyle\stackrel{{\scriptstyle l^{\prime}=2l+\mu}}{{=}}\sum_{\mu=0,1}\
\ \sum_{l,m^{\prime}\in\mathbb{Z}}2\pi
i(\beta\sigma+\theta\alpha)(2m^{\prime}-2l-\mu)\exp\\{4\pi i\xi(l+\mu/2)$
$\displaystyle\ \ \ \ \ -2\pi
i[m^{\prime}\varepsilon^{\prime}-(m^{\prime}-2l-\mu)\varepsilon]-\pi[m^{\prime
2}+(m^{\prime}-2l-\mu)^{2}]{\tau}\\}$
Finally letting $m^{\prime}=n+l$, we conclude that
$\displaystyle\Delta=\sum_{\mu=0,1}\left[\sum_{n\in\mathbb{Z}}4\pi
i(\beta\sigma+\theta\alpha)[n-\mu/2]\exp\\{-2\pi
i(n-\mu/2)(\varepsilon^{\prime}-\varepsilon)-2\pi{\tau}(n-\mu/2)^{2}\\}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \times\left[\sum_{l\in\mathbb{Z}}\exp\\{2\pi
i(l+\mu/2)(2\xi+\varepsilon^{\prime}+\varepsilon)-2\pi{\tau}(l+\mu/2)^{2}\right]$
$\displaystyle=\left[\sum_{\mu=0,1}\mathfrak{D}_{x}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,-\mu/2|2{\tau})|_{\xi=0}\right]\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,\mu/2|2\tau),$
by using the following relations
$\displaystyle n+l=(n-\mu/2)+(l+\mu/2),\ \ n-l-\mu=(n-\mu/2)-(l+\mu/2).$
In a similar way, we can prove the formula (3.5). The formula (3.7) follows
from (3.5) and (3.6). $\Box$
Remark 2. The formulae (3.7) and (3.8) show that if the following equations
are satisfied
$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=0,$
for all possible combinations $\mu_{1}=0,1;\mu_{2}=0,1;\cdots;\mu_{N}=0,1$, in
other word, all such combinations are solutions of equation (3.9), then
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})$
and
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
are $N$-periodic wave solutions of the bilinear equation
$F(S_{t},D_{x},D_{t})\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=0.$
We call the formula (3.9) constraint equations, whose number is $2^{N}$. This
formula actually provides us an unified approach to construct multi-periodic
wave solutions for supersymmetric equations. Once a supersymmetric equation is
written bilinear forms, then its multi-periodic wave solutions can be directly
obtained by solving system (3.9).
Theorem 2. Let
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
and $F(S_{x},D_{x},D_{t})$ be given in Theorem 1, and make a choice such that
$\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\ j=1,\cdots,N$. Then
(i) If $F(S_{x},D_{x},D_{t})$ is an even function in the form
$F(-S_{x},-D_{x},-D_{t})=F(S_{x},D_{x},D_{t}),$
then
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
vanishes automatically for the case when $\sum_{j=1}^{N}\mu_{j}$ is an odd
number, namely
$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})|_{\boldsymbol{\mu}}=0,\
\ {\rm for}\ \ \ \sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2.$
(ii) If $F(S_{x},D_{x},D_{t})$ is an odd function in the form
$F(-S_{x},-D_{x},-D_{t})=-F(S_{x},D_{x},D_{t}),$
then
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
vanishes automatically for the case when $\sum_{j=1}^{N}\mu_{j}$ is an even
number, namely
$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})|_{\boldsymbol{\mu}}=0,\
\ {\rm for}\ \sum_{j=1}^{N}\mu_{j}=0,\ {\rm mod}\ 2.$
Proof. We are going to consider the case where $F(S_{x},D_{x},D_{t})$ is an
even function and prove the formula (3.9). The formula (3.11) is analogous.
Making transformation $\boldsymbol{n}=-\boldsymbol{\bar{n}}+\boldsymbol{\mu}\
(\boldsymbol{\bar{n}}=(\bar{n}_{1},\cdots,\bar{n}_{N}),\
\bar{n}_{j}\in\mathbb{Z},\ \ j=1,\cdots,N.$ ), and noting
$F(S_{x},D_{x},D_{t})$ is even, we then deduce that
$\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{\bar{n}}\in\mathbb{Z}^{N}}F(-\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2),\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2\rangle+2\pi
i\langle\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\exp\left(4\pi
i\langle\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon}\rangle\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\exp\left(\pm
2\pi i\sum_{j=1}^{N}\bar{n}_{j}\right)\exp\left(\pm\pi
i\sum_{j=1}^{N}\mu_{j}\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\exp(\pm\pi
i)=-C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),$
which proves the formula (3.10). $\square$
Corollary 1. Let $\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\
j=1,\cdots,N$. Assume $F(S_{x},D_{x},D_{t})$ is a linear combination of even
and odd functions
$F(S_{x},D_{x},D_{t})=F_{1}(S_{x},D_{x},D_{t})+F_{2}(S_{x},D_{x},D_{t}),$
where $F_{1}(S_{x},D_{x},D_{t})$ is even and $F_{2}(S_{x},D_{x},D_{t})$ is
odd. In addition,
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
corresponding (3.8) is given by
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})+C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),$
where
$C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}F_{1}(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right],$
$C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}F_{2}(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right].$
Then
$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\
\ {\rm for}\ \ \ \sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2,$ $None$
$\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),\
\ {\rm for}\ \sum_{j=1}^{N}\mu_{j}=0,\ {\rm mod}\ 2.$
Proof. In a similar to the proof of Theorem 2, shifting sum index as
$\boldsymbol{n}=-\boldsymbol{\bar{n}}+\boldsymbol{\mu}$, and using
$F_{1}(S_{x},D_{x},D_{t})$ even and $F_{2}(S_{x},D_{x},D_{t})$ odd, we have
$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})+C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=\left[C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})-C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\right]\exp\left(\pm\pi
i\sum_{j=1}^{N}\mu_{j}\right).$
Then for $\sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2$, the equation (3.15) gives
$C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=0,$
which implies the formula (3.12). The formula (3.13) is analogous. $\square$
The theorem 2 and corollary 1 are very useful to deal with coupled super-
Hirota’s bilinear equations, which will be seen in the following section 5.
By introducing differential operators
$\displaystyle\nabla=(\partial_{\xi_{1}},\partial_{\xi_{2}},\cdots,\partial_{\xi_{N}}),$
$\displaystyle\partial_{x}=\alpha_{1}\partial_{\xi_{1}}+\alpha_{2}\partial_{\xi_{2}}+\cdots+\alpha_{N}\partial_{\xi_{N}}=\boldsymbol{\alpha}\cdot\nabla,$
$\displaystyle\partial_{t}=\omega_{1}\partial_{\xi_{1}}+\omega_{2}\partial_{\xi_{2}}+\cdots+\omega_{N}\partial_{\xi_{N}}=\boldsymbol{\omega}\cdot\nabla,$
$\displaystyle\mathfrak{D}=(\sigma_{1}+\theta\alpha_{1})\partial_{\xi_{1}}+(\sigma_{2}+\theta\alpha_{2})\partial_{\xi_{2}}+\cdots+(\sigma_{N}+\theta\alpha_{N})\partial_{\xi_{N}}=(\boldsymbol{\sigma}+\theta\boldsymbol{\alpha})\cdot\nabla,$
then we have
$None$
$\displaystyle\mathfrak{D}\partial_{x}^{k}\partial_{t}^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=[(\boldsymbol{\sigma}+\theta\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\alpha}\cdot\nabla)^{k}(\boldsymbol{\omega}\cdot\nabla)^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$
$\displaystyle=(\boldsymbol{\sigma}\cdot\nabla)(\boldsymbol{\alpha}\cdot\nabla)^{k}(\boldsymbol{\omega}\cdot\nabla)^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})+\theta(\boldsymbol{\alpha}\cdot\nabla)^{k+1}(\boldsymbol{\omega}\cdot\nabla)^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$
$\displaystyle j,k=0,1,\cdots.$
4\. $\mathcal{N}=1$ supersymmetric Sawada-Kotera-Ramani equation
The supersymmetric Sawada-Kotera-Ramani equation takes the form
$None$
$\displaystyle\phi_{t}+\mathfrak{D}^{2}\left[10(\mathfrak{D}\phi)\mathfrak{D}^{4}\phi+5(\mathfrak{D}^{5}\phi)\phi+15(\mathfrak{D}\phi)^{2}\phi\right]+\mathfrak{D}^{10}\phi=0,$
where
$\phi=\phi(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is fermionic superfield depending on usual independent variable $x$, $t$ and
Grassmann variable $\theta$. This equation was first proposed by Carstea [43].
The soliton solutions, Lax representation and infinite conserved quantities of
the equation have been further obtained recently [56, 57]. Here we are
interested in quasi-periodic wave solutions to the supersymmetric equation
(4.1). We will show that the soliton solutions can be obtained as limiting
case of these quasi-periodic solutions.
To apply the Hirota bilinear method in superspace for constructing multi-
periodic wave solutions of the equation (4.1), we hope more fee variables and
consider a general variable transformation
$None$ $\phi=\phi_{0}+2\mathfrak{D}^{3}\ln f(x,t,\theta),$
where
$f(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{1,0}$
is an even superfield and
$\phi_{0}=\phi_{0}(\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is an odd special solution of the equation (4.1). Substituting (4.2) into
(4.1) and integrating with respect to $x$, we then get the following super
Hirota’s bilinear form
$None$ $\displaystyle F(S_{x},D_{x},D_{t})f\cdot
f=(S_{x}D_{t}+S_{x}D_{x}^{5}+15\phi_{0}^{2}D_{x}^{2}+5\phi_{0}D_{x}^{4}+c)f\cdot
f=0,$
where
$c=c(\theta,t):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is an odd integration constant. In the special case when $\phi_{0}=c=0$,
starting from the bilinear equation (4.3), it is easy to find that the
equation (4.1) admits one-soliton solution (also called one-supersoliton
solution) in superspace $\mathbb{R}_{\Lambda}^{2,1}$ over two-dimensional
Grassmann algebra $G_{1}(\sigma)$
$None$ $\phi_{1}=2\mathfrak{D}^{3}\ln(1+e^{\eta}),$
with phase variable $\eta=px-p^{5}t+q\theta\sigma+r$ with
$p,q,r\in\Lambda_{0}$. While two-soliton solution (super two-soliton solution)
reads
$None$
$\phi_{2}=2\mathfrak{D}^{3}\ln(1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}}),$
with $\eta_{j}=p_{j}x-p_{j}^{3}t+q_{j}\theta\sigma+r_{j},\ \ j=1,2$ and
$\displaystyle
e^{A_{12}}=\frac{(p_{1}-p_{2})^{2}(p_{1}^{2}-p_{1}p_{2}+p_{2}^{2})}{(p_{1}+p_{2})^{2}(p_{1}^{2}+p_{1}p_{2}+p_{2}^{2})}\left(1+2\theta\sigma\frac{p_{1}q_{2}-p_{2}q_{1}}{p_{1}-p_{2}}\right),$
and here $p_{j},q_{j},r_{j}\in\Lambda_{0},j=1,2$ are free constants.
Next, we turn to see the periodicity of the solution (4.2), the function $f$
is chosen to be a Riemann theta function, namely,
$f(x,t,\theta)=\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$
where phase variable $\xi$ is taken as the form
$\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T}$,
$\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j},\ \
j=1,2,\cdots,N.$ With Proposition 3, we refer to
$\phi=\phi_{0}+2\sum_{k,l=1}^{N}\alpha_{k}(\beta_{l}\sigma+\theta\alpha_{l})\partial_{\xi_{k}\xi_{l}}^{2}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$
which shows that the solution $\phi$ is indeed a quasi-periodic function with
$2N$ fundamental periods $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau}_{j},\ \ j=1,\cdots,N\\}$. The quasi-periodic means that
$\phi$ is periodic in each of the $N$ phases $\\{\xi_{j},\ \ j=1,\cdots,N\\}$,
if the other $N-1$ phases are fixed.
3.1. One-periodic waves and asymptotic analysis
We first consider one-periodic wave solutions of the equation (4.1). As a
simple case of the theta function (3.1) when $N=1,s=\varepsilon=0$, we take
$f$ as
$None$ $f(x,t,\theta)=\vartheta(\xi,\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi
in\xi-\pi n^{2}\tau}),$
where the phase variable $\xi=\alpha x+\omega t+\beta\theta\sigma+\delta$, and
the parameter $\tau>0$.
Next, we let the Riemann theta function (4.6) be a solution of the bilinear
equation (4.3). By using Theorem 1 and the formula (3.9), the following two
equations ( corresponding to $\mu=0$ and $1$ respectively) should be satisfied
$None$
$\displaystyle\sum_{n\in\mathbb{Z}}\\{-(4\pi(n-\mu/2))^{2}(\beta\sigma+\theta\alpha)\omega-(4\pi(n-\mu/2))^{6}(\beta\sigma+\theta\alpha)\alpha^{5}-15(4\pi(n-\mu/2))^{2}\alpha^{2}\phi_{0}^{2}$
$\displaystyle+5(4\pi(n-\mu/2))^{4}\alpha^{4}\phi_{0}+c\\}\exp(-2\pi\tau(n-\mu/2)^{2})=0,\
\ \mu=0,1.$
We introduce the notations by
$\begin{aligned}
&\lambda=e^{-\pi\tau/2},\quad\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},0,0|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{4n^{2}}\exp(4i\pi
n\xi),\\\
&\vartheta_{2}(\xi,\lambda)=\vartheta(2\xi,0,-1/2|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{(2n-1)^{2}}\exp[2i\pi(2n-1)\xi],\end{aligned})$
the equation (4.7) can be written as a linear system about $\omega,c$
$None$
$\displaystyle(\beta\sigma+\theta\alpha)\vartheta_{j}^{\prime\prime}\omega+(\beta\sigma+\theta\alpha)\alpha^{5}\vartheta_{j}^{(6)}+3\alpha^{2}\vartheta_{j}^{\prime\prime}\phi_{0}^{2}+5\alpha^{4}\vartheta_{j}^{(4)}\phi_{0}+\vartheta_{j}c=0,\
\ j=1,\ 2,$
where $\omega\in\Lambda_{0}$ is even and $c,\phi_{0}\in\Lambda_{1}$ are odd,
and we have denoted the derivative value of $\vartheta_{j}(\xi,\lambda)$ at
$\xi=0$ by simple notations
$\vartheta_{j}^{\prime}=\vartheta_{j}^{\prime}(0,\lambda)=\frac{d\vartheta_{j}(\xi,\lambda)}{d\xi}|_{\xi=0},\
\ j=1,2.$
Moreover, we see that the functions $\vartheta_{j}$ and their derivatives are
independent of Grassmann variable $\theta$ and anticommuting number $\sigma$.
We take $\phi_{0}=0$ for the simplicity. It is obvious that the coefficient
determinant of the system (4.8) is nonzero and
$(\alpha^{5}\vartheta_{1}^{(6)},\alpha^{5}\vartheta_{2}^{(6)})^{T}\not=0$,
therefore the system (4.8) admits a solution
$None$
$\displaystyle\omega=\frac{\alpha^{5}(\vartheta_{2}^{(6)}\vartheta_{1}-\vartheta_{1}^{(6)}\vartheta_{2})}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}},\
\ \
b_{1}=\frac{\alpha^{5}(\beta\sigma+\theta\alpha)(\vartheta_{2}^{(6)}\vartheta_{1}^{\prime\prime}-\vartheta_{1}^{(6)}\vartheta_{2}^{\prime\prime})}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}},$
where $\omega$ is independent of Grassmann variable $\theta$ and auticommuting
number $\sigma$, and parameter $\alpha$ is free.
In this way, a one-periodic wave solution of the equation (4.1) is explicitly
obtained by
$None$ $\phi=2\mathfrak{D}^{3}\ln\vartheta(\xi,\tau),$
with the theta function $\vartheta(\xi,\tau)$ given by (4.6) and parameters
$\omega$, $c$ by (4.9), while other parameters
$\alpha,\beta,\tau,\delta\in\Lambda_{0}$ are free. Among them, the three
parameters $\alpha$ and $\tau$ completely dominate a one-periodic wave.
In summary, one-periodic wave (4.10) possesses the following features:
(i) It is one-dimensional, i.e. there is a single phase variable $\xi$.
Moreover, it has two fundamental periods $1$ and $i\tau$ in phase variable
$\xi$, but it need not to be periodic in $x$, $t$ and $\theta$ directions.
(ii) It can be viewed as a parallel superposition of overlapping one-soliton
waves, placed one period apart ( see $(a)$ and $(b)$ in Figure 1 ).
(iii) Different form the purely bosonic case, it is observed shows that there
is an influencing band among the one-periodic waves under the presence of the
Grassmann variable (in contour plot, the bright hexagons are crests and the
dark hexagons are troughs). The one-periodic waves are symmetric about the
band but collapse along with the band. Furthermore, the amplitudes of the
quasi-periodic waves increase as the waves move away from the band ( see $(a)$
and $(b)$ in Figure 1 ).
(iv) The quasi-periodic wave will degenerate to pure bosonic quasi-periodic
wave when $\theta$ becomes small ( see Figure 2 ).
$(b)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 1. A one-periodic wave for $\mathcal{N}=1$ supersymmetric Sawada-
Kotera-Ramani equation with parameters: $\alpha=0.1,$ $\tau=32,\sigma=0.013$.
(a) Perspective view of wave. (b) Overhead view of wave, with contour plot
shown.
$(b)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 2. A purely bosonic case of one-periodic wave to $\mathcal{N}=1$
supersymmetric Sawada-Kotera-Ramani equation with parameters: $\alpha=0.1,$
$\tau=32,\ \sigma=0.013,\ \theta=0.00000001$. (a) Perspective view of wave.
(b) Overhead view of wave, with contour plot shown.
In the following, we further consider asymptotic properties of the one-
periodic wave solution. Interestingly, the relation between the one-periodic
wave solution (4.10) and the one-soliton solution (4.4) can be established as
follows.
Theorem 3. Suppose that $\omega$ and $c$ are given by (4.9), and for the one-
periodic wave solution (4.10), we let
$None$ $\alpha=\frac{p}{2\pi i},\ \ \beta=\frac{q}{2\pi i},\ \
\delta=\frac{r+\pi\tau}{2\pi i},$
where the $p,q$ and $r$ are given in (4.4). Then we have the following
asymptotic properties
$\displaystyle c\longrightarrow 0,\ \ 2\pi
i\xi-\pi\tau\longrightarrow\eta=px-p^{5}t+q\theta\sigma+r,$
$\displaystyle\vartheta(\xi,\tau)\longrightarrow 1+e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$
In other words, the one-periodic solution (4.10) tends to the soliton solution
(4.4) under a small amplitude limit, that is,
$None$ $\phi\longrightarrow\phi_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$
Proof. Here we will directly use the system (4.8) to analyze asymptotic
properties of one-periodic solution (4.10), which is more simple and effective
than our original method by solving the system [30]-[34]. Since the
coefficients of system (4.8) are power series about $\lambda$, its solution
$(\omega,c)$ also should be a series about $\lambda$. We explicitly expand the
coefficients of system (4.8) as follows
$None$ $\displaystyle\vartheta_{1}=1+2\lambda^{4}+\cdots,\quad\
\vartheta_{1}^{\prime\prime}=-32\pi^{2}\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{1}^{(6)}=-8192\pi^{6}\lambda^{4}+\cdots,\ \
\vartheta_{2}=2\lambda+2\lambda^{9}+\cdots$
$\displaystyle\vartheta_{2}^{\prime\prime}=-8\pi^{2}\lambda+\cdots,\ \ \
\vartheta_{1}^{(6)}=-128\pi^{6}\lambda+\cdots.$
Let the solution of the system (4.8) be of the form
$None$
$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$\displaystyle c=c_{0}+c_{1}\lambda+c_{2}\lambda^{2}+\cdots=c_{0}+o(\lambda).$
Substituting the expansions (4.13) and (4.14) into the system (4.8) (the
second equation is divided by $\lambda$ ) and letting $\lambda\longrightarrow
0$, we immediately obtain the following relations
$\displaystyle c_{0}=0,\ \
-8\pi^{2}(\beta\sigma+\theta\alpha)\omega_{0}+2c_{0}-128\pi^{6}(\beta\sigma+\theta\alpha)\alpha^{5}=0,$
which has a solution
$None$ $c_{0}=0,\ \ w_{0}=-16\pi^{4}\alpha^{5}.$
Combining (4.14) and (4.15) then yields
$c\longrightarrow 0,\ \ 2\pi
i\omega\longrightarrow-32i\pi^{5}\alpha^{5}=-p^{5},\ \ {\rm as}\ \
\lambda\rightarrow 0.$
Hence we conclude
$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=px+2\pi i\omega
t+q\theta\sigma+r$ $\displaystyle\quad\longrightarrow
px-p^{5}t+q\theta\sigma+r=\eta,\ \ {\rm as}\ \ \lambda\rightarrow 0.$
It remains to consider asymptotic properties of the one-periodic wave solution
(4.10) under the limit $\lambda\rightarrow 0$. By expanding the Riemann theta
function $\vartheta(\xi,\tau)$ and by using (4.16), it follows that
$\displaystyle\vartheta(\xi,\tau)=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots$ $\displaystyle\ \ \ \
\
=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\ \ \ \ \ \quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow
1+e^{\eta},\ \ {\rm as}\ \ \lambda\rightarrow 0,$
which together with (4.10) lead to (4.12). Therefore we conclude that the one-
periodic solution (4.10) just goes to the one-soliton solution (4.4) as the
amplitude $\lambda\rightarrow 0$. $\square$
3.2. Two-periodic wave solutions and asymptotic analysis
We proceed to the construction of the two-periodic wave solutions to the
supersymmetric Sawada-Kotera-Ramani equation (4.1), which are a two-
dimensional generalization of one-periodic wave solutions. The two-periodic
waves of interest here have three-dimensional velocity fields and two-
dimensional surface patterns.
For the case when $N=2,\boldsymbol{s}=\boldsymbol{\varepsilon}=\boldsymbol{0}$
in the Riemann theta function (3.1), we takes $f$ as
$None$
$f=\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\},$
where $\boldsymbol{n}=(n_{1},n_{2})^{T}\in\mathbb{Z}^{2},\ \
\boldsymbol{\xi}=(\xi_{1},\xi_{2})^{T}\in\mathbb{C}^{2},\ \
\xi_{i}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j},\ \ j=1,2$;
The matrix $\boldsymbol{\tau}$ is a positive definite and real-valued
symmetric ${2\times 2}$ matrix, which can takes of the form
$\boldsymbol{\tau}=(\tau_{ij})_{2\times 2},\ \ \tau_{12}=\tau_{21},\ \
\tau_{11}>0,\ \ \tau_{22}>0,\ \ \tau_{11}\tau_{22}-\tau_{12}^{2}>0.$
Next, we explore the conditions to make the Riemann theta function (4.17)
satisfy the bilinear equation (4.3). Theorem 1 and the formula (3.9) give rise
to the following four constraint equations
$None$ $\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}\left[-4\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\sigma}+\theta\boldsymbol{\alpha}\rangle\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\omega}\rangle-64\pi^{6}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{5}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\sigma}+\theta\boldsymbol{\alpha}\rangle\right.$
$\displaystyle-60\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{2}\phi_{0}^{2}+80\pi^{4}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{4}\phi_{0}+c]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle\\}=0,$
where $\boldsymbol{\mu}=(\mu_{1},\mu_{2})$ takes all possible combinations of
$\mu_{1},\mu_{2}=0,1$.
By introducing the notations
$\displaystyle\lambda_{kl}=e^{-\pi\tau_{kl}/2},k,l=1,2,\boldsymbol{\lambda}=(\lambda_{11},\lambda_{12},\lambda_{22})$
$\displaystyle\vartheta_{j}(\boldsymbol{\xi},\boldsymbol{\lambda})=\vartheta(2\xi,\boldsymbol{0},-\boldsymbol{s}_{j}/2|2\tau)=\sum_{n_{1},n_{2}\in
Z}\exp\\{4\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}-\boldsymbol{s_{j}}/2\rangle\\}\prod_{k,l=1}^{2}\lambda_{kl}^{(2n_{k}-s_{j,k})(2n_{j}-s_{j,l})},$
$\displaystyle\boldsymbol{s_{j}}=(s_{j,1},s_{j,2}),\quad j=1,2,3,4,\ \
\boldsymbol{s_{1}}=(0,0),\ \ \boldsymbol{s_{2}}=(1,0),\ \
\boldsymbol{s_{3}}=(0,1),\ \ \boldsymbol{s_{4}}=(1,1),$
then by using (3.15), the system (4.18) can be written as a linear system
$None$
$\displaystyle[(\boldsymbol{\beta}\sigma+\theta\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\omega}\cdot\nabla)\vartheta_{j}(0,\boldsymbol{\lambda})+[(\boldsymbol{\beta}\sigma+\theta\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\alpha}\cdot\nabla)^{5}\vartheta_{j}(0,\boldsymbol{\lambda})$
$\displaystyle+15(\boldsymbol{\alpha}\cdot\nabla)^{2}\vartheta_{j}(0,\boldsymbol{\lambda})\phi_{0}^{2}+5(\boldsymbol{\alpha}\cdot\nabla)^{4}\vartheta_{j}(0,\boldsymbol{\lambda})\phi_{0}+\vartheta_{j}(0,\boldsymbol{\lambda})c=0.$
This system is easy to be solved in such a way: $\phi_{0}$ by solving a
quadratic equation with one unknown; $\omega_{1},\omega_{2}$ and $c$ by
solving a linear system. With such a solution
$(\omega_{1},\omega_{2},\phi_{0},c)$, we then get an exact two-periodic wave
solution
$None$
$\phi=\phi_{0}+2\mathfrak{D}^{3}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$
with $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ and
$\omega_{1},\omega_{2},\phi_{0},c$ given by (4.17) and (4.19), respectively,
while other parameters $\beta_{1},\beta_{2},\alpha_{1},$
$\alpha_{2},\tau_{11},\tau_{22},\tau_{12},\delta_{1},\delta_{2}\in\Lambda_{0}$
are free.
In summary, two-periodic wave (4.20), which is a direct generalization of one-
periodic wave, has the following features:
(i) The two-periodic wave solution is genuinely two-dimensional. Its surface
pattern is two-dimensional, namely, there are two phase variables $\xi_{1}$
and $\xi_{2}$.
(ii) It has two independent spatial periods in two independent horizontal
directions. It has $4$ fundamental periods $\\{e_{1},e_{2}\\}$ and
$\\{i\tau_{1},i\tau_{2}\\}$ in $(\xi_{1},\xi_{2})$. It is spatially periodic
in two directions $\xi_{1},\xi_{2}$, but it does not need periodic in the all
$x$-, $t$\- and $\theta$-directions.
(iii) As in the case of on-periodic waves, there is an influencing band among
the two-periodic waves under the presence of the Grassmann variable. ( see
Figure 3 ).
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 3. A two-super periodic wave for $\mathcal{N}=1$ supersymmetric Sawada-
Kotera-Ramani equation with parameters: $\alpha_{1}=0.1,\alpha_{2}=-0.1,$
$\tau_{11}=2,\tau_{12}=0.2,\tau_{22}=2,\sigma_{1}=0.1,\sigma_{2}=0.1$. (a)
Perspective view of wave. (b) Overhead view of wave, with contour plot shown.
At last, we consider the asymptotic properties of the two-periodic solution
(4.20). In a similar way to Theorem 3, we can establish the relation between
the two-periodic solution (4.20) and the two-soliton solution as follows.
Theorem 4. Assume that $(\omega_{1},\omega_{2},\phi_{0},c)^{T}$ is a solution
of the system (4.19). We choose the parameters in the two-periodic wave
solution (4.20) as follows
$None$ $\displaystyle\alpha_{j}=\frac{p_{j}}{2\pi i},\ \
\beta_{j}=\frac{q_{j}}{2\pi i},\ \ \delta_{j}=\frac{r_{j}+\pi\tau_{jj}}{2\pi
i},\ \ \tau_{12}=-\frac{A_{12}}{2\pi},\ \ j=1,2,$
with the $p_{j},q_{j},r_{j},j=1,2$ as those given in (4.5). Then under
constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$, we have the following
asymptotic relations
$None$ $\displaystyle\phi_{0}\longrightarrow 0,\ \ \ c\longrightarrow 0,\ \
2\pi i\xi_{j}-\pi\tau_{jj}\longrightarrow
p_{j}x-p_{j}^{5}t+q_{j}\theta\sigma+r_{j}=\eta_{j},\ \ j=1,2,$
$\displaystyle\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})\longrightarrow
1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},\ \ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$
So the two-periodic wave solution (5.20) tends to the two-soliton solution
(4.5), namely,
$\phi\longrightarrow\phi_{2},\ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$
Proof. From (4.21), the constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$
leads to $p_{1}q_{2}-p_{2}q_{1}=0$, which implies that
$\tau_{12}=-A_{12}/2\pi$ is independent of Grassmann variable $\theta$
according to (4.5).
In the same way as the proof of Theorem 3, we expand the Riemann function
$\vartheta(\xi_{1},\xi_{2},\boldsymbol{\tau})$ in the following form
$\displaystyle\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=1+(e^{2\pi
i\xi_{1}}+e^{-2\pi i\xi_{1}})e^{-\pi{\tau}_{11}}+(e^{2\pi i\xi_{2}}+e^{-2\pi
i\xi_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ +(e^{2\pi
i(\xi_{1}+\xi_{2})}+e^{-2\pi
i(\xi_{1}+\xi_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$
Further by using (4.21) and making a transformation $\hat{\omega}_{j}=2\pi
i\omega_{j},j=1,2$, we get
$\displaystyle\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}-2\pi\tau_{12}}+\lambda_{11}^{4}e^{-\hat{\xi}_{1}}+\lambda_{22}^{4}e^{-\hat{\xi}_{2}}+\lambda_{11}^{4}\lambda_{22}^{4}e^{-\hat{\xi}_{1}-\hat{\xi}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}+A_{12}},\
\ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$
where $\hat{\xi}_{j}=p_{j}x+\hat{\omega}_{j}t+p_{j}\theta\sigma+r_{j},\ \
j=1,2.$
It remains to prove that
$None$ $\displaystyle c\longrightarrow 0,\ \ \
\hat{\omega}_{j}\longrightarrow-p_{j}^{5},\ \
\hat{\xi}_{j}\longrightarrow\eta_{j},\ \ j=1,2,\ \ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$
As in the case when $N=1$, we let the solution of the system (4.19) be the
form
$None$
$\displaystyle\omega_{1}=\omega_{1,0}+\omega_{1,1}\lambda_{11}+\omega_{1,2}\lambda_{22}+\cdots=\omega_{1,0}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle\omega_{2}=\omega_{2,0}+\omega_{2,1}\lambda_{11}+\omega_{2,2}\lambda_{22}+\cdots=\omega_{2,0}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle\phi_{0}=\phi_{0,0}+\phi_{0,1}\lambda_{11}+\phi_{0,2}\lambda_{22}+\cdots=\phi_{0,0}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle
c=c_{0}+c_{1}\lambda_{11}+c_{2}\lambda_{22}+\cdots=c_{0}+o(\lambda_{11},\lambda_{22}).$
Expanding functions $\vartheta_{j},j=1,2,3,4$ in the system (4.19), together
with substitution of assumption (4.24), the second and third equation is
divided by $\lambda_{11}$ and $\lambda_{22}$, respectively; the fourth
equation is divided by $\lambda_{11}\lambda_{22}$, and letting
$\lambda_{11},\lambda_{22}\longrightarrow 0$ , we then obtain
$\displaystyle c_{0}=0,\
-8\pi(\beta_{1}\sigma+\theta\alpha_{1})\omega_{1}-128\pi^{6}\alpha_{1}^{5}(\beta_{1}\sigma+\theta\alpha_{1})-120\pi\alpha_{1}^{2}\phi_{0,0}^{2}+2\phi_{0,0}=0,$
$\displaystyle-8\pi(\beta_{2}\sigma+\theta\alpha_{2})\omega_{2}-128\pi^{6}\alpha_{2}^{5}(\beta_{2}\sigma+\theta\alpha_{2})-120\pi\alpha_{2}^{2}\phi_{0,0}^{2}+2\phi_{0,0}=0,$
which has solution
$None$ $\displaystyle c_{0}=0,\ \ \phi_{0,0}=0,\ \ \
\omega_{1,0}=-16\pi^{4}\alpha_{1}^{5},\ \
\omega_{2,0}=-16\pi^{4}\alpha_{2}^{5}.$
The expressions (4.24) and (4.25) implies that
$\displaystyle\phi_{0}=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
c=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
\omega_{1}=-16\pi^{4}\alpha_{1}^{5}+o(\lambda_{11},\lambda_{22})\longrightarrow-16\pi^{4}\alpha_{1}^{5},$
$\displaystyle\omega_{2}=-16\pi^{4}\alpha_{2}^{5}+o(\lambda_{11},\lambda_{22})\longrightarrow-16\pi^{4}\alpha_{2}^{5},\
\ \ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$
thus proving (4.23). We conclude that the two-periodic wave solution (4.20)
tends to the two-soliton solution (4.5) as
$\lambda_{11},\lambda_{22}\rightarrow 0$. $\square$
4\. $\mathcal{N}=2$ supersymmetric KdV equation
We consider $\mathcal{N}=2$ supersymmetric KdV equation
$None$
$\displaystyle\phi_{t}=-\phi_{xxx}+3(\phi\mathfrak{D}_{1}\mathfrak{D}_{2}\phi)_{x}+\frac{1}{2}(a-1)(\mathfrak{D}_{1}\mathfrak{D}_{2}\phi^{2})_{x}+3a\phi^{2}\phi_{x}^{2},$
which was originally introduced by Laberge and Mathieu [58, 59]. In the
equation (5.1),
$\phi=\phi(x,t,\theta_{1},\theta_{2}):\mathbb{R}_{\Lambda}^{2,2}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is a superboson function depending on temporal variable $t$, spatial variable
$x$ and its fermionic counterparts $\theta_{1},\theta_{2}$. The operators
$\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ are the super derivatives defined by
$\mathfrak{D}_{1}=\partial_{\theta_{1}}+\theta_{1}\partial_{x},\ \
\mathfrak{D}_{2}=\partial_{\theta_{2}}+\theta_{2}\partial_{x}$ and $a$ is a
parameter. The equation (5.1) is called supersymmetric KdVa equation [60]. For
the cases when $a=1$ and $a=4$, the Lax representation, Hamiltonian structure,
Painleve analysis and soliton solutions of the equation (5.1) can refer to,
for instance, papers [58]–[60].
Here we are interested in quasi-periodic wave solutions to the supersymmetric
equation (5.1) by using Theorem 1 and 5. We only consider the case when $a=1$,
so the equation (5.1) reduces to
$None$
$\displaystyle\phi_{t}=-\phi_{xxx}+3(\phi\mathfrak{D}_{1}\mathfrak{D}_{2}\phi)_{x}+3\phi^{2}\phi_{x}^{2}.$
To apply the Hirota bilinear method for constructing multi-periodic wave
solutions of the equation (5.2), we add two variables and consider a general
variable transformation
$None$ $\displaystyle\phi=u+\theta_{2}v,\ \ u=i\partial_{x}\ln\frac{f}{g},\ \
v=v_{0}-\partial_{x}\mathfrak{D}\ln(fg),$
where
$u(x,t,\theta_{1}),f(x,t,\theta_{1}),g(x,t,\theta_{1}):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{1,0}$,
and
$v(x,t,\theta_{1}),v_{0}=v_{0}(\theta_{1}):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is a special solution of the equation (5.2). Hereafter we use
$\mathfrak{D}=\mathfrak{D}_{1}$ for simplicity, Substituting (5.3) into (5.2),
we then get the following bilinear form
$None$ $\displaystyle F(D_{t},D_{x})f\cdot g=(D_{t}+D_{x}^{3})f\cdot g=0,$
$\displaystyle G(S_{x},D_{t},D_{x})f\cdot
g=(S_{x}D_{t}+S_{x}D_{x}^{3}+3v_{0}D_{x}^{2}+c)f\cdot g=0,$
where
$c=c(\theta_{1},t):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is an odd integration constant to variable $x$; The equation (5.4) is a type
of coupled bilinear equations, which is more difficult to be dealt with than
the bilinear equation (4.3) due to appearance of two functions and two
equations. We will take full advantages of Theorem 2 to reduce the number of
constraint equations.
Now we take into account the periodicity of the solution (5.3), in which we
take $f$ and $g$ as
$f=\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau}),\ \
g=\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau}),\ \
\boldsymbol{e},\boldsymbol{h}\in\mathbb{Z}^{N},$
where phase variable $\xi$ is taken as the form
$\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T}$,
$\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta_{1}\sigma+\delta_{j},\ \
j=1,2,\cdots,N.$ By means of Proposition 3, we deduce that
$\displaystyle\phi=u+\theta_{2}v,\ \
u=i\sum_{k=1}^{N}\alpha_{k}\partial_{\xi_{k}}\ln\frac{\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})},$
$\displaystyle
v=v_{0}-2\sum_{k,l=1}^{N}\alpha_{k}(\beta_{l}\sigma+\theta\alpha_{l})\partial_{\xi_{k}\xi_{l}}^{2}\ln[{\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})}],$
which indicates that the solution $\phi$ is a quasi-periodic function with
$2N$ fundamental periods $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau_{j}},\ \ j=1,\cdots,N\\}$.
In the special case of $v_{0}=c=0$, starting from the bilinear equation (5.4),
Zhang et al. found that the equation (5.2) admits one-soliton solution [60]
$None$
$\phi_{1}=i\partial_{x}\ln\frac{f_{1}}{g_{1}}+\theta_{2}[v_{0}-\partial_{x}\mathfrak{D}\ln(f_{1}g_{1})],$
with
$f_{1}=1+e^{\eta},\ \ g_{1}=1-e^{\eta}$
and phase variable $\eta=px-p^{3}t+q\theta_{1}\sigma+r$ with
$p,q,r\in\Lambda_{0}$. While two-soliton solution takes the form
$None$
$\phi_{1}=i\partial_{x}\ln\frac{f_{2}}{g_{2}}+\theta_{2}[v_{0}-\partial_{x}\mathfrak{D}\ln(f_{2}g_{2})],$
with
$None$ $\displaystyle
f_{2}=1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},$
$\displaystyle
g_{2}=1-+e^{\eta_{1}}-e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},$
and $\eta_{j}=p_{j}x-p_{j}^{3}t+q_{j}\theta_{1}\sigma+r_{j},\ \ j=1,2,$
$\displaystyle
e^{A_{12}}=\frac{(p_{1}-p_{2})^{2}}{(p_{1}+p_{2})^{2}}+2\theta_{1}\sigma\frac{(p_{1}-p_{2})(p_{1}q_{2}-p_{2}q_{1})}{(p_{1}+p_{2})^{2}},$
here $p_{j},q_{j},r_{j}\in\Lambda_{0},j=1,2$ are free constants.
4.1. One-periodic waves and asymptotic analysis
We first construct one-periodic wave solutions of the equation (5.2). As a
simple case of the theta function (3.2) when $N=1,s=0$, we take $f$ and $g$ as
$None$ $\displaystyle
f=\vartheta(\xi,0,0|\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in\xi-\pi
n^{2}\tau}),$ $\displaystyle
g=\vartheta(\xi,1/2,0|\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in(\xi+1/2)-\pi
n^{2}\tau})$ $\displaystyle\ \ \ =\sum_{n\in\mathbb{Z}}(-1)^{n}\exp({2\pi
in\xi-\pi n^{2}\tau}),$
where the phase variable $\xi=\alpha x+\omega t+\beta\theta_{1}\sigma+\delta$,
and the parameter $\tau>0$.
Due to the fact that $F(D_{t},D_{x})$ is an odd function, its constraint
equations in the formula (3.10) vanish automatically for $\mu=0$. Similarly
the constraint equations associated with $G(S_{x},D_{t},D_{x})$ also vanish
automatically for $\mu=1$. Therefore, the Riemann theta function (5.8) is a
solution of the bilinear equation (5.4), provided the following equations
$None$ $\displaystyle\sum_{n\in\mathbb{Z}}\\{2\pi
i(2n-\mu)\omega-i(2\pi\alpha)^{3}(2n-\mu)^{3}\\}\exp[-2\pi\tau(n-\mu/2)^{2}+\pi
i(n-\mu/2)]|_{\mu=1}=0,$
$\displaystyle\sum_{n\in\mathbb{Z}}\\{-[2\pi(2n-\mu)]^{2}(\beta\sigma+\theta_{1}\alpha)\omega+(2\pi(2n-\mu)]^{4}(\beta\sigma+\theta_{1}\alpha)\alpha^{3}-(2\pi(2n-\mu)\alpha]^{2}v_{0}+c\\}$
$\displaystyle\times\exp(-2\pi\tau(n-\mu/2)^{2}+\pi i(n-\mu/2))|_{\mu=0}=0.$
We introduce the notations by
$\displaystyle\lambda=e^{-\pi\tau/2},$
$\displaystyle\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},1/4,-1/2|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{(2n-1)^{2}}\exp[4i\pi(n-1/2)(\xi+1/4)],$
$\displaystyle\vartheta_{2}(\xi,\lambda)=\vartheta(2\xi,1/4,0|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{4n^{2}}\exp[4i\pi
n(\xi+1/4)],$
the equation (5.9) can be written as a linear system about $\omega,c$
$None$
$\displaystyle\vartheta_{1}^{\prime}\omega+\alpha^{3}\vartheta_{1}^{\prime\prime\prime}=0,$
$\displaystyle(\beta\sigma+\theta_{1}\alpha)\vartheta_{2}^{\prime\prime}\omega+\vartheta_{2}c+(\beta\sigma+\theta_{1}\alpha)\alpha^{3}\vartheta_{2}^{(4)}+\alpha^{2}\vartheta_{2}^{\prime\prime}v_{0}=0.$
where $\omega\in\Lambda_{0}$ is even and $c,v_{0}\in\Lambda_{1}$ are odd, and
we have denoted the derivative value of $\vartheta_{j}(\xi,\lambda)$ at
$\xi=0$ by simple notations
$\vartheta_{j}^{\prime}=\vartheta_{j}^{\prime}(0,\lambda)=\frac{d\vartheta_{j}(\xi,\lambda)}{d\xi}|_{\xi=0},\
\ j=1,2.$
Moreover, we see that the functions $\vartheta_{j}$ and their derivatives are
independent of Grassmann variable $\theta$ and anticommuting number $\sigma$.
We take $v_{0}=\gamma\alpha(\sigma+\theta\alpha),\ \gamma\in\Lambda_{0}$ for
the simplicity, then the system (5.10) admits a solution
$None$
$\displaystyle\omega=-\frac{\alpha^{3}\vartheta_{1}^{\prime\prime\prime}}{\vartheta_{1}^{\prime}},\
\ \
c=\frac{\alpha^{3}(\beta\sigma+\theta_{1}\alpha)}{\vartheta_{1}^{\prime}\vartheta_{2}}(\vartheta_{1}^{\prime\prime\prime}\vartheta_{2}^{\prime\prime}-\vartheta_{1}^{\prime}\vartheta_{2}^{(4)}-\gamma\vartheta_{1}^{\prime}\vartheta_{2}^{\prime\prime}),$
where $\omega$ is independent of Grassmann variable $\theta$ and auticommuting
number $\sigma$. In this way, a one-periodic wave solution reads
$None$
$\phi=i\partial_{x}\ln\frac{\vartheta(\xi,0,0|\tau)}{\vartheta(\xi,1/2,0|\tau)}+\theta_{2}\\{v_{0}-\partial_{x}\mathfrak{D}\ln[\vartheta(\xi,0,0|\tau)\vartheta(\xi,1/2,0|\tau)]\\},$
where parameters $\omega$ and $c$ are given by (5.11), while other parameters
$\alpha,\beta,\tau,\delta\in\Lambda_{0}$ are free. Among them, the three
parameters $\alpha$ and $\tau$ completely dominate a one-periodic wave.
In summary, one-periodic wave (5.12) has the following features:
(i) It is one-dimensional and has two fundamental periods $1$ and $i\tau$ in
phase variable $\xi$. It can be viewed as a parallel superposition of
overlapping one-soliton waves, placed one period apart (see Figure 5-7).
(ii) As in the case of the supersymmetric Sawada-Kotera-Ramani equation, there
is also an influencing band among the real part of one-periodic waves for the
supersymmetric KdV equation under the presence of the Grassmann variable (see
Figure 4).
(iii) It was not observed that influencing band appears among the imaginary
part and modulus of the one-periodic wave. Moreover, they seem to have the
same shape from the observation of their plots (see Figures 5 and 6).
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 4. Real part of one-periodic wave for $\mathcal{N}=2$ supersymmetric
KdV equation with parameters: $\alpha=0.1,$ $\tau=3,\sigma_{1}=0.01$. (a)
Perspective view of wave. (b) Overhead view of wave, with contour plot shown.
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 5. Imaginary part of one-periodic wave for $\mathcal{N}=2$
supersymmetric KdV equation with parameters: $\alpha=0.1,$
$\tau=3,\sigma_{1}=0.01$. (a) Perspective view of wave. (b) Overhead view of
wave, with contour plot shown.
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 6. Modulus of one-periodic wave for $\mathcal{N}=2$ supersymmetric KdV
equation with parameters: $\alpha=0.1,$ $\tau=3,\sigma_{1}=0.01$. (a)
Perspective view of wave. (b) Overhead view of wave, with contour plot shown.
In the following, we further consider asymptotic properties of the one-
periodic wave solution. The relation between the one-periodic wave solution
(5.12) and the one-soliton solution (5.5) can be established as follows.
Theorem 5. Suppose that $\omega$ and $c$ are given by (5.11). In the one-
periodic wave solution (5.12), we choose parameters as
$None$ $\gamma=0,\ \ \alpha=\frac{p}{2\pi i},\ \ \beta=\frac{q}{2\pi i},\ \
\delta=\frac{r+\pi\tau}{2\pi i},$
where the $p,q$ and $r$ are the same as those in (5.5). Then we have the
following asymptotic properties
$c\longrightarrow 0,\ \ \xi\longrightarrow\frac{\eta+\pi\tau}{2\pi i},\ \
f\longrightarrow 1+e^{\eta},\ \ g\longrightarrow 1-e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$
In other words, the one-periodic solution (5.12) tends to the one-soliton
solution (5.5) under a small amplitude limit , that is,
$None$ $\phi\longrightarrow\phi_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$
Proof. Here we will directly use the system (5.10) to analyze asymptotic
properties of one-periodic solution (5.12). We explicitly expand the
coefficients of system (5.10) as follows
$None$
$\displaystyle\vartheta_{1}^{\prime}=-4\pi\lambda+12\pi\lambda^{9}+\cdots,\quad\vartheta_{1}^{\prime\prime\prime}=16\pi^{3}\lambda+432\pi^{3}\lambda^{9}+\cdots,$
$\displaystyle\vartheta_{2}=-2+2\lambda^{4}+\cdots,\quad\vartheta_{2}^{\prime\prime}=32\pi^{2}\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{2}^{(4)}=512\pi^{4}\lambda^{4}+\cdots$
Suppose that the solution of the system (5.10) is of the form
$None$
$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$\displaystyle c=c_{0}+c_{1}\lambda+c_{2}\lambda^{2}+\cdots=c_{0}+o(\lambda).$
Substituting the expansions (5.15) and (5.16) into the system (5.11) and
letting $\lambda\longrightarrow 0$, we immediately obtain the following
relations
$\displaystyle-4\pi\omega_{0}+16\pi^{3}\alpha^{3}=0,\ \ c_{0}=0,$
which has a solution
$None$ $c_{0}=0,\ \ w_{0}=4\pi^{2}\alpha^{3}.$
Combining (5.13) and (5.17) leads to
$c\longrightarrow 0,\ \ 2\pi i\omega\longrightarrow
8\pi^{3}i\alpha^{3}=-p^{3},\ \ {\rm as}\ \ \lambda\rightarrow 0,$
or equivalently
$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=px+2\pi i\omega
t+q\theta_{1}\sigma+r$ $\displaystyle\quad\longrightarrow
px-p^{3}t+q\theta_{1}\sigma+r=\eta,\ \ {\rm as}\ \ \lambda\rightarrow 0.$
It remains to identify that the one-periodic wave (5.12) possesses the same
form with the one-soliton solution (5.5) under the limit $\lambda\rightarrow
0$. For this purpose, we start to expand the functions $f$ and $g$ in the form
$f=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi i\xi})+\lambda^{8}(e^{4\pi
i\xi}+e^{-4\pi i\xi})+\cdots.$ $g=1-\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots.$
By using (5.13) and (5.17), it follows that
$None$ $\displaystyle
f=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow 1+e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0;$ $\displaystyle
g=1-e^{\hat{\xi}}+\lambda^{4}(e^{2\hat{\xi}}-e^{-\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}-e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1-e^{\hat{\xi}}\longrightarrow 1-e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0.$
The expression (5.13) follows from (5.19), and thus we conclude that the one-
periodic solution (5.12) just goes to the one-soliton solution (5.5) as the
amplitude $\lambda\rightarrow 0$. $\square$
4.2. Two-periodic waves and asymptotic properties
We now consider two-periodic wave solutions to the supersymmetric KdV equation
(5.2). For the case when $N=2,\ \boldsymbol{s}=\boldsymbol{0},\
\boldsymbol{\varepsilon}=\boldsymbol{1/2}=(1/2,1/2)$ in the Riemann theta
function (3.2), we choose $f$ and $g$ to be
$None$ $\displaystyle
f=\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\},$
$\displaystyle
g=\vartheta(\boldsymbol{\xi},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi}+\boldsymbol{1/2},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$
$\displaystyle\ \
=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}(-1)^{n_{1}+n_{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$
where we denote $\boldsymbol{n}=(n_{1},n_{2})\in Z^{2},\ \
\boldsymbol{\xi}=(\xi_{1},\xi_{2})\in\mathcal{C}^{2},\ \
\xi_{i}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta_{1}\sigma+\delta_{j},\ \
j=1,2$, and $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2}),\
\boldsymbol{\omega}=(\omega_{1},\omega_{2}),\
\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathcal{C}^{2},$; The matrix
$\boldsymbol{\tau}$ is a positive definite and real-valued symmetric ${2\times
2}$ matrix, which can take the form
$\boldsymbol{\tau}=(\tau_{ij})_{2\times 2},\ \ \tau_{12}=\tau_{21},\ \
\tau_{11}>0,\ \ \tau_{22}>0,\ \ \tau_{11}\tau_{22}-\tau_{12}^{2}>0.$
According to Theorem 5, constraint equations associated with $F(D_{t},D_{x})$
vanish automatically for $(\mu_{1},\mu_{2})=(0,0),(1,1)$, and the constraint
equations associated with $G(S_{x},D_{t},D_{x})$ vanish automatically for
$(\mu_{1},\mu_{2})=(1,0),(0,1)$. Hence, making the theta functions $f$ and $g$
satisfy the bilinear equation (5.4) gives to the following constraint
equations
$None$ $\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}\left[2\pi i\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\omega}\rangle-8\pi^{3}i\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{3}\right]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle$
$\displaystyle+\pi
i\sum_{j=1}^{2}(n_{j}-\mu_{j}/2)\\}|_{\boldsymbol{\mu}=(\mu_{1},\mu_{2})}=0,\
\ {\rm for}\ \ (\mu_{1},\mu_{2})=(0,1),\ (1,0).$
and
$None$ $\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}[-4\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha}\rangle\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\omega}\rangle+16\pi^{4}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha}\rangle\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{3}$
$\displaystyle\ \ \ \ -4\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{2}v_{0}+c]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle$
$\displaystyle\ \ \ \ +\pi
i\sum_{j=1}^{2}(n_{j}-\mu_{j}/2)\\}|_{\boldsymbol{\mu}=(\mu_{1},\mu_{2})}=0,\
\ {\rm for}\ \ (\mu_{1},\mu_{2})=(0,0),\ (1,1).$
Next, introducing the following notations
$\displaystyle\lambda_{kl}=e^{-\pi\tau_{kl}/2},k,l=1,2,\boldsymbol{\lambda}=(\lambda_{11},\lambda_{12},\lambda_{22})$
$\displaystyle\vartheta_{j}(\boldsymbol{\xi},\boldsymbol{\lambda})=\vartheta(2\xi,\boldsymbol{1/4},-\boldsymbol{s}_{j}/2|2\tau)=\sum_{n_{1},n_{2}\in
Z}\exp\\{4\pi
i\langle\boldsymbol{\xi}+\boldsymbol{1/4},\boldsymbol{n}-\boldsymbol{s_{j}}/2\rangle\\}\prod_{k,l=1}^{2}\lambda_{kl}^{(2n_{k}-s_{j,k})(2n_{j}-s_{j,l})},$
$\displaystyle\boldsymbol{s_{j}}=(s_{j,1},s_{j,2}),\quad j=1,2,3,4,\ \
\boldsymbol{s_{1}}=(0,1),\ \ \boldsymbol{s_{2}}=(1,0),\ \
\boldsymbol{s_{3}}=(0,0),\ \ \boldsymbol{s_{4}}=(1,1),$
then by using (3.15), the system (5.21) and (5.22) can be written as a linear
system
$None$
$\displaystyle(\boldsymbol{\omega}\cdot\nabla)\vartheta_{j}(0,\boldsymbol{\lambda})+(\boldsymbol{\alpha}\cdot\nabla)^{3}\vartheta_{j}(0,\boldsymbol{\lambda})=0,\
j=1,2,$
$\displaystyle[(\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\omega}\cdot\nabla)\vartheta_{j}(0,\boldsymbol{\lambda})+[(\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\alpha}\cdot\nabla)^{3}\vartheta_{j}(0,\boldsymbol{\lambda})$
$\displaystyle+(\boldsymbol{\alpha}\cdot\nabla)^{2}\vartheta_{j}(0,\boldsymbol{\lambda})v_{0}+\vartheta_{j}(0,\boldsymbol{\lambda})c=0,\
j=3,4.$
This system can be solved in such a way: After we obtain
$\omega,_{1},\omega_{2}$ form the first two equations, We substitute them into
last two equations to get $v_{0},c$. With the solution
$(\omega_{1},\omega_{2},v_{0},c)$, we get a two-periodic wave solution to the
supersymmetric equation (5.2)
$None$
$\phi=i\partial_{x}\ln\frac{\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})}+\theta_{2}\\{v_{0}-\partial_{x}\mathfrak{D}\ln[\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})\vartheta(\boldsymbol{\xi},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})]\\},$
where parameters $\omega_{1},\omega_{2},v_{0}$ and $c$ are given by (5.22),
while other parameters $\sigma_{1},\sigma_{2},\alpha_{1},$
$\alpha_{2},\tau_{11}$, $\tau_{22},\tau_{12},\delta_{1}$ and $\delta_{2}$ are
free.
In summary, the two-periodic wave (5.24), which is a direct generalization of
one-periodic waves, has the following features:
(i) Its surface pattern is two-dimensional, namely, there are two phase
variables $\xi_{1}$ and $\xi_{2}$. It has $4$ fundamental periods
$\\{e_{1},e_{2}\\}$ and $\\{i\tau_{1},i\tau_{2}\\}$ in $(\xi_{1},\xi_{2})$,
and is spatially periodic in two directions $\xi_{1},\xi_{2}$. Its real part
is not periodic in $\theta$ direction, while its real part, imaginary part and
modulus are all periodic in $x$ and $t$ directions.
(iii) There is also an influencing band among the Real part of two-periodic
waves for the supersymmetric KdV equation under the presence of the Grassmann
variable ( see Figure 7 ).
(iv) It was not found that influencing band appears among the imaginary part
and modulus of two-periodic waves to the supersymmetric KdV equation ( see
Figures 8 and 9 ).
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 7. Real part of two-periodic wave for $\mathcal{N}=2$ supersymmetric
KdV equation with parameters: $\alpha_{1}=0.1,\ \alpha_{2}=0.2$ $\tau_{11}=3,\
\tau_{12}=0.2,\tau_{22}=3,\sigma_{1}=0.2,\ \sigma_{2}=-0.1$. (a) Perspective
view of wave. (b) Overhead view of wave, with contour plot shown.
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 8. Imaginary part of two-periodic wave for $\mathcal{N}=2$
supersymmetric KdV equation with parameters: $\alpha_{1}=0.1,\ \alpha_{2}=0.2$
$\tau_{11}=3,\ \tau_{12}=0.2,\tau_{22}=3,\sigma_{1}=0.2,\ \sigma_{2}=-0.1$.
(a) Perspective view of wave. (b) Overhead view of wave, with contour plot
shown.
$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$
Figure 9. Modulus of two-periodic wave for $\mathcal{N}=2$ supersymmetric KdV
equation with parameters: $\alpha_{1}=0.1,\ \alpha_{2}=0.2$ $\tau_{11}=3,\
\tau_{12}=0.2,\tau_{22}=3,\sigma_{1}=0.2,\ \sigma_{2}=-0.1$. (a) Perspective
view of wave. (b) Overhead view of wave, with contour plot shown.
Finally, we consider the asymptotic properties of the two-periodic solution
(5.24). In a similar way to Theorem 5, we can establish the relation between
the two-periodic solution (5.24) and the two-soliton solution (5.6) as
follows.
Theorem 6. Assume that $(\omega_{1},\omega_{2},v_{0},c)$ is a solution of the
system (5.23). In the two-periodic wave solution (5.24), a choice of
parameters is given by
$None$ $\displaystyle\alpha_{j}=\frac{p_{j}}{2\pi i},\ \
\beta_{j}=\frac{q_{j}}{2\pi i},\ \ \delta_{j}=\frac{r_{j}+\pi\tau_{jj}}{2\pi
i},\ \ \tau_{12}=-\frac{A_{12}}{2\pi},\ \ j=1,2,$
with the $p_{j},q_{j},r_{j}\in\Lambda_{0},j=1,2$ and $A_{12}$ as those given
in (5.6). Then under the constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$,
we have the following asymptotic relations
$None$ $\displaystyle v_{0}\longrightarrow 0,\ \ \ c\longrightarrow 0,\ \
\xi_{j}\longrightarrow\frac{\eta_{j}+\pi\tau_{jj}}{2\pi i},\ \ j=1,2,$
$\displaystyle f\longrightarrow
1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},\ \ g\longrightarrow
1-e^{\eta_{1}}-e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},$ $\displaystyle\ \ \
{\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0.$
So the two-periodic wave solution (5.24) just tends to the two-soliton
solution (5.6) under a certain limit
$\phi\longrightarrow\phi_{2},\ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$
Proof. From (5.25), the constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$
leads to $p_{1}q_{2}-p_{2}q_{1}=0$, which implies that
$\tau_{12}=-A_{12}/2\pi$ is independent of Grassmann variable $\theta$
according to (5.7).
Using (5.20), we explicitly expand the functions $f$ and $g$ in the following
form
$\displaystyle f=1+(e^{2\pi i\xi_{1}}+e^{-2\pi
i\xi_{1}})e^{-\pi\tau_{11}}+(e^{2\pi i\xi_{2}}+e^{-2\pi
i\xi_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ +(e^{2\pi
i(\xi_{1}+\xi_{2})}+e^{-2\pi
i(\xi_{1}+\xi_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$
$\displaystyle g=1-(e^{2\pi i\xi_{1}}+e^{-2\pi
i\xi_{1}})e^{-\pi\tau_{11}}-(e^{2\pi i\xi_{2}}+e^{-2\pi
i\xi_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ +(e^{2\pi
i(\xi_{1}+\xi_{2})}+e^{-2\pi
i(\xi_{1}+\xi_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$
Further using (4.5) and making a transformation $\hat{\omega}_{j}=2\pi
i\omega_{j},j=1,2$, we infer that
$\displaystyle
f=1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}-2\pi\tau_{12}}+\lambda_{11}^{4}e^{-\hat{\xi}_{1}}+\lambda_{22}^{4}e^{-\hat{\xi}_{2}}+\lambda_{11}^{4}\lambda_{22}^{4}e^{-\hat{\xi}_{1}-\hat{\xi}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}+A_{12}},\
\ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$ $\displaystyle
g=1-e^{\hat{\xi}_{1}}-e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}-2\pi\tau_{12}}-\lambda_{11}^{4}e^{-\hat{\xi}_{1}}-\lambda_{22}^{4}e^{-\hat{\xi}_{2}}+\lambda_{11}^{4}\lambda_{22}^{4}e^{-\hat{\xi}_{1}-\hat{\xi}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1-e^{\hat{\xi}_{1}}-e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}+A_{12}},\
\ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$
where $\hat{\xi}_{j}=p_{j}x+\hat{\omega}_{j}t+q_{j}\theta_{1}\sigma+r_{j},\ \
j=1,2.$
It remains to prove that
$None$ $\displaystyle c\longrightarrow 0,\ \ \
\hat{\omega}_{j}\longrightarrow-p_{j}^{3},\ \
\hat{\xi}_{j}\longrightarrow\eta_{j},\ \ j=1,2,\ \ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$
As in the case when $N=1$, we let the solution of the system (5.23) be the
form
$None$
$\displaystyle\omega_{1}=\omega_{1,0}+\omega_{1,1}\lambda_{11}+\omega_{2,2}\lambda_{22}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle\omega_{2}=\omega_{2,0}+\omega_{2,1}\lambda_{11}+\omega_{2,2}\lambda_{22}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle
v_{0}=v_{0,0}+v_{0,1}\lambda_{11}+v_{0,2}\lambda_{22}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle
c=c_{0}+c_{1}\lambda_{11}+c_{2}\lambda_{22}+o(\lambda_{11},\lambda_{22}).$
Expanding functions $\vartheta_{j},j=1,2,3,4$ in the system (5.23), together
with substitution of assumption (5.28), the second and third equation is
divided by $\lambda_{11}$ and $\lambda_{22}$, respectively; the fourth
equation is divided by $\lambda_{11}\lambda_{22}$, and letting $v_{0,0}=0,\
\lambda_{11},\lambda_{22}\longrightarrow 0$ , we then obtain
$None$ $\displaystyle c_{0}=0,$
$\displaystyle-8\pi\omega_{1,0}+32\pi^{4}\alpha_{1}^{3}=0,$
$\displaystyle-8\pi\omega_{2,0}+32\pi^{4}\alpha_{2}^{3}=0,$
$\displaystyle[-8\pi^{2}(\omega_{1,0}+\omega_{2,0})+32\pi^{4}(\alpha_{1}+\alpha_{2})^{3}]\lambda_{12}$
$\displaystyle\ \ \ \
+[-8\pi^{2}(\omega_{1,0}-\omega_{2,0})+32\pi^{4}(\alpha_{1}-\alpha_{2})^{3}]\lambda_{12}^{-1}=0.$
The first three equations in the system (5.9) have a solution
$None$ $\displaystyle c_{0}=0,\ \ v_{0,0}=0,\ \
\omega_{1,0}=4\pi^{2}\alpha_{1}^{3},\ \ \omega_{2,0}=4\pi^{2}\alpha_{2}^{3},$
The fourth equation in the system (5.29) satisfied automatically by using
(5.25) and (5.30), thus we have
$None$ $\displaystyle c_{0}=c_{1}=c_{2}=0,\ \ v_{0,0}=0,\ \
\omega_{1,0}=4\pi^{2}\alpha_{1}^{3},\ \ \omega_{2,0}=4\pi^{2}\alpha_{2}^{3}.$
Using (5.28) and (5.31), we conclude that
$\displaystyle v_{0}=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
c=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
\omega_{1}=4\pi^{2}\alpha_{1}^{3}+o(\lambda_{11},\lambda_{22})\longrightarrow
4\pi^{2}\alpha_{1}^{3},$
$\displaystyle\omega_{2}=4\pi^{2}\alpha_{2}^{3}+o(\lambda_{11},\lambda_{22})\longrightarrow
4\pi^{2}\alpha_{2}^{3},\ \ \ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow
0,$
and therefore we have (5.26). So the two-periodic wave solution (5.23) tends
to the two-supersoliton solution (5.6) as
$\lambda_{11},\lambda_{22}\rightarrow 0$. $\square$
6\. Conclusion and future work
Following the procedure described in this paper, we are able to construct
quasi-periodic wave solutions for other supersymmetric equations also can be
dealt with by the same way. For instance,
(1) Supersymmetric KdV equation [40, 42, 43]
$\displaystyle\Phi_{t}+3\left(\Phi\mathfrak{D}\Phi\right)_{x}+\Phi_{xxx}=0,$
(2) Supersymmetric MKdV equation [41, 49, 65]
$\displaystyle\phi_{t}-3\phi\mathfrak{D}(\phi_{x})\mathfrak{D}\phi-3(\mathfrak{D}\phi)^{2}\phi_{x}+\phi_{xxx}=0,$
(3) Supersymmetric It’s equation [66]
$\displaystyle\mathfrak{D}_{t}F_{t}+6(F_{x}(\mathfrak{D}_{t}F))_{x}+\mathfrak{D}_{t}F_{xxx}=0,$
(4) Supersymmetric two-boson equation [67, 68]
$\displaystyle\phi_{1,t}=\mathfrak{D}((\mathfrak{D}\phi_{1})^{2})+2\phi_{2,x}-\phi_{1,xx},$
$\displaystyle\phi_{2,t}=2((\mathfrak{D}\phi_{1})\phi_{2})_{x}+\phi_{2,xx}.$
The system (3.10) indicates that constructing multi-periodic wave solutions
depends on the solvability of the system. We consider the number of constraint
equations and some unknown parameters. Obviously, the number of constraint
equations of the type (3.10) is $2^{N}$. On the other hand we have parameters
$\tau_{ii},\tau_{ij},\alpha_{i},\omega_{i},\phi_{0},c$, whose total number is
$\frac{1}{2}N(N+1)+3N+2$. Among them, $2N$ parameters $\tau_{ii},\omega_{i}$
are taken to be the given parameters related to the amplitudes and wave
numbers of $N$-periodic waves. Hence, the number of the unknown parameters is
$\frac{1}{2}N(N+1)+N+2$. while $\frac{1}{2}N(N+1)$ parameters $\tau_{ij}$
implicitly appear in series form, which is general can not to be solved
explicit. Hence, the number of the explicit unknown parameters is only $N+2$.
The number of equations is larger than the unknown parameters in the case when
$N>2$. This fact means that if equation (3.10) is satisfied by the unknowns,
we have at least $N$-periodic wave solutions ($N\leq 2$). It is still possible
to construct multi-periodic wave solutions for $N\geq 3$ by adding the number
of parameters (for example, using constant solution and integration constant)
or decreasing the number of equations (for example, using odd and even
properties of considered equations). In this paper, we consider one-periodic
wave solution of the equation (1.2), which belongs to the cases when $N=1$ and
$N=2$ in the Riemann theta function (3.1). There are still certain
difficulties in the calculation for the case $N\geq 3$.
We expect the proposed method to be extended to $\mathcal{N}=1$ supersymmetric
sine-Gordon equation and $\mathcal{N}=1$ supersymmetric KP equation, as well
as other discrete supersymmetric equations (like supersymmetric Toda lattice).
For the $\mathcal{N}=2$ supersymmetric equations with ordinary variables $x,t$
and Grassmann variables $\theta_{1},\theta_{2}$, their corresponding
superspace is
$\mathbb{R}_{\Lambda}^{2,2}=\Lambda_{0}^{2}\times\Lambda_{1}^{2}$. In this
case, the Grassmann algebra $G_{2}(\sigma_{1},\sigma_{2})$ whose dimension is
four. The $\tau$ matrix will be dependent on the odd parameters
$\sigma_{1},\sigma_{2}$. In superspace $\mathbb{R}_{\Lambda}^{2,2}$, the super
bilinear forms of $\mathcal{N}=2$ supersymmetric equations, their quasi-
periodic solutions and asymptotic properties remain open. We intend to return
to these question in some future publications.
Acknowledgment
The work described in this paper was supported by grants from the National
Science Foundation of China (No.10971031), Shanghai Shuguang Tracking Project
(No.08GG01) and Innovation Program of Shanghai Municipal Education Commission
(No.10ZZ131).
## References
* [1]
* [2] S. P. Novikov, Funct. Anal. Appl. 8(1974): 236-246.
* [3] B. A. Dubrovin, Funct. Anal. Appl. 9(1975): 265-277.
* [4] A. Its and V. Matveev, Funct. Anal. Appl. 9(1975): 65-66.
* [5] P. D. Lax, Comm. Pure Appl. 28(1975): 141-188.
* [6] H. P. Mckean and P. Moerbeke, Invent. Math. 30(1975): 217-274.
* [7] E. Belokolos, A. Bobenko, V. Enol’skij, A. Its and V. Matveev, Algebro-Geometrical Approach to Nonlinear Integrable Equations ( Springer, Berlin, 1994).
* [8] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions, (Cambridge University Press, 2003).
* [9] F. Gesztesy and H Holden. _Phil Trans R Soc A_ 366 (2008): 1025-1054.
* [10] Z. J. Qiao, _Comm Math Phys_ , 239 (2003): 309-341.
* [11] L. Zampogni, _Advanced Nonl Studies_ , 3 (2007): 345-380.
* [12] R. G. Zhou, J. Math. Phys. 38(1997): 2535-2346.
* [13] C. W. Cao, Y. T. Wu and X. G. Geng. J. Math. Phys. 40(1999): 3948-3970.
* [14] X. G. Geng, Y. T. Wu and C. W. Cao. J. Phys. A 32 (1999): 3733-3742.
* [15] X. G. Geng, and C. W. Cao. Nonlinearity, 14 (2001): 1433-1452.
* [16] X. G. Geng, H. H. Dai, J. Y. Zhu and H. Y. Wang. Stud. Appl. Math. 118 (2007): 281-312.
* [17] Y. C. Hon and E. G. Fan, J. Math.Phys. 46 (2005): 032701-21.
* [18] J. Hammack, N. Scheffner and H. Segur, J. Fluid Mech. 209 567-589 (1989)
* [19] J. Hammack, D. McCallister, N. Scheffner and H. Segur, J. Fluid Mech. 285 95-112 (1995)
* [20] R. Hirota and J. Satsuma: Prog. Theor. Phys. 57 (1977) 797.
* [21] R. Hirota: Direct methods in soliton theory (Springer-verlag, Berlin, 2004).
* [22] X. B. Hu and P. A. Clarkson, J. Phys. A 28 (1995) 5009.
* [23] X. B. Hu, C X Li, J. J. C. Nimmo and G. F. Yu, J. Phys. A, 38 (2005) 195.
* [24] R Hirota and Y. Ohta: J. Phys. Soc. Jpn. 60 (1991) 798\.
* [25] D. J. Zhang, J. Phys. Soc. Jpn. 71 (2002) 2649.
* [26] K. Sawada and T Kotera: Prog. Theor. Phys. 51 (1974) 1355.
* [27] A. Nakamura, J. Phys. Soc. Jpn. 47, 1701-1705 (1979).
* [28] A. Nakamura, J. Phys. Soc. Jpn. 48(1980), 1365.
* [29] R. Hirota and M. Ito, J. Phys. Soc. Jpn. 48(1980), 1365.
* [30] H. H. Dai, E. G. Fan and X. G. Geng, arxiv.org/pdf/nlin/0602015
* [31] Y. Zhang, L. Y. Ye, Y. N. Lv and H. Q. Zhao, J. Phys A, 40 (2007), 5539.
* [32] Y. C. Hon, E. G. Fan and Z. Y. Qin, Modern Phys Lett B, 22 (2008), 547.
* [33] E. G. Fan and Y. C. Hon, Phys Rev E, 78 (2008), 036607.
* [34] E. G. Fan, J. Phys A, 42 (2009), 095206.
* [35] W. X. Ma, R. G. Zhou, J. Math. Phys, 24 (2009), 1677.
* [36] P. Ramond, Phys. Rev. D, 3 (1971), 2415.
* [37] A. Neveu and J. H. Schwarz, Nucl. Phys. B, 31 (1971), 86.
* [38] J. Wess and B. Zumino, Nucl. Phys. B, 70 (1974), 39.
* [39] F. A. Berezin, Introduction to super-Analysi, Dordrecht: Reidel, 1987.
* [40] Yu. I. Manin and A. O. Radul, Commun. Math. Phys. 98 (1985), 65.
* [41] P. Mathieu, J. Math. Phys. 29 (1988), 2499.
* [42] W. Oevel and Z. Popowicz, Commun. Math. Phys. 139 (1991), 441.
* [43] A. S. Carstea, Nonlinearity 13 (2000), 1645.
* [44] A. S. Carstea, A. Ramani and B Grammaticos, Nonlinearity 14 (2000), 1419.
* [45] D. Sarma, Nucl. Phys. B 681 (2004), 351.
* [46] Q. P. Liu, Lett Math. Phys. 35 (1995), 115.
* [47] Q. P. Liu and Y. F. Xie, Phys. Lett. A 325 (2004), 139.
* [48] Q. P. Liu and X. B. Hu, J. Phys. A 38 (2005), 6371.
* [49] Q. P. Liu, X. B. Hu and M. X. Zhang, Nonlinearity 18 (2005), 1597.
* [50] S. Ghosh, J. Nonl. Math. Phys. 10 (2003), 526.
* [51] V. S. Vladimirov, Theor. Math. Phys. 59 (1984), 3.
* [52] V. S. Vladimirov, Theor. Math. Phys. 60 (1984), 169\.
* [53] D. Mumford, Tata Lectures on Theta II Progress in Mathmatics, Vol. 43 ( Boston: Birkhäuser, 1984).
* [54] R. Hirota and M Ito, J. Phys. Soc. Jpn. 52 (1983) 744
* [55] K. Konno, J. Phys. Soc. Jpn. 61 (1992) 51
* [56] Y. X. Yu, Commun. Theor. Phys. 49 (2008), 685.
* [57] K. Tian and Q. P. Liu, Phys. Lett. A 373 (2009), 169.
* [58] C. A. Laberge, P. Mathieu, Phys. Lett. B 215 (1988), 718.
* [59] C. A. Laberge, P. Mathieu, J. Math. Phys. 32 (1991), 923.
* [60] M. X. Zhang, Q. P. Liu, Y. L. Shen and K. Wu, Sic. in China. 52 (2009), 1973.
* [61] Z. Popowicz. Phys. Lett. A 174 (1993), 411.
* [62] S. Bourque and P. Mathieu, J. Math. Phys. 42 (2001), 3517.
* [63] T. Inami and H. Kanno, Nuclear Phys. B 359 (1991), 201.
* [64] Q. P. Liu and Y. F. Xie, Phys. Lett. A 235 (1997), 335.
* [65] S. Ghosh and D. Sarma, Nonlinearity, 16 (2003), 411.
* [66] S. Q. Liu, X. B. Hu and Q. P. Liu, J. Phys. Soc. Jpn 75 (2006), 064004\.
* [67] J. C. Brunelli, A. Das, Phys. Lett. B, 337 (1994), 303\.
* [68] Q. P. Liu, X. X. Yang, Phys. Phys A. 351 (2006), 131.
|
arxiv-papers
| 2010-01-10T01:55:11 |
2024-09-04T02:49:07.619282
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1001.1402"
}
|
1001.1460
|
# Gromov Conjecture on Surface Subgroups: Computational Experiments
Anastasia V. Kisil Trinity College, Cambridge, CB2 1TQ ak528@cam.ac.uk
(Date: 2nd October 2008 )
###### Abstract.
In this paper we investigate Gromov’s question: whether every one-ended word
hyperbolic group contains a surface subgroup. The case of double groups is
considered by studying the associated one relator groups. We show that the
majority ($96\%$) of the randomly selected double groups with three generators
have the property. The experiments are performed on MAGMA software.
###### Key words and phrases:
one relator groups, surface group, Gromov, word hyperbolic group, MAGMA
###### 2000 Mathematics Subject Classification:
Primary 20F67; Secondary 20Fxx.
This project was sponsered by Trinity College, Cambridge.
## 1\. Introduction
In this paper we are going to investigate the following question:
###### Question 1.1 (Gromov).
[problems] Does every one-ended word hyperbolic group contains a surface
group?
Here a “surface subgroup” means a subgroup isomorphic to the fundamental group
of a closed surface with non-positive Euler characteristic. This question of
Gromov is of interest partly because it is a natural generalisation of famous
Surface Subgroup Conjecture. In the case of the fundamental groups of
hyperbolic 3-manifolds it is exactly the Conjecture. The question of finding
subgroups is studied from different angles and it has proved to be a highly
nontrivial problem [sub].
To define what is meant by the number of ends of a finitely generated group
take $S\subseteq G$ a finite generating set of $G$ and let $\Gamma(G,S)$ be
the Cayley graph of $G$ with respect to $S$. Then the number of ends is
$e(\Gamma(G,S))$ ($e$ stands for edges) which does not depend on the choice of
a finite generating set $S$ of $G$ hence it is well-defined. Stallings’
theorem about ends of groups states that a finitely generated group $G$ has
more than one end if and only if the group $G$ admits a nontrivial
decomposition as an amalgamated free product or an HNN extension over a finite
subgroup [end]. A word hyperbolic group roughly speaking, is a finitely
generated group equipped with a word metric satisfying certain properties
characteristic of hyperbolic geometry.
The famous Gromov’s question has been much speculated about but it is still
very much open even for very concrete groups. It is not even quite clear which
answer to expect. One of the few classes of groups the answer is know to is in
the case of Coxeter groups and some Artin groups, where it is true [coxeter].
It is not even know for one-relator groups
$G_{n}(w)=\frac{F_{n}}{\langle\langle w\rangle\rangle}$ where $w$ is an
element of a free group of rank $n$, $F_{n}$ that is not a proper power.
In this paper we focus on doubles $D_{n}(w)=F_{n}*_{\langle w\rangle}F_{n}$
where $F_{n}$ is a free group of rank $n$ and $w\in F_{n}$. The useful recent
reduction of the question in the case of the doubles is:
###### Theorem 1.2 (Gordon, Wilton).
[simple] Let $w\in F_{n}$. If $G_{n}(w)$ has an index-k subgroup $G^{\prime}$
with $\beta_{1}(G^{\prime})>1+k(n-2)$ then the double
$D_{n}(w)=F_{n}*_{\langle w\rangle}F_{n}$ contains a surface subgroup.
The above result allowed Gordon and Wilton to exhibit several infinite
families of new examples of doubles with surface subgroups [simple]. This
result reduces the Gromov’s question for doubles to virtual homology. The only
difficulty is that general approach to computing the virtual homology is not
fully developed. But for each particular group $\beta_{1}$ of a subgroup can
be attempted to be calculated using a computer. This is the approach taken to
gather evidence for Gromov’s question.
We will be using this to investigate mainly doubles with $n=3$. It will be
also shown how this method works for $n=4$. The Question 1.1 was already
studied with success by Button in the case of $n=2$ [2rel] using similar
methods.
## 2\. Algorithm
We will be looking at groups of the form $G_{3}(w)=\langle a,b,c\mid
w(a,b,c)\rangle$ where $w(a,b,c)$ is a cyclically reduced word in three
letters of length up to 18. The reason 18 is chosen is that the longer the
word is the more computational time is needed. Cyclically reduced means that
cyclic permutations are reduced. Reduced simply means that all obvious
cancellations like $a^{-1}$ followed by $a$ are made.
In general there is no algorithm to decided weather a group is hyperbolic of
not. But in the one relator setting there is a number of theorems we will need
to use later on.
The trivial corollary to the above Theorem 1.2 that we will be using in the
remainder of the paper is:
###### Corollary 2.1.
Let $w\in F_{3}$. If $G_{3}(w)$ has an index-k subgroup $G^{\prime}$ with
$\beta_{1}(G^{\prime})>n+1$ then the double $D_{3}(w)=F_{3}*_{\langle
w\rangle}F_{3}$ contains a surface subgroup.
A randomly chosen finitely presented group is almost surely word-hyperbolic
with an appropriate definition of “almost surely”. That is why initially we
did not included any checks weather the group is hyperbolic or not. Double
groups are one-ended if $w$ is not in a proper free factor of $F_{n}$
[simple].
The algorithm is as follows:
1. (1)
Generate a random word $w(a,b,c)$. It is done by choosing randomly 18
characters from $a,a^{-1},b,b^{-1},c,c^{-1}$ and then cyclically reduce the
word. Since we do not know how much cancellation will take place we only know
that the resulting word will be of length smaller then 18 typically around 14.
2. (2)
Calculate the index $i=1$ subgroups for $G_{3}(w)=\langle a,b,c\mid
w(a,b,c)\rangle$.
3. (3)
Checking for the condition in Corollary 2.1 for each subgroup. So for each
subgroup we calculate the first Betti number. In other words the abelinisation
of the subgroup is calculated and the first betti number is the rank of it.
4. (4)
If the condition is satisfied to step 6.
5. (5)
If the condition is not satisfied go back to step 2 and increase $i$ by $1$.
Do this until $i<10$ then move to the next step. The reason why the index is
chosen to be 10 is that it is the highest average computer will calculate in
reasonable time for a generic group.
6. (6)
Record the result weather the condition is satisfied for all $i$ and go to
step 1. The output is $w(a,b,c)$ and either the program found that the
condition is satisfied and if so at which index or that it failed.
7. (7)
Then calculate the number of successes and fails over the number of groups
tried.
This algorithm was implemented in MAGMA software for symbolic calculations.
The main limitation of this method is the speed of the computer.
## 3\. Analysis of results
It became clear after running the program that $F_{2}$ appears quite often
which produces double that are not hyperbolic (and not one-ended). To filter
it out we used three methods.
The first one is linked to the Nielsen’s moves. Let $G$ be a group and let
$M=(g_{1},...,g_{n})\in G_{n}$ be an $n$-tuple of elements of $G$. The
following moves are called elementary Nielsen moves on $M$, for generators
$g_{i}$, $1<i<n$ :
1. (1)
For some $i$, replace $g_{i}$ by $g^{-1}$ in $M$.
2. (2)
For some $i\neq j$, $1<i,j<n$ replace $g_{i}$ by $g_{i}g_{j}$ in $M$.
3. (3)
For some $i\neq j$, $1<i$, $j<n$ interchange $g_{i}$ and $g_{j}$ in $M$.
We say that two n-tuples are Nielsen equivalent if there is a chain of
elementary Nielsen moves which transforms one into another. In fact if they
are Nielsen equivalent if and only if they generate the same group. So if $w$
has only one of $a,a^{-1},b,b^{-1},c,c^{-1}$ then $G_{3}(w)$ is Nielsen
equivalent to $F_{2}$. Hence this is the first thing to check for, since it is
the least computationally expensive.
Secondly there is a function in MAGMA which looks for isomorphisms between
groups. So the next job is to find isomorphisms of $G_{3}(w)$ and $F_{2}$
which we do with parameter 9. The parameter in the function isomorphism
indicated how hard it looks in the sense the higher the parameter the longer
it will try to look for before giving up.
Finally we can gather evidence that the group is $F_{2}$ or at least disprove
that it is not by looking at the number of subgroups of a all index (up to
conjugacy the way it is counted in MAGMA). If the group has the same number of
subgroups for all indexes up to $9$ it is likely to be either $F_{2}$ or it is
indistinguishable (by looking at subgroups) from it.
The second class of groups which will not satisfy the condition in Corollary
2.1 is: when it can be written as $G=F_{1}*H$ where $H=BS(n,m)$
Baumslag–Solitar groups or very close to them like $\langle a,b^{p}\mid
b^{-p}a^{n}b^{p}=a^{m\pm 1}\rangle$. The later are due to Higman [hig] called
Baumslag–Brunner–Gersten in [2rel]. Baumslag–Solitar groups are of the form:
$BS(n,m)\cong\langle a,b\mid b^{-1}a^{n}b=a^{m}\rangle.$
We have $\beta_{1}(G^{\prime})=i+1$ for all subgroups $G^{\prime}$ of $G$ with
index $i$ see section 5.5 in [pro] for the proof.
###### Proposition 3.1.
The linked one relator groups which do not give a surface subgroups for the
doubles in light of Corollary 2.1 are either
1. (1)
The free group on two generators $F_{2}$.
2. (2)
The groups of the form $G=F_{1}*H$ where $H=BS(n,m)$ Baumslag–Solitar or
Baumslag–Brunner–Gersten groups.
Note that for all subgroups $F^{\prime}$ of a free group on two generators
$F_{2}$ we have that $\beta_{1}(F^{\prime})=i+1$.
It appears that those can be very not trivial to spot even using a computer.
What is easier is to prove that the group is not of the a certain form. In the
next section we display four examples where our computer did not find any
surface subgroups but which are not of the above kind.
The above algorithm was run for 1000 random groups showing that it either
contains a surface subgroup or are of the above form for $96\%$ of all group.
## 4\. Open Questions
The relator of four $G_{3}(w)$ groups which are not of the above form are:
* •
$ba^{-1}c^{-1}b^{-1}ab^{-2}a^{-1}b^{-1}a^{-1}c$
* •
$b^{-1}ab^{-3}c^{-2}b^{-2}ca^{-1}$
* •
$ac^{-1}ac^{-1}ac^{2}b^{2}ca$
* •
$a^{-1}bc^{2}a^{-1}ca^{-1}ba^{3}b^{-1}$
One way to see that those relators do not give rise to $F_{2}$ is to compare
the number of subgroups of index say $9$ which is different for each one. This
actually indicates that no two of the above groups are isomorphic to each
other. To see that it is not $G=F_{1}*H$ where $H=BS(n,m)$ we prove that all
four of the above groups are word hyperbolic. To do that we use the paper
[hyper], which has a nice criteria in the case of one relator groups. The
approach works if we can find a presentation which has one letter appearing no
more then three times, which we have in all of the above. Then it is simply
the matter of an easy check. Note that hyperbolic groups cannot contain a
Baumslag–Solitar groups as a subgroup, this implies that the above groups are
not of the form as in the Proposition 3.1.
The above groups are intriguing: could it be the case that they are
decomposable but not with Baumslag–Solitar groups? If this is the case then
$G=F_{1}*H$, where $H$ will be two generator one relator group and the relator
could be either of height 1 or not. One relator groups are well studied and
all of the counter-examples seem to come from height 1 relators. Without loss
of generality height one word is:
$w=ba^{i_{1}}ba^{i_{2}}...ba^{i_{l-1}}ba^{i_{l}}$
If it is not height 1 we cannot say anything about that at the moment. But if
$H$ is then there is a theorem of J. Button in [hight] which says that those
groups are either large or are indistinguishable from Baumslag-Solitar groups
from looking at subgroups. If $H$ is large then certainly the $\beta_{1}>1+i$
for some $i$. And also the above groups do not have the same number of
subgroups as any Baumslag–Solitar groups.
The way to see that is to note that we can work out the $n-m$ from the
abelianisation and bound $n+m<16$ by the fact that Nielsen’s moves preserve
the highest powers. Then there is only a few possible Baumslag–Solitar groups
to check, and none of them work for any of the above groups. In fact the
number of subgroups is strictly in between that of $F_{2}$ and Baumslag-
Solitar groups. Hence if it is decomposable then $H$ is not of height $1$.
It maybe it might be the case that a higher index is needed to detect the
required property. Or do there exist doubles which have a surface subgroup but
this is not detectable by Theorem 1.2 for arbitrary index? The property that
the above groups seem to share is very little torsion in the abelianisation of
subgroups. Also up to index $9$ there is no abelianisation of a subgroup which
has the repeating torsion, which could have been used to try the method
described in Section 7.
## 5\. Decomposable into the two generators one relator group and a free
group
In this section we will be testing the groups of the form $F(b,c\mid
w(b,c))*F_{1}$ where $F_{1}$ is the free one generator group. It is
interesting to see for which $F(b,c\mid w(b,c))$ we cannot find a surface
subgroup in the associated doubles.
Question 1.1 does not apply in this senario (the above is not one-ended) but
it is still of interest to see how many of them actually satisfy the above
property. Using this approach we were able to come up with examples of groups
which will not terminate using the below program but which nevertheless
satisfy the condition in Result 2.1. Since the group is a free product it is
enough to study $F(a,b\mid w(b,c))$ which is much smaller and so the computer
can go to a much higher index.
For example, with $w(b,c)=c^{-1}*b^{-2}*c^{2}*b^{-3}*c^{-2}$ the property is
only detected at index $13$. One would need a very powerful computer to go
that high for $n=3$. Furthermore for $w(b,c)=c*b^{2}*c*b*c*b^{-1}*c^{-2}*b$
the index the property is detected is $30$. It is not possible to calculate
all subgroups up to index 30 even with the most powerful computer. So we use a
trick that was used in [2rel] by spotting that at index $15$ there is a
subgroup with an abelianisation which had three cyclic groups of the same
order. So take this subgroup as the group and repeat the process with it,
where it works already at index $2$.
## 6\. Four Generators
We also tried this method in the case of $n=4$. The program below dealt with
about $87\%$ of the random double groups. The reason why less of them is dealt
with is that with more generators the algorithms become more expensive and the
index up to which it is possible to go is only 6\. Also the index we might
need to go up to might be bigger.
The way the algorithm worked is as follows:
1. (1)
We pick a random relator in the same sense as in the three generator case.
2. (2)
Cyclically reduce it.
3. (3)
Check if there is a letter which occurs only once or not at all.
4. (4)
If it occurs only once then it is isomorphic to $F_{3}$ by the Nelson’s moves
same as in the $n=3$ case.
5. (5)
If there is a letter absent then $G_{4}=F_{1}*G_{3}$ so it can be recovered
from the $n=3$ case. It is important to see if it is decomposable since
$\beta_{1}(K*L)=\beta_{1}(K)*\beta_{1}(L)$.
6. (6)
If neither of the two happens we search for isomorphisms with $F_{3}$ this
time with parameter 7 (smaller one had to be chosen due to more time consuming
search). Then we follow exactly the same procedure as in the case of $n=3$.
## 7\. Acknowledgement
I am very grateful to Dr Jack Button for suggesting this project and for the
very helpful discussion along the way.
## References
## Appendix A The code for three generator groups
F$<$a, b, c$>$:= FreeGroup(3); F1$<$a1,b1$>$:= FreeGroup(2); kon:=0; free2:=0;
sub:=[1, 3, 7, 26, 97, 624, 4163, 34470, 314493]; //number of // subgroups of
$\backslash$(F 2$\backslash$) for indexes up to 9 to check against for i 1
:= 1 to 50 do //numbers of groups checked rel:=Id(F); c1:=0; 10 c2:=0; c3:=0;
for i := 1 to 18 do j:=Random(3, 6); if j eq 1 then rel:=rel*a; elif j eq 2
then rel:=rel*a^$-$1; elif j eq 3 then rel:=rel*b; elif j eq 4 then
rel:=rel*b^$-$1; elif j eq 5 then rel:=rel*c; else rel:=rel*c^$-$1; 20 end if;
end for;
for i:=0 to #rel do //cyclically reducing l1:=LeadingGenerator(rel);
rel1:=rel*l1; if #rel gt #rel1 then rel:=l1^$-$1*rel*l1; else break; end if;
end for; seq:=Eltseq(rel); 10 k:=1; for i := 1 to #rel do //counting the
number of each relator if seq[i] eq 1 then c1:=c1+1; elif seq[i] eq $-$1 then
c1:=c1+1; elif seq[i] eq 2 then c2:=c2+1; elif seq[i]eq $-$2 then c2:=c2+1;
elif seq[i] eq 3 then c3:=c3+1; else c3:=c3+1; end if; end for; 20 if c1 eq 1
then k:=2; print "Isomorphic to F2 trivially"; free2:=free2+1; elif c3 eq 1
then k:=2; print "Isomorphic to F2 trivially"; free2:=free2+1; elif c2 eq 1
then k:=2; print "Isomorphic to F2 trivially"; free2:=free2+1; end if; rel; G
$<$e, f, g$>$ := quo$<$F $|$ rel $>$; ab:=0; nu:=0; sB:=0; if k eq 1 then 30
isiso, f1, f2 := SearchForIsomorphism(G,F1,9); isiso; if isiso then k:=2;
print "Isomorphic to F2"; free2:=free2+1; end if; end if;
for i := 1 to for i := 1 to 9 do //the index up to which it is going up if k
eq 2 then break; end if; t:=LowIndexSubgroups(G, $<$i, i$>$); if #t ne sub[i]
then sB:=1; end if; for j:= 1 to #t do l:=AQInvariants(t[j]); con:=0;
//calculating the number of zero’s in the abelinisation 10 for m:=1 to #l do
if 1 gt l[m] then con:=con+1; else ab:=1; end if; end for; if con gt i+1 then
print i; k:=2; kon:=kon+1; //checking condition end if; if con ne i+1 then
nu:=1; end if; 20 if k eq 2 then break; end if; end for; if k eq 2 then break;
end if; end for; if k eq 1 then print "Did not find surface subgroups"; end
if; 30 if ab eq 0 and nu eq 0 and sB eq 0 then print "Looks like F2"; end if;
end for; print "Free 2"; free2; print "Done"; kon;
## Appendix B The code for four generator groups
F$<$a, b, c, d$>$:= FreeGroup(4); F1$<$a1,b1, c1$>$:= FreeGroup(3); kon:=0;
free3:=0; free2:=0; sub:=[1, 7, 41, 604, 13753, 504243]; //number of //
subgroups of // subgroups of $\backslash$(F 3$\backslash$) for indexes up to
6 to check against for i 1 := 1 to 50 do rel:=Id(F); 10 c1:=0; c2:=0; c3:=0;
c4:=0; for i := 1 to 14 do j:=Random($-$1, 6); if j eq 1 then rel:=rel*a; elif
j eq $-$1 then rel:=rel*d^$-$1; elif j eq 0 then rel:=rel*d; elif j eq 2 then
rel:=rel*a^$-$1; 20 elif j eq 3 then rel:=rel*b; elif j eq 4 then
rel:=rel*b^$-$1; elif j eq 5 then rel:=rel*c; else rel:=rel*c^$-$1; end if;
end for; for i:=0 to #rel do l1:=LeadingGenerator(rel); rel1:=rel*l1; 30 if
#rel gt #rel1 then rel:=l1^$-$1*rel*l1; else break; end if; end for;
seq:=Eltseq(rel); k:=1; for i := 1 to #rel do if seq[i] eq 1 then c1:=c1+1; 40
elif seq[i] eq $-$1 then c1:=c1+1; elif seq[i] eq 2 then c2:=c2+1; elif
seq[i]eq $-$2 then c2:=c2+1; elif seq[i] eq 3 then c3:=c3+1; elif seq[i] eq 4
then c4:=c4+1; elif seq[i] eq $-$4 then c4:=c4+1; else c3:=c3+1; end if; end
for;
if c1 eq 1 then k:=2; print "Isomorphic to F3 trivially"; free3:=free3+1; elif
c3 eq 1 then k:=2; print "Isomorphic to F3 trivially"; free3:=free3+1; elif c2
eq 1 then k:=2; print "Isomorphic to F3 trivially"; free3:=free3+1; elif c4 eq
1 then k:=2; print "Isomorphic to F3 trivially"; free3:=free3+1; elif c1 eq 0
then k:=2; print "Back to 3 generator case"; free2:=free2+1; elif c3 eq 0 then
k:=2; print "Back to 3 generator case"; free2:=free2+1; elif c2 eq 0 then
k:=2; print "Back to 3 generator case"; free2:=free2+1; elif c4 eq 0 then
k:=2; print "Back to 3 generator case"; free2:=free2+1; end if; rel; 10 G
$<$e, f, g$>$ := quo$<$F $|$ rel $>$; ab:=0; nu:=0; sB:=0; if k eq 1 then
isiso, f1, f2 := SearchForIsomorphism(G,F1,7); isiso; if isiso then k:=2;
print "Isomorphic to F2"; free3:=free3+1; end if; end if; 20 for i := 1 to for
i := 1 to 6 do if k eq 2 then break; end if; t:=LowIndexSubgroups(G, $<$i,
i$>$); if #t ne sub[i] then sB:=1; end if; for j:= 1 to #t do
l:=AQInvariants(t[j]); 30 con:=0; for m:=1 to #l do if 1 gt l[m] then
con:=con+1; else ab:=1; end if; end for; if con gt 2*i+1 then print i; k:=2;
kon:=kon+1; end if; if con ne i+1 then nu:=1; 40 end if; if k eq 2 then break;
end if; end for; if k eq 2 then break; end if; end for; if k eq 1 then print
“Did not find surface subgroups’’; end if; 50 if ab eq 0 and nu eq 0 and sB eq
0 then print "Looks like F3"; end if; end for; print "Free 3:"; free3; print
"Back to 3 generator case:"; free2; print "Done:"; kon;
|
arxiv-papers
| 2010-01-09T23:36:03 |
2024-09-04T02:49:07.632842
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anastasia V. Kisil",
"submitter": "Anastasia Kisil",
"url": "https://arxiv.org/abs/1001.1460"
}
|
1001.1492
|
# Explicit quasi-periodic wave solutions and asymptotic analysis to the
supersymmetric Ito’s equation
Engui Fana111 Electronic mail: faneg@fudan.edu.cn., Y. C. Honb,
a. School of Mathematics Sciences, Fudan University, Shanghai 200433, PR China
b. Department of Mathematics, City University of Hong Kong, Hong Kong, PR
China
Abstract: Based on a Riemann theta function and the super-Hirota bilinear
form, we propose a key formula for explicitly constructing quasi-periodic wave
solutions of the supersymmetric Ito’s equation in superspace
$\mathbb{C}_{\Lambda}^{2,1}$. Once a nonlinear equation is written in bilinear
forms, then the quasi-periodic wave solutions can be directly obtained from
our formula. The relations between the periodic wave solutions and the well-
known soliton solutions are rigorously established. It is shown that the
quasi-periodic wave solutions tends to the soliton solutions under small
amplitude limits.
## 1\. Introduction
The Ito’s equation takes the form
$u_{tt}+6(u_{x}u_{t})_{x}+u_{xxxt}=0,$ $None$
which was first proposed by Ito, and its bilinear Bäcklund transformation, Lax
representation and multi-soliton solutions were obtained [1] . The other
integrable properties of this equation such as the nonlinear superposition
formula, Kac-Moody algebra, bi-Hamiltonian structure have been further found
[2]-[5] . Recently, Liu, Hu and Liu proposed the following supersymmetric
Its’s equation [6]
$\displaystyle\mathfrak{D}_{t}F_{t}+6(F_{x}(\mathfrak{D}_{t}F))_{x}+\mathfrak{D}_{t}F_{xxx}=0,$
$None$
and obtained its one-, two- and three-soliton solutions, where
$F=F(x,t,\theta)$ is fermionic superfield depending on usual even independent
variable $x$, $t$ and odd Grassman variable $\theta$. The differential
operator $\mathfrak{D}_{t}=\partial_{\theta}+\theta\partial_{t}$ is the super
derivative.
The bilinear derivative method developed by Hirota is a powerful approach for
constructing exact solution of nonlinear equations[7]–[13] . Based on the
Hirota bilinear form and the Riemann theta functions, Nakamura presented an
approach to directly construct a kind of quasi-periodic solutions of nonlinear
equation [14, 15] , where the periodic wave solutions of the KdV equation and
the Boussinesq equation were obtained. This method not only conveniently
obtains periodic solutions of a nonlinear equation, but also directly gives
the explicit relations among frequencies, wave-numbers, phase shifts and
amplitudes of the wave. Recently, this method is further developed to
investigate the discrete Toda lattice, (2+1)-dimensional Kadomtsev-
Petviashvili equation and Bogoyavlenskii’s breaking soliton equation[16]-[20]
.
Our present paper will considerably improve the key steps of the above method
so as to make the method much more lucid and straightforward for applying a
class of nonlinear supersymmetric equations. First, the above method will be
generalized into the supersymmetric context. The quasi-periodic solutions of
supersymmetric equations still seem not investigated to our acknowledge.
Second, we a formula that the Riemannn theta functions satisfy a super-Hirota
bilinear equation. This formula actually provides us an uniform method which
can be used to construct quasi-periodic wave solutions of nonlinear
differential, difference and supersymmetric equations. Once a nonlinear
equation is written in bilinear forms, then the quasi-periodic wave solutions
of the nonlinear equation can be obtained directly by using the formula. As
illustrative example, we shall construct quasi-periodic wave solutions to the
supersymmetric Ito’s equation (1.2). Moreover, we also establish the relations
between our quasi-periodic wave solutions and the soliton solutions that were
obtained by Liu and Hu [5] .
The organization of this paper is as follows. In section 2, we briefly
introduce a super-Hirota bilinear that will be suitable for constructing
quasi-periodic solutions of the equation (1.2). And then introduce a general
Riemann theta function and provide a key formula for constructing periodic
wave solutions. In section 3, as application of our formula, we construct one-
periodic wave solutions to the equation (1.2). We further present a simple and
effective limiting procedure to analyze asymptotic behavior of the one-
periodic wave solutions. It is rigorously shown that the quasi-periodic wave
solutions tends to the known soliton solutions obtained by Liu and Hu under
“small amplitude” limits. At last, we briefly discuss the conditions on the
construction of multi-periodic wave solutions of the equation (1.2) in section
4.
## 2\. The superspace, Hirota bilinear form and the Riemann theta functions
To fix the notations and make our presentation self-contained, we briefly
recall some properties about superanalysis and super-Hirota bilinear
operators. The details about superanalysis refer, for instance, to
Vladimirov’s work [22, 23].
A linear space $\Lambda$ is called $Z_{2}$-graded if it represented as a
direct sum of two subspaces
$\Lambda=\Lambda_{0}\oplus\Lambda,$
where elements of the spaces $\Lambda_{0}$ and $\Lambda_{1}$ are homogeneous.
We assume that $\Lambda_{0}$ is a subspace consisting of even elements and
$\Lambda_{1}$ is a subspace consisting of odd elements. For the element
$f\in\Lambda$ we denote by $f_{0}$ and $f_{1}$ its even and odd components. A
parity function is introduced on the $\Lambda$, namely,
$|f|=\left\\{\begin{matrix}0,\ \ {\rm if}\ \ f\in\Lambda_{0},\\\ \ 1,\ \ {\rm
if}\ \ f\in\Lambda_{1}.\end{matrix}\right.$
We introduce an annihilator of the set of odd elements by setting
${}^{\perp}\Lambda_{1}=\\{\lambda\in\Lambda:\lambda\Lambda_{1}=0\\}.$
A superalgebra is a $Z_{2}$-graded space $\Lambda=\Lambda_{0}\oplus\Lambda$ in
which, besides usual operations of addition and multiplication by numbers, a
product of elements is defined with the usual distribution law:
$a(\alpha b+\beta c)=\alpha ab+\beta ac,\ \ (\alpha b+\beta c)a=\alpha
ba+\beta ca,$
where $a,b,c\in\Lambda$ and $\alpha,\beta\in\mathbb{C}.$ Moreover, a structure
on $\Lambda$ is introduced of an associative algebra with a unite $e$ and even
multiplication i.e., the product of two even and two odd elements is an even
element and the product of an even element by an odd one is an odd element:
$|ab|=|a|+|b|$ mod (2).
A commutative superalgebra with unit $e=1$ is called a finite-dimensional
Grassmann algebra if it contains a system of anticommuting generators
$\sigma_{j},j=1,\cdots,n$ with the property:
$\sigma_{j}\sigma_{k}+\sigma_{k}\sigma_{j}=0,\ j,k=1,2,\cdots,n$, in
particular, $\sigma_{j}^{2}=0$. The Grassmann algebra will be denote by
$G_{n}=G_{n}(\sigma_{1},\cdots,\sigma_{n})$.
The monomials $\\{e_{0},e_{i}=\sigma_{j_{1}}\cdots\sigma_{j_{n}}\\}$,
$j=(j_{1}<\cdots<j_{n})$ form a basis in the Grassmann algebra $G_{n}$, $\dim
G_{n}=2^{n}$. Then it follows that any element of $G_{n}$ is a linear
combination of monomials $\sigma_{j_{1}}\cdots\sigma_{j_{k}},\
j_{1}<\cdots<j_{k}$, that is,
$f=f_{0}+\sum_{k\geq 0}\sum_{j_{1}<\cdots<j_{k}}f_{j_{1}\cdots
j_{k}}\sigma_{j_{1}}\cdots\sigma_{j_{k}},$
where the coefficients $f_{j_{1}\cdots j_{k}}\in\mathbb{C}$.
Definition 1. Let $\Lambda=\Lambda_{0}\oplus\Lambda$ be a commutative Banach
superalgebra, then the Banach space
$\mathbb{C}_{\Lambda}^{m,n}=\Lambda_{0}^{m}\times\Lambda_{1}^{n}$
is called a superspace of dimension $(m,n)$ over $\Lambda$. In particular, if
$\Lambda_{0}=\mathbb{C}$ and $\Lambda_{1}=0$, then
$\mathbb{C}_{\Lambda}^{m,n}=\mathbb{C}^{m}.$
A function $f(\boldsymbol{x}):\mathbb{C}_{\Lambda}^{m,n}\rightarrow\Lambda$ is
said to be superdifferentiable at the point $x\in\mathbb{C}_{\Lambda}^{m,n}$,
if there exist elements $F_{j}(\boldsymbol{x})$ in $\Lambda,\ j=1,\cdots,m+n$,
such that
$f(\boldsymbol{x}+\boldsymbol{h})=f(\boldsymbol{x})+\sum_{j=1}^{m+n}\langle
F_{j}(\boldsymbol{x}),h_{j}\rangle+o(\boldsymbol{x},\boldsymbol{h}),$
where $\boldsymbol{x}=(x_{1},\cdots,x_{m},x_{m+1},\cdots,x_{n})$ with
components $x_{j},j=1,\cdots,m$ being even variable and
$x_{m+j}=\theta_{j},j=1,\cdots,n$ being Grassmann odd ones. The vector
$\boldsymbol{h}=(h_{1},\cdots,h_{m}$, $h_{m+1},\cdots,h_{m+n})$ with
$(h_{1},\cdots,h_{m})\in\Lambda_{0}^{m}$ and
$(h_{m+1},\cdots,h_{m+n})\in\Lambda_{1}^{n}$. Moreover,
$\lim_{\parallel\boldsymbol{h}\parallel\rightarrow 0}\frac{\parallel
o(\boldsymbol{x},\boldsymbol{h})\parallel}{\parallel\boldsymbol{h}\parallel}\longrightarrow
0.$
The $F_{j}(\boldsymbol{x})$ are called the super partial derivative of $f$
with respect to $x_{j}$ at the point $\boldsymbol{x}$ and are denoted,
respectively, by
$\frac{\partial f(\boldsymbol{x})}{\partial x_{j}}=F_{j}(\boldsymbol{x}),\
j=1,\cdots,m+n.$
The derivatives $\frac{\partial f(\boldsymbol{x})}{\partial x_{j}}$ with
respect to even variables $x_{j},\ j=1,2,\cdots n$ are uniquely defined. While
the derivatives $\frac{\partial f(\boldsymbol{x})}{\partial\theta_{j}}$ to odd
variables $\theta_{j}=x_{j+n},\ j=1,2,\cdots m$ are not uniquely defined, but
with an accuracy to within an addition constant
$c\sigma_{1}\cdots\sigma_{n},c\in\mathbb{C}$ from an annihilator
${}^{\perp}G_{n}$ of finite-dimensional Grassmann algebra $G_{n}$.
The super derivative also satisfies Leibniz formula
$\frac{\partial(f(\boldsymbol{x})g(\boldsymbol{x}))}{\partial
x_{j}}=\frac{\partial f(\boldsymbol{x})}{\partial
x_{j}}g(\boldsymbol{x})+(-1)^{|x_{j}||f|}f(\boldsymbol{x})\frac{\partial
g(\boldsymbol{x})}{\partial x_{j}},\ j=1,\cdots,m+n.$ $None$
Denote by $\mathcal{P}(\Lambda_{1}^{n},\Lambda)$ the set of polynomials
defined on $\Lambda_{1}^{n}$ with value in $\Lambda$. We say that a super
integral is a map $I:\mathcal{P}(\Lambda_{1}^{n},\Lambda)\rightarrow\Lambda$
satisfying the following condition is an super integral about Grassmann
variable
(1) A linearity: $I(\mu f+\nu g)=\mu I(f)+\nu I(g),\ \mu,\nu\in\Lambda,\
f,g\in\mathcal{P}(\Lambda_{1}^{n},\Lambda);$
(2) translation invariance: $I(f_{\xi})=I(f)$, where
$f_{\xi}=f(\boldsymbol{\theta}+\boldsymbol{\xi})$ for all
$\boldsymbol{\xi}\in\Lambda_{1}^{n}$,
$f\in\mathcal{P}(\Lambda_{1}^{n},\Lambda).$
We denote $I(\theta^{\varepsilon})=I_{\varepsilon}$, where $\varepsilon$
belongs to the set of multiindices
$N_{n}=\\{\boldsymbol{\epsilon}=(\varepsilon_{1},\cdots,\varepsilon_{n}),\varepsilon_{j}=0,1,\boldsymbol{\theta}^{\varepsilon}=\theta_{1}^{\varepsilon_{1}}\cdots\theta_{n}^{\varepsilon_{n}}\not\equiv
0\\}$. In the case when $I_{\varepsilon}=0,\varepsilon\in
N_{n},|\varepsilon|\leq n=n-1$, such kind of integral has the form
$I(f)=J(f)I(1,\cdots,1),$
where
$J(f)=\frac{\partial^{n}f(0)}{\partial\theta_{1}\cdots\partial\theta_{n}}.$
Since the derivative is defined with an accurcy to with an additive constant
form the annihilator ${}^{\perp}L_{n}$,
$L_{n}=\\{\theta_{1}\cdots\theta_{n},\boldsymbol{\theta}\in\Lambda_{1}^{n}\\}$,
it follows that $J:\mathcal{P}\rightarrow\Lambda/^{\perp}L_{n}$ is single-
valued mapping. This mapping also satisfies the conditions 1 and 2, and
therefore we shall call it an integral and denote
$J(f)=\int
f(\boldsymbol{\theta})d\boldsymbol{\theta}=\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots
d\theta_{n},$
which has properties:
$\displaystyle\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots d\theta_{n}=1,$
$None$ $\displaystyle\int\frac{\partial
f}{\partial\theta_{j}}d\theta_{1}\cdots d\theta_{n}=0,\ j=1,\cdots,n.$
$\displaystyle\int f(\boldsymbol{\theta})\frac{\partial
g(\boldsymbol{\theta})}{\partial\theta_{j}}d\boldsymbol{\theta}=(-1)^{1+|g|}\int\frac{\partial
f(\boldsymbol{\theta})}{\partial\theta_{j}}g(\boldsymbol{\theta})d\boldsymbol{\theta}.$
In this paper, we consider functions with two ordinary even variables $x,t$
and a Grassmann odd variable $\theta$. The associated space
$\mathbb{C}_{\Lambda}^{2,1}=\Lambda_{0}^{2}\times\Lambda_{1}$ (we may take
$\Lambda_{0}=\mathbb{R}$ or $\mathbb{C}$) is a superspace over Grassmann
algebra $G_{1}(\sigma)=G_{1,0}\oplus G_{1,1}$, whose elements have the form
$f=f_{0}+f_{1}\sigma.$
where $e=1$ is a unit, $\sigma$ is anticommuting generator. The monomials
$\\{1,\sigma\\}$ form a basis of the $G_{1}(\sigma)$, dim$G_{1}(\sigma)=2$.
Under traveling wave frame in space $\mathbb{C}_{\Lambda}^{2,1}$, the phase
variable should have the form
$\xi=\alpha x+\omega t+\theta\sigma.$
Now we consider the bilinear form of the equation (1.2). By the dependent
variable transformation
$F=\partial_{x}\ln f(x,t,\theta),$ $None$
where
$f(x,t,\theta):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{1,0}$
is a superdifferential function, the equation (1.2) is then transformed into a
bilinear form
$(S_{t}D_{t}+S_{t}D_{x}^{3})f(x,t,\theta)\cdot f(x,t,\theta)=0.$ $None$
where the Hirota bilinear differential operators $D_{x}$ and $D_{t}$ are
defined by
$\displaystyle D_{x}^{m}D_{t}^{n}f(x,t,\theta)\cdot
g(x,t,\theta)=(\partial_{x}-\partial_{x^{\prime}})^{m}(\partial_{t}-\partial_{t^{\prime}})^{n}f(x,t,\theta)g(x^{\prime},t^{\prime},\theta^{\prime})|_{x^{\prime}=x,t^{\prime}=t,\theta^{\prime}=\theta}.$
The super-Hirota bilinear operator is defined as [21]
$S_{t}^{N}f(x,t,\theta)\cdot
g(x,t,\theta)=\sum_{j=0}^{N}(-1)^{j|f|+\frac{1}{2}j(j+1)}\left[\begin{matrix}N\\\
j\end{matrix}\right]\mathfrak{D}_{t}^{N-j}f(x,t,\theta)\mathfrak{D}_{t}^{j}g(x,t,\theta),$
where the super binomial coefficients are defined by
$\left[\begin{matrix}N\\\
j\end{matrix}\right]=\left\\{\begin{matrix}\left(\begin{matrix}[N/2]\cr[j/2]\end{matrix}\right),{\rm
if}\ \ (N,j)\not=(0,1)\ \ {\rm mod}\ \ 2,\\\ 0,\ \ {\rm
otherwise}.\end{matrix}\right.$
$[k]$ is the integer part of the real number $k$ ($[k]\leq k\leq[k]+1$).
Following the Hirota bilinear theory, It is easy to find that the equation
(1.2) admits one-soliton solution (also called one-supersoliton solution)
$F_{1}=\partial_{x}\ln(1+e^{\eta}),$ $None$
with phase variable $\eta=kx-k^{3}t+\theta\zeta+\gamma$ and
$k,\gamma\in\Lambda_{0}$, $\zeta\in\Lambda_{1}$.
To apply the Hirota bilinear method for constructing periodic wave solutions
of the equation (1.2), we hope to add two odd variables $F_{0}$, $c$ and
consider a more general form than the bilinear equation (1.4)
$F=\partial_{\theta}^{-1}F_{0}+\partial_{x}\ln f(x,t,\theta),$ $None$
where
$F_{0}=F_{0}(\theta):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{0,1}$
is an odd special solution of the equation (1.2). Substituting (2.1) into
(1.2) and integrating with respect to $x$, we then get the following bilinear
form
$\displaystyle G(S_{t},D_{x},D_{t})f\cdot
f=(S_{t}D_{t}+S_{t}D_{x}^{3}+3F_{0}D_{x}^{2}+c)f\cdot f=0,$ $None$
where
$c=c(\theta,t):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{0,1}$
is an odd integration constant. For the bilinear equation (2.2), we are
interested in its multi-periodic solutions in terms of the Riemann theta
functions.
In the following, we introduce a super one-dimensional Riemann theta function
on super space $\mathbb{C}_{\Lambda}^{2,1}$ and discuss its quasi-periodicity,
which plays a central role in this paper. The Riemann theta function reads
$\vartheta(\mathbf{\xi},\varepsilon,s|\tau)=\sum_{n\in\mathbb{Z}}\exp[2\pi
i(\xi+\varepsilon)(n+s)-\pi\tau({n}+{s})^{2}].$ $None$
Here the integer value $n\in\mathbb{Z}$, $s,\varepsilon\in\mathcal{C}$, and
complex phase variables $\xi=\alpha x+\omega t+\theta\sigma+\delta$ is
dependent of even variable $x,t$ and odd $\theta$; The $\tau>0$ is called the
period matrix of the Riemann theta function. It is obvious that the Riemann
theta function (2.3) converges absolutely and superdifferentiable on
superspace $\mathbb{C}_{\Lambda}^{2,1}$. For the simplicity, in the case when
$s=\omega=0$, we denote
$\vartheta(\mathbf{\xi},\tau)=\vartheta(\mathbf{\xi},0,0|\tau).$
Definition 2. A function
$f(\xi):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{1,0}$ is
said to be quasi-periodic in $\xi=\alpha x+\omega t+\theta\sigma+\delta$ with
fundamental periods $T$, if there exist certain constants $a,b\in\Lambda_{0}$,
such that
$f(\xi+T)=f(\xi)+a\xi+b.$
An example of this is the ordinary Weierstrass zeta function, where
$\zeta(\xi+\omega)=\zeta(\xi)+\eta,$
for a fixed constant $\eta$ when $\omega$ is a period of the corresponding
Weierstrass elliptic $\wp$ function.
Proposition 2. [24] The Riemann theta function $\vartheta(\xi,\tau)$ defined
above has the periodic properties
$\displaystyle\vartheta(\xi+1+i\tau,\tau)=\exp(-2\pi
i\xi+\pi\tau)\vartheta(\xi,\tau).$ $None$
Now we turn to see the periodicity of the solution (2.4), we take
$f(x,t,\theta)$ in the bilinear equation (2.2) as
$f(x,t,\theta)=\vartheta(\xi,\tau),$
where phase variable $\xi=\alpha x+\omega t+\theta\sigma+\delta$. By using
(2.4), it is easy to see that
$\displaystyle\frac{\vartheta^{\prime}_{\xi}(\xi+i\tau,\tau)}{\vartheta(\xi+i\tau,\tau)}=-2\pi
i+\frac{\vartheta^{\prime}_{\xi}(\xi,\tau)}{\vartheta(\xi,\tau)},$
that is,
$\displaystyle\partial_{\xi}\ln\vartheta(\xi+i\tau,\tau)=-2\pi
i+\partial_{\xi}\ln\vartheta(\xi,\tau).$ $None$
According to the differential relation, we have
$F(x,t,\theta)=F(\xi)=\partial_{\theta}^{-1}F_{0}+\alpha\partial_{\xi}\ln\vartheta(\xi,\tau).$
$None$
The equations (2.5) and (2.6) demonstrate that
$F(\xi+1+i\tau)=\partial_{\theta}^{-1}F_{0}+\alpha\partial_{\xi}\ln\vartheta(\xi+1+i\tau,\tau)=-2\pi
i\alpha+F(\xi).$
Therefore the solution $F(\xi)$ is a quasi-periodic function with two
fundamental periods $1$ and $i\tau$.
In following, we establish uniform formula on the Riemannn theta function,
which will play a key role in the construction of the periodic wave solutions.
Proposition 2. [21] Suppose that $f(x,t,\theta),g(x,t,\theta)$ are super
differentiable on space $\mathbb{C}_{\Lambda}^{2,1}$. Then the Hirota bilinear
operators $D_{x},D_{t}$ and super-Hirota bilinear operator $S_{x}$ have
properties
$\displaystyle S_{x}^{2N}f\cdot g=D_{x}^{N}f\cdot g,$ $None$ $\displaystyle
D_{x}^{m}D_{t}^{n}e^{\xi_{1}}\cdot
e^{\xi_{2}}=(\alpha_{1}-\alpha_{2})^{m}(\omega_{1}-\omega_{2})^{n}e^{\xi_{1}+\xi_{2}},$
$\displaystyle S_{x}e^{\xi_{1}}\cdot
e^{\xi_{2}}=[\sigma_{1}-\sigma_{2}+\theta(\alpha_{1}-\alpha_{2})]e^{\xi_{1}+\xi_{2}},$
where $\xi_{j}=\alpha_{j}x+\omega_{j}t+\theta\sigma_{j}+\delta_{j},j=1,2$.
More generally, we have
$\displaystyle F(S_{x},D_{x},D_{t})e^{\xi_{1}}\cdot
e^{\xi_{2}}=F(\sigma_{1}-\sigma_{2}+\theta(\alpha_{1}-\alpha_{2}),\alpha_{1}-\alpha_{2},\omega_{1}-\omega_{2})e^{\xi_{1}+\xi_{2}},$
$None$
where $G(S_{t},D_{x},D_{t})$ is a polynomial about $S_{t},D_{x}$ and $D_{t}$.
This properties will be utilized later to explore the quasi-periodic wave
solutions of the equation (1.2).
Proposition 3. The Hirota bilinear operators $D_{x},D_{t}$ and super-Hirota
bilinear operator $S_{x}$ have properties when they act on the Riemann theta
functions
$D_{x}\vartheta(\xi,\varepsilon^{\prime},s^{\prime}|\tau)\cdot\vartheta(\xi,\varepsilon,s|\tau)=\sum_{\mu=0,1}\partial_{x}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,(s^{\prime}-s-\mu)/2|2\tau)|_{\xi=0}\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,(s^{\prime}+s+\mu)/2|2\tau),$
$None$
$S_{t}\vartheta(\xi,\varepsilon^{\prime},s^{\prime}|\tau)\cdot\vartheta(\xi,\varepsilon,s|\tau)=\sum_{\mu=0,1}\mathfrak{D}_{t}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,(s^{\prime}-s-\mu)/2|2\tau)|_{\xi=0}\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,(s^{\prime}+s+\mu)/2|2\tau),$
$None$
where $\sum_{\mu=0,1}$ indicates sum with respective to $\mu=0,1$.
In general, for a polynomial operator $G(S_{t},D_{x},D_{t})$ about
$S_{t},D_{x}$ and $D_{t}$, we have
$\displaystyle
G(S_{t},D_{x},D_{t})\vartheta(\xi,\tau)\cdot\vartheta(\xi,\tau)=\sum_{\mu=0,1}C(\alpha,\omega,\sigma,\mu)\vartheta(2\xi,\mu/2|2\tau),$
$None$
where
$\displaystyle\xi=\alpha x+\omega t+\theta\sigma+\gamma.$ $None$
$\displaystyle C(\alpha,\omega,\sigma,\mu|\tau)=\sum_{n\in Z}G\left\\{4\pi
i(n-\mu/2)\alpha,4\pi i(n-\mu/2)\omega,\right.$ $\displaystyle\left.4\pi
i(n-\mu/2)(\sigma+\theta\omega)\right\\}\times\exp\left[-2\pi\tau(n-\mu/2)^{2}\right].$
Proof. By using (2.7), we have
$\displaystyle\Gamma=S_{t}\vartheta(\xi,\varepsilon^{\prime},s^{\prime}|\tau)\cdot\vartheta(\xi,\varepsilon,s|\tau)$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}S_{t}\exp\\{2\pi
i(m^{\prime}+s^{\prime})(\xi+\varepsilon^{\prime})-\pi(m^{\prime}+s^{\prime})^{2}\tau\\}\cdot\exp\\{2\pi
i(m+s)(\xi+\varepsilon)-\pi(m+s)^{2}\tau\\},$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}2\pi
i(\sigma+\theta\omega)(m^{\prime}-m+s^{\prime}-s)\exp\left\\{2\pi
i(m^{\prime}+m+s^{\prime}+s)\xi-2\pi
i[(m^{\prime}+s^{\prime})\varepsilon^{\prime}+(m+s)\varepsilon]\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\pi\tau[(m^{\prime}+s^{\prime})^{2}+(m+s)^{2}]\right\\}$
$\displaystyle\stackrel{{\scriptstyle
m=l^{\prime}-m^{\prime}}}{{=}}\sum_{l^{\prime},m^{\prime}\in\mathbb{Z}}2\pi
i(\sigma+\theta\omega)(2m^{\prime}-l^{\prime}+s^{\prime}-s)\exp\left\\{2\pi
i(l^{\prime}+s^{\prime}+s)\xi-2\pi
i[(m^{\prime}+s^{\prime})\varepsilon^{\prime}\right.$ $\displaystyle\ \ \ \ \
\ \ \ \ \ \ \ \ \
\left.+(l^{\prime}-m^{\prime}+s)\varepsilon]-\pi[(m^{\prime}+s^{\prime})^{2}+(l^{\prime}-m^{\prime}+s)^{2}]\tau\right\\}$
$\displaystyle\stackrel{{\scriptstyle
l^{\prime}=2l+\mu}}{{=}}\sum_{\mu=0,1}\sum_{l,m^{\prime}\in\mathbb{Z}}2\pi
i(\sigma+\theta\omega)(2m^{\prime}-2l+s^{\prime}-s-\mu)\exp\\{4\pi
i\xi[l+(s^{\prime}+s+\mu)/2]$ $\displaystyle\ \ \ \ \ -2\pi
i[(m^{\prime}+s^{\prime})\varepsilon^{\prime}-(m^{\prime}-2l-s-\mu)\varepsilon]-\pi[(m^{\prime}+s^{\prime})^{2}+(m^{\prime}-2l-s-\mu)^{2}]\tau\\}$
Let $m^{\prime}=n+l$, and using the relations
$\displaystyle
n+l+s^{\prime}=[n+(s^{\prime}-s-\mu)/2]+[l+(s^{\prime}+s+\mu)/2],$
$\displaystyle n-l-s-\mu=[n+(s^{\prime}-s-\mu)/2]-[l+(s^{\prime}+s+\mu)/2],$
we finally obtain that
$\displaystyle\Gamma=\sum_{\mu=0,1}\left[\sum_{n\in\mathbb{Z}}4\pi
i(\sigma+\theta\omega)[n+(s^{\prime}-s-\mu)/2]\exp\\{-2\pi
i[n+(s^{\prime}-s-\mu)/2](\varepsilon^{\prime}-\varepsilon)-2\pi\tau[n+(s^{\prime}-s-\mu)/2]^{2}\\}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \times\left[\sum_{l\in\mathbb{Z}}\exp\\{2\pi
i[l+(s^{\prime}+s+\mu)/2](2\xi+\varepsilon^{\prime}+\varepsilon)-2\pi\tau[l+(s^{\prime}+s+\mu)/2]^{2}\right]$
$\displaystyle=\sum_{\mu=0,1}\mathfrak{D}_{t}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,(s^{\prime}-s-\mu)/2|2\tau)|_{\xi=0}\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,(s^{\prime}+s+\mu)/2|2\tau).$
In a similar way, we can prove the formulae (2.9). As a special case when
$\varepsilon=s=0$ of the Riemann theta function (2.3), by using (2.9) an
(2.10), we can prove the formula (2.11). $\Box$
From the formulae (2.11) and (2.12), it is seen that if the following
equations are satisfied
$C(\alpha,\omega,\sigma,\mu|\tau)=0,$
for $\mu=0,1$, then $\vartheta(\xi,\tau)$ is a solution of the bilinear
equation
$G(S_{t},D_{x},D_{t})\vartheta(\xi,\tau)\cdot\vartheta(\xi,\tau)=0.$
## 3\. Quasi-periodic waves and asymptotic properties
In this section, we consider periodic wave solutions of the equation (1.2). As
a simple case of the theta function (2.3) when $N=1,s=0$, we take
$f(x,t,\theta)$ as
$f(x,t,\theta)=\vartheta(\xi,\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in\xi-\pi
n^{2}\tau}),$ $None$
where the phase variable $\xi=\alpha x+\omega t+\theta\sigma+\delta$, and the
parameter $\tau>0$.
To let the Riemann theta function (3.1) be a solution of the bilinear equation
(2.2), according to the formula (2.11), the following equations only need to
be satisfied
$\displaystyle\sum_{n\in\mathbb{Z}}\left[-16\pi^{2}(n-\mu/2)^{2}(\sigma+\theta\omega)\omega+256\pi^{4}(n-\mu/2)^{4}(\sigma+\theta\omega)\alpha^{3}\right.$
$None$ $\displaystyle\left.\ \ \ \ \ \ \ \ \
-48\pi^{2}(n-\mu/2)^{2}\alpha^{2}F_{0}+c\right]\exp[-2\pi(n-\mu/2)^{2}\tau]=0,\
\mu=0,1.$
We introduce the notations by
$\displaystyle\lambda=e^{-\pi\tau/2},\quad\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{4n^{2}}\exp(4i\pi
n\xi),$ $None$
$\displaystyle\vartheta_{2}(\xi,\lambda)=\vartheta(2\xi,0,-1/2,2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{(2n-1)^{2}}\exp[2i\pi(2n-1)\xi].$
By using formula (3.3), the equation (3.2) can be written as a linear system
$\displaystyle\theta\vartheta_{1}^{\prime\prime}\omega^{2}+(\sigma\vartheta_{1}^{\prime\prime}+\alpha^{3}\theta\vartheta_{1}^{(4)})\omega+\vartheta_{1}c+\sigma\alpha^{3}\vartheta_{1}^{(4)}+3\alpha^{2}F_{0}\vartheta_{1}^{\prime\prime}=0,$
$None$
$\displaystyle\theta\vartheta_{2}^{\prime\prime}\omega^{2}+(\sigma\vartheta_{2}^{\prime\prime}+\alpha^{3}\theta\vartheta_{2}^{(4)})\omega+\vartheta_{2}c+\sigma\alpha^{3}\vartheta_{2}^{(4)}+3\alpha^{2}F_{0}\vartheta_{2}^{\prime\prime}=0,$
where $\omega\in\Lambda_{0}$ is even and
$c,F_{0}:\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{0,1}$ are
odd. In addition, we have denoted derivatives of $\vartheta_{j}(\xi,\lambda)$
at $\xi=0$ by simple notations
$\vartheta_{j}^{(k)}=\vartheta_{j}^{(k)}(0,\lambda)=\frac{d^{k}\vartheta_{j}(\xi,\lambda)}{d\xi^{k}}|_{\xi=0},\
\ j=1,2,k=0,1,2,\cdots$
Moreover, these functions are independent of Grassmann variable $\theta$ and
$\sigma$.
We show there existence real solutions to the system (3.4). Since
$c=c(\theta,t)$ and $F=F_{0}(\theta)$ are function of Grassmann variable
$\theta$, we can expand them in the form
$c=c_{1}+c_{2}\theta,\ \ F_{0}=f_{1}+f_{2}\theta,$ $None$
where $c_{1},f_{1}\in\Lambda_{1}$ are odd and $c_{2},f_{2}\in\Lambda_{0}$ are
even. Substituting (3.5) into (3.4) leads to
$\displaystyle(\sigma\vartheta_{1}^{\prime\prime}\omega+\vartheta_{1}c_{1}+\sigma\alpha^{3}\vartheta_{1}^{(4)}+3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{1})+\theta(3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{2}+\vartheta_{1}c_{2}+\vartheta_{1}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{1}^{(4)}\omega)=0,$
$None$
$\displaystyle(\sigma\vartheta_{2}^{\prime\prime}\omega+\vartheta_{2}c_{1}+\sigma\alpha^{3}\vartheta_{2}^{(4)}+3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{1})+\theta(3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{2}+\vartheta_{2}c_{2}+\vartheta_{2}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{2}^{(4)}\omega)=0,$
where $\omega,\ c_{1},\ c_{2},f_{1}$ and $f_{2}$ are parameters to be
determined.
Since $\theta$ is a Grassmann variable, the system (3.6) will be satisfied
provided that
$\displaystyle\sigma\vartheta_{1}^{\prime\prime}\omega+\vartheta_{1}c_{1}+\sigma\alpha^{3}\vartheta_{1}^{(4)}+3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{1}=0,$
$None$
$\displaystyle\sigma\vartheta_{2}^{\prime\prime}\omega+\vartheta_{2}c_{1}+\sigma\alpha^{3}\vartheta_{2}^{(4)}+3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{1}=0$
and
$\displaystyle
3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{2}+\vartheta_{1}c_{2}+\vartheta_{1}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{1}^{(4)}\omega=0,$
$None$ $\displaystyle
3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{2}+\vartheta_{2}c_{2}+\vartheta_{2}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{2}^{(4)}\omega=0.$
In the systems (3.7) and (3.8), it is obvious that vectors
$(\vartheta_{1},\vartheta_{2})^{T}$ and
$(\vartheta_{1}^{\prime\prime},\vartheta_{2}^{\prime\prime})^{T}$ are linear
independent, and $(\vartheta_{1}^{(4)},\vartheta_{2}^{(4)})^{T}\not=0$.
Therefore the system (3.7) admits a solution
$\displaystyle\omega=-3\beta\alpha^{2}+\frac{(\vartheta_{2}^{(4)}\vartheta_{1}-\vartheta_{1}^{(4)}\vartheta_{2})\alpha^{3}}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}}\in\Lambda_{0},\
\ \
c_{1}=\frac{(\vartheta_{2}^{(4)}\vartheta_{1}^{\prime\prime}-\vartheta_{1}^{(4)}\vartheta_{2}^{\prime\prime})\alpha^{3}\sigma}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}}\in\Lambda_{1},$
$None$
here we have taken $f_{1}=\beta\sigma,\ \beta\in R$ for simplicity, and other
parameters $\alpha,\tau,\sigma$, $\beta$ are free.
By using (3.9) and solving system (3.8), we obtain that
$\displaystyle f_{2}=-\beta\omega\in\Lambda_{0},\ \ \
c_{2}=\frac{(\vartheta_{1}^{(4)}\vartheta_{2}^{\prime\prime}-\vartheta_{2}^{(4)}\vartheta_{1}^{\prime\prime})\alpha^{3}\omega}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}}\in\Lambda_{0}.$
$None$
Noting that $\int\theta\ d\theta=1$ and $\ \int d\theta=0$, we have
$\partial^{-1}F_{0}=\int(\beta\sigma-\beta\omega\theta)\
d\theta=-\beta\omega.$
In this way, we indeed can get an explicit periodic wave solution of the
equation (1.12)
$F=-\beta\omega+\partial_{x}\ln\vartheta(\xi,\tau),$ $None$
with the theta function $\vartheta(\xi,\tau)$ given by (3.1) and parameters
$\omega$, $c_{1},\ c_{2}$ by (3.9) and (3.10), while other parameters
$\alpha,\sigma,\tau,\delta,\ \beta$ are free. Among them, the three parameters
$\alpha,\sigma$ and $\tau$ completely dominate a periodic wave. In summary,
the periodic wave (3.11) is real-valued and bounded for all complex variables
$(x,t,\theta)$. It is one-dimensional, i.e. there is a single phase variable
$\xi$, and has two fundamental periods $1$ and $i\tau$ in phase variable
$\xi$.
In the following, we further consider asymptotic properties of the periodic
wave solution. Interestingly, the relation between the one-periodic wave
solution (3.11) and the one-super soliton solution (1.5) can be established as
follows.
Theorem 1. Suppose that the $\omega\in\Lambda_{0}$ and $c\in\Lambda_{1}$ are
given given by (3.5), (3.9) and (3.10). For the one-periodic wave solution
(3.11), we let
$\alpha=\frac{k}{2\pi i},\ \ \sigma=\frac{\zeta}{2\pi i},\ \
\delta=\frac{\gamma+\pi\tau}{2\pi i},$ $None$
where the $k,\zeta$ and $\gamma$ are the same as those in (1.5). Then we have
the following asymptotic properties
$c\longrightarrow 0,\ \ \xi\longrightarrow\frac{\eta+\pi\tau}{2\pi i},\ \
\vartheta(\xi,\tau)\longrightarrow 1+e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$ $None$
In other words, the periodic solution (3.11) tends to the one-soliton solution
(1.5) under a small amplitude limit , that is,
$F\longrightarrow F_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$ $None$
Proof. Here we will directly use the system (3.4) to analyze asymptotic
properties of one-periodic solution, which is more simple and effective than
our original method by solving the system [16]-[20] . Since the coefficients
of system (3.4) are power series about $\lambda$, its solution
$(\omega,c)^{T}$ also should be a series about $\lambda$.
We explicitly expand the coefficients of system (3.4) as follows
$\displaystyle\vartheta_{1}(0,\lambda)=1+2\lambda^{4}+\cdots,\quad\vartheta_{1}^{\prime\prime}(0,\lambda)=-32\pi^{2}\lambda^{4}+\cdots,$
$None$
$\displaystyle\vartheta_{1}^{(4)}(0,\lambda)=512\pi^{4}\lambda^{4}+\cdots,\ \
\vartheta_{2}(0,\lambda)=2+2\lambda^{8}+\cdots$
$\displaystyle\vartheta_{2}^{\prime\prime}(0,\lambda)=-8\pi^{2}-72\pi^{2}\lambda^{8}+\cdots,\
\vartheta_{2}^{(4)}(0,\lambda)=32\pi^{4}+2592\pi^{4}\lambda^{8}+\cdots.$
Let the solution of the system (3.4) be in the form
$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$None$ $\displaystyle
c=b_{0}+b_{1}\lambda+b_{2}\lambda^{2}+\cdots=b_{0}+o(\lambda),$
where $\omega_{j}\in\Lambda_{0},\ b_{j}\in\Lambda_{1},\ j=0,1,2\cdots$
Substituting the expansions (3.11) and (3.12) into the system (3.5) and
letting $\lambda\longrightarrow 0$, we immediately obtain the following
relations
$\displaystyle b_{0}=0,\ \
-8\pi^{2}\sigma\omega_{0}+2b_{0}+32\pi^{4}\sigma\alpha^{3}=0,$
which has a solution
$b_{0}=0,\ \ w_{0}=4\pi^{2}\alpha^{3}.$
Then from the relations (3.12) and (3.16), we have
$c\longrightarrow 0,\ \ 2\pi i\omega\longrightarrow
8\pi^{3}i\alpha^{3}=-k^{3},\ \ {\rm as}\ \ \lambda\rightarrow 0,$
and thus
$\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=kx+2\pi i\omega
t+\theta\zeta+\gamma$ $None$ $\displaystyle\quad\longrightarrow
kx-k^{3}t+\theta\zeta+\gamma=\eta,\ \ {\rm as}\ \ \lambda\rightarrow 0,$
It remains to show that the one-periodic wave (3.11) possesses the same form
with the one-soliton solution (1.5) under the limit $\lambda\rightarrow 0$.
For this purpose, we first expand the Riemann theta function
$\vartheta(\xi,\tau)$ in the form
$\vartheta(\xi,\tau)=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots.$
By using the (3.12) and (3.17), it follows that
$\displaystyle\vartheta(\xi,\tau)=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow 1+e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0,$
which implies (3.13) and (3.14). Therefore we conclude that the one-periodic
solution (3.11) just goes to the one-soliton solution (1.5) as the amplitude
$\lambda\rightarrow 0$. $\square$
## 4\. Discussion on the conditions of $N$-periodic wave solutions
In this section, we consider condition for $N$-periodic wave solutions of the
equation (1.2). The theta function is taken the form
$\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=\vartheta(\xi_{1},\cdots,\xi_{N},\tau)=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}\exp\\{2\pi
i<\boldsymbol{\xi},\boldsymbol{n}>-\pi<\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}>\\},$
$None$
where $\boldsymbol{n}=(n_{1},\cdots,n_{N})^{T}\in\mathbb{Z}^{N},\ \
\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T}\in\mathcal{C}^{N},\ \
\xi_{i}=\alpha_{j}x+\omega_{j}t+\theta\sigma_{j}+\delta_{j},\ \ j=1,\cdots,N$,
$\tau$ is a ${N\times N}$ symmetric positive definite matrix.
To make the theta function (4.1) satisfy the bilinear equation (2.2), we
obtain that according to the formula (2.11)
$\displaystyle\sum_{\mu=0,1}\ \
\sum_{n_{1},\cdots,n_{N}=-\infty}^{\infty}G\left\\{4\pi
i\sum_{j=1}^{N}(n_{j}-\mu_{j}/2)\alpha_{j},4\pi
i\sum_{j=1}^{N}(n_{j}-\mu_{j}/2)\omega_{j},\right.$ $None$
$\displaystyle\left.4\pi
i\sum_{j=1}^{N}(n_{j}-\mu_{j}/2)(\sigma_{j}+\theta\omega_{j})\right\\}\times\exp\left[-2\pi\sum_{j,k=1}^{N}(n_{j}-\mu_{j}/2)\tau_{jk}(n_{k}-\mu_{k}/2)\right]=0.$
Now we consider the number of equation and some unknown parameters. Obviously,
in the case of supersymmetric equations, the number of constraint equations of
the type (4.2) is $2^{N+1}$, which is two times of the constraint equations
needed in the case of ordinary equations [16]-[20] . On the other hand we have
parameters
$\tau_{ij}=\tau_{ji},c_{1},c_{2},f_{1},f_{2},\alpha_{i},\omega_{i}$, whose
total number is $\frac{1}{2}N(N+1)+2N+4$. Among them, $2N$ parameters
$\tau_{ii},\omega_{i}$ are taken to be the given parameters related to the
amplitudes and wave numbers (or frequencies) of $N$-periodic waves;
$\frac{1}{2}N(N+1)$ parameters $\tau_{ij}$ implicitly appear in series form,
which is general can not to be solved explicit. Hence, the number of the
explicit unknown parameters is only $N+4$. The number of equations is larger
than the unknown parameters in the case when $N\geq 2$. In this paper, we
consider one-periodic wave solution of the equation (1.2), which belongs to
the cases when $N=1$. There are still certain difficulties in the calculation
for the case $N\geq 2$, which will be considered in our future work.
## Acknowledgment
I would like to express my special thanks to the referee for constructive
suggestions which have been followed in the present improved version of the
paper. The work described in this paper was supported by grants from the
Research Grants Council of Hong Kong (No.9041473), the National Science
Foundation of China (No.10971031), Shanghai Shuguang Tracking Project
(No.08GG01) and Innovation Program of Shanghai Municipal Education Commission
(No.10ZZ131).
## References
* [1] M. Ito, J. Phys. Soc. Jpn. 49 (1980) 771
* [2] X. B. Hu and Y Li: J. Phys. A 24 (1991) 1979.
* [3] M. Jimbo and T. Miwa: Publ. RIMS Kyoto Univ. 19 (1983) 943.
* [4] V. G. Drinfeld and V. Sokolov: J. Sov. Math. 30 (1985) 1975.
* [5] Q. P. Liu: Phys. Lett. A 277 (2000), 31.
* [6] S. Q. Liu, X. B. Hu and Q. P. Liu, J. Phys. Soc. Jpn 75 (2006), 064004
* [7] R. Hirota and J. Satsuma: Prog. Theor. Phys. 57 (1977) 797.
* [8] R. Hirota: Direct methods in soliton theory (Springer-verlag, Berlin, 2004).
* [9] X. B. Hu: P. A. Clarkson, J. Phys. A 28 (1995) 5009.
* [10] X. B. Hu: C X Li, J. J. C. Nimmo, G. F. Yu, J. Phys. A, 38 (2005) 195.
* [11] R Hirota and Y. Ohta: J. Phys. Soc. Jpn. 60 (1991) 798\.
* [12] D. J. Zhang: J. Phys. Soc. Jpn. 71 (2002) 2649.
* [13] K Sawada and T Kotera: Prog. Theor. Phys. 51 (1974) 1355.
* [14] A. Nakamura, J. Phys. Soc. Jpn. 47, 1701-1705 (1979).
* [15] A. Nakamura, J. Phys. Soc. Jpn. 48(1980), 1365.
* [16] H. H. Dai, E. G. Fan and X. G. Geng, arxiv.org/pdf/nlin/0602015
* [17] Y. C. Hon and E. G. Fan, Modern Phys Lett B, 22 (2008), 547.
* [18] E. G. Fan and Y. C. Hon, Phys Rev E, 78 (2008), 036607.
* [19] W. X. Ma, R. G. Zhou and L. Gao, Modern Phys. Lett. A, 21 (2009), 1677.
* [20] E. G. Fan, J. Phys A, 42 (2009), 095206.
* [21] A. S. Carstea, Nonlinearity 13 (2000), 1645.
* [22] V. S. Vladimirov, Theor. Math. Phys. 59 (1984), 3.
* [23] V. S. Vladimirov, Theor. Math. Phys. 60 (1984), 169\.
* [24] D. Mumford, Tata Lectures on Theta II Progress in Mathmatics, Vol. 43 ( Boston: Birkhäuser, 1984).
|
arxiv-papers
| 2010-01-10T10:14:48 |
2024-09-04T02:49:07.639661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Engui Fan, Y. C. Hon",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1001.1492"
}
|
1001.1579
|
# PT-Rotations, PT-Spherical Harmonics and the PT-Hydrogen Atom
Juan M. Romero jromero@correo.cua.uam.mx R. Bernal-Jaquez
rbernal@correo.cua.uam.mx O. González-Gaxiola ogonzalez@correo.cua.uam.mx
Departamento de Matemáticas Aplicadas y Sistemas
Universidad Autónoma Metropolitana-Cuajimalpa, México 01120 DF, México
###### Abstract
We have constructed a set of non-Hermitian operators that satisfy the
commutation relations of the $SO(3)$-Lie algebra. It is shown that this
operators generate rotations in the configuration space and not in the
momentum space but in a modified non-Hermitian momentum space. This generators
are related with a new type of spherical harmonics that result to be
$\mathcal{PT}$-orthonormal. Additionally, we have shown that this operators
represent conserved quantities for a non-Hermitian Hamiltonian with an
additional complex term. As a particular case, the solutions of the
corresponding $\mathcal{PT}$-Hydrogen atom that includes a complex term are
obtained, and it is found that a non-Hermitian Runge-Lenz vector is a
conserved quantity. In this way, we obtain a set of non-Hermitian operators
that satisfy the $SO(4)$-Lie algebra.
###### pacs:
11.30.Er, 11.30.-j, 03.65.-w.
## I Introduction
Quantum mechanics is considered as one the most solid and well established
theories in physics. Different experiments have corroborated its predictions.
However, at a theoretical level there are different facts that make us think
that it could be necessary to modify or extend this theory. For example, it
has not been possible to find a consistent quantum-mechanical formulation of
general relativity, then quantum theory maybe modified in order to make it
compatible with general relativity.
As it is well known, quantum mechanics has been formulated in terms of
Hermitian operators in order to obtain real spectra. However, it has become
clear that Hermiticity is not a necessary condition to obtain real spectra.
This opens the possibility for quantum mechanics to be extended using non-
Hermitian operators, this is the so called $\mathcal{PT}$-symmetry theory, see
review bender-0:gnus and its references.
The $\mathcal{PT}$-version of quantum mechanics has strongly attracted
attention because it gives a way to deal with some problems that are out of
the scope of conventional quantum mechanics. For example, we can solve certain
kind of problems in which the potentials are given by complex-valued functions
and whose spectra results to be real Ben-1:gnus . In the same way, using this
formulation it has been possible to achieve a consistent quantization of a
system with high order derivatives: the so called the Pais-Uhlenbeck
oscillator model bender-uhlenbeck:gnus . This opens the possibility to
construct in a consistent way high order derivatives field theories. This fact
is important because different theories with high order derivatives have been
recently proposed, for example, in extensions of the standard model
pospelov:gnus , in the noncommutative spaces szabo:gnus and gravity theories
gravedad:gnus . In this way, it becomes possible that an extension of
$\mathcal{PT}$-symmetry theory applied to field theory can give a consistent
description of these systems.
There is not a finished version of the theory, however, a growing number of
themes are under study in the $\mathcal{PT}$-framework, some of them can be
found das:gnus . An aspect that has been scarcely treated in the
$\mathcal{PT}$-context is the study of symmetries and conserved quantities. In
this work, we will study some aspects of this topic. We will obtain a non-
Hermitian set of operators that satisfy the commutation relations of the Lie
$SO(3)$ rotation group. It will be shown that these operators generate
rotations in the configuration space $x_{i}$, and not in the momentum space
$\vec{p}=-i\nabla$ but in a modified non-Hermitian momentum space
$\vec{p}_{f}=\vec{p}+i\vec{\nabla{f}}$, originally considered by Dirac in his
seminal book Dirac:gnus . Also, we will show that the Casimir of the algebra
has real spectra and that its eigenfunctions, under the $\mathcal{PT}$-inner
product, form a complete basis. This eigenfunctions will be called
$\mathcal{PT}$-spherical harmonics.
Additionally we will study a central potential Hamiltonian with an additional
complex term. It will be shown that the conventional angular momentum is not a
conserved quantity anymore and we will have a modified non-Hermitian angular
momentum operator. As a particular case, we obtain the solutions of the
corresponding $\mathcal{PT}$-Hydrogen atom that includes a complex term, and
it will be found that a non-Hermitian Runge-Lenz vector is a conserved
quantity. Then we will have the non-Hermitian generators of the $SO(4)$-Lie
algebra.
This work is organized as follows: section II make a brief review of
$\mathcal{PT}$-theory and of conventional spherical harmonics, in section III
we study the $\mathcal{PT}$-rotations, in section IV we study the completeness
relation and some examples, section V is devoted to the study of symmetry
transformations, in section VI we deal with the central potential problem, in
section VII we will study the Hydrogen atom and at last we conclude with a
summary of our results.
## II $\mathcal{PT}$-Symmetry and Spherical Harmonics
In this section, a review of some well known facts of $\mathcal{PT}$-symmetry
theory and spherical harmonics is made before consider the
$\mathcal{PT}$-transformed version of this functions
### II.1 $\mathcal{PT}$-Inner product
$\mathcal{PT}$-theory considers the transformations under the parity operator
$\mathcal{P}$ and the time reversal operator $\mathcal{T}$. Under the
$\mathcal{P}$-operator we have the transformation
$\displaystyle x,y,z\to-x,-y,-z$ (1)
and under $\mathcal{T}$
$\displaystyle i\to-i$ (2)
In this way, any function $f(\vec{x})$ can be transform as
$\displaystyle\mathcal{PT}\left(f(\vec{x})\right)=f^{*}(-\vec{x}).$ (3)
Note that, in spherical coordinates $\mathcal{P}$ produces the transformation
$\displaystyle(r,\theta,\varphi)\to(r,\pi-\theta,\varphi+\pi),$ (4)
under $\mathcal{P}$ a $f$ function transforms as
$\displaystyle\mathcal{P}\left(f(r,\theta,\varphi)\right)=f(r,\pi-\theta,\varphi+\pi),$
(5)
therefore
$\displaystyle\mathcal{PT}\left(f(r,\theta,\varphi)\right)=f^{*}(r,\pi-\theta,\varphi+\pi).$
(6)
Now, the $\mathcal{PT}$-inner product is defined as
$\displaystyle\langle f|g\rangle=\int d\vec{x}[\mathcal{PT}f(x)]g(x).$ (7)
This expressions will be used in sections bellow. An exhaustive study of the
$\mathcal{PT}$-theory can be found in bender-0:gnus .
### II.2 Spherical Harmonics
The angular momentum components are given by the Hermitian operators
landau:gnus
$\displaystyle L_{x}=-i\left(y\frac{\partial}{\partial
z}-z\frac{\partial}{\partial
y}\right),\;L_{y}=-i\left(z\frac{\partial}{\partial
x}-x\frac{\partial}{\partial
z}\right),\;L_{z}=-i\left(x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}\right).$
Its algebra is given by
$\displaystyle\left[L_{x},L_{y}\right]=iL_{z},\qquad\left[L_{z},L_{x}\right]=iL_{y},\qquad\left[L_{y},L_{z}\right]=iL_{x}.$
(8)
Considering $L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$ and Eq. (8), we have
$\displaystyle\left[L^{2},L_{x}\right]=\left[L^{2},L_{y}\right]=\left[L^{2},L_{z}\right]=0.$
(9)
An important equation in mathematical-physics is the eigenvalue equation
$L^{2}Y_{lm}=l(l+1)Y_{lm},l=0,1,2\cdots,$ that in spherical coordinates is
written
$\displaystyle L^{2}Y_{lm}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle-\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
Y_{lm}(\theta,\varphi)}{\partial\theta}\right)+\frac{1}{\sin\theta^{2}}\frac{\partial^{2}Y_{lm}(\theta,\varphi)}{\partial\varphi^{2}}\right]$
(10) $\displaystyle=$ $\displaystyle l(l+1)Y_{lm}(\theta,\varphi).$
If $\varphi\in(0,2\pi)$ and $\theta\in(0,\pi),$ the solutions of this equation
are given by the spherical harmonics fesbach:gnus
$\displaystyle Y_{lm}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}}e^{im\varphi}P_{l}^{m}(\cos\theta),\qquad-l\leq
m\leq l,$ (11)
with
$\displaystyle l=0,1,2,3\cdots,\qquad P_{l}^{m}(u)$ $\displaystyle=$
$\displaystyle(-)^{m}(1-u^{2})^{\frac{m}{2}}\frac{d^{m}}{du^{m}}P_{l}(u),\qquad
P_{l}(u)=\frac{1}{2^{l}l!}\frac{d^{l}}{du^{l}}\left(u^{2}-1\right)^{l}$ (12)
where $P_{l}^{m}(u)$ denoted the associated Legendre polynomials and
$P_{l}(u)$ the Legendre polynomials respectively. The spherical harmonics
satisfy the orthonormality relation
$\displaystyle<Y_{l^{\prime}m^{\prime}}(\theta,\varphi)|Y_{lm}(\theta,\varphi)>=\int
d\Omega
Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi)=\delta_{mm^{\prime}}\delta_{l^{\prime}l}.$
(13)
Given that the spherical harmonics constitute an orthonormal basis, we can
write any function $F(\theta,\varphi)$ as a linear combination of them, that
is
$\displaystyle F(\theta,\varphi)=\sum_{l\geq
0}\sum_{m=-l}^{l}C_{lm}Y_{lm}(\theta,\varphi).$ (14)
Using Eq. (13), we find
$\displaystyle C_{lm}=\int d\Omega
Y^{*}_{lm}(\theta,\varphi)F(\theta,\varphi).$ (15)
Substituting $C_{lm}$ in Eq. (14) and making the change of variables
$u^{\prime}=\cos\theta^{\prime},u=\cos\theta$ we obtain
$\displaystyle
F(\theta,\varphi)=\int_{0}^{2\pi}d\varphi\int_{-1}^{1}du^{\prime}F(u^{\prime},\varphi^{\prime})\left(\sum_{l\geq
0}\sum_{m=-l}^{l}Y^{*}_{lm}(u^{\prime},\varphi^{\prime})Y_{lm}(u,\varphi)\right).$
Therefore, the expression inside the parenthesis must be equal to
$\delta(\varphi-\varphi^{\prime})\delta(u-u^{\prime}),$ that is
$\displaystyle\sum_{l\geq
0}\sum_{m=-l}^{l}Y^{*}_{lm}(\theta^{\prime},\phi^{\prime})Y_{lm}(\theta,\phi)=\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime}),$
(16)
this expression is called completeness relation.
Under the parity operator the spherical harmonics transform as
$\displaystyle\mathcal{P}\left(Y_{lm}(\theta,\varphi)\right)=Y_{lm}(\pi-\theta,\varphi+\pi)=(-)^{l}Y_{lm}(\theta,\varphi),$
(17)
and under the $\mathcal{PT}$-operator, we have
$\displaystyle\mathcal{PT}\left(Y_{lm}(\theta,\varphi)\right)=Y^{*}_{lm}(\pi-\theta,\varphi+\pi)=(-)^{l}Y^{*}_{lm}(\theta,\varphi).$
(18)
This results will be used below.
## III $\mathcal{PT}$ and Rotations
Given any $f=f(r,\theta,\varphi)$, we can define
$\displaystyle L_{fi}=e^{f}L_{i}e^{-f}.$ (19)
In general $L_{fi}$ is a non-Hermitian operator, however
$\displaystyle\left[L_{fx},L_{fy}\right]=iL_{fz},\qquad\left[L_{fz},L_{fx}\right]=iL_{fy},\qquad\left[L_{fy},L_{fz}\right]=iL_{fx},$
(20)
that we can identify as the $SO(3)$-Lie algebra commutation relations.
Considering $L_{f}^{2}=L_{fx}^{2}+L_{fy}^{2}+L_{fz}^{2}$ and Eq. (20), we have
$\displaystyle\left[L_{f}^{2},L_{fi}\right]=0,$ (21)
therefore, we have the same algebra as the one satisfied by $L_{i}.$ Besides
$\displaystyle
L_{f}^{2}Y_{flm}(\theta,\varphi)=l(l+1)Y_{flm}(\theta,\varphi),\qquad
Y_{flm}(\theta,\varphi)=e^{f}Y_{lm}(\theta,\varphi),$ (22)
that will be called $\mathcal{PT}$-spherical harmonics. In this case, the
$\mathcal{PT}$-inner product is given by
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$ $\displaystyle\int
d\Omega\mathcal{PT}\left(Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)\right)Y_{flm}(\theta,\varphi).$
(23)
Under a $\mathcal{PT}$-transformation, we have
$\displaystyle\mathcal{PT}(Y_{flm})$ $\displaystyle=$ $\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)}(-)^{l}Y^{*}_{lm}(\theta,\varphi).$ (24)
therefore we can write
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$ $\displaystyle(-)^{l}\int d\Omega
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi).$
It becomes clear that, under this inner product not any function $f$ allows
the set $Y_{flm}$ to be an orthogonal set. However, if the following condition
is fulfilled
$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=\lambda,\qquad\lambda={\rm
const}$ (25)
then we have
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}\lambda\delta_{l^{\prime}l}\delta_{m^{\prime}m}.$ (26)
In this way, the spherical harmonics $Y_{flm}(\theta,\varphi)$ are orthogonal
under the $\mathcal{PT}$-inner product only if Eq. (25) is satisfy. It is
worthy to mention that due to the parity of the wave functions, in some
$\mathcal{PT}$-symmetry systems the following orthogonality relations
$\displaystyle\langle\phi_{m}|\phi_{n}\rangle=(-1)^{n}\delta_{m,n},$ (27)
may be obtained bender-0:gnus .
## IV Completeness Relation
Using the $\mathcal{PT}$-spherical harmonics $Y_{flm}(\theta,\varphi)$, we can
have the expansion
$\displaystyle F(\theta,\varphi)=\sum_{l\geq
0}\sum_{m=-l}^{l}a_{lm}Y_{flm}(\theta,\varphi).$ (28)
Appeling to the orthonormality relations Eq. (26), we find
$\displaystyle
a_{lm}=\frac{(-)^{l}}{\lambda}<Y_{flm}(\theta,\varphi)|F(\theta,\varphi)>_{f}=\frac{(-)^{l}}{\lambda}\int
d\Omega\mathcal{PT}\left(Y_{flm}(\theta,\varphi)\right)F(\theta,\varphi).$
(29)
substituting this result into Eq.(28), we obtain
$\displaystyle F(\theta,\varphi)$ $\displaystyle=$ $\displaystyle\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{(-)^{l}}{\lambda}\int
d\Omega^{\prime}\mathcal{PT}\left(Y_{flm}(\theta^{\prime},\varphi^{\prime})\right)F(\theta^{\prime},\varphi^{\prime})Y_{flm}(\theta,\varphi)$
$\displaystyle=$ $\displaystyle\int
d\Omega^{\prime}F(\theta^{\prime},\varphi^{\prime})\left[\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{(-)^{l}}{\lambda}\mathcal{PT}\left(Y_{flm}(\theta^{\prime},\varphi^{\prime})\right)Y_{flm}(\theta,\varphi)\right]$
$\displaystyle=$ $\displaystyle\int
d\Omega^{\prime}F(\theta^{\prime},\varphi^{\prime})\left[\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{e^{f^{*}(r,\pi-\theta^{\prime},\varphi^{\prime}+\pi)+f(r,\theta,\varphi)}}{\lambda}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi)\right],$
therefore
$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{e^{f^{*}(r,\pi-\theta^{\prime},\varphi^{\prime}+\pi)+f(r,\theta,\varphi)}}{\lambda}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(30)
This is the completeness relation for the $\mathcal{PT}$-spherical harmonics.
A similar completeness relation is found for different systems in
$\mathcal{PT}$-quantum mechanics bender-0:gnus .
### IV.1 Examples on Completeness
In this subsection we will see some examples of functions that satisfy Eq.
(25).
Let us suppose that $a$ is a real constant, then we can define
$\displaystyle f(r,\theta,\varphi)=a\theta$ (31)
therefore
$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{a(\pi-\theta)+a\theta}=e^{a\pi},$
(32)
where
$\displaystyle\lambda=e^{a\pi}.$ (33)
In this case, the orthonormality relations are given by
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}e^{a\pi}\delta_{l^{\prime}l}\delta_{m^{\prime}m}$ (34)
and the completeness relation is given by
$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{a(\theta-\theta^{\prime})}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(35)
Now consider
$\displaystyle f(r,\theta,\varphi)=ai\sin\theta,$ (36)
then we obtain
$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{-ai\sin\theta+ia\sin\theta}=1=\lambda$
(37)
with the orthonormality relations
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}\delta_{l^{\prime}l}\delta_{m^{\prime}m}.$ (38)
In this case the completeness relation is given by
$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{ai(\sin\theta-\sin\theta^{\prime})}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(39)
Using
$\displaystyle f(r,\theta,\varphi)=a\cos\theta,$ (40)
we have
$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{-a\cos\theta+a\cos\theta}=\lambda=1$
(41)
in this case the orthogonality relations are given by
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}\delta_{l^{\prime}l}\delta_{m^{\prime}m},$ (42)
and the completeness relation is given by
$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{a(\cos\theta-\cos\theta^{\prime})}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(43)
With the function
$\displaystyle f(r,\theta,\varphi)=ai\varphi$ (44)
we obtain
$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{-ai(\varphi+\pi)+ia\varphi}=e^{-ia\pi}=\lambda,$
(45)
with the orthogonality relation given by
$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}e^{-ia\pi}\delta_{l^{\prime}l}\delta_{m^{\prime}m},$ (46)
and the completeness relation
$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{ai(\varphi-\varphi^{\prime})}Y^{*}_{lm}(\theta^{\prime}.\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(47)
Considering the above examples, it is clear that many functions satisfy Eq.
(25).
## V Symmetry Transformations
Consider $A,B,C$ that satisfy the following commutation relations
$\displaystyle[A,B]=C.$ (48)
Transforming the $A,B,C$ operators, we obtain
$A_{f}=e^{f}Ae^{-f},B_{f}=e^{f}Be^{-f}$ y $C_{f}=e^{f}Ce^{-f},$ then
$\displaystyle[A_{f},B_{f}]=C_{f}.$ (49)
We know that $[L_{i},x_{j}]=i\epsilon_{ijk}x_{k}$. If we consider the
transformation $x_{fi}=e^{f}x_{i}e^{-f}=x_{i}$, we arrive to
$\displaystyle[L_{fi},x_{j}]=i\epsilon_{ijk}x_{k},$ (50)
therefore the operators $L_{fi}$ generate infinitesimal rotations in the space
$x_{i}$. However, as $p_{i}=-i\frac{\partial}{\partial x^{i}},$ then
$\displaystyle[L_{fi},p_{j}]\not=i\epsilon_{ijk}p_{k},$ (51)
and we say that the operators $L_{fi}$ does not generate infinitesimal
rotations in the space $p_{i}$. Now, if $p_{fi}$ is given by
$p_{fi}=e^{f}p_{i}e^{-f}$, then we have
$\displaystyle[L_{fi},p_{fj}]=i\epsilon_{ijk}p_{fk}.$ (52)
Note that
$\displaystyle\vec{p}_{f}=e^{f}\vec{p}e^{-f}=\vec{p}+i\vec{\nabla}f.$ (53)
this operator was studied by Dirac in his seminal book Dirac:gnus . If the
Hamiltonian operator is given by
$\displaystyle H=\frac{\vec{p}^{\;2}}{2m}+V(r),$ (54)
then
$\displaystyle[L_{i},H]=0.$ (55)
Defining $H_{f}$ by
$\displaystyle H_{f}=e^{f}He^{-f},$ (56)
we have that, in general
$\displaystyle[L_{i},H_{f}]\not=0,$ (57)
therefore the angular momentum $L_{i}$ is not a conserved quantity for
Hamiltonians of the form $H_{f}.$ However
$\displaystyle[L_{fi},H_{f}]=0,$ (58)
then the modified angular momentum $L_{fi}$ is conserved. Note that the
Hamiltonian $H_{f}$ is given by
$\displaystyle
H_{f}=\frac{\vec{p}_{f}^{\;2}}{2m}+V(r)=\frac{m}{2}\left(\vec{p}+i\vec{\nabla}f\right)^{2}+V(r)$
(59)
In the next section we will consider one important example.
## VI The Central Problem
Consider the Hamiltonian
$\displaystyle H=\frac{m}{2}\vec{p}^{\;2}+V(x,y,z),$ (60)
then
$\displaystyle H_{f}$ $\displaystyle=$ $\displaystyle
e^{f}He^{-f}=\frac{m}{2}\left(\vec{p}+i\vec{\nabla}f\right)^{2}+V(x,y,z)$ (61)
$\displaystyle=$
$\displaystyle\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}+(\nabla^{2}f)-\left(\vec{\nabla}f\right)^{2}\right)+V(x,y,z),$
that is a non-Hermitian Hamiltonian. If the potential is given by
$\displaystyle
V(x,y,z)=-\frac{m}{2}\left(\nabla^{2}f-\left(\vec{\nabla}f\right)^{2}\right)+U(x,y,z),$
(62)
then we can write
$\displaystyle H_{f}$ $\displaystyle=$
$\displaystyle\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}\right)+U(x,y,z).$
(63)
This kind of Hamiltonians naturally arise in some statistical models
reichl:gnus .
Note that if $\psi$ is an eigenfunction in the equation
$\displaystyle H\psi=E\psi$ (64)
then we can define the $f$-states $\psi_{f}=e^{f}\psi$ that satisfy
$\displaystyle H_{f}\psi_{f}=E\psi_{f}.$ (65)
It follows that although $H_{f}$ is a non-Hermitian operator it does has a
real spectrum.
As an example, let us consider the central potential problem $V(r)$ whose
Schrodinger equation is given by landau:gnus
$\displaystyle H\psi=\left(\frac{m}{2}\vec{p}^{\;2}+V(r)\right)\psi=E\psi$
(66)
and whose solutions
$\displaystyle\psi_{E}(r,\theta,\varphi)=\phi_{E}(r)Y_{lm}(\theta,\varphi)$
(67)
satisfy the orthogonality relations
$\displaystyle<\psi_{E^{\prime}}(r,\theta,\varphi)|\psi_{E}(r,\theta,\varphi)>=\int
drr^{2}d\Omega\psi^{*}_{E^{\prime}}(r,\theta,\varphi)\psi_{E}(r,\theta,\varphi)=\delta_{EE^{\prime}}.$
(68)
Then the solutions of the equation
$\displaystyle H_{f}\psi_{f}$ $\displaystyle=$
$\displaystyle\left[\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}+\nabla^{2}f-\left(\vec{\nabla}f\right)^{2}\right)+V(r)\right]\psi_{f}$
(69) $\displaystyle=$ $\displaystyle E\psi_{f}$
are given by
$\displaystyle\psi_{Ef}(r,\theta,\varphi)=e^{f(r,\theta,\varphi)}\phi_{E}(r)Y_{lm}(\theta,\varphi).$
(70)
The $\mathcal{PT}$-inner product for the $\psi_{f}$-states is given by
$\displaystyle<\psi_{E^{\prime}f}(r,\theta,\varphi)|\psi_{Ef}(r,\theta,\varphi)>_{f}=\int
drd\Omega\mathcal{PT}\left(\psi_{E^{\prime}f}(r,\theta,\varphi)\right)\psi_{Ef}(r,\theta,\varphi)$
$\displaystyle=$ $\displaystyle\int
drd\Omega(-)^{l}e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}\phi_{E^{\prime}}^{*}(r)\phi_{E}(r)Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi).$
Besides, if $f(r,\theta,\varphi)$ satisfies Eq. (25) and considering Eq. (68),
we have
$\displaystyle<\psi_{E^{\prime}f}(r,\theta,\varphi)|\psi_{Ef}(r,\theta,\varphi)>_{f}$
$\displaystyle=$ $\displaystyle(-)^{l}\lambda\int
drd\Omega\phi_{E^{\prime}}^{*}(r)\phi_{E}(r)Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi)$
(71) $\displaystyle=$ $\displaystyle\lambda(-)^{l}\delta_{EE^{\prime}}.$
Given that $[L_{i},H_{f}]\not=0$, it follows that $L_{i}$ is not a conserved
quantity. However, as
$\displaystyle[L_{fi},H_{f}]=0,$ (72)
then $L_{fi}$ is conserved.
## VII The Hydrogen Atom
In the case of the Hydrogen atom, we have the potential
$\displaystyle V(r)=-\frac{Ze^{2}}{r},$ (73)
where the solutions are given by
$\displaystyle\psi_{Nlm}(\rho,\theta,\varphi)$ $\displaystyle=$
$\displaystyle\frac{2}{N^{2}}\sqrt{\frac{Z^{3}}{a_{RB}^{3}}\frac{(N-l-1)!}{(N+l)!}}\rho^{l}L_{N-(l+1)}^{2l+1}(\rho)e^{-\frac{\rho}{2}}Y_{lm}(\theta,\varphi),$
$\displaystyle E_{N}$ $\displaystyle=$
$\displaystyle-\frac{Ze^{2}}{a_{RB}N^{2}},\qquad N=n+l+1,\qquad
n=0,1,2\cdots,$
where $a_{RB}$ is the Bohr radius and
$\displaystyle\rho=\alpha
r,\qquad\qquad\alpha=2\sqrt{\frac{-2mE}{\hbar^{2}}}.$ (74)
Taking into account Eq. (69), we have the equation
$\displaystyle H_{f}\psi_{f}$ $\displaystyle=$
$\displaystyle\left[\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}+\nabla^{2}f-\left(\vec{\nabla}f\right)^{2}\right)-\frac{Ze^{2}}{r}\right]\psi_{f}=E\psi_{f}$
(75)
whose solutions are given by
$\displaystyle\psi_{fNlm}(\rho,\theta,\varphi)$ $\displaystyle=$
$\displaystyle e^{f(r,\theta,\varphi)}\psi_{Nlm}(\rho,\theta,\varphi)$ (76)
this are orthogonal functions if equation (25) is satisfied. A remarkable fact
is that $L_{fi}$ is a conserved quantity. In the conventional Hydrogen atom,
the Runge-Lenz vector is also conserved landau:gnus
$\displaystyle
R_{i}=\frac{1}{2}\left(\vec{L}\times\vec{p}-\vec{p}\times\vec{L}+\frac{Ze^{2}\vec{r}}{r}\right)_{i}.$
(77)
In the case of the Hamiltonian $H_{f}$, we have
$\displaystyle[R_{fi},H_{f}]=0,$ (78)
and we can say that the transformed non-Hermitian Runge-Lenz vector is
conserved.
Note that in this case, we have obtained a set of conserved quantities
$L_{fi},L_{f}^{2},R_{fi}$, that are the non-Hermitian generators of the
$SO(4)$ algebra.
## VIII Summary
In this work we have constructed a set of non-Hermitian operators $L_{fi}$
that satisfy the commutation relations of the $SO(3)$-Lie algebra. We have
shown that this operators generate rotations in the configuration space and
not in the conventional momentum space but in a modified non-Hermitian
momentum space $\vec{p}_{f}=\vec{p}+i\vec{\nabla{f}}.$ It is worthy to mention
that this operator was originally considered by Dirac in his seminal book.
Besides, the $L_{if}$ generators are related with a new type of spherical
harmonics that result to be $\mathcal{PT}$-orthonormal. Additionally, we have
shown that this quantities are conserved for mechanical systems described by a
central potential Hamiltonian with an additional complex term. As a particular
case, we have obtained the solutions of the corresponding
$\mathcal{PT}$-Hydrogen atom that includes a complex term, and we have found
that a non-Hermitian Runge-Lenz vector is a conserved quantity. Considering
this case, one remarkable result is that, as we have obtained the non-
Hermitian generators of the $SO(3)$-Lie algebra and also a non-Hermitian
Runge-Lenz vector, then we have the non-Hermitian generators of the
$SO(4)$-Lie algebra. In a future work we will study the non-Hermitian
generators corresponding to others symmetry groups.
## References
* (1) C. M. Bender, Introduction to PT-Symmetric Quantum Theory, Contemp. Phys. 46 277 (2005).
* (2) C. Bender, S. Boettcher, Phys. Rev. Lett; 80, 5243 (1998).
* (3) C. M. Bender, P. D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100 110402 (2008).
* (4) R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003).
* (5) R. J. Szabo, Quantum Field Theory on Noncommutative Spaces, Phys.Rept. 378 207 (2003).
* (6) E. S. Fradkin, A. A. Tseytlin, Conformal supergravity, Phys. Rept. 119 233 (1985).
* (7) C. M. Bender, D. W. Hook, P. N. Meisinger, Q. Wang, Probability Density in the Complex Plane, arXiv:0912.4659 [hep-th]; D. Bazeia, A. Das, L. Greenwood, L. Losanoa, The structure of supersymmetry in PT symmetric quantum mechanics, Phys.Lett.B 673 283 (2009); A. Mostafazadeh, Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies,; Phys. Rev. Lett. 102 220402 (2009); S. Klaiman, N. Moiseyev, U. Gunther Visualization of Branch Points in PT-Symmetric Waveguides, Phys. Rev. Lett. 101 080402 (2008); Z. H. Musslimani, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Optical Solitons in PT Periodic Potentials, Phys. Rev. Lett. 100 030402 (2008).
* (8) P.A. M. Dirac, The Principles of Quantum Mechanics,Oxford (1930).
* (9) L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, (Pergamon Press, UK, 1989).
* (10) P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. I, Vol. II, McGraw-Hill, (1953).
* (11) L. E. Reichl, A Modern Course in Statistical Physics, University of Texas Press, 1980.
|
arxiv-papers
| 2010-01-11T20:06:54 |
2024-09-04T02:49:07.648881
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan M. Romero, R. Bernal-Jaquez, O. Gonzalez-Gaxiola",
"submitter": "Roberto Bernal-Jaquez",
"url": "https://arxiv.org/abs/1001.1579"
}
|
1001.1595
|
# Centrality dependence of forward-backward multiplicity correlation
in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV
Yu-Liang Yan1, Dai-Mei Zhou2, Bao-Guo Dong1,3, Xiao-Mei Li1, Hai-Liang Ma1,
Ben-Hao Sa1,2,4111Corresponding author: sabh@ciae.ac.cn 1 China Institute of
Atomic Energy, P.O. Box 275(18), Beijing 102413, China
2 Institute of Particle Physics, Huazhong Normal University, Wuhan 430079,
China
3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion
Accelerator of Lanzhou, Lanzhou 730000,China
4 China Center of Advanced Science and Technology, World Laboratory, P. O. Box
8730 Beijing 100080, China
###### Abstract
We have studied the centrality dependence of charged particle forward-backward
multiplicity correlation strength in Au+Au collisions at $\sqrt{s_{NN}}$=200
GeV with a parton and hadron cascade model, PACIAE, based on PYTHIA. The
calculated results are compared with the STAR data. The experimentally
observed correlation strength characters: (1) the approximately flat
pseudorapidity dependence in central collisions and (2) the monotonous
decrease with decreasing centrality are well reproduced. However the
theoretical results are larger than the STAR data for the peripheral
collisions. A discussion is given for the comparison among the different
models and STAR data. A prediction for the forward-backward multiplicity
correlation in Pb+Pb collisions at $\sqrt{s_{NN}}$=5500 GeV is also given.
###### pacs:
24.10.Lx, 24.60.Ky, 25.75.Gz
## I INTRODUCTION
The study of fluctuations and correlations has been suggested as a useful
means for revealing the mechanism of particle production and Quark-Gluon-
Plasma (QGP) formation in relativistic heavy ion collisions hwa2 ; naya .
Correlations and fluctuations of the thermodynamic quantities and/or the
produced particle distributions may be significantly altered when the system
undergoes phase transition from hadronic matter to quark-gluon matter because
of the very different degrees of freedom between two matters.
The experimental study of fluctuations and correlations becomes a hot topic in
relativistic heavy ion collisions with the availability of high multiplicity
event-by-event measurements at the CERN-SPS and BNL-RHIC. An abundant
experimental data have been reported star2 ; phen2 ; phob ; star where a lot
of new physics arise and are urgent to be studied. A lot of theoretical
investigations have been reported as well paja ; hwa1 ; brog ; yan ; fu ; bzda
; konc ; yan2 .
Recently STAR collaboration have measured the charged particle forward-
backward (FB) multiplicity correlation strength $b$ in a given centrality bin
size at different centralities from the most central to peripheral Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV star . The outstanding features of STAR
data are:
1. 1.
In most central collisions, the correlation strength is approximately
independent of the distance between the centers of forward and backward
pseudorapidity bins, $\Delta\eta$.
2. 2.
The correlation strength monotonously decreases with decreasing centrality.
3. 3.
In the peripheral collisions, the correlation strength approaches to an
exponential function of $\Delta\eta$.
A lot of theoretical interests fu ; bzda ; konc ; yan2 has been stimulated.
The wounded nucleon model was used in bzda to study the correlation strength,
the first two characters of STAR data were reproduced, but the third one was
not. In Ref. konc , the Glauber Monte Carlo code (GMC) with a “toy” wounded-
nucleon model and the Hadron-String Dynamics (HSD) transport approach have
been used to analyze the STAR data. They used three different centrality
determinations: the impact parameter $b_{i}$, the number of participant
(wounded) nucleons $N_{part}$, and the charged particle multiplicity $N_{ch}$
in midrapidity $|\eta|<1$. The first two characters of STAR data can be
reproduced by the $N_{part}$ and $N_{ch}$ centrality determinations, while the
third character can not.
We have used a parton and hadron cascade model, PACIAE, to investigate the
centrality bin size dependence of charged particle multiplicity correlation in
most central (5, 0-5, and 0-10%) Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200
GeV yan2 . It turned out that the correlation strength increases with
increasing bin size. In yan2 we follow cape defining the charged particle FB
multiplicity correlation strength $b$ as
$\ b=\frac{\langle n_{f}n_{b}\rangle-\langle n_{f}\rangle\langle
n_{b}\rangle}{\langle n_{f}^{2}\rangle-\langle
n_{f}\rangle^{2}}=\frac{{\rm{cov}}(n_{f},n_{b})}{{\rm{var}}(n_{f})},$ (1)
where $n_{f}$ and $n_{b}$ are, respectively, the number of charged particles
in forward and backward pseudorapidity bins defined relatively and
symmetrically to a given pseudorapidity $\eta$. $\langle n_{f}\rangle$ refers
to the mean value of $n_{f}$ for instance. cov($n_{f}$,$n_{b}$) and
var($n_{f}$) are the FB multiplicity covariance and forward multiplicity
variance, respectively.
In this paper we use the PACIAE model to study the charged particle FB
multiplicity correlation strength $b$ in a given centrality bin size at
different centralities from the most central to peripheral Au+Au collisions at
$\sqrt{s_{NN}}$=200 GeV. A discussion is given for the comparison among the
models and STAR data as well as the STAR’s convention of different centrality
determination for different measured $\Delta\eta$ points. A prediction for the
forward-backward multiplicity correlation in Pb+Pb collisions at
$\sqrt{s_{NN}}$=5500 GeV is also given.
## II THE PACIAE MODEL
The parton and hadron cascade model, PACIAE sa , is based on PYTHIA soj2
which is a model for hadron-hadron ($hh$) collisions. The PACIAE model is
composed of four stages: the parton initialization, parton evolution
(rescattering), hadronization, and hadron evolution (rescattering).
1\. PARTON INITIALIZATION: In the PACIAE model, a nucleus-nucleus collision is
decomposed into the nucleon-nucleon collisions based on the collision
geometry. A nucleon-nucleon ($NN$) collision is described with the PYTHIA
model, where a $NN$ (hadron-hadron, $hh$) collision is decomposed into the
parton-parton collisions. The hard parton-parton collision is described by the
lowest-leading-order (LO) pQCD parton-parton cross section comb with
modification of parton distribution function in the nucleon. And the soft
parton-parton interaction is considered empirically. The semihard, between
hard and soft, QCD $2\rightarrow 2$ processes are also involved in the PYTHIA
(PACIAE) model. Because of the initial- and final-state QCD radiation added to
the parton-parton collision processes, the PYTHIA (PACIAE) model generates a
partonic multijet event for a $NN$ ($hh$) collision. That is followed, in the
PYTHIA model, by the string-based fragmentation scheme (Lund string model
and/or Independent Fragmentation model), thus a hadronic final state is
reached for a $NN$ ($hh$) collision. However, in the PACIAE model the above
fragmentation is switched off temporarily, so the result is a partonic
multijet event (composed of quark pairs, diquark pairs and gluons) instead of
a hadronic final state. If the diquarks (anti-diquarks) are split forcibly
into quarks (anti-quarks) randomly, the consequence of a $NN$ ($hh$) collision
is its initial partonic state composed of quarks, anti-quarks, and gluons.
2\. PARTON EVOLUTION: The next stage in the PACIAE model is the parton
evolution (parton rescattering). Here the $2\rightarrow 2$ LO-pQCD
differential cross sections comb are employed. The differential cross section
of a subprocess $ij\rightarrow kl$ reads
$\frac{d\sigma_{ij\rightarrow
kl}}{d\hat{t}}=K\frac{\pi\alpha_{s}^{2}}{\hat{s}}\sum_{ij\rightarrow kl},$ (2)
where the factor $K$ is introduced considering the higher order pQCD and non-
perturbative QCD corrections as usual, $\alpha_{s}$ stands for the strong
(running) coupling constant. Taking the process $q_{1}q_{2}\rightarrow
q_{1}q_{2}$ as an example one has
$\sum_{q_{1}q_{2}\rightarrow
q_{1}q_{2}}=\frac{4}{9}\frac{\hat{s}^{2}+\hat{u}^{2}}{\hat{t}^{2}},$ (3)
where $\hat{s}$, $\hat{t}$, and $\hat{u}$ are the Mandelstam variables. Since
it diverges at $\hat{t}$=0 one has to regularize it with the parton color
screen mass $\mu$
$\sum_{q_{1}q_{2}\rightarrow
q_{1}q_{2}}=\frac{4}{9}\frac{\hat{s}^{2}+\hat{u}^{2}}{(\hat{t}-\mu^{2})^{2}}.$
(4)
The total cross section of the parton collision, $i+j$, then reads
$\sigma_{ij}(\hat{s})=\sum_{k,l}\int_{-\hat{s}}^{0}d\hat{t}\frac{d\sigma_{ij\to
kl}}{d\hat{t}}.$ (5)
With the total and differential cross sections above the parton evolution
(parton rescattering) can be simulated by the Monte Carlo method.
3\. HADRONIZATION: The parton evolution stage is followed by the hadronization
at the moment of partonic freeze-out (no any more parton collision). In the
PACIAE model, the Lund string fragmentation model and phenomenological
coalescence model are supplied for the hadronization of partons after
rescattering. The Lund string fragmentation model is adopted in this paper. We
refer to sa for the details of the hadronization stage.
4\. HADRON EVOLUTION: After hadronization the rescattering among produced
hadrons is dealt with the usual two-body collision model. Only the
rescattering among
$\pi,k,p,n,\rho(\omega),\Delta,\Lambda,\Sigma,\Xi,\Omega,J/\Psi$ and their
antiparticles are considered in the calculations. An isospin averaged
parametrization formula is used for the $hh$ cross section koch ; bald . We
also provide an option for the constant total, elastic, and inelastic cross
sections ($\sigma_{\rm{tot}}^{NN}=40$ mb, $\sigma_{\rm{tot}}^{\pi N}=25$ mb,
$\sigma_{\rm{tot}}^{kN}=35$ mb, $\sigma_{\rm{tot}}^{\pi\pi}=10$ mb) and the
assumed ratio of inelastic to total cross section of 0.85. More details of
hadronic rescattering can see sa1 .
## III CALCULATIONS AND RESULTS
In our calculations the default values given in the PYTHIA model are adopted
for all model parameters except the parameters $K$ and $b_{s}$ (in the Lund
string fragmentation function). $K$=3 is assumed and $b_{s}$=6 is tuned to the
PHOBOS data of charged particle multiplicity in 0-6% most central Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV phob2 , as shown in Tab. 1.
Table 1: Total charged particle multiplicity in three $\eta$ fiducial ranges in 0-6% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. | $N_{\rm{ch}}(|\eta|<4.7)$ | $N_{\rm{ch}}(|\eta|<5.4)$ | $N_{\rm{ch}}$(total)
---|---|---|---
PHOBOSa | 4810 $\pm$ 240 | 4960 $\pm$ 250 | 5060 $\pm$ 250
PACIAE | 4819 | 4983 | 5100
a The experimental data are taken from phob2 .
In the theoretical calculation it is convenient to define the centrality by
impact parameter $b_{i}$. The mapping relation between centrality definition
in theory and experiment
$b_{i}=\sqrt{g}b_{i}^{\rm{max}},\qquad b_{i}^{\rm{max}}=R_{A}+R_{B},$ (6)
is introduced sa2 . In the above equation, $g$ stands for the geometrical
(total) cross section percentage (or charged multiplicity percentage) used in
the experimental determination of centrality. $R_{A}=1.12A^{1/3}+0.45$ fm is
the radius of nucleus $A$. We have also used the centrality determination
based on the charged multiplicity $N_{ch}$ in the central pseudorapidity
window $|\eta|<1$ to study the FB multiplicity correlation strength as a
function of $\Delta\eta$.
Figure 1: (Color online) The charged particle pseudorapidity distribution for
the specified centrality bin in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV.
The solid and open symbols are the PHOBOS data phob2 and PACIAE results,
respectively.
We compare the calculated charged particle pseudorapidity distribution in
specified centrality bin in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV with
the corresponding PHOBOS data phob2 in Fig. 1. Here the theoretical
centrality are defined by impact parameter $b_{i}$ via Eq.(6). One sees that
the PHOBOS data are well reproduced.
Figure 2: (Color online) The FB multiplicity correlation strength $b$ in Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV for centrality determination of (a)
impact parameter $b_{i}$ and (b) charged particle multiplicity $N_{ch}$
($|\eta|<1$). The open symbols with line and solid symbols are the PACIAE
results and STAR data star , respectively. Figure 3: (Color online) The FB
multiplicity correlation strength $b$ calculated by different models are
compared with the STAR datastar for (a) 0-10% and (b) 40-50% central Au+Au
collisions at $\sqrt{s_{NN}}$=200 GeV.
The PACIAE model results of FB multiplicity correlation strength are shown in
Fig. 2 (a) and (b) for the centrality determinations of the impact parameter
$b_{i}$ and charged multiplicity $N_{ch}$ ($|\eta|<1$), respectively. One sees
in Fig. 2 (a) that the theoretical results can reproduce the STAR data for
0-10% most central collisions. However the theoretical results are all higher
than the corresponding STAR data for the 10-20%, 20-30%, 30-40%, and 40-50%
most central collisions and those theoretical results are closed to each
other. That is because the FB multiplicity correlation in Au+Au collisions is
mainly the statistical correlation steaming from the multiplicity fluctuation
yan2 . And the multiplicity fluctuation in those centralities defined by
impact parameter are similar to each other.
Figure 2 (b) gives the FB correlation strength calculated with the charged
particle multiplicity ($|\eta|<1$) centrality determination together with the
STAR data. We see in this panel that the STAR data are well reproduced by the
PACIAE model for the 0-10%, 10-20%, and 20-30% central collisions. However,
the PACIAE results are higher than the corresponding STAR data for 30-40%, and
40-50% central collisions. The correlation strength approaches an exponential
function of $\Delta\eta$ observed by STAR in 40-50% central collisions can not
be reproduced especially.
In Fig. 3 we compare the FB multiplicity correlation strength as a function of
$\Delta\eta$ calculated by the wounded nucleon model bzda , GMC code with a
“toy” wounded-nucleon model, HSD transport approach konc , and PACIAE model
with the STAR data for the 0-10% (panel(a)) and 40-50% central collisions
(panel (b)). We see in Fig. 3 (a) that all of the four models can nearly
reproduce the STAR data for 0-10% central collisions and the PACIAE model is
somewhat better than others. However all of the model results are higher than
the STAR data for the 40-50% central collisions, therefore can not reproduce
$b$ as an exponential function of $\Delta\eta$ observed by STAR as shown in
Fig. 3 (b).
Figure 4: (Color online) The same as Fig. 2 (b) but the PACIAE results are
calculated with STAR’s convention of different centrality determination for
different measured $\Delta\eta$ points (see text for the details).
We have noted the mentions in star that “To avoid a bias in the FB
correlation measurements, care was taken to use different pseudorapidity
selection for the centrality determination which is also based on
multiplicity. Therefore, the centrality determination for the FB correlation
strength for $\Delta\eta$=0.2, 0.4, and 0.6 is based on the multiplicity in
$0.5<|\eta|<1.0$, while for $\Delta\eta$=1.2, 1.4, 1.6, and 1.8, the
centrality is obtained from $|\eta|<0.5$. For $\Delta\eta$=0.8 and 1.0, the
sum of multiplicities from $|\eta|<0.3$ and $0.8<|\eta|<1.0$ is used for the
centrality determination.” So we follow STAR’s centrality determination
convention to repeat all of the PACIAE calculations and draw Fig. 4 with these
results. Comparing Fig. 4 with Fig. 2 (b) we see this complicated centrality
determination does not improve but even worsens the agreement between
experiment and theory. That means STAR’s centrality determination convention
may be needed for the experimental measurement of FB correlation strength but
not for the theoretical calculation. In this kind of theoretical calculations,
each definite centrality curve in Fig. 4 is composed of $b$ calculated at
different $\Delta\eta$ with different centrality determination, such kind of
curve is not reasonable in theoretical physics.
Figure 5: (Color online) The PACIAE calculated FB multiplicity correlation
strength $b$ in Pb+Pb collisions at $\sqrt{s_{NN}}$=5500 GeV with centrality
determination of charged particle multiplicity $N_{ch}$ ($|\eta|<1$).
Figure 5 gives the PACIAE model prediction for the charged particle FB
multiplicity correlation strength in Pb+Pb collisions at $\sqrt{s_{NN}}$=5500
GeV. We can see that the FB multiplicity correlation strength in Pb+Pb
collisions is smaller than in Au+Au collisions. One reason may be that the
multiplicity in Pb+Pb collisions at LHC energy is much larger than in Au+Au
collisions at RHIC energy (the $N_{ch}/d\eta$ at mid-rapidity, $|\eta|<$0.5,
is about 600 in 0-10% most central Au+Au collisions at RHIC energy but it is
around 1200 in 0-10% most central Pb+Pb collisions at LHC energy sa3 ). This
observation is similar to the report that the elliptic flow parameter $v_{2}$
in Pb+Pb collisions at LHC energy is significantly smaller than in Au+Au
collisions at RHIC energy eyyu . That is attributed to the fact that the hard
process is more influential at the LHC energy than RHIC energy in eyyu .
Whether the competition between the hard and soft processes is also the reason
of the FB multiplicity correlation decreasing from RHIC energy to LHC energy
is beyond this paper scope, and it would be studied later.
## IV CONCLUSION
We have used a parton and hadron cascade model, PACIAE, to study the
centrality dependence of charged particle FB multiplicity correlation strength
in 0-10%, 10-20%, 20-30%, 30-40%, and 40-50% central Au+Au collisions at
$\sqrt{s_{NN}}$=200 GeV. For the 0-10%, 10-20%, and 20-30% central collisions,
the STAR data are well reproduced. The STAR observed characters of (1) $b$ as
a function of $\Delta\eta$ is approximately flat for central collisions and
(2) $b$ decreases with decreasing centrality are reproduced as well. However
the PACIAE results are higher than the STAR data for the 30-40% and 40-50%
central collisions and can not obtain $b$ as an exponential function of
$\Delta\eta$ for the 40-50% central collisions, especially.
It turned out that the PACIAE model is somewhat better than the wounded
nucleon model, the GMC code with a “toy” wounded-nucleon model, or the HSD
transport model in comparing with the STAR correlation data. However all the
models can not reproduce $b$ as an exponential function of $\Delta\eta$ in the
40-50% central collisions observed by STAR. That should be studied further.
The PACIAE calculations are repeated using the STAR’s centrality determination
convention mentioned above. The results not improve but even worsens the
agreement between theory and experiment. This means STAR’s centrality
determination convention may be needed for the experimental measurement but
not for the theoretical calculations. Because in such theoretical
calculations, each definite centrality curve is composed of $b$ calculated at
different $\Delta\eta$ with different centrality determination, this kind of
curve is not reasonable in theoretical physics.
A prediction for the charged particle FB multiplicity correlation strength in
0-10% Pb+Pb collisions at $\sqrt{s_{NN}}$=5500 GeV is also given. The charged
particle FB multiplicity correlation strength in Pb+Pb collisions at LHC
energy is much smaller than in Au+Au collisions at RHIC energy. The further
study is out of present paper scope and has to be investigated in another
paper.
ACKNOWLEDGMENT
Finally, the financial support from NSFC (10635020, 10605040, 10705012,
10475032, 10975062, and 10875174) in China is acknowledged.
## References
* (1) R. C. Hwa, Int. J. Mod. Phys. E 16, 3395 (2008).
* (2) T. K. Nayak, J. of Phys. G 32, S187 (2006).
* (3) J. Adams, et al., STAR Collaboration, Phys. Rev. C 75, 034901 (2007).
* (4) A. Adare, et al., PHENIX Collaboration, Phys. Rev. Lett. 98, 232302 (2007).
* (5) Zheng-Wei Chai , et al., PHOBOS Collaboration, J. of Phys.: Conference Series 27, 128 (2005).
* (6) B. I. Abelev, STAR Collaboration, Phys. Rev. Lett. 103, 172301 (2009).
* (7) N. S. Amelin, N. Armesto, M. A. Braun, E. G. Ferreiro, and C. Pajares, Phys. Rev. Lett. 73, 2813 (1994); N. Armesto, M. A. Braun, and C. Pajares, Phys. Rev. C 75, 054902 (2007).
* (8) R. C. Hwa and C. B. Yang , arXiv:nucl-th/0705.3073.
* (9) P.Brogueira, J. Dias de Deus, and J. G. Milhano, Phys. Rev. C 76, 064901 (2007).
* (10) Yu-Liang Yan, Bao-Guo Dong, Dai-Mei Zhou, Xiao-Mei Li, and Ben-Hao Sa, Phys. Lett. B 660, 478 (2008).
* (11) Jinghua Fu, Phys. Rev. C 77, 027902 (2008).
* (12) A. Bzdak, Phys. Rev. C 80, 024906 (2009).
* (13) V. P. Konchakovski, M. Hauer, G. Torrieri, M. I. Gorenstein, and E. L. Bratkovskaya, Phys. Rev. C 79, 034910 (2009).
* (14) Yu-Liang Yan, Dai-Mei Zhou, Bao-Guo Dong, Xiao-Mei Li, Hai-Liang Ma, and Ben-Hao Sa, Phys. Rev. C 79, 054902 (2009).
* (15) A. Capella, U. Sukhatme, C.-I. Tan, and J. Tran Thanh Van, Phys. Rep. 236, 225 (1994).
* (16) Dai-Mei Zhou, Xiao-Mei Li, Bao-Guo Dong, and Ben-Hao Sa, Phys. Lett. B 638, 461 (2006); Ben-Hao Sa, Xiao-Mei Li, Shou-Yang Hu, Shou-Ping Li, Jing Feng, and Dai-Mei Zhou, Phys. Rev. C 75, 054912 (2007).
* (17) T. Söjstrand, S. Mrenna, and P. Skands, J. High Energy Phys. JHEP05, 026 (2006).
* (18) B. L. Combridge, J. Kripfgang, and J. Ranft, Phys. Lett. B 70, 234 (1977).
* (19) P. Koch, B. Müller, and J. Rafelski Phys. Rep. 142, 167 (1986).
* (20) A. Baldini, et al., “Total cross sections for reactions of high energy particles”, Springer-Verlag, Berlin, 1988.
* (21) Ben-Hao Sa and Tai An, Comput. Phys. Commun. 90, 121 (1995); Tai An and Ben-Hao Sa, Comput. Phys. Commun. 116, 353 (1999).
* (22) B. B. Back, et al., PHOBOS Collaboration, Phys. Rev. Lett. 91, 052303 (2003).
* (23) Ben-Hao Sa, A. Bonasera, An Tai and Dai-Mei Zhou, Phys. Lett. B 537, 268 (2002).
* (24) Ben-Hao Sa, Dai-Mei Zhou, Bao-Guo Dong, Yu-Liang Yan, Hai-Liang Ma, and Xiao-Mei Li, J. Phys. G 36, 025007 (2009).
* (25) G. Eyyubova, L. V. Bravina, E. Zabrodin, V. L. Korotkikh, I. P. Lokhtin, L. V. Malinina, S. V. Petrushanko, and A. M. Snigirev, Phys. Rev. C 80, 064907 (2009).
|
arxiv-papers
| 2010-01-11T06:25:55 |
2024-09-04T02:49:07.655990
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu-Liang Yan, Dai-Mei Zhou, Bao-Guo Dong, Xiao-Mei Li, Hai-Liang Ma,\n Ben-Hao Sa",
"submitter": "Yuliang Yan",
"url": "https://arxiv.org/abs/1001.1595"
}
|
1001.1599
|
$Id:espcrc2.tex,v1.22004/02/2411:22:11speppingExp$ M. Bhuyan and S.K. Patra
# Effects of density and parametrization on scattering observables
M. Bhuyanand S. K. Patra School of Physics, Sambalpur University,
Jyotivihar-768 019, India Institute of Physics, Sachivalaya Marg,
Bhubaneswar-751 005, India
###### Abstract
We calculate the density distribution of protons and neutrons for
${}^{40,42,44,48}Ca$ in the frame-work of relativistic mean field (RMF) theory
with NL3 and G2 parameter sets. The microscopic proton-nucleus optical
potential for $p+^{40}Ca$ system is evaluted from Dirac NN-scattering
amplitude and the density of the target nucleus using Relativistic-Love-Franey
and McNeil-Ray-Wallace parametrizations. Then we estimate the scattering
observables, such as elastic differential scattering cross-section, analysing
power and the spin observables with relativistic impulse approximation. We
compare the results with the experimental data for some selective cases and
found that the use of density as well as the scattering matrix parametrization
is crucial for the theoretical prediction.
Explaining the nuclear structure by taking the tool of nuclear reaction is one
of the most curious and challenging solution for Nuclear Physics both in
theory and laboratory. So far the elastic scattering reaction of Neucleon-
Nucleus is more interesting than that of Nucleus-Nucleus at laboratory energy
$E_{lab}\simeq$ 1000 MeV. The Neucleon-Nucleus interaction provides a fruitful
source to determine the nuclear structure and a clear path toward the
formation of exotic nuclei in laboratory. One of the theoritical method to
study such type of reaction is the Relativistic Impulse Approximation (RIA).
In a wide range of energy interval, the conventional impulse approximation [1,
2] reproduces quantitatively the main features of quasi-elastic scattering for
medium mass nuclei [3, 4]. The observables of the elastic scattering reaction
not only depend on the energy of the incident particle but also on the
kinematic parameter as well as the density discributions of the target
nucleus. In the present letter, our motivation is to calculate the nucleon-
nucleus elastic differential scattering cross-section
($\frac{d\sigma}{d\Omega}$) and other quantities, like optical potential
($U_{opt}$), analysing power ($A_{y}$) and spin observables ($Q-$value) taking
input as relativistic mean field (RMF) and recently proposed effective field
theory motivated relativistic mean field (E-RMF) density. The RMF and E-RMF
densities are obtained from the most successful NL3 [5] and advanced G2 [6]
parameter sets, respectively. As representative cases, we used these target
densities folded with the NN-aplitude of 1000 MeV energetic proton projectile
with Relativistic-Love-Franey (RLF) and McNeil-Ray-Wallace (MRW)
parametrizations [7] for ${}^{40,42,44,48}Ca$ in our calculations.
The RMF and E-RMF theories are well documented [6, 8, 9] and for completeness
we outline here very briefly the formalisms for finite nuclei. The energy
density functional of the E-RMF model for finite nuclei is written as [10,
11],
$\displaystyle\mathcal{E}(\mathbf{r})=\sum_{\alpha}\varphi_{\alpha}^{\dagger}\Bigg{\\{}-i\mbox{\boldmath$\alpha$}\\!\cdot\\!\mbox{\boldmath$\nabla$}+\beta(M-\Phi)+W+$
$\displaystyle\frac{1}{2}\tau_{3}R+\frac{1+\tau_{3}}{2}A-\frac{i}{2M}\beta\mbox{\boldmath$\alpha$}\\!\cdot\\!(f_{v}\mbox{\boldmath$\nabla$}W+\frac{1}{2}f_{\rho}\tau_{3}\mbox{\boldmath$\nabla$}$
$\displaystyle
R+\lambda\mbox{\boldmath$\nabla$}A)+\frac{1}{2M^{2}}\left(\beta_{s}+\beta_{v}\tau_{3}\right)\Delta
A\Bigg{\\}}\varphi_{\alpha}$
$\displaystyle\hbox{}+\left(\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi}{M}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}}{M^{2}}\right)\frac{m_{s}^{2}}{g_{s}^{2}}\Phi^{2}-\frac{\zeta_{0}}{4!}\frac{1}{g_{v}^{2}}W^{4}$
$\displaystyle\hbox{}+\frac{1}{2g_{s}^{2}}\left(1+\alpha_{1}\frac{\Phi}{M}\right)\left(\mbox{\boldmath$\nabla$}\Phi\right)^{2}-\frac{1}{2g_{v}^{2}}\left(1+\alpha_{2}\frac{\Phi}{M}\right)$
$\displaystyle\left(\mbox{\boldmath$\nabla$}W\right)^{2}\hbox{}-\frac{1}{2}\left(1+\eta_{1}\frac{\Phi}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M^{2}}\right)\frac{{m_{v}}^{2}}{{g_{v}}^{2}}W^{2}-\frac{1}{2g_{\rho}^{2}}$
$\displaystyle\left(\mbox{\boldmath$\nabla$}R\right)^{2}-\frac{1}{2}\left(1+\eta_{\rho}\frac{\Phi}{M}\right)\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}$
$\displaystyle\hbox{}-\frac{1}{2e^{2}}\left(\mbox{\boldmath$\nabla$}A\right)^{2}+\frac{1}{3g_{\gamma}g_{v}}A\Delta
W+\frac{1}{g_{\gamma}g_{\rho}}A\Delta R,$ (1)
where the index $\alpha$ runs over all occupied states
$\varphi_{\alpha}(\mathbf{r})$ of the positive energy spectrum, $\Phi\equiv
g_{s}\phi_{0}(\mathbf{r})$, $W\equiv g_{v}V_{0}(\mathbf{r})$, $R\equiv
g_{\rho}b_{0}(\mathbf{r})$ and $A\equiv eA_{0}(\mathbf{r})$. The terms with
$g_{\gamma}$, $\lambda$, $\beta_{s}$ and $\beta_{v}$ take care of the effects
related with the electromagnetic structure of the pion and the nucleon (see
Ref. [11]). The energy density contains tensor couplings, and scalar-vector
and vector-vector meson interactions, in addition to the standard scalar self
interactions $\kappa_{3}$ and $\kappa_{4}$. Thus, the E-RMF formalism can be
interpreted as a covariant formulation of density functional theory as it
contains all the higher order terms in the Lagrangian, obtained by expanding
it in powers of the meson fields. The terms in the Lagrangian are kept finite
by adjusting the parameters. Further insight into the concepts of the E-RMF
model can be obtained from Ref. [11]. It may be noted that the standard RMF
Lagrangian is obtained from that of the E-RMF by ignoring the vector-vector
and scalar-vector cross interactions, and hence does not need a separate
discussion. In each of the two formalisms (E-RMF and RMF), the set of coupled
equations are solved numerically by a self-consistent iteration method and the
baryon, scalar, isovector, proton, neutron and tensor densities are
calculated.
The numerical procedure of calculation and the detailed equations for the
ground state properties of finite nuclei, we refere the reader to Refs. [9,
8]. The densities obtained from RMF (NL3) [5] and E-RMF (G2) [6] are used for
folding with the NN-sacttering amplitude at $E_{lab}=1000MeV$, which gives the
proton-nucleus complex optical potential for RMF and E-RMF formalisms. RIA
involves mainly two steps [12, 13] of calculations for the evaluation of the
NN-scattering amplitude. In this case, five Lorentz covariant function [7]
multiply with the so called Fermi invariant Dirac matrix (NN-scattering
amplitudes). This NN-amplitudes are folded with the target densities of
protons and neutrons to produced a first order complex optical potential
$U_{opt}$. The invariant NN-scattering operater ${\cal F}$ can be written in
terms of five complex functions (the five terms involves in the proton-proton
pp and neutron-neutron pn scattering) as follows:
$\displaystyle{\cal F(q,E)}={\cal F}^{S}+{\cal
F}^{V}\gamma^{\mu}_{(0)}\gamma_{(1)\mu}+{\cal
F}^{PS}\gamma^{5}_{(0)}\gamma^{5}_{(1)}$ $\displaystyle+{\cal
F}^{T}\sigma^{\mu\nu}_{(0)}\sigma_{(1)\mu\nu}+{\cal
F}^{A}\gamma^{5}_{(0)}\gamma^{\mu}_{(0)}\gamma^{5}_{(1)}\gamma_{(1)\mu},$ (2)
where (0) and (1) are the incident and struck nucleons respectively. The
amplitude for each ${\cal F}^{L}$ is a complex function of the Lorentz
invariants T and S with ${\it E}=E_{lab}$ and q is the four momentum. We
recommend the redears for detail expressions to Refs. [14, 15, 16, 17, 18, 19,
20, 21, 22]. Then the Dirac optical potential ${\it U}_{opt}(q,E)$ can be
written as,
$\displaystyle{\it U}_{opt}(q,E)=\frac{-4\pi
ip}{M}\langle\psi|\sum_{n=1}^{A}exp^{iq.x(n)}{\cal F}(q,E;n)|\psi\rangle,$ (3)
where ${\cal F}$ is the scattering operator, ${\it p}$ is the momentum of the
projectile in the nucleon-nucleus center of mass frame, $|\psi\rangle$ is the
nuclear ground state wave function for A-particle. Finally using the Numerov
algorithm the obtained wave function is matched with the coulomb scattering
solution for a boundary condition at $r\rightarrow\infty$ and we get the
scattering observables from the scattering amplitude, which are defined as:
$\displaystyle\frac{d\sigma}{d\Omega}\equiv|A(\theta)|^{2}+|B(\theta)|^{2}$
(4) $\displaystyle
A_{y}\equiv\frac{2Re[A^{*}(\theta)B(\theta)]}{d\sigma/d\Omega}$ (5)
$\displaystyle and$ $\displaystyle
Q\equiv\frac{2Im[A(\theta)B^{*}(\theta)]}{d\sigma/d\Omega}.$ (6)
Now we present our calculated results of neutrons and protons density
distribution obtained from the RMF and E-RMF formalisms [8]. Then we evaluate
the scattering observables using these densities in the relativistic impulse
approximation, which involves the following two steps: in the first step we
generate the complex NN-interaction from the Lorentz invariant matrix ${\cal
F}^{L}(q,E)$ as defined in Eq. (2). Then the interaction is folded with the
ground state target nuclear density for both the RLF and MRW parameters [7]
separately and obtained the nucleon-nucleus complex optical potential
$U_{opt}(q,E)$ for the parametrisations. It is to be noted that pairing
interaction is taken care using the Pauli blocking approximation. In the
second step, we solve the wave function of the scattering state utilising the
optical potential prepared in the first step by well known Numerov algorithm
[23]. The result approxumated with the non-relativistic Coulomb scattering for
a longer range of radial component which results the scattering amplitude and
other observables [24]. In thr present paper we calculate the density
distribution of protons and neutrons for 40,42,44,48Ca in NL3 and G2 parameter
sets. From the density we evalute the optical potential and other scattering
observables and some representative cases are presented in Figures $1-3$.
Figure 1: (upper panel): The neutrons and protons density distribution for
${}^{40}Ca$ with NL3 and G2 parameter sets. (lower panel) (a) the Dirac
optical potential for $p+^{40}Ca$ system using RMF (NL3) and E-RMF (G2)
densities with RLF parametrisation, (b) same as (a), but for MRW
parametrisation. The projectile proton with $E_{lab}=1000$ MeV is taken.
In Fig. 1, the protons and neutrons density distribution for ${}^{40}Ca$ using
NL3 and G2 parameter sets (upper panel) and the optical potential obtained
with RLF and MRW parametrisation for $p+^{40}Ca$ at 1000 MeV proton energy
(lower panel) are shown. From the figure, it is noticed that, there is no
significant difference in desities for RMF and E-RMF parameter sets. However,
a careful inspection shows a small enhancement in central density (0-1.6 fm)
for NL3 set. On the otherhand the densities obtained from G2 elongated to a
larger distance towards the tail part of the density distribution. As the
optical potential is a complex function which constitute both real and
imaginary part for both scalar and vector, we have displyed those values in
the lower panel of Fig. 1. Unlike to the (upper panel) of protons and neutrons
density distribution, here we find a large difference of $U_{opt}(q,E)$
between the RLF and MRW parametrisation. Further, the $U_{opt}(q,E)$ value of
either RLF or MRW differs significantly depending on the NL3 or G2 force
parameters. That means, the optical potential not only sensitive to RLF or MRW
but also to the use of NL3 or G2 parameter sets. Investigating the figure it
is clear that, the extrimum magnitude of real and imaginary part of the scalar
potential are -442.2 and 113.6 MeV for RLF (G2) and -372.4 and 109.1 MeV for
RLF (NL3). The same values for the MRW parametrisation are -219.8 and 32.8 MeV
with G2 and -175.1 and 33.2 MeV with NL3 sets. In case of vector potential,
the extremum values for real and imaginary parts are 361.3 and -179.2 MeV for
RLF (G2) and 279.2 and -164.8 MeV for RLF (NL3) but with MRW parametrisation
these are appeared at 128.1 and -87.4 MeV in G2 and 99.2 and -76.6 MeV in NL3.
From these large variation in magnitude of scalar and vector potentials, it is
clear that the predicted results not only depend on the input target density,
but also highly sensitive with the kinematic of the reaction dynamics. A
further analysis of the results for the optical potential with NL3 and G2, it
suggest that the $U_{opt}$ value extends for a larger distance in NL3 than G2.
For example, with RLF the central part of $U_{opt}$ with G2 is more expanded
than with NL3 and ended at $r\sim 6fm$, whereas the optical potential persists
till $r\sim 8fm$ in NL3. Similar situation is also valid in MRW
parametrisation. This nature of the potential suggests the applicability of
NL3 over G2 force parameter. This is because in case of NL3 the soft-core
interaction between the projectile and the target nucleon is more effective.
Figure 2: The elastic differential scattering cross-section
($\frac{d\sigma}{d\Omega}$) as a function of scattering angle
$\theta_{cm}$(deg) for ${}^{40,42,44,48}Ca$ using both RLF and MRW
parametrisations. The value of $\frac{d\sigma}{d\Omega}$ is shown for RMF
(NL3) and E-RMF (G2) densities.
In Fig. 2., we have plotted the elastic scattering cross-section of the proton
with ${}^{40,42,44,48}Ca$ at laboratory energy $E_{lab}=$1000 MeV using both
densities obtained in the NL3 and G2 parameter sets with RLF and MRW
parametrisations. The experimental data [25] are also given for comparison. It
is reported in Refs. [7, 26] the superiority of RLF over MRW for lower energy
($E_{lab}\leq 400$ MeV), however the MRW shows better results at energy
$E_{lab}>400$ MeV. In the present case, our incident energy is 1000 MeV which
matches better (MRW) with experimental values. This is consistent with the
optical potential also (see Fig. 1). From the differential cross-section for
both NL3 and G2 densities with MRW parametrization, it is clearly seen that
$\frac{d\sigma}{d\Omega}$ with NL3 desity is more closer to experimental data
which insist not only the importance of parametrization (RLF or MRW) but also
to choose proper density input for the reaction dynamics. Analysing the
elastic differential cross-section along the isotopic chain of Ca from A=40 to
48, the calculated results improve with increasing mass number of the target.
Figure 3: (a) The calculated values of analysing power $A_{y}$ as a function
of scattering angle $\theta_{cm}$(deg) for ${}^{40}Ca$ (b) The spin observable
$Q-$value as a function of scattering angle $\theta_{cm}$(deg) for
${}^{40}Ca$. In both (a) and (b), the RLF and MRW parametrisations are used
with RMF (NL3) and E-RMF (G2) densities.
The analysing power for $p+^{40}Ca$ composite system is calculated from the
general formulae given in eqns. (4) and (5) and are shown in Fig. 3 with RLF
and MRW. The $A_{y}$ and $Q-$values obtained by NL3 and G2 sets almost matches
with each other both in RLF and MRW. But if we compare the results with RLF
and MRW it differs significantly. Again, we get a small oscillation of $A_{y}$
and $Q$ in G2 set with increasing scattering angle $\theta_{c.m.}^{0}$ for RLF
which does not appear in NL3 set. There is a rotation of $Q-$value from
positive to negative direction when we calculate with MRW parametrization,
which does not appear in case of RLF parametrization. This rotation shows a
shining path towards the formation of exotic nuclei in the laboratory.
In summary, we calculate the density distribution of protons and neutrons for
${}^{40,42,44,48}Ca$ by using RMF (NL3) and E-RMF (G2) parameter sets. We
found similar density distribution for protons and neutrons in both the sets
with a small difference at the central region. This small difference in
densities make a significant influence in the prediction of optical potential,
elastic differential cross-section, analysing power and the spin observable
for $p+Ca$ systems. The effect of kinematic parameters for reaction dynamics,
RLF and MRW, are also highly sensitive to the predicted results. That means,
the differential scattering cross-section and scattering observables are
highly depent on the input density and the choice of parametrisation.
## References
* [1] Faddeev in: Trudy matematicheskogo institute im. V. A. Steklova, Akad. Nauk SSSR, Moscow 69 (1963) 369.
* [2] C. Mahux, Proc. Conf. on Microscopic optical potentials, (1978) Hamurg p-1.
* [3] V. V. Balashov and J. V. Meboniya, Nucl. Phys. A 107 (1968) 369.
* [4] R. J. Glauber, Phys. Rev. 100 (1955) 242.
* [5] G. A. Lalazissis, J. König, and P. Ring, Phys. Rev. C 55 (1997) 540.
* [6] R. J. Furnstahl, B. D. Serot, and H. B. Tang, Nucl. Phys. 615 (1997) 441; R. J. Furnstahl, and B. D. Serot, Nucl. Phys. A 671 (2000) 447.
* [7] J. A. McNeil, L. Ray, and S. J. Wallace, Phys. Rev. C 27 (1983) 2123.
* [8] M. Del Estal, M. Centelles, X. Viñas, and S. K. Patra, Phys. Rev. C 63 (2001) 044321; M. Del Estal, M. Centelles, X. Viñas, and S. K. Patra, Phys. Rev. C 63 (2001) 044314; S. K. Patra, M. Del Estal, M. Centelles, and X. Viñas, Phys. Rev. C 63 (2001) 024311; P. Arumugam, B. K. Sharma, P. K. Sahu, S. K. Patra, Tapas Sil, M. Centelles, and X. Viñas, Phys. Lett. B601 (2004) 51.
* [9] S. K. Patra, and C. R. Praharaj, Phys. Rev. C 44 (1991) 2552; Y. K. Gambhir, P. Ring, and A. Thimet, Ann. Phys. (N.Y.) 198 (1990) 132.
* [10] B. D. Serot, and J. D. Walecka, Int. J. Mod. Phys. E 6 (1997) 515.
* [11] R. J. Furnstahl, B. D. Serot, and H. B. Tang, Nucl. Phys. A 598 (1996) 539.
* [12] R. J. Fernstahl, C. E. Price, and G. E. Walker, Phys. Rev. C 36 (1987) 2590.
* [13] J. A. McNeil, J. R. Shepard, and S. J. Wallace, Phys. Rev. Lett. 50 (1983) 1439.
* [14] M. Bhuyan and S. K. Patra Phys. Rev. C on preparation.
* [15] R. J. Perry, Phys. Lett. 182B (1986) 269.
* [16] W. R. Fox, Nucl. Phys. A 495 (1989) 463.
* [17] Murdock Proton scattering as a probe of Relativity Nuclei Ph.D. Thesis, MIT, (1987).
* [18] F. A. Brieva, and J. R. Rook, Nucl. Phys. A 291 (1977) 317.
* [19] C. J. Horowitz, and B. D. Serot, Phys. Lett. 137B (1984) 287.
* [20] R. Machleidt, and R. Brockmann, Phys. Lett. 149B (1984) 283.
* [21] B. Haar ter and R. Malfliet, Phys. Lett. 172B (1986) 10; Phys. Rev. Lett. 56 (1986) 1237.
* [22] C. J. Horowitz, and B. D. Serot, Nucl. Phys. A 464 (1987) 613; Phys. Rev. Lett. 86 (1986) 760 (E).
* [23] Koonin Computational Physics Benjamin, Reading, MA (1986).
* [24] McCarthy Introduction to Nuclear Theory Wiley, New York (1968).
* [25] G. Bruge, International Report D.Ph-N/ME/78-1 CEN, Salay, (1978).
* [26] C. J. Horowitz, D. P. Murdock and B. D. Serot, Indina University Report No. IU/NTC 90-01.
|
arxiv-papers
| 2010-01-11T07:21:35 |
2024-09-04T02:49:07.662168
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Bhuyan and S. K. Patra",
"submitter": "Mrutunjaya Bhuyan Mr",
"url": "https://arxiv.org/abs/1001.1599"
}
|
1001.1760
|
# High Energy Radiation from Black Holes: A Summary
C. D. Dermer Space Science Division, Code 7653, Naval Research Laboratory,
Washington, DC 20375-5352 G. Menon Department of Mathematics and Physics,
Troy University, Troy, AL 36082
###### Abstract
Bright $\gamma$-ray flares observed from sources far beyond our Galaxy are
best explained if enormous amounts of energy are liberated by black holes. The
highest-energy particles in nature—the ultra-high energy cosmic rays—cannot be
confined by the Milky Way’s magnetic field, and must originate from sources
outside our Galaxy. Here we summarize the themes of our book, “High Energy
Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos,” just
published by Princeton University Press. In this book, we develop a
mathematical framework that can be used to help establish the nature of
$\gamma$-ray sources, to evaluate evidence for cosmic-ray acceleration in
blazars, GRBs and microquasars, to decide whether black holes accelerate the
ultra-high energy cosmic rays, and to determine whether the Blandford-Znajek
mechanism for energy extraction from rotating black holes can explain the
differences between $\gamma$-ray blazars and radio-quiet AGNs.
## I SCIENTIFIC HYPOTHESES
In our book dm09 , a mathematical formulation of high-energy radiation
processes and strong-field gravity is presented. This framework provides a
starting point to investigate the following hypotheses:
1. 1.
The most energetic and powerful radiations are made by processes taking place
in black-hole jets.
2. 2.
Ultra-high energy cosmic rays (UHECRs) are accelerated by radio and
$\gamma$-ray loud blazars and GRBs.
3. 3.
Turbulence and shocks accelerate particles to high energies through Fermi
processes.
4. 4.
The energy source of black holes with relativistic jets is black-hole
rotation.
## II ESTABLISHING THE HYPOTHESES
We do not attempt to present a comprehensive summary of the observational
background to be tested in our book, which is impossible given the rapidly
expanding empirical data base arising from multiple facilities, in particular,
the Fermi Gamma ray Space Telescope, the MAGIC, HESS, and VERITAS ground-based
$\gamma$-ray telescopes, the Pierre Auger Cosmic Ray Observatory, and the
IceCube Neutrino Observatory. Instead we present a mathematical formalism to
help researchers establish or refute the hypotheses listed above. Here we
mention some observations that drive the study, and related subjects dealt
with in the book.
### II.1 Black-Hole Jets as Sources of Energetic Radiations
The Fermi telescope, building on the success of the Energetic Gamma Ray
Experiment Telescope (EGRET) on the Compton Gamma Ray Observatory, found that
$\sim 100$ MeV – GeV $\gamma$ rays are produced by the blazar AGN class. More
than 30 blazar AGNs, principally of the X-ray selected BL Lac class, have now
been detected with ground-based $\gamma$-ray telescopes. At least 12 GRBs of
both the long soft and short hard class have been detected with the Large Area
Telescope (LAT) on Fermi. Before that, five spark chamber events and several
TASC (Total Absorption Shower Counter)/BATSE (Burst and Transient Source
Experiment) events on CGRO, showed that GRBs produce extremely luminous multi-
MeV/GeV emissions. Galactic microquasars like LS 5039 and Cyg X-1 are also
found to be $\gamma$-ray sources. Powerful extragalactic $\gamma$-ray sources
are thought to be formed by black-hole jets, with the jets nearly aligned to
the observer.
To establish whether black-hole jets are the sources of luminous $\gamma$
radiations, we develop the theory of relativistic flows, starting with
relativistic kinematics and special relativity. Compton and synchrotron
processes are treated with the goal of presenting a framework from which to
model the multiwavelength spectral energy distributions of high-energy
radiation sources. Starting from the Compton cross section, relations are
derived to model external Compton scattering involving surrounding isotropic
radiation fields and anisotropic accretion disk radiation fields. The
formalism applies to scattering throughout the Thomson and Klein-Nishina
regime, though $\delta$-function approximations are provided to make simple
back-of-the-envelope estimates.
Synchrotron radiation formulae are presented, and the synchrotron self-Compton
and synchrotron self-absorption processes are developed. The $\gamma\gamma$
pair production process, starting from the elementary $\gamma\gamma$ cross
section, is applied to $\gamma$-ray attenuation by target photons found in
sources of high-energy radiation, and by target photons of the extragalactic
background light.
With this formalism in hand, the nature of high-energy radiation sources can
be examined. This includes tests for relativistic beaming such as the Compton
catastrophe, whereby the size scale of the source determined directly from
radio observations or indirectly from the variability timescale of the
radiation implies the level of the Compton-scattered X-ray and $\gamma$-ray
flux if the radio-through-optical/UV emission is nonthermal synchrotron
radiation. Minimum bulk Lorentz factors of relativistic outflows from
$\gamma\gamma$ opacity arguments are derived. Moreover, minimum power
requirements for synchrotron sources can be determined from equipartition
arguments. Together, these arguments point toward relativistic outflows from
highly compact sources, which almost certainly implicate black holes as the
engines of the luminous radiation.
### II.2 UHECRs from Blazars and GRBs
Not even the highest energy cosmic rays point back to their sources due to
deflections by the Galactic and intergalactic magnetic fields, which has made
the birth of charged-particle astronomy difficult. In the meantime, the
sources of the UHECRs can be established indirectly by identifying
$\gamma$-ray signatures of hadronic acceleration, and directly by detecting
neutrinos from their sources with, for example, the IceCube neutrino
telescope. Discriminating leptonic from hadronic emission signatures in the
absence of neutrino detection represents an important challenge in the Fermi
era. Identification of features peculiar to hadronic processes, such as orphan
flares in blazars, represents one approach, and the interpretation of unusual
temporal and spectral behaviors as indicated in delayed onset or extended
radiation from GRBs, represents another.
A treatment of photohadronic processes is given. This includes photomeson
production from proton-photon interactions, ion-photon photopair (Bethe-
Heitler) processes, and ion photodisintegration. To illustrate the use of the
photohadronic physics presented in the book, a calculation is made of the
UHECR spectrum from the superposition of long-duration GRB sources whose rate
density is assumed to follow various star-formation rate functions. (This
calculation is far more difficult if UHECRs are accelerated by blazars,
insofar as supermassive black holes increase in mass with time, their fueling
is episodic, and they come from various classes of objects, for example, BL
Lac objects and flat spectrum radio quasars.) The associated GZK neutrino
spectrum is calculated for an assumed long-duration GRB origin of the UHECRs.
The calculations are specific to UHECR protons, but photodisintegration cross
sections and approximations are also presented, including giant dipole
resonance and nonresonant channels. Calculations of neutrino production from
nuclear photodisintegration is briefly described.
For completeness, binary collision processes useful for calculating radiation
signatures from cosmic-ray interactions are given, including the processes of
secondary nuclear production, bremsstrahlung, electron-positron annihilation,
and Coulomb thermalization and knock-on electron processes.
### II.3 Particle Acceleration in Black-Hole Jets
One of the important questions in high-energy astrophysics is the method by
which cosmic rays are accelerated to $\approx$ PeV energies by Galactic
sources, and to ultra-high energies in extragalactic sources. The empirical
data base that must be explained is extensive. This includes cosmic-ray
anisotropy, ionic composition, and UHECR arrival directions. Features in the
cosmic-ray spectrum include the knee at $\approx 3$ PeV, the second knee at
$\approx 4\times 10^{17}$ eV, the ankle at $\approx 5\times 10^{18}$ eV, and
the cutoff at $\approx 6\times 10^{19}$ eV, probably due to GZK effects.
A complete understanding of cosmic-ray origin requires that a mechanism for
particle acceleration be identified. For this purpose, an introduction to
Fermi acceleration is given, starting from the two distinct types of Fermi
acceleration mechanisms due to shock acceleration resulting from particle
diffusion across a shock front and convection downstream of the shock, and
stochastic acceleration by plasma wave turbulence. Dimensional arguments
leading to the Kraichnan and Kolmogorov wave turbulence spectra are given. The
Hillas criterion, whereby allowed sites for particle acceleration to energy
$E$ are restricted to those which satisfy the condition that the Larmor radius
is smaller than the size scale of the system, is presented.
First-order and second-order Fermi acceleration mechanisms are developed to
the point where maximum particle energies in different astrophysical
environments can be derived. These maximum particle energies then implicate
different classes of sources as plausible sites for cosmic-ray acceleration.
Acceleration of cosmic rays by supernova remnant shocks remains the favored
explanation for the sources which accelerate galactic cosmic rays, and may
very well be demonstrated through Fermi measurements of the $\gamma$-ray
spectra of supernova remnants. UHECR acceleration in the shocks and turbulent
plasma formed in the relativistic jetted outflows of black-hole jets is the
most likely explanation for their origin, compared to, for example, highly
magnetized neutron stars or accretion or merger shocks in clusters of
galaxies.
The blast-wave physics developed over the past two decades to explain the
extraordinarily luminous radiation from GRBs is also presented, including
spectral and temporal SEDs from external ahocks and colliding shells,
photospheric emission, and neutron decoupling in the expanding fireball.
### II.4 Energy Extraction through Blandford-Znajek Processes
If jetted black holes make the most energetic radiations in nature, with
particle acceleration occurring via Fermi processes, a remaining question of
fundamental importance is the energy source for these radiations. Accretion
onto supermassive black holes obviously powers the luminous quasi-thermal
emissions in active galaxies, but less obvious are the reasons for the
differences between radio-quiet and radio-loud AGNs, or for the different
classes of GRBs, including X-ray flashes, low-luminosity GRBs, and long
duration GRBs.
We consider whether these differences may be due to the spin of the black
hole, with the energy extracted through the Blandford-Znajek process. A
treatment of general relativity and black-hole electrodynamics is given based
on a “3+1” formalism that provides a more compact approach than found in the
original treatment. On this basis, we have recast the constraint equation for
a force-free black-hole magnetosphere in a form that allows one to search for
analytic solutions. We derive an exact solution which does not give energy
extraction, and an approximation solution which generalizes the Blandford-
Znajek split monopole solution to arbitrary values of the black-hole spin
parameter. This solution is considered in light of evidence for jetted
outflows from black-hole sources.
## III Summary
The science covered in our book and, in particular, the hypotheses outlined in
the first section, will be addressed by the ongoing observational activity in
high-energy astronomy. The hypothesis that the highest energy and most
luminous radiations in nature, including the bright $\gamma$-ray flares from
blazars and GRBs, and the UHECRs from unknown extragalactic sources, are
energized by the rotational energy of spinning black holes, can be tested with
observatories now in operation. This assertion will be established or refuted
as new data is collected and analyzed.
###### Acknowledgements.
This work has been supported by the Office of Naval Research and NASA.
## References
* (1) Dermer, C. D., & Menon, G. 2009, “High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos,” (Princeton University Press: Princeton, NJ)
|
arxiv-papers
| 2010-01-11T22:45:07 |
2024-09-04T02:49:07.670908
|
{
"license": "Public Domain",
"authors": "Charles D. Dermer (1), Govind Menon (2) ((1)NRL, (2)Troy University)",
"submitter": "Charles Dermer",
"url": "https://arxiv.org/abs/1001.1760"
}
|
1001.1773
|
# Strong Decays of the Radial Excited States $B(2S)$ and $D(2S)$
Jin-Mei Zhang111jinmei_zhang@tom.com and Guo-Li Wang222gl_wang@hit.edu.cn
Department of Physics, Harbin Institute of Technology, Harbin 150001, China
> ABSTRACT
>
> The strong OZI allowed decays of the first radial excited states $B(2S)$ and
> $D(2S)$ are studied in the instantaneous Bethe-Salpeter method, and by using
> these OZI allowed channels we estimate the full decay widths:
> $\Gamma_{B^{0}(2S)}=24.4$ MeV, $\Gamma_{B^{+}(2S)}=23.7$ MeV,
> $\Gamma_{D^{0}(2S)}=11.3$ MeV and $\Gamma_{D^{+}(2S)}=11.9$ MeV. We also
> predict the masses of them: $M_{B^{0}(2S)}=5.777$ GeV, $M_{B^{+}(2S)}=5.774$
> GeV, $M_{D^{0}(2S)}=2.390$ GeV and $M_{D^{+}(2S)}=2.393$ GeV.
In the past few years, there are many new states observed in experiments.
Among them, the new states $D^{*}_{s0}(2317)$, $D_{s1}(2460)$ [1],
$B_{s1}(5830)$ and $B_{s2}(5840)$ [2] are orbitally excited states, which are
also called $P$ wave states. So far, great progress has been made on the
physics of orbital excited states $D^{*}_{s0}(2317)$ and $D_{s1}(2460)$ [3],
and there are already exist some investigations of $B_{s1}(5830)$ and
$B_{s2}(5840)$ [4]. Around the energy of these hadrons, according to
constitute quark model, there may be the radial excited $S$ wave states
$B(2S)$ and $D(2S)$. But due to their absence, the experimental and
theoretical studies for the radial excited 2$S$ states $B(2S)$ and $D(2S)$ are
still missing in the literature.
We know that the first radial excited 2$S$ state has a node structure in its
wave function, which means relativistic correction of 2$S$ state is much
larger than the one of corresponding basic state, even the 2$S$ state is a
heavy meson, so to consider the physics of radial excited state a relativistic
method is needed. Bethe-Salpeter equation [5] and its instantaneous one,
Salpeter equation [6], are famous relativistic methods to describe the
dynamics of a bound state. In a previous letter [7], we have solved the full
Salpeter equations for pseudoscalar mesons, the masses of first radial excited
2$S$ states are obtained, they are $M_{B^{0}(2S)}=5.777$ GeV,
$M_{B^{+}(2S)}=5.774$ GeV, $M_{D^{0}(2S)}=2.390$ GeV and $M_{D^{+}(2S)}=2.393$
GeV.
The mass of $B(2S)$ is 310 MeV higher than the threshold of mass scale of
$B^{*}\pi$, but lower than the threshold of $B_{s}^{*}K$, and the mass of
$D(2S)$ is 240 MeV higher than the threshold of mass scale of $D^{*}\pi$, but
lower than the threshold of $D_{s}^{*}K$, so the strong decays
$B(2S)\rightarrow B^{*}+\pi$ and $D(2S)\rightarrow D^{*}+\pi$ are OZI allowed
strong decays, and they are dominate decay channels of $B(2S)$ and $D(2S)$,
respectively. In this letter, we calculate the strong decay widths of
$B(2S)\rightarrow B^{*}+\pi$ and $D(2S)\rightarrow D^{*}+\pi$ in the framework
of Bethe-Salpeter method.
Since one of the final state is $\pi$ meson in the OZI allowed $B(2S)$ or
$D(2S)$ strong decay, we use the reduction formula, PCAC relation and low
energy theorem, so for the strong decays (considering the
$B^{0}(2S)\rightarrow B^{*+}\pi^{-}$ as an example) shown in Fig. 1, the
transition matrix element can be written as [8]:
$T=\frac{P_{f_{2}}^{\mu}}{f_{P_{f_{2}}}}\langle
B^{*+}(P_{f_{1}})|\bar{u}\gamma_{\mu}\gamma_{5}d|B^{0}(P)\rangle,$ (1)
where $P$, $P_{f_{1}}$ and $P_{f_{2}}$ are the momenta of the initial state
$B^{0}(2S)$, final states $B^{*+}$ and $\pi^{-}$, respectively, and
$f_{P_{f_{2}}}$ is the decay constant of $\pi^{-}$ meson.
To evaluate Eq. (1), we need to calculate the hadron matrix element $\langle
B^{*+}(P_{f_{1}})|\bar{u}\gamma_{\mu}\gamma_{5}d|B^{0}(P)\rangle$. It is well
known that the Mandelstam formalism [9] is one of proper approaches to compute
the hadron matrix elements sandwiched by the Bethe-Salpeter or Salpeter wave
functions of two bound-state. With the help of this method, in leading order,
the hadron matrix elements in the center of mass system of initial meson can
be written as [8, 10]:
$\displaystyle\langle
B^{*+}(P_{f_{1}})|\bar{u}\gamma_{\mu}\gamma_{5}d|B^{0}_{2S}(P)\rangle=\int\frac{d{\vec{q}}}{(2\pi)^{3}}Tr\left[\bar{\varphi}^{++}_{{}_{P_{f_{1}}}}(\vec{q^{\prime}})\gamma_{\mu}\gamma_{5}{\varphi}^{++}_{{}_{P}}({\vec{q}})\frac{\not\\!P}{M}\right].$
(2)
Figure 1: Feynman diagram corresponding to the strong decays
$B^{0}(2S)\rightarrow B^{*+}\pi^{-}$.
where $\vec{q}$ is the relative three-momentum of the quark-anti-quark in the
initial meson $B^{0}(2S)$ and
$\vec{q^{\prime}}={\vec{q}}+\frac{m_{2}^{\prime}}{m_{1}^{\prime}+m_{2}^{\prime}}{\vec{r}}$,
$M$ is the mass of $B^{0}(2S)$, ${\vec{r}}$ is the three dimensional momentum
of the final meson $B^{*+}$, ${\varphi}^{++}_{{}_{P}}$ is the positive energy
B.S. wave function for the relevant mesons and
$\bar{\varphi}^{++}_{{}_{P_{f_{1}}}}=\gamma_{0}({\varphi}^{++}_{{}_{P_{f_{1}}}})^{+}\gamma_{0}$.
For the initial state pseudoscalar meson $B^{0}(2S)$ ($J^{P}=0^{-}$), the
positive energy wave function takes the general form [7]:
$\displaystyle\varphi_{0^{-}}^{++}(\vec{q})$ $\displaystyle=$
$\displaystyle\frac{M}{2}\left\\{\left[f_{1}(\vec{q})+f_{2}(\vec{q})\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}\right]\left[\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}+\frac{\not\\!{P}}{M}-\frac{\not\\!{q_{{}_{\bot}}}(m_{1}-m_{2})}{m_{2}\omega_{1}+m_{1}\omega_{2}}\right]\right.$
(3) $\displaystyle+$
$\displaystyle\left.\frac{\not\\!{q_{{}_{\bot}}}\not\\!P(\omega_{1}+\omega_{2})}{M(m_{2}\omega_{1}+m_{1}\omega_{2})}\right\\}\gamma_{5}.$
where $q_{{}_{\bot}}=(0,\vec{q})$ and
$\omega_{i}=\sqrt{m_{i}^{2}+\vec{q}^{2}}$, $f_{i}(\vec{q})$ are eigenvalue
wave functions which can be obtained by solving the full $0^{-}$ state
Salpeter equations. For the final state vector meson $B^{*+}$ ($J^{P}=1^{-}$),
the positive energy wave function takes the general form [11]:
$\displaystyle\varphi_{1^{-}}^{++}(\vec{q^{\prime}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[A\not\\!\epsilon_{{}_{\bot}}^{\prime\lambda}+B\not\\!\epsilon_{{}_{\bot}}^{\prime\lambda}\not\\!{P_{{}_{f_{1}}}}+C(\not\\!{q_{{}_{\bot}}^{\prime}}{\not\\!\epsilon}_{{}_{\bot}}^{\prime\lambda}-q_{{}_{\bot}}^{\prime}\cdot\epsilon_{{}_{\bot}}^{\prime\lambda})+D(\not\\!{P_{{}_{f_{1}}}}\not\\!\epsilon_{{}_{\bot}}^{\prime\lambda}\not\\!{q_{{}_{\bot}}^{\prime}}-\not\\!{P_{{}_{f_{1}}}}q_{{}_{\bot}}^{\prime}\cdot\epsilon_{{}_{\bot}}^{\prime\lambda})\right.$
(4) $\displaystyle+$
$\displaystyle\left.q_{{}_{\bot}}^{\prime}\cdot\epsilon_{{}_{\bot}}^{\prime\lambda}(E+F\not\\!{P_{{}_{f_{1}}}}+G\not\\!{q_{{}_{\bot}}^{\prime}}+H\not\\!{P_{{}_{f_{1}}}}\not\\!{q_{{}_{\bot}}^{\prime}})\right],$
where $\epsilon$ is the polarization vector of meson, and $A,\ B,\ C,\ D,\ E,\
F,\ G,\ H$ are defined as:
$\displaystyle A$ $\displaystyle=$ $\displaystyle
M^{\prime}\left[f_{5}(\vec{q^{\prime}})-f_{6}(\vec{q^{\prime}})\frac{\omega_{1}^{\prime}+\omega_{2}^{\prime}}{m_{1}^{\prime}+m_{2}^{\prime}}\right],$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\left[f_{6}(\vec{q^{\prime}})-f_{5}(\vec{q^{\prime}})\frac{m_{1}^{\prime}+m_{2}^{\prime}}{\omega_{1}^{\prime}+\omega_{2}^{\prime}}\right],$
$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{M^{\prime}(\omega_{2}^{\prime}-\omega_{1}^{\prime})}{m_{2}^{\prime}\omega_{1}^{\prime}+m_{1}^{\prime}\omega_{2}^{\prime}}\left[f_{5}(\vec{q^{\prime}})-f_{6}(\vec{q^{\prime}})\frac{\omega_{1}^{\prime}+\omega_{2}^{\prime}}{m_{1}^{\prime}+m_{2}^{\prime}}\right],$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{\omega_{1}^{\prime}+\omega_{2}^{\prime}}{\omega_{1}^{\prime}\omega_{2}^{\prime}+m_{1}^{\prime}m_{2}^{\prime}+\vec{q^{\prime}}^{2}}\left[f_{5}(\vec{q^{\prime}})-f_{6}(\vec{q^{\prime}})\frac{\omega_{1}^{\prime}+\omega_{2}^{\prime}}{m_{1}^{\prime}+m_{2}^{\prime}}\right],$
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{\prime}+m_{2}^{\prime}}{M^{\prime}(\omega_{1}^{\prime}\omega_{2}^{\prime}+m_{1}^{\prime}m_{2}^{\prime}-\vec{q^{\prime}}^{2})}\left\\{M^{\prime
2}\left[f_{5}(\vec{q^{\prime}})-f_{6}(\vec{q^{\prime}})\frac{m_{1}^{\prime}+m_{2}^{\prime}}{\omega_{1}^{\prime}+\omega_{2}^{\prime}}\right]-\vec{q^{\prime}}^{2}\left[f_{3}(\vec{q^{\prime}})+f_{4}(\vec{q^{\prime}})\frac{m_{1}^{\prime}+m_{2}^{\prime}}{\omega_{1}^{\prime}+\omega_{2}^{\prime}}\right]\right\\},$
$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{\omega_{1}^{\prime}-\omega_{2}^{\prime}}{M^{\prime
2}(\omega_{1}^{\prime}\omega_{2}^{\prime}+m_{1}^{\prime}m_{2}^{\prime}-\vec{q^{\prime}}^{2})}\left\\{M^{\prime
2}\left[f_{5}(\vec{q^{\prime}})-f_{6}(\vec{q^{\prime}})\frac{m_{1}^{\prime}+m_{2}^{\prime}}{\omega_{1}^{\prime}+\omega_{2}^{\prime}}\right]-\vec{q^{\prime}}^{2}\left[f_{3}(\vec{q^{\prime}})+f_{4}(\vec{q^{\prime}})\frac{m_{1}^{\prime}+m_{2}^{\prime}}{\omega_{1}^{\prime}+\omega_{2}^{\prime}}\right]\right\\},$
$\displaystyle G$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{M^{\prime}}\left[f_{3}(\vec{q^{\prime}})+f_{4}(\vec{q^{\prime}})\frac{m_{1}^{\prime}+m_{2}^{\prime}}{\omega_{1}^{\prime}+\omega_{2}^{\prime}}\right]-\frac{2f_{6}(\vec{q^{\prime}})M^{\prime}}{m_{2}^{\prime}\omega_{1}^{\prime}+m_{1}^{\prime}\omega_{2}^{\prime}}\right\\},$
$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{M^{\prime
2}}\left\\{\left[f_{3}(\vec{q^{\prime}})\frac{\omega_{1}^{\prime}+\omega_{2}^{\prime}}{m_{1}^{\prime}+m_{2}^{\prime}}+f_{4}(\vec{q^{\prime}})\right]-2f_{5}(\vec{q^{\prime}})\frac{M^{\prime
2}(\omega_{1}^{\prime}+\omega_{2}^{\prime})}{(m_{1}^{\prime}+m_{2}^{\prime})(\omega_{1}^{\prime}\omega_{2}^{\prime}+m_{1}^{\prime}m_{2}^{\prime}+\vec{q^{\prime}}^{2})}\right\\}.$
(5)
where $M^{\prime}$ is the mass of $B^{*+}$, eigenvalue wave functions
$f_{i}(\vec{q^{\prime}})$ can be obtained by solving the full $1^{-}$ state
Salpeter equations.
In calculation of transition matrix element and solving the full Salpeter
equation, there are some parameters have to be fixed, the input parameters are
chosen as follows [7]: $m_{b}=5.224$ GeV, $m_{c}=1.7553$ GeV, $m_{d}=0.311$
GeV, $m_{u}=0.307$ GeV. The values of the decay constants we use in this
letter are $f_{\pi^{\pm}}=0.1307$ GeV, $f_{\pi^{0}}=0.13$ GeV [12]. With the
parameters, the masses of the radial excited 2$S$ states are present:
$M_{B^{0}(2S)}=5.777$ GeV, $M_{B^{+}(2S)}=5.774$ GeV, $M_{D^{0}(2S)}=2.390$
GeV and $M_{D^{+}(2S)}=2.393$ GeV. The numerical strong decay widths of
$B(2S)$ and $D(2S)$ mesons are shown in Table 1.
In our results only the $1^{-}0^{-}$ final states are calculated ($B^{*}\pi$
and $D^{*}\pi$), in our estimate of mass spectra, there are no other OZI
allowed strong decay channels. For example, from the analysis of quantum
number, there may be the decay channels with $P$ wave in the final states, for
example, the final state can be $0^{+}0^{-}$ $B(1P)\pi$ states, but due to our
estimate the mass of lightest $P$ wave $0^{+}$ state $m_{B(1P)}=5.665$ GeV
[13], which is larger than the threshold of $B(2S)$ (the same results for
$D(2S)$ cases), so there is no phase space for this channel, if later
experimental discovery of mass of this state is lower than theoretical
estimate like happened to $D_{s0}(2317)$, which has been hoped much higher
than $2317$ MeV, this channel become a OZI allowed one, but because the phase
space is very small, and it is a $P$ wave, the transition decay width should
be smaller than the case when it is $S$ wave, so we can ignore the
contributions of these channels and other electroweak channels, and we use
these OZI allowed decay widths to estimate the full decay width of this 2$S$
state.
Table 1: The strong decay widths of the 2$S$ state $B$ and $D$ mesons. Mode | $\Gamma$ (MeV) | Mode | $\Gamma$ (MeV)
---|---|---|---
$B^{0}(2S)\rightarrow B^{*+}\pi^{-}$ | 12.3 | $D^{0}(2S)\rightarrow D^{*+}\pi^{-}$ | 5.48
$B^{0}(2S)\rightarrow B^{*0}\pi^{0}$ | 12.1 | $D^{0}(2S)\rightarrow D^{*0}\pi^{0}$ | 5.85
$B^{+}(2S)\rightarrow B^{*0}\pi^{+}$ | 11.7 | $D^{+}(2S)\rightarrow D^{*0}\pi^{+}$ | 6.05
$B^{+}(2S)\rightarrow B^{*+}\pi^{0}$ | 12.0 | $D^{+}(2S)\rightarrow D^{*+}\pi^{0}$ | 5.80
This work was supported in part by the National Natural Science Foundation of
China (NSFC) under Grant No. 10875032, and in part by SRF for ROCS, SEM.
## References
* [1] BABAR Collaboration, B. Aubert $et~{}al$, Phys. Rev. Lett. 90 (2003) 242001; Belle Collaboration, P. Krokovny $et~{}al$, Phys. Rev. Lett. 91 (2003) 262002.
* [2] CDF Collaboration, T. Aaltonen $et~{}al$, Phys. Rev. Lett. 100 (2008) 082001; D0 Collaboration, V. M. Abazov $et~{}al$, arXiv: 0711.0319.
* [3] R.N. Cahn, J.D. Jackson, Phys. Rev. D68 (2003) 037502; T. Barnes, F.E. Close, H.J. Lipkin, Phys. Rev. D68 (2003) 054006; E. Beveren, G. Rupp, Phys. Rev. Lett. 91 (2003) 012003; H.-Y. Cheng, W.-S. Hou, Phys. Lett. B566 (2003) 193; W.A. Bardeen, E.J. Eichten, C.T. Hill, Phys. Rev. D68 (2003) 054024; A.P. Szczepaniak, Phys. Lett. B567 (2003) 23; S. Godfrey, Phys. Lett. B568 (2003) 254; P. Clangelo, F. De Fazio, Phys. Lett. B570 (2003) 180; Y.-B. Dai, C.-S. Huang, C. Liu, S.-L. Zhu, Phys. Rev. D68 (2003) 114011; T.E. Browder, S. Pakvasa, A.A. Petrov, Phys. Lett. B578 (2004) 365; E.E. Kolomeitsev, M.F.M. Lutz, Phys. Lett. B582 (2004) 39\.
* [4] Z.-G. Luo, X.-L. Chen, X. Liu, S.-L. Zhu, Eur. Phys. J. C60 (2009) 403; Z.-G. Luo, X.-L. Chen, X. Liu, Phys. Rev. D79 (2009) 074020; Z.-G. Wang, Chin. Phys. Lett. 25 (2008) 3908; Z.-G. Wang, Phys. Rev. D77 (2008) 054024; X.-H. Zhong, Q. Zhao, Phys. Rev. D78 (2008) 014029.
* [5] E.E. Salpeter, H.A. Bethe, Phys. Rev. 84 (1951) 1232.
* [6] E.E. Salpeter, Phys. Rev. 87 (1952) 328.
* [7] C.S. Kim, G.-L. Wang, Phys. Lett. B584 (2004) 285.
* [8] C.-H. Chang, C.S. Kim, G.-L. Wang, Phys. Lett. B623 (2005) 218\.
* [9] S. Mandelstam, Proc. R. Soc. London 233 (1955) 248.
* [10] C.-H. Chang, J.-K. Chen, G.-L. Wang, Commun. Theor. Phys. 46 (2006) 467.
* [11] G.-L. Wang, Phys. Lett. B633 (2006) 492.
* [12] Particle Data Group, S. Eidelman $et\ al$., Phys. Lett. B592 (2004) 1.
* [13] G.-L. Wang, Phys. Lett. B650 (2007) 15.
|
arxiv-papers
| 2010-01-12T01:19:34 |
2024-09-04T02:49:07.676510
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jin-Mei Zhang and Guo-Li Wang",
"submitter": "Guo-Li Wang",
"url": "https://arxiv.org/abs/1001.1773"
}
|
1001.1796
|
# A theory for magnetic-field effects of nonmagnetic organic semiconducting
materials
X. R. Wang Physics Department, The Hong Kong University of Science and
Technology, Clear Water Bay, Hong Kong SAR, China S. J. Xie School of
Physics, National Key Laboratory of Crystal Materials, Shandong University,
Jinan, P. R. China
###### Abstract
A universal mechanism for strong magnetic-field effects of nonmagnetic organic
semiconductors is presented. A weak magnetic field (less than hundreds mT) can
substantially change the charge carrier hopping coefficient between two
neighboring organic molecules when the magnetic length is not too much longer
than the molecule-molecule separation and localization length of electronic
states involved. Under the illumination of lights or under a high electric
field, the change of hopping coefficients leads also to the change of polaron
density so that photocurrent, photoluminescence, electroluminescence,
magnetoresistance and electrical-injection current become sensitive to a weak
magnetic field. The present theory can not only explain all observed features,
but also provide a solid theoretical basis for the widely used empirical
fitting formulas.
###### pacs:
72.80.Le, 73.43.Qt, 75.47.-m, 85.65.+h
One of the long-termmerrifield ; williams ; wohlgenannt ; wohlgenannt1
unsolved fundamental issues in organic physics is the mechanism behind the
strong (a few per cent) responses of electrical and optical properties of
nonmagnetic organic semiconductors to a weak magnetic field (less than
hundreds mT), known as organic magnetic-field effect (OMFE). The recent
revival interest in OMFE on the magnetoresistance, photoluminescence,
photocurrent, electroluminescence, and electrical-injection current in
nonmagnetic organic semiconductors is largely due to its importance in
fundamental science and technology applicationshu . Firstly, there is a belief
that the OMFE can be used as a powerful experimental tool to probe useful and
non-useful excited processes of organic materials. Secondly, the OMFE can be
used to develop new multifunctional organic devices for information, sensing
and energy technologiesshi . Experiments showed that OMFE has following
surprising yet universal features. 1) The OMFE appears in vast different
organic semiconductors without any magnetic elements at room temperature
although the possible energy level shifts due to the presence of a magnetic
field are orders magnitude smaller than the thermal energy and other energy
scales. 2) The electroluminescence, photocurrent, photoluminescence, and
electrical-injection current are very sensitive to weak magnetic field with
both positive and negative OMFE though positive OMFE (or negative
magnetoresistance (MR) in the convention terminology) at very weak field is
typically observed. 3) The OMFE can often be fitted by two empirical formulas:
$[B/(B+B_{0})]^{2}$ and $B^{2}/(B^{2}+B_{0}^{2})$wohlgenannt1 , where $B$ is
the applied magnetic field. In the theoretical side, it is knownwohlgenannt ;
wohlgenannt1 that familiar MR mechanisms such as Lorentz force, conventional
hopping MR, electron-electron interaction and weak localization are highly
unlikely to be the cause behind the OMFE. The current belief in the community
is that the MEFS is intimately tied to spin physics involving spin
configuration, spin correlation, and spin fliphu . However, there is no
convincing arguments why an extremely small Zeeman energy can beat other much
larger energy scales in controlling electron spin dynamics to generate this
OMFE. Debate on whether excitons, biexcitons, polarons, or bipolarons are the
cause of the OMFE has been going on for many years with more confusions than
any conclusionKalinowski ; Prigodin ; Hu2 ; Robbert ; Majumdar . Even the
studies on the possible roles of the hyperfine and spin-orbital interactions
in the OMFE can hardly provide any clue to the answer to this standing puzzle
why a field of less than hundreds mT (including contributions from hyperfine
and spin-orbital interactions) can produce such big magnetic-field effects at
room temperature. Both extensive experimental and theoretical studies so far
are suggesting a novel explanation is needed. This new MR mechanism should
explain not only all OMFE features, but also why the similar effects does not
appear in the usual inorganic nonmagnetic semiconductors. In this report, we
present such a theory that does not explicitly rely on the electron spin
degrees of freedom. It is showed that the OMFE originates from the substantial
change of electron hopping coefficient in a magnetic field because of narrow
bandwidth of nonmagnetic organic semiconductors and large (comparable to the
magnetic length) molecule-molecule separation and localization length.
Nonmagnetic organic semiconductors have a few distinct properties that their
inorganic counterparts do not have. Instead of covalent chemical bonding,
organic molecules in nonmagnetic organic semiconductors are bonded by the Van
der Waals force so that the electron hopping coefficient between nearby
molecules is small at tenths eV, resulting in very narrow band instead of
order of 10eV bandwidth for their inorganic counterparts Troisi . The
intramolecular excitons have strong binding energies of order of eVhu . On the
other hand, the electron and hole become polaron pair when they are located on
different molecules because the intermolecular exciton binding energy is
smaller than the thermal energyhu . The electrical properties of an organic
semiconductor are mainly determined by the motion of polarons since the motion
of excitons does not contribute to the electric current. The singlet excitons
are, on the other hand, responsible to the luminescence due to the spin-
selection rule. A weak field should not change much of energy levels of
various excitation states so that their populations at thermal equilibrium
should not be sensitive to a magnetic field since they are given by the Fermi-
Dirac distribution that depends only on the energy level distribution and the
temperature. Any significant change in magnetoresistance near the
quasiequilibrium state must come from the mobility change. The question is
whether a weak field of 100mT can change the mobility of polarons in organic
semiconductors. In the usual inorganic crystals with lattice constant of order
of angstroms, the answer is no. However, we are going to argue below that with
molecule-molecule separation of tens $nm$ in organic conjugated materials,
this can indeed happen.
Figure 1: Schematical draw of two organic molecules (Alq3) separated by a
distance $d$ and aligned along x-direction. Molecule 1 is centered at the
origin, and molecule 2 is centered at $(d,0,0)$. The field is assumed to be
along z-direction. $\psi_{1}$ and $\psi_{2}$ are two localized states with
localization lengths $\xi_{1}$ and $\xi_{2}$ on molecules 1 and 2,
respectively.
In order to understand why a weak magnetic field can change charge carrier
(electron and hole or polaron) mobility in an organic conjugated material, we
consider a system with two molecules separated by a distance $d$ as
schematically shown in Fig. 1. One-electron Hamiltonian in a magnetic field
can in general be described by
$H=-\frac{1}{2m}(\vec{p}-\frac{e}{c}\vec{A})^{2}+V_{1}+V_{2}$ (1)
where $V_{1}$ and $V_{2}$ are the potential created by molecules 1 and 2,
respectively. $\vec{A}$ is the vector potential due to magnetic field
$\vec{B}$. For the simplicity and clarity, we shall assume that the two
molecules are aligned along x-direction, the field is along the z-direction
(pointing out of the paper). The important quantity for electron transport is
the tunneling matrix element between two molecules. When an electron tunnels
from an initially occupied state, say $\psi_{1}$ of molecule $\bf 1$, to empty
state $\psi_{2}$ of molecule $\bf 2$ with tunneling matrix $t$ , it will
contribute to the hopping probability $P$ (per unit time), proportional to
$|t|^{2}\exp(-\Delta\epsilon_{12}/(KT))$, where $\Delta\epsilon_{12}$
describes the relative energy level with respect to the Fermi levelboris ;
bottger . The hopping conduction can be regarded as an electron diffusion
process in which an electron undergoes a Brownian motion from one molecule to
another, and the diffusion constant $D$ relates to $P$ as $D=Pd^{2}$, where
$d$ should be regarded as the average distance between two neighboring
molecules. According to the Einstein relation, the electron mobility $\mu$ is
given by $\mu=eD/(KT)$ which is related to the conductivity in the
conventional waybottger . Therefore, we can concentrate on how the tunneling
matrix element depends on the magnetic field in order to study the
magnetoresistance of the system.
In the tight-binding approximationlandau , one of the authors in an early
publicationxrw has generalized the Bardeen’s transfer matrix formalism to
high dimension and in the presence of a magnetic field. In 3D, it is
$t=\frac{\hbar^{2}}{m}\int[(\psi^{\star}_{1}\frac{\partial\psi_{2}}{\partial
x}-\psi_{2}\frac{\partial\psi^{\star}_{1}}{\partial
x})-\frac{2i}{\phi_{0}}(\vec{A}\cdot\hat{x})\psi^{\star}_{1}\psi_{2}]|_{x=\frac{d}{2}}dydz,$
(2)
where $\phi_{0}=c\hbar/e$ is the flux quanta. The integration is over the
plane of $x=d/2$. For small $\vec{A}$ when the magnetic length
$l_{B}=\sqrt{\phi_{0}/(\pi B)}$ is bigger than $d$, magnetic confinement that
is responsible for the exponential increase of resistance in the usual hopping
conduction can be neglected and $\psi_{1}$ and $\psi_{2}$ do not depends on
$B$ to the zero order approximation. Then the magnitude of the field-
independent part of $t$ is order of
$\frac{\hbar^{2}}{m\xi}\int\psi^{\star}_{1}\psi_{2}|_{x=\frac{d}{2}}dydz$
while that of the field dependent part is about
$\frac{\hbar^{2}}{m}\frac{d}{l_{B}^{2}}\int\psi^{\star}_{1}\psi_{2}|_{x=\frac{d}{2}}dydz.$
$\xi^{-1}=\xi_{2}^{-1}-\xi_{1}^{-1}$, and $\xi_{1}$ and $\xi_{2}$ are the
localization lengths of wavefunction $\psi_{1}$ and $\psi_{2}$, respectively.
The two terms are comparable if $\xi d/l_{B}^{2}$ is not too small. For an
organic semiconductor with $d=20nm$ and $\xi\simeq d$ in a field of $B=10mT$,
$d/l_{B}$ is about $0.1$. Thus one shall expect an increase of $t$ in the
field by $1\%$, same order of experimentally observed OMFE. For $B=100mT$,
$d/l_{B}$ is about $0.33$ and $t$ increases by $10\%$! In reality, $\xi$
should be much bigger than $d$, especially when the hopping involves higher
excited states as it is the case in photon-involved processes. Then the field-
induced hopping coefficient could be even bigger than above estimated value,
resulting in even bigger OMFE.
Due to the Van der Waals bonding, the electric and optical signals of organic
semiconductors are too small to be detected without the illumination of a
light or applying a high electric field. This is why the OMFE are measured
under an optical injection of carriers (photoluminescence and photocurrent
measurements) or an electric field above a threshold (electric injection
current and electroluminescence measurements). When an organic semiconductor
is under the illumination of a light or under a high electric field, the field
dependent $t$ will also lead to a field dependence of polaron density. Take
optical injection of carriers as an example, under the illumination of a
light, an electron in a highest occupied molecular orbit (HOMO) absorbs a
photon and jumps to a higher empty molecular orbit of the same molecule. As
schematically illustrated in Fig. 2, the excited electron can either dump its
excessive kinetic energy to the other degrees of freedom of the system and
form an exciton with the hole left behind or jumps to neighboring molecules
and become polarons. Depending on the relative probabilities of excited
electrons (holes) staying in the same molecules and jumping to different
molecules, the polaron density shall vary with the illumination intensity. Let
us denote the probability (per unit time) of a pair of electron and hole
forming an exciton in the same molecule by $P_{0}\sim\hbar/\tau$, where $\tau$
is the typical time for a pair of electron and hole to form an exciton.
$P_{0}$ is not sensitive to a weak field since the field cannot change much
molecule orbits that determine $P_{0}$. Then the polaron generation rate per
unit volume is $JP/(P_{0}+P)$ where $J$ is the photon absorption rate per unit
volume and $P\propto|t|^{2}$ is the intermolecular hopping probability.
Without the illumination of a light, polaron density shall reach its
equilibrium density $n_{0}^{\prime}$ at a rate of $\gamma(n-n_{0}^{\prime})$,
where $\gamma$ is polaron decay rate. At balance,
$JP/(P_{0}+P)=\gamma(n-n_{0}^{\prime})$, thus the photon-generated polaron
density $n$ should be $n_{0}^{\prime}+JP/[\gamma(P_{0}+P)]$. Clearly,
$B$-dependence of $P$ results in a $B-$dependence of polaron density.
Figure 2: Schematic illustration of polaron and exciton formation after a pair
of electron and hole is created by a photon absorption. The excited electron-
hole pair has probability $P$ jumping to the neighboring molecules to form
positive-charged and negative-charged polarons, and probability $P_{0}$ to
form an exciton.
It is an experimental fact that the OMFE can often be fitted by two empirical
functions $B^{2}/(B^{2}+B_{0}^{2})$ and $[B/(B+B_{0})]^{2}$ wohlgenannt1 . A
correct theory should be able to explain why this is the case. A natural
question is whether the present picture can provide a base for these
functions. According to Eq. (2), $t$ take a form of $B_{0}+iaB$ with $B_{0}$
and $a$ real and field-independent parameters if $\psi_{1}$ and $\psi_{2}$ are
real functions. This is the case when the molecule orbits involved in hopping
are localized or not degeneratedlandau . In this case,
$P\propto|t|^{2}=a(B^{2}+B_{0}^{2})$ and the polaron density shall depend on
the magnetic field as $\frac{P}{\gamma(P_{0}+P)}J+n_{0}^{\prime}=n_{0}+\alpha
B^{2}/(B^{2}+B_{0}^{2})$, where $n_{0}$, $n_{0}^{\prime}$, $\alpha$ and
$B_{0}$ are B-independent parameters that depend on the molecule orbits
involved. Thus, $B^{2}/(B^{2}+B_{0}^{2})$ is a natural OMFE function for
$t=B_{0}+iaB$. Interestingly, this function also appears in the magnetization
expectation value involving the hyperfine interactions when only quantum spin
precession is considered and all other processes are neglected wohlgenannt2 .
However, it is puzzle to us how one relates z-component spin to the resistance
and how the spin polarization can survive under the huge thermal interaction.
Surprisingly, the second type of empirical function can also naturally appear
in when $t$ takes a form of $i(B_{0}+aB)$. According to Eq. (2), this can
happen when the spatial derivatives of $\psi_{1}$ or $\psi_{2}$ are the
functions multiplied by pure imaginary numbers. Of course, this must
correspond to degenerated states. In this case, the leading term in the
polaron density takes a form of $[B/(B+B_{0})]^{2}$ in a similar argument when
$P\gg P_{0}$. In reality, electron (polaron) hopping between two organic
molecules should involve many molecule orbits, especially in photophysical
processes and in a high electric field. One then needs to add contributions
from all hopping events. Thus, it is likely that both
$B^{2}/(B^{2}+B_{0}^{2})$ and $[B/(B+B_{0})]^{2}$ processes are presented, and
OMFE should then be fitted by the linear combinations of these two functions,
consistent with experimental findings.
The novel mechanism is very robust against the temperature and other
variations. At the room temperature, the transport of charge carriers will
involve many different molecule orbits. Each hopping event will subject to the
influence of this mechanism as long as magnetic confinement is negligible
($l_{B}>d$) and $d\xi/l_{B}^{2}$ is not too small (order of 1). Of course, one
needs to take thermal average over all hopping events. In real experiments,
organic semiconductors are highly unlikely to have well defined crystal
structures due to the nature of organic molecules. It is then reasonable to
assume that molecule-molecule orientation of samples with a large number of
molecules is quasi-random, meaning isotropic at large length scale, and the
magnetic field can be along any direction with respect to the molecule-
molecule bond instead of perpendicular direction as assumed in above
discussion. This explains why OMFE is not sensitive to the field direction in
devices. According to Eq. (2), different angle between the field and molecule-
molecule bond leads to different hopping coefficient. It should also be
emphasized that the mechanism present here does not depend on electron spins,
and it does not require large energy splits of different spin configurations.
Obviously, the picture is equally applicable to both bipolar and hole-only (or
electron-only) devices. Differ from the previous theories that try to relate
the OMFE with the electronic structure (either charge or spin state) changes,
the present theory attributes the OMFE to the change of electron hopping
coefficient in a field. Thus, it does not have all the troubles as those spin-
dynamics related theories involve concepts of excitons and bipolarons
Kalinowski ; Prigodin ; Hu2 ; Robbert ; Majumdar ; Sheng ; Rybicki .
The strong OMFE in nonmagnetic organic materials is the consequences of
combined effects of the Van der Waals bonding between organic molecules, a
large molecule-molecule separation and localization length. In the usual
inorganic semiconductor, all these conditions are not satisfied. Firstly, the
bonding between atoms are covalent so that the field-independent hopping
coefficient is order of several eV, much bigger than the field-related
contribution. Secondly, in the hopping conduction of the conventional doped
semiconductors, the localization length of a localized state and the hopping
distance are order of angstroms so that $\xi d/l_{B}^{2}$ is many orders
magnitude smaller than that in the organic semiconductors in the weak field.
In a strong field $l_{B}<d$, the opposite regime of OMFE phenomenon, magnetic
confinement dominates electron hopping, and the resistance increases
exponentially with the magnetic field. This is why the similar magnetic-field
effects had not been observed in inorganic semiconductors. Whether the
mechanism presented here is genuine or not is subject to the experimental
tests. Thus, we are proposing to design and manufacture devices with various
$\xi d/l_{B}^{2}$ by using different materials and molecule structures. It
would be a definite proof of present theory if all devices show OMFE when
$d/l_{B}<1$ and $\xi d/l_{B}^{2}$ is order of 1.
In conclusion, we present a novel mechanism for the OMFE for nonmagnetic
organic semiconductors. The mechanism is very general and robust for organic
semiconductors, but is normally not important for usual covalently bonded
inorganic semiconductors. The mechanism can not only explain all
experimentally observed OMFE, but also provide a solid theoretical bases for
the empirical OMFE formulas. New experiments are needed to firmly establish
this mechanism as the genuine cause of the OMFE.
This work is supported by Hong Kong UGC grants (#604109, HKUST17/CRF/08, and
RPC07/08.SC03). S.J. Xie is supported by the National Basic Research Program
of China (Grant No.2009CB929204 and No.2010CB923402) and the National Natural
Science Foundation of the People’s Republic of China (Grant No. 10874100)
## References
* (1) V. Ern and R. E. Merrifield, Phys. Rev. Lett. 21, 609 (1968).
* (2) H. P. Schwob and D. F. Williams, Bull Am. Phys. Soc. 17, 661 (1972).
* (3) T. L. Francis, Ö. Mermer, G. Veeraraghavan, and M. Wohlgenannt, New J. of Phys. 6, 185 (2004).
* (4) Ö. Mermer, G. Veeraraghavan, T. L. Francis, Y. Sheng, D. T. Nguyen, M. Wohlgenannt, A. Kohler, M. K. Al-Suti, and M. S. Khan, Phys. Rev. B 72, 205202 (2005).
* (5) B. Hu, L. Yan and M. Shao, Adv. Mater. 21, 1500 (2009).
* (6) Z. H. Xiong, D. Wu, Z. V. Vardeny, and J. Shi, Nature 427, 821 (2004).
* (7) J. Kalinowski, M. Cocchi, D. Virgili, P. Di Marco and V. Fattori, Chem. Phys. Lett.380, 710 (2003).
* (8) V. Prigodin,J. Bergeson, D. Lincoln and A. Epstein, Synth. Metals 156, 757 (2006).
* (9) B. Hu and Y. Wu, Nature Mater. 6, 985 (2007).
* (10) P. A. Robbert, T. D. Nguyen, F. W. A. van Oost, B. Koopmans and M. Wohlgenannt, Phys. Rev. Lett. 99, 216801 (2007).
* (11) S. Majumdar, H. S. Majumdar, H. Aarnio, D. Vanderzande, R. Laiho and R. Österbacka, Phys. Rev. B 79, 201201(R) (2009).
* (12) A. Troisi and G. Orlandi, Phys. Rev. Lett. 96, 086601 (2006).
* (13) B. I. Shklovskii and A. L. Efros, in Electronic Properties of Doped Semiconductors (Springer, Berlin, 1984).
* (14) H. Böttger and V. V. Bryksin, Hopping Conduction in Solids (VCH Publisher, 1985).
* (15) L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977).
* (16) X. R. Wang and X. C. Xie, Europhys. Lett. 38, 55 (1997); X. R. Wang, S. C. Ma, and X. C. Xie, ibid. 45, 368 (1999).
* (17) M. Wohlgenannt, arXiv:cond-mat/0609592v2.
* (18) Y. Sheng, T. D. Nguyen, G. Veeraraghavan, Ö. Mermer, M. Wohlgenannt, S. Qiu and U. Schert, Phys. Rev. B 74, 045213 (2006).
* (19) J. Rybicki and M. Wohlgenannt, Phys. Rev. B 79, 153202 (2009).
|
arxiv-papers
| 2010-01-12T05:58:36 |
2024-09-04T02:49:07.681661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X. R. Wang and S. J. Xie",
"submitter": "X.R. Wang",
"url": "https://arxiv.org/abs/1001.1796"
}
|
1001.2032
|
# Koszul duality in algebraic topology
Dev P. Sinha Department of Mathematics
University of Oregon
Eugene, OR 97403 dps@math.uoregon.edu
The most prevalent examples of Koszul duality of operads are the self-duality
of the associative operad and the duality between the Lie and commutative
operads. At the level of algebras and coalgebras, the former duality was first
noticed as such by Moore, as announced in his ICM talk at Nice [13]. Thus this
particular duality has typically been called Moore duality, and some prefer to
call the general phenomenon Koszul-Moore duality. The second duality at the
level of algebras was realized in the seminal work of Quillen on rational
homotopy theory [15]. Our aim in these notes based on our talk at the Luminy
workshop on Operads in 2009 is to try to provide some historical, topological
context for these two classical algebraic dualities.
We first review the original cobar and bar constructions used to study loop
spaces and classifying spaces, emphasizing the less-familiar geometry of the
cobar construction. Then, after some elementary topology, we state duality
between bar and cobar complexes in that setting. Before explaining Quillen’s
work, we also share some other ideas - calculations of Cartan-Serre and
Milnor-Moore and philosphy of Eckmann-Hilton - which may have influenced him.
After stating Quillen’s duality, we share some recent work which relates these
constructions to geometry through Hopf invariants and in particular linking
phenomena.
I would like to thank my collaborator Ben Walter. This material is a union of
standard material which either I taught him or he taught me along with new
theorems which we have figured out together.
## 1\. Bar and cobar constructions
### 1.1. $\Omega X$ and the cobar construction
Studying mapping spaces is one of the central tasks of topology, and loop
spaces are the simplest and most fundamental examples (unless one counts maps
from finite sets, which yield products). We require a model for loops where
the loop sum is associative exactly, not up to homotopy. Thus, for us $\Omega
X$ denotes the Moore loop space which consists of pairs $f:{\mathbb{R}}\to X$
and a “curfew” $a>0$ such that $f(x)$ is the basepoint if $x\leq 0$ or if
$x\geq a$. Loop sum adds these curfews, which makes multiplication
associative.
The cobar construction of Adams and Hilton [2] was informed by the almost
concurrent work of James [10] who studied $\Omega\Sigma X$, the loop space on
the reduced suspension of $X$, namely $\Sigma X=X\times{\mathbb{I}}/(X\times
0\smallsmile*\times{\mathbb{I}}\smallsmile 1\times X)$. There is a canonical
inclusion of $J:X\hookrightarrow\Omega\Sigma X$ sending $x$ to $J(x)(t)$, the
path which sends $t$ to the image in $\Sigma X$ of $(x,t)$. Because
$\Omega\Sigma X$ is a topological monoid, this map extends to a map from the
free moinoid (with unit) on $X$ to $\Omega\Sigma X$ which we call the James
map $\hat{J}$. For example, the formal product $y*x*z$ goes to a loop with
coordinates $(x,t)$ for $t\in[0,1]$ then $(y,t-1)$ for $t\in[1,2]$, then
$(z,t-2)$ for $t\in[2,3]$ – see the figure below.
An illustration of $\hat{J}$ of $y*x*z$ (traversing the path through $y$
first, etc).
###### Theorem 1.1 (James [10]).
The James map $\hat{J}$ from the free monoid on $X$ to $\Omega\Sigma X$ is a
homotopy equivalence.
Recall that the homology of any space with an associative multiplication, or
even a homotopy associative multiplication, is an associative algebra.
###### Corollary 1.2.
The homology of $\Omega\Sigma X$ with field coefficient is isomorphic as an
algebra to the tensor (that is, free associative) algebra on the homology of
$X$.
###### Exercise 1.
Explicitly define the free topological monoid on a (well-based) topological
space $X$. Show that its homology with field coefficients is isomorphic to the
tensor algebra on the homology of $X$. [Hint: make heavy use of the Künneth
theorem.]
Interestingly, the corollary is typically proven in the course of proof of the
theorem. Details in a well-digested form are in Section 4.J in Hatcher’s
textbook [9] or the survey paper of Carlsson and Milgram [4], whose treatment
of the Adams-Hilton construction heavily influences our treatment below. There
are however more geometric proofs which build on the fact that the space of
paths in the cone on $X$ with endpoints in the image of $X$ is homotopy
equivalent to $X\times X$ through the projection onto the endpoints.
For the Adams-Hilton construction, we start with a simply-connected simplicial
complex $\hat{X}$ and then contract the union of the $1$-skeleton along with
enough of the two-skeleton so that the quotient map $\hat{X}\to X$ is a
homotopy equivalence. Then $X$ is a CW-complex, and its cellular chains are a
quotient of the simplicial chains on $\hat{X}$. By abuse of notation, we
denote these cellular chains by $C^{\Delta}_{*}(X)$.
Next, consider the cubical singular chain complex of the loop space
$C^{\square}_{*}(\Omega X)$, which is an associative differential graded
algebra. On generators, the product of $\sigma_{1}:{\mathbb{I}}^{n}\to\Omega
X$ and $\sigma_{2}:{\mathbb{I}}^{m}\to\Omega X$ is the composite
${\mathbb{I}}^{n+m}\cong{\mathbb{I}}^{n}\times{\mathbb{I}}^{m}\overset{\sigma_{1}\times\sigma_{2}}{\to}\Omega
X\times\Omega X\to\Omega X$.
The Adams-Hilton construction defines a map of associative algebras from the
free associative algebra on $C^{\Delta}_{*}(X)$ to $C^{\square}_{*}(\Omega
X)$. The first key observation is that any choice of map
$\gamma_{n}:{\mathbb{I}}^{n}\to\Delta^{n}$ defines a map
$AH_{\gamma_{n}}:C^{\Delta}_{n}(X)\to C^{\square}_{n-1}(\Omega X)$. Let
$\chi_{\sigma}$ denote the characteristic map $\Delta^{n}\to X$ of a simplex
$\sigma$ of $X$. Then $AH_{\gamma_{n}}(\sigma)$ is basically given by the
composite $\gamma_{n}\circ\chi_{\sigma}:{\mathbb{I}}^{n}\to X$. From this
composite we by adjointness (choosing say the last coordinate as the loop
coordinate) produce a map ${\mathbb{I}}^{n-1}\to{\rm Map}({\mathbb{I}},X)$,
which then is identified with a generator of $C^{\square}_{n-1}(\Omega X)$
through viewing ${\rm Map}({\mathbb{I}},X)$ as Moore loops with curfew one.
The game is to define $\gamma_{n}$ appropriately so that we can calculate
boundaries, and more importantly so that the Adams-Hilton map yields a quasi-
isomorphism. By abuse, we suppress $\gamma_{n}$ from notation and write
$AH_{\gamma_{n}}(\sigma)$ as $|\sigma|$. For the first case when $n=2$, a good
way to choose $\gamma_{2}$ is to to have
$\gamma_{2}:{\mathbb{I}}^{2}\to\Delta^{2}$ send the boundary of
${\mathbb{I}}^{2}$ to that of $\Delta^{2}$. In any way this is done, we would
have that $d|\sigma_{2}|=|d\sigma_{2}|=0$, since the one-skeleton of $X$ is
has been identified to a point. Looking forward, it is much better to choose
$\gamma_{2}$ to be a “degree one” map ${\mathbb{I}}^{2}\to\Delta^{2}$ which
when we consider the adjoint $\hat{\gamma}_{2}:{\mathbb{I}}\to{\rm
Map}({\mathbb{I}},\Delta^{2})$ interpolates between the direct path from
vertex $0$ to vertex $2$ of $\Delta^{2}$ along the edge between them and the
“long” path from $0$ to $2$ which first traverses the $0$-$1$-edge and then
the $1$-$2$ edge. When composed with the characteristic map into $X$ these
edge paths will yield constant loops, but the choice of the paths in between
is important.
One possible choice for $\gamma_{2}$.
At the next stage, building on some such choice of degree one $\gamma_{2}$, we
can define a $\gamma_{3}$ such that $d|\sigma_{3}|=|d\sigma_{3}|$. There are
four faces of ${\mathbb{I}}^{2}$ and four faces of $\Delta^{3}$, and this
equality identifies those faces. For example, one face of ${\mathbb{I}}^{2}$
will be mapped by $\hat{\gamma}_{3}:{\mathbb{I}}^{2}\to{\rm
Map}({\mathbb{I}},\Delta^{3})$ to paths from the $0$ vertex to the vertex $3$
(reminder: such a path in $\hat{X}$ will project to a loop in $X$) which first
go to the vertex $1$ directly along the $0$-$1$ edge and then go to $3$ along
paths compatible with the choice made of $\gamma_{2}$. On another face of
${\mathbb{I}}^{2}$, paths go only along the $0$-$2$-$3$ face of $\Delta^{3}$,
again compatibly with $\gamma_{2}$, and so on.
At $n=4$ the construction there is a surprise. Assume $\gamma_{3}$ has been
defined, and start defining $\hat{\gamma_{4}}:{\mathbb{I}}^{3}\to\Delta^{4}$
by setting its restriction to various faces as before, for example sending one
face of ${\mathbb{I}}^{3}$ to paths on the $0$-$2$-$3$-$4$ face of
$\Delta^{4}$. But, there are six faces of ${\mathbb{I}}^{3}$ and only five
faces of $\Delta^{4}$! What is the natural last term? When one does the
geometry carefully, one see that on the last face of ${\mathbb{I}}^{3}$ should
map to paths from $0$ to $4$ which first go along the $0$-$1$-$2$ face and
then along the $2$-$3$-$4$ face. These two faces appear in the standard
definition of the coproduct on simplicial chains dual to cup product.
Moreover, such composites are given by the product in $C^{\square}_{*}(\Omega
X)$. That is, we can construct $\gamma_{4}$ such that
$d|\sigma_{4}|=|d\sigma_{4}|+|\alpha_{2}|*|\beta_{2}|,$
where the coproduct of $\sigma_{4}$ is $\alpha_{2}\otimes\beta_{2}$ plus terms
in bidegrees $(1,3)$, $(0,4)$, etc. (These other terms in the coproduct yield
trivial chains with the $1$-skeleton of $X$ collapsed.)
In general, let us denote products of Adams-Hilton chains by
$|\sigma|*|\tau|=|\sigma|\tau|$. Let ${\rm Cobar}(C^{\Delta}_{*}(X))$ denote
the sub-algebra of $C^{\square}_{*}(\Omega X)$ generated by the Adams-Hilton
chains (in positive degrees).
###### Theorem 1.3 (Adams-Hilton).
There are degree-one choices for the maps $\gamma_{n}$ such that the boundary
on ${\rm Cobar}(C^{\Delta}_{*}(X))$ is the cofree extension of the map with
$d|\sigma|=\pm|d\sigma|+\sum_{\bar{\Delta}\sigma=\sum\alpha_{i}\otimes\beta_{i}}\pm|\alpha_{i}|\beta_{i}|.$
Here $\bar{\Delta}$ denotes the reduced cup coproduct including terms of only
positive bidgrees.
The inclusion of any such ${\rm Cobar}\left(C^{\Delta}_{*}(X)\right)$ in
$C^{\square}_{*}(\Omega X)$ is a quasi-ismorphism of differential graded
associative algebras.
###### Exercise 2.
Try to write down $\gamma_{n}$ for $n\leq 4$ as an explicit piecewise-linear
map.
###### Exercise 3.
The cobar construction is defined for any differential graded coalgebra.
Compute it for the coalgebra given by the homology of
${\mathcal{C}}P^{\infty}$.
###### Exercise 4.
Deduce the James theorem from the Adams-Hilton theorem.
The algebraic cobar construction (often denoted $\Omega$ but not at the moment
because of potential confusion) has become part of the standard toolkit for
algebraic topologists, and there are more algebraic approaches which can yield
similar theorems. A more geometric approach to the topology of iterated loop
spaces was extended by Milgram who studied $\Omega^{n}\Sigma^{n}X$ in [11]
(see also [4]). But the geometry and formalism of PROPs and operads, in
particular the elegance of the little disks construction of Boardman and Vogt
[3], became more popular than this intricate geometry. Perhaps there could be
something gained by revisiting these ideas.
### 1.2. Classifying spaces and the bar construction
We will be more brief about the bar construction, whose topology is better
known. The topological bar construction provides a model for the classifying
space $BG$, which when $G$ is discrete is just the Eilenberg-MacClane space
$K(G,1)$.
Topologists are often ambiguous and refer to any quotient of a contractible
space with free $G$-action as its classifying space $BG$. We resolve this
issue by only saying that a space is homotopy equivalent (rather than equal
to) $BG$. As we remark below, there is a choice which is useful to call $BG$
unambiguously.
###### Example 1.4.
* •
$B{\mathbb{Z}}\simeq S^{1}$.
* •
$B{\mathbb{Z}}/2\simeq{\mathbb{R}}P^{\infty}$.
* •
$B{\mathbb{Z}}/n\simeq S^{\infty}/({\mathbb{Z}}/n)$, called an infinite Lens
space.
* •
If $G=\pi_{1}(S)$ where $S$ is a surface of positive genus, then $BG\simeq S$.
###### Theorem 1.5.
If $G$ is discrete, then $BG$ is homotopy equivalent to a simplicial complex
whose $n$-simplices are in one-to-one correspondence with $n$-tuples of
elements of $G$, which we denote $|g_{1}|g_{2}|\cdots|g_{n}|$. The $(n+1)$
faces of an $n$-simplex are given by
$d_{i}(|g_{1}|\cdots|g_{n}|)=\begin{cases}|g_{2}|\cdots|g_{n}|&i=0\\\
|g_{1}|\cdots|g_{i}g_{i+1}|\cdots|g_{n}|&0<i<n\\\
|g_{1}|\cdots|g_{n-1}|&i=n.\end{cases}$
To prove this, one constructs $EG$ in a similar fashion.
###### Corollary 1.6.
The homology of $BG$ is given by the homology of the algebraic bar
construction applied to the group ring ${\bf k}[G]$, an associative algebra.
###### Exercise 5.
Do the simple unraveling of definitions to check that this corollary follows.
We obtain a better model if we quotient by identifying each $n$-simplex of the
form $|g_{1}|\cdots|e|g_{i+1}|\cdots|g_{n}|$ with the $(n-1)$-simplex
$|g_{1}|\cdots|g_{i-1}|g_{i+1}|\cdots|g_{n}|$ through the appropriate standard
projection of $\Delta^{n}\to\Delta^{n-1}$. The following exercise is a must
for any topology student.
###### Exercise 6.
Show that this reduced construction for ${\mathbb{Z}}/2$ is homeomorphic to
${\mathbb{R}}P^{\infty}$.
Thus ${\mathbb{R}}P^{\infty}$ has ${\mathbb{Z}}/2$ as its DNA, so to speak.
Theorem 1.5 is true in greater generality in particular when $G$ has a
topology (with some mild assumptions) which gets incorporated in the topology
on $BG$, or when $G$ is just a monoid. Indeed, this construction is a special
case of the nerve of a category.
### 1.3. Relating the bar and cobar constructions
We defined the homotopy type of $BG$ through the fiber sequence
$G\subset EG\to BG.$
Let $PX$ denote the path space on $X$, which is contractible, and let $ev$
denote the map which sends a path $\gamma$ to $\gamma(1)\in X$. Then the
sequence
$\Omega X\to PX\overset{ev}{\to}X$
is a fibration. Consider as well the map $PEG\to BG$ defined by evaluation
composed by the quotient. This map is equivalent to both the projection $EG\to
BG$ and the evaluation $PBG\to BG$, which are thus equivalent to each other.
We deduce that their fibers are equivalent, so that $\Omega BG\simeq G$.
Similarly, if $X$ is connected then $B\Omega X\simeq X$ (the content of this
statement depends on the definition of classifying space for $\Omega X$; some
say its classifying space is $X$ by definition).
These homotopy equivalences are reflected in the following algebra, which is
now viewed as a consequence of Koszul duality of the associative operad.
Recall that the cobar construction was defined in terms of a free associative
algebra (and indeed computed the homology of $\Omega X$ as an algebra). We can
view the bar construction as based on the free coassociative coalgebra
generated by ${\bf k}[G]$, with the coproduct defined by breaking bar
expressions in two and differential defined using the product of $G$.
###### Theorem 1.7.
The bar construction $B$ and the cobar construction $\Omega$ define an adjoint
pair of functors between differential graded associative algebras dgaa and
differential graded associative coalgebras dgac.
dgac$\scriptstyle{\Omega}$dgaa$\scriptstyle{B}$
Moreover, there are natural transformations $\Omega BA\to A$ and $B\Omega C\to
C$ which if are quasi-isomorphisms if $A$ is positively graded and if $C$ is
$1$-connected respectively.
This theorem was announced by Moore [13], so it has historically been referred
to as Moore duality. In topology, this equivalence reflects the bijection
between homotopy classes of monoid maps from some $M$ to $\Omega X$ and
homotopy classes of maps from $BM$ to $X$.
Not only is it the first example of adjoint functors giving equivalences
between categories of algebras and coalgebras over an operad and its Koszul
dual, but it played a central role in Priddy’s definition of Koszul quadratic
algebras [14]. A graded augmented algebra $A$ can be given a zero
differential. Over a field ${\bf k}$ and with finiteness degree-wise, the
homology of the bar complex of $A$ is the linear dual of ${\rm Ext}_{A}({\bf
k},{\bf k})$, compatible with their coalgebra and algebra structures. (In the
case of $A={\bf k}[G]$, this is reflected by Corollary 1.6 and the fact that
the cohomology of $BG$ is coincides with ${\rm Ext}_{{\bf k}[G]}({\bf k},{\bf
k})$.) If $A$ is a Koszul algebra, then we can replace the bar complex with a
much smaller resolution, which leads to an explicit presentation of this ${\rm
Ext}$-algebra. Moreover, the theory applies to this ${\rm Ext}$-algebra as
well and replaces the cumbersome quasi-isomophism of $A\simeq\Omega BA$ with
an isomorphism $A\cong{\rm Ext}_{{\rm Ext}_{A}({\bf k},{\bf k})}({\bf k},{\bf
k})$.
## 2\. Other ideas in the air
Following up on his thesis, Serre along with Cartan considered the rational
homotopy groups of a simply connected space. When shifted down, as best done
by considering the homotopy groups of $\Omega X$, those groups form a graded
Lie algebra. Typically the Hurewicz homomorphism from homotopy to homology
captures little information. But rationally for loop spaces, this map gives a
clear picture. Building on calculations of Cartan and Serre [5], Milnor and
Moore in [12] prove the following.
###### Theorem 2.1.
If $X$ is simply connected, the Hurewicz map $\pi_{*}(\Omega
X)\otimes{\mathbb{Q}}\to H_{*}(\Omega X;{\mathbb{Q}})$ is an injection,
mapping the rational homotopy Lie algebra of $X$ to the primitives in the Hopf
algebra $H_{*}(\Omega X;{\mathbb{Q}})$.
Another influential idea at that time was Eckmann-Hilton “Duality,” which
draws attention to parallel structures in cohomology and homotopy. See the
table below.
Cohomology | Homotopy
---|---
L.E.S of a cofibration $A\hookrightarrow X\to X/A$ | L.E.S of a fibration $F\to E\to B$
Spheres and Moore spaces | Eilenberg-MacClane spaces
Suspension / desuspension | Loop space / classifying space
CW structures | Postnikov tower
Graded commutative ring structure | Graded Lie algebra structure
co-$H$-space (comonoid) | $H$-space (monoid)
pushout square / homotopy colimit | pull-back square / homotopy limit
Steenrod algebra | Stable homotopy groups of spheres
Leray-Serre spectral sequence | Blakers-Massey theorems
This duality is more of a philosophy than a theory. There are no theorems of
the form “Given a true statement about homotopy groups, there is a true
statement about cohomology groups obtained by…” or “Given a space $X$ there is
a dual space $\hat{X}$ whose cohomology groups are the homotopy groups of $X$
and…” Nonetheless, the duality can point to interesting directions of study.
For example, looking at our table one notices a significant difference between
CW structures, which are not canonical in any sense, and the Postnikov tower,
which is. This leads to finding the homology decomposition of a space (see
Chapter 4.H of [9] for a basic treatment).
## 3\. Quillen functors and rational homotopy theory
Quillen, influenced by Kan, took the step in [15] of proving theorems not
about homotopy groups but about all of homotopy theory. He must have taken
Theorems 2.1 and 1.3 as an important starting point. Indeed, if the rational
homology of the cobar construction computes the homology of the loop space,
and one is to then take primitives to get rational homotopy groups, why not
take primitives first at the level of the cobar complex itself (see exercise
below)? The great advantage is that in the cobar construction one is
considering the free associative algebra, whose primitives are known to be the
free Lie algebra, so one can just use the free Lie algebra functor as a
starting point. Quillen was also aware of Chevalley-Eilenberg cohomology of
Lie algebras [6], and probably knew of some cases in which applying this
functor to the rational homotopy Lie algebra of a space recovered its
cohomology (an easy case being wedges of spheres, whose rational homotopy Lie
algebra is free). Once again, a refinement is needed, going from applying a
functor at the level of algebras (in the previous case primitives, in the
current case Lie algebra cohomology) to applying it at the level of chain
complexes. Quillen’s adaption of the Chevalley-Eilenberg construction now
bears his name as well.
Quillen put these two constructions together in the following theorem.
###### Theorem 3.1.
The Lie algebraic cobar construction $\Omega_{\mathcal{L}\\!{\it ie\/}}$ and a
commutative coalgebraic bar construction $B_{\mathcal{C}\\!{\it omm\/}}$,
which generalizes the Chevalley-Eilenberg construction, form an adjoint pair
of functors
dgcc$\scriptstyle{\Omega_{\mathcal{L}\\!{\it
ie\/}}}$dgla$\scriptstyle{B_{\mathcal{C}\\!{\it omm\/}}}$
Here dgcc are $1$-connected differential graded cocommutative coalgebras and
dgla are connected differential graded Lie algebras. These functors preserve
all notions relevant to homotopy theory (fibrations, cofibrations, weak
equivalences).
Any simply-connected space $X$ has functorial models $C_{X}$ and $L_{X}$ in
dgcc and dgla respectively such that the homology of $C_{X}$ is the rational
homology coalgebra of $X$ and the homology of $L_{X}$ is the rational homotopy
Lie algebra of $X$.
In current language, we would say that $\Omega_{\mathcal{L}\\!{\it ie\/}}$ and
$B_{\mathcal{C}\\!{\it omm\/}}$ form a Quillen adjoint pair of functors on the
model categories dgcc and dgla, reflecting the Koszul duality of the operads
$\mathcal{L}\\!{\it ie\/}$ and $\mathcal{C}\\!{\it omm\/}$. This theorem gives
a precise manifestation of Eckmann-Hilton duality, through the fact that these
functors preserve model structures along with the symmetries of the model
structure axioms. What complicates [15] significantly is that there is, to
this day, no simple way to construct a commutative cochain algebra of a space
and thus easily land in this picture. Quillen has to walk for forty days
through the desert, producing a long chain of functors in order to produce
$L_{X}$ and $C_{X}$. That difficulty led Sullivan to find a simple way to
produce commutativity, not on chains but on cochains. Additionally, instead of
using bar or cobar constructions Sullivan studied cofibrant replacements with
some additional smallness property, the famous minimal models of [18].
###### Exercise 7.
Check directly in some cases that the primitives of differential graded Hopf
algebra form a split sub-complex, so that the primitives of the homology of
$C_{\bullet}$ is isomorphic to the homology of the complex obtained by taking
the primitives of $C_{\bullet}$.
###### Exercise 8.
Compute the Chevalley-Eilenberg cohomology of the graded Lie algebra with
three generators $x,y,z$ in degree three with the only relation being
$[x,y]=[y,z]$.
## 4\. Koszul duality and Hopf invariants
We have recently found [17] that Koszul duality and Quillen functors allow one
to give a definitive treatment of rational homotopy functionals through Hopf
invariants. The basic idea can be seen as using the bar complex to understand
a map $f:S^{n}\to X$ by first passing to $\Omega f:\Omega S^{n}\to\Omega X$
and then evaluating cohomology classes of $\Omega X$ on the image of the
fundamental class of $\Omega S^{n}$. By Theorem 2.1, such invariants are
complete.
We must pause to make a choice in notation. If one is studying the cohomology
of $\Omega X$ using the cochains on $X$, one could either denote the
construction you use by $\Omega$ to reflect topology or $B$ to denote a bar
construction which is applied to an algebra (rather than a cobar construction
applied to a coalgebra). The algebraists seem to have won this notational
conflict, so we consider $BC^{*}(X)$, the bar construction on the cochains of
$X$ with their associative cup product. We let $H_{B}^{*}(X)$ denote the
homology of $BC^{*}(X)$. Define the weight of a generator of a bar complex to
be the number of elements appearing.
###### Lemma 4.1.
$H^{n-1}_{B}(S^{n})$ is rank one, generated by an element of weight one
corresponding to a generator of $H^{n}(S^{n})$.
###### Exercise 9.
Prove this. Hint: you’ll need the Künneth theorem to put yourself in a
position to do some “weight reduction,” as we use below.
###### Definition 4.2.
Let $\gamma\in B^{n-1}(C^{*}S^{n})$ be a cocycle. Define
$\tau(\gamma)\simeq\gamma$ to be a choice of weight one cocycle to which
$\gamma$ is cohomologous.
Define $\int_{B(S^{n})}$ to be the map from cocyles in $B^{n-1}(C^{*}S^{n})$
to $\mathbb{Z}$ given by $\int_{B(S^{n})}\gamma=\int_{S^{n}}\tau(\gamma)$,
where $\int_{S^{n}}$ denotes evaluation on the fundamental class of $S^{n}$.
Define $\eta(\gamma)$, the Hopf invariant associated to $\gamma$ by
$\eta(\gamma)(f)=\int_{B(S^{n})}f^{*}\gamma$.
The choice of Hopf cochain is not unique, but the corresponding Hopf
invariants are. It is immediate that the Hopf invariants are functorial.
Moreover, note that the definitions hold with any ring cofficients.
Topologically we have the following interpretation.
###### Proposition 4.3.
$\eta(\gamma)(f)$ coincides with the evaluation of the cohomology class given
by $\gamma$ in $H^{n-1}(\Omega X)$ on the image under $\Omega f$ of the
fundamental class in $H_{n-1}(\Omega S^{n})$.
### 4.1. Examples
###### Example 4.4.
A cocycle of weight one in $B(C^{*}X)$ is just a closed cochain on $X$, which
may be pulled back and immediately evaluated. Decomposable elements of weight
one in $B(X)$ are null-homologous, consistent with the fact that products
evaluate trivially on the Hurewicz homomorphism.
###### Example 4.5.
Let $\omega_{1}$ and $\omega_{2}$ be generating $2$-cocycles on $S^{2}$ and
$f:S^{3}\to S^{2}$. Then $\gamma=-|\omega_{1}|\omega_{2}|$ is a cocycle in
$B(C^{*}S^{2})$ which $f$ pulls back to $-|f^{*}\omega_{1}|f^{*}\omega_{2}|$,
a weight two cocycle of total degree two on $S^{3}$. Because $f^{*}\omega_{1}$
is closed and of degree two on $S^{3}$, it is exact. Let
$d^{-1}f^{*}\omega_{1}$ be a choice of a cobounding cochain. Then
$d\left(|d^{-1}f^{*}\omega_{1}|f^{*}\omega_{2}|\right)=|f^{*}\omega_{1}|f^{*}\omega_{2}|\
+\ |d^{-1}f^{*}\omega_{1}\smallsmile f^{*}\omega_{2}|.$
Thus $f^{*}\gamma$ is homologous to $|d^{-1}f^{*}\omega_{1}\smallsmile
f^{*}\omega_{2}|$, and the corresponding Hopf invariant is
$\int_{S^{3}}{d^{-1}f^{*}\omega_{1}\smallsmile f^{*}\omega_{2}}$, which when
choosing $\omega_{1}=\omega_{2}$ is the classical formula for Hopf invariant
given by Whitehead [19]. It is a direct translation of the linking number
definition of Hopf invariant into the language of cochains. Understanding this
formula from the point of view of the bar construction has, to our knowledge,
only come over fifty years since all of these concepts were introduced.
Whitehead’s integral, viewed through intersections of supports of cochains.
###### Example 4.6.
For an arbitrary $X$ and cochains $x_{i},y_{i}$ and $\theta$ on $X$ with
$dx_{i}=dy_{i}=0$ and $d\theta=\sum(-1)^{|x_{i}|}x_{i}\smallsmile y_{i}$, the
cochain $\gamma=\sum|x_{i}|y_{i}|+|\theta|\in B(C^{*}X)$ is closed. The
possible formulae for the Hopf invariant are all of the form
$\int_{S^{n}}\left(f^{*}\theta-\sum\left((-1)^{|x_{i}|}t_{i}\cdot
d^{-1}f^{*}x_{i}\smallsmile f^{*}y_{i}+(1-t_{i})\cdot f^{*}x_{i}\smallsmile
d^{-1}f^{*}y_{i}\right)\right),$
for some real numbers $t_{i}$. This generalizes formulae given in [18], [7]
and [8].
By choosing $t=\frac{1}{2}$ we see that reversing the order to consider
$\sum|y_{i}|x_{i}|$ will yield the same Hopf invariant, up to sign. Thus
$\sum|x_{i}|y_{i}|\mp|y_{i}|x_{i}|$ yields a zero Hopf invariant. There are
many Hopf invariants of the classical bar construction which are zero, a
defect remedied by using the Lie coalgebra cobar construction.
###### Exercise 10.
Suppose $x$ and $y$ are cochains supported on codimension two submanifolds $X$
and $Y$ of $W$ and $\theta$ satisfies $d\theta=x\smallsmile y$ and is
supported on a codimension three submanifold which cobounds $X\cap Y$. Draw
pictures of how the Hopf invariant associated to $|x|y|\mp|\theta|$ evaluates
some map $f:S^{3}\to X$. Moreover, draw pictures of what can happen in
$S^{3}\times{\mathbb{I}}$ if one has a homotopy between $f$ and $g$. [Hint:
Start with the picture in the figure, but then draw in the preimage of the
support of $\theta$; then, think about what can happen with the preimage of
$X\cap Y$ through a homotopy.]
One can do similar calculations in higher weight, and interpret them all when
one chooses cochains supported on submanifolds in terms of linking behavior of
the preimages of those submanifolds. See [17].
### 4.2. The cokernel and kernel of the Hopf invariant map
Our Hopf invariant construction defines a homomorphism
$\eta:H_{*}(B(C^{*}(X;{\mathbb{Z}})))\to{\rm Hom}(\pi_{*}X,{\mathbb{Z}})$. It
follows from Proposition 4.3 and Theorem 2.1 that this map is surjective when
tensored with the rational numbers, and thus is full rank.
###### Problem 11.
Compute the cokernel of $\eta$. By Adams’ celebrated result [1], this cokernel
is trivial for $X$ an odd sphere and for $S^{2}$, $S^{4}$ and $S^{8}$, and it
is ${\mathbb{Z}}/2$ for other even spheres.
The proofs in [17] show that one might be able to directly understand the
relation of this cokernel to lack of commutativity of cup product. Though this
cokernel is clearly a very subtle homotopy invariant, we do not see any
applications of its calculation.
Also, $\eta$ has a very large kernel, explained from the operadic viewpoint as
the fact that we are taking the wrong bar construction. The rational PL
cochains on a simplicial set are commutative, so we should be taking a bar
construction over the Koszul dual cooperad, namely the Lie cooperad, rather
than associative cooperad. The homology of such a bar construction
$B_{\mathcal{L}\\!{\it ie}}$ is known as Harrison homology. Using a graphical
model for the Lie cooperad developed in [16] which makes calculations
possible, we prove the following.
###### Theorem 4.7.
[17] There is a Hopf invariant map $\eta^{\mathcal{L}\\!{\it ie}}$ which
factors the map $\eta$ such that $\eta^{\mathcal{L}\\!{\it
ie}}:H^{*}_{B_{\mathcal{L}\\!{\it ie}}}(X)\to{\rm
Hom}(\pi_{*}(X),\;{\mathbb{Q}})$ is an isomorphism of Lie coalgebras.
It is almost immediate that similar Hopf invariants can be used to concretely
realize similar isomorphisms arising for Koszul pairs in general.
To summarize, in homology theory it has been helpful to have geometry attached
not only to homology but cohomology. In particular, homology classes are often
represented by closed submanifolds and cohomology classes are represented by
either forms or proper submanifolds. The geometry of homotopy groups arising
from their definition is almost too simple. To reflect on the geometry of
Theorem 2.1, we notice that the Lie algebra generators of
$\pi_{*}(X)\otimes{\mathbb{Q}}$ have non-trivial Hurewicz image, as noticed by
Cartan-Serre. That is, the rational homotopy groups of $X$ are spanned by
Whitehead products of spherical homology classes. Our work on Hopf invariants
shows that the geometry of homotopy functionals is given by linking
invariants, as perfectly governed by the Lie cooperad, completing the
geometric understanding of these basic functors in the rational setting. We
hope these ideas can be extended to the non-simply connected setting, and
perhaps - at least in part - in characteristic $p$.
## References
* [1] J. F. Adams, _On the non-existence of elements of Hopf invariant one_ , Ann. of Math. (2) 72 (1960), 20–104. MR MR0141119 (25 #4530)
* [2] J. F. Adams and P. J. Hilton, _On the chain algebra of a loop space_ , Comment. Math. Helv. 30 (1956), 305–330. MR MR0077929 (17,1119b)
* [3] J. M. Boardman and R. M. Vogt, _Homotopy invariant algebraic structures on topological spaces_ , Springer-Verlag, Berlin, 1973, Lecture Notes in Mathematics, Vol. 347. MR MR0420609 (54 #8623a)
* [4] Gunnar Carlsson and R. James Milgram, _Stable homotopy and iterated loop spaces_ , Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 505–583. MR MR1361898 (97j:55007)
* [5] Henri Cartan and Jean-Pierre Serre, _Espaces fibrés et groupes d’homotopie. II. Applications_ , C. R. Acad. Sci. Paris 234 (1952), 393–395. MR MR0046046 (13,675b)
* [6] Claude Chevalley and Samuel Eilenberg, _Cohomology theory of Lie groups and Lie algebras_ , Trans. Amer. Math. Soc. 63 (1948), 85–124. MR MR0024908 (9,567a)
* [7] Phillip A. Griffiths and John W. Morgan, _Rational homotopy theory and differential forms_ , Progress in Mathematics, vol. 16, Birkhäuser Boston, Mass., 1981. MR MR641551 (82m:55014)
* [8] André Haefliger, _Whitehead products and differential forms_ , Differential topology, foliations and Gelfand-Fuks cohomology, Lecture Notes in Math., vol. 652, Springer, Berlin, 1978, pp. 13–24. MR MR505648 (80a:55012)
* [9] Allen Hatcher, _Algebraic topology_ , Cambridge University Press, Cambridge, 2002. MR MR1867354 (2002k:55001)
* [10] I. M. James, _Reduced product spaces_ , Ann. of Math. (2) 62 (1955), 170–197. MR MR0073181 (17,396b)
* [11] R. James Milgram, _Iterated loop spaces_ , Ann. of Math. (2) 84 (1966), 386–403. MR MR0206951 (34 #6767)
* [12] John W. Milnor and John C. Moore, _On the structure of Hopf algebras_ , Ann. of Math. (2) 81 (1965), 211–264. MR MR0174052 (30 #4259)
* [13] John C. Moore, _Differential homological algebra_ , Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 335–339. MR MR0436178 (55 #9128)
* [14] Stewart B. Priddy, _Koszul resolutions_ , Trans. Amer. Math. Soc. 152 (1970), 39–60. MR MR0265437 (42 #346)
* [15] Daniel Quillen, _Rational homotopy theory_ , Ann. of Math. (2) 90 (1969), 205–295. MR MR0258031 (41 #2678)
* [16] Dev P. Sinha and Ben Walter, _Lie coalgebras and rational homotopy theory, I_ , math.AT/0610437.
* [17] by same author, _Lie coalgebras and rational homotopy theory, II: Hopf invariants._ , arXiv:0809.5084.
* [18] Dennis Sullivan, _Infinitesimal computations in topology_ , Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR MR0646078 (58 #31119)
* [19] J. H. C. Whitehead, _An expression of Hopf’s invariant as an integral_ , Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 117–123. MR MR0020255 (8,525h)
|
arxiv-papers
| 2010-01-12T22:09:01 |
2024-09-04T02:49:07.692359
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dev Sinha",
"submitter": "Dev Sinha",
"url": "https://arxiv.org/abs/1001.2032"
}
|
1001.2035
|
# The Decays of $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
$B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$
Guo-Li Wang111gl_wang@hit.edu.cn, Jin-Mei Zhang and Zhi-Hui Wang
Department of Physics, Harbin Institute of Technology, Harbin 150001, China
> ABSTRACT
>
> We analyzed the decays of $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
> $B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$ by naive factorization method
> and model dependent calculation based on the Bethe-Salpeter method, the
> branching ratios are $Br({\bar{D}^{0}}D^{+}_{sJ}(2S))=(0.72\pm 0.12)\%$ and
> $Br({\bar{D}^{0}}D^{+}_{sJ}(1D))=(0.027\pm 0.007)\%$. The branching ratio of
> decay
> $B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S)\rightarrow{\bar{D}^{0}}D^{0}K^{+}$
> consist with the data of Belle Collaboration, so we conclude that the new
> state $D^{+}_{sJ}(2700)$ is the first excited state $D^{+}_{sJ}(2S)$.
BABAR Collaboration reported the observation of a new charmed meson called
$D^{+}_{sJ}(2690)$ with a mass $2688\pm 4\pm 3$ MeV and with a broad width
$\Gamma=112\pm 7\pm 36$ MeV [1], Belle Collaboration also reported a charmed
state $D^{+}_{sJ}(2700)$, $M=2715\pm 11^{+11}_{-14}$ MeV with width
$\Gamma=115\pm 20^{+36}_{-32}$ MeV [2], later modified to $M=2708\pm
9^{+11}_{-10}$ MeV and $\Gamma=108\pm 23^{+36}_{-31}$ MeV [3]. Most authors
believe that these two states should be one state, and the most possible
candidate is the $2S$ or the $S-D$ ($2~{}^{3}S_{1}-1~{}^{3}D_{1}$) mixing
$c\bar{s}$ $1^{-}$ state [4, 5, 6, 7, 8, 9, 10, 11].
Recently, we have analyzed the decays of the $D^{+}_{sJ}(2S)$ and
$D^{+}_{sJ}(1D)$, we concluded that one can not distinguish them from their
decays, because they have the similar decay channels and the corresponding
decay widths are comparable [12]. In this paper, we give the calculations of
the decays $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
$B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$, we find that we can separate
them, and according to the current experimental data of Belle Collaboration,
we conclude that the new state $D^{+}_{sJ}(2700)$ is the first excited state
$D^{+}_{sJ}(2S)$. These channels have also been considered in literatures, for
example, Close et al [4, 23] give branching ratios and possible mixing of $2S$
and $1D$; Colangelo et al [7] assume the new state is the $2S$ state, and
extract the decay constant.
The decay amplitude for $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}$ can be
described in the naive factorization approach:
$\displaystyle T=\frac{G_{F}}{\sqrt{2}}V_{cs}V_{cb}a_{1}\langle
D_{{}_{sJ}}^{+}|J_{\mu}|0\rangle\langle{\bar{D}^{0}}|J^{\mu}|B^{+}\rangle,$
(1)
where the CKM matrix element $V_{cs}=0.97334$ and $V_{cb}=0.0415$ [13], since
we focus on the difference between $D^{+}_{sJ}(2S)$ and $D^{+}_{sJ}(1D)$, not
on the careful study, we have chosen $a_{1}=c_{1}+\frac{1}{3}c_{2}=1$, where
$c_{1}$ and $c_{2}$ are the Wilson coefficients [14]. We also delete other
higher order contributions like the contributions from penguin operators. And
$\displaystyle\langle
D_{sJ}^{+}|J_{\mu}|0\rangle=iF^{*}_{V}M_{{D_{{}_{sJ}}^{+}}}\epsilon^{\lambda}_{\mu}~{},$
(2)
$F_{V}$ and $\epsilon^{\lambda}_{\mu}$ are the decay constant and polarization
vector of the meson $D_{{}_{sJ}}^{+}$, respectively.
We have solved the exact instantaneous Bethe-Salpeter equations [15] (or the
Salpeter equations [16]) for $1^{-}$ states, the general form for the
relativistic Salpeter wave function of vector $1^{-}$ state can be written as
[17, 18]:
$\varphi_{1^{-}}^{\lambda}(q_{\perp})=q_{\perp}\cdot{\epsilon}^{\lambda}_{\perp}\left[f_{1}(q_{\perp})+\frac{\not\\!P}{M}f_{2}(q_{\perp})+\frac{{\not\\!q}_{\perp}}{M}f_{3}(q_{\perp})+\frac{{\not\\!P}{\not\\!q}_{\perp}}{M^{2}}f_{4}(q_{\perp})\right]+M{\not\\!\epsilon}^{\lambda}_{\perp}f_{5}(q_{\perp})$
$+{\not\\!\epsilon}^{\lambda}_{\perp}{\not\\!P}f_{6}(q_{\perp})+({\not\\!q}_{\perp}{\not\\!\epsilon}^{\lambda}_{\perp}-q_{\perp}\cdot{\epsilon}^{\lambda}_{\perp})f_{7}(q_{\perp})+\frac{1}{M}({\not\\!P}{\not\\!\epsilon}^{\lambda}_{\perp}{\not\\!q}_{\perp}-{\not\\!P}q_{\perp}\cdot{\epsilon}^{\lambda}_{\perp})f_{8}(q_{\perp}),$
(3)
where the $P$, $q$ and ${\epsilon}^{\lambda}_{\perp}$ are the momentum,
relative inner momentum and polarization vector of the vector meson,
respectively; $f_{i}(q_{\perp})$ is the function of $-q_{\perp}^{2}$, and we
have used the notation $q^{\mu}_{\perp}\equiv q^{\mu}-(P\cdot q/M^{2})P^{\mu}$
(which is $(0,~{}\vec{q})$ in the center of mass system).
The $8$ wave functions $f_{i}$ are not independent, there are only $4$
independent wave functions [18], and we have the relations
$f_{1}(q_{\perp})=\frac{\left[q_{\perp}^{2}f_{3}(q_{\perp})+M^{2}f_{5}(q_{\perp})\right](m_{1}m_{2}-\omega_{1}\omega_{2}+q_{\perp}^{2})}{M(m_{1}+m_{2})q_{\perp}^{2}},~{}~{}~{}f_{7}(q_{\perp})=\frac{f_{5}(q_{\perp})M(-m_{1}m_{2}+\omega_{1}\omega_{2}+q_{\perp}^{2})}{(m_{1}-m_{2})q_{\perp}^{2}},$
$f_{2}(q_{\perp})=\frac{\left[-q_{\perp}^{2}f_{4}(q_{\perp})+M^{2}f_{6}(q_{\perp})\right](m_{1}\omega_{2}-m_{2}\omega_{1})}{M(\omega_{1}+\omega_{2})q_{\perp}^{2}},~{}~{}~{}f_{8}(q_{\perp})=\frac{f_{6}(q_{\perp})M(m_{1}\omega_{2}-m_{2}\omega_{1})}{(\omega_{1}-\omega_{2})q_{\perp}^{2}}.$
In our method, the $S-D$ mixing automatically exist in the wave function of
$1^{-}$ state, because we give the whole wave function Eq. (3) which is
$J^{P}=1^{-}$, but some of the wave function are $S$ wave, some are $D$ wave,
for example, see Figure 1-3, we show wave functions. One can see that, for
$1S$ and $2S$, the wave function $f_{5}$ and $f_{6}$ are dominate, and they
are $S$ wave, but there is little $D$ wave mixing in these two states which
come from the terms $f_{3}$ and $f_{4}$. But for the third state, we labeled
as $1D$ state, the terms $f_{3}$ and $f_{4}$ are dominate, but these two terms
are not pure $D$ wave, they will give contribute as a $S$ wave [17]. So we
conclude that the $1S$ and $2S$ states are $S$ wave dominate states, mixed
with a little $D$ wave (come from the terms $f_{3}$ and $f_{4}$), while the
$1D$ state is a $D$ wave dominate state ($f_{3}$ and $f_{4}$), mixed with a
valuable part of $S$ wave (still come from the terms $f_{3}$ and $f_{4}$).
Figure 1: wave functions of $D_{s}^{*}(1S)$. Figure 2: wave functions of
$D_{s}^{*}(2S)$. Figure 3: wave functions of $D_{s}^{*}(1D)$.
For $c\bar{s}$ vector $1^{-}$ state, our mass prediction for the first radial
excited $2S$ state is $2673$ MeV, and for $1D$, our result is $2718$ MeV [18],
so there are two states around $2700$ MeV.
In Ref. [18], we only give the leading order calculation for decay constant,
which is
$F_{V}=4\sqrt{N_{c}}\int\frac{d{\vec{q}}}{(2\pi)^{3}}f_{5}({\vec{q}})$, the
whole equation should be
$F_{V}=4\sqrt{N_{c}}\int\frac{d{\vec{q}}}{(2\pi)^{3}}(f_{5}-\frac{{\vec{q}}^{2}f_{3}}{3M^{2}}),$
(4)
and our results of the decay constants for vector $c\bar{s}$ system are:
$F_{V}(1S)=353\pm 21~{}~{}\rm MeV,$ $F_{V}(2S)=295\pm 13~{}~{}\rm MeV,$
$F_{V}(1D)=57.1\pm 5.1~{}~{}\rm MeV.$
Where the uncertainties are given by varying all the input parameters
simultaneously within $\pm 5\%$ of the central values in our model, and we
will calculate all the uncertainties this way in this letter. The center
values of $S$ waves are little smaller than the estimates in Ref. [18], where
$F_{V}(1S)=375\pm 24~{}\rm MeV$ and $F_{V}(2S)=312\pm 17~{}\rm MeV$, but the
value of $D$ wave decay constant is much smaller than the predict in Ref. [18]
where we did not shown it, this is because we should not ignore the
contribution from the term of $f_{3}$ when consider a $D$ wave state. Our
estimate of $F_{V}(1S)=353\pm 23~{}\rm MeV$ is close to the newer result
$F_{D^{*}_{s}}=268\sim 290~{}\rm MeV$ by Choi [19]. Our estimate of
$F_{V}(2S)=312\pm 17~{}\rm MeV$ is a little larger than the estimate
$F_{V}(2S)=243\pm 41~{}\rm MeV$ in Ref. [7], where they extracted it from the
decay $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2700)$ assuming the new state
$D^{+}_{sJ}(2700)$ is the first radial excited state $D^{+}_{sJ}(2S)$.
According to the Mandelstam formalism [20], at the leading order, the
transition amplitude for $B^{+}\rightarrow\bar{D^{0}}$ can be written as [21]:
$\displaystyle\langle{\bar{D}^{0}}|J^{\mu}|B^{+}\rangle=\int\frac{d{\vec{q}}}{(2\pi)^{3}}Tr\left[\bar{\varphi}^{++}_{{}_{\bar{D}^{0}}}({\vec{q}}-\frac{m_{u}}{m_{c}+m_{u}}{\vec{r}})\frac{\not\\!P}{M}{\varphi}^{++}_{{}_{B^{+}}}({\vec{q}})\gamma^{\mu}(1-\gamma_{5})\right],$
(5)
where $\vec{r}$ is the recoil three dimensional momentum of the final state
$\bar{D}^{0}$ meson, ${\varphi}^{++}$ is called the positive energy wave
function, and
$\bar{\varphi}^{++}_{{}_{\bar{D}^{0}}}=\gamma_{0}({\varphi}^{++}_{{}_{\bar{D}^{0}}})^{+}\gamma_{0}$.
The wave function forms for pseudoscalar $B^{+}$ and $\bar{D^{0}}$ are
similar, for example, the wave function for $B^{+}$ can be written as [22]
$\varphi^{++}_{{}_{B^{+}}}(\stackrel{{\scriptstyle\rightarrow}}{{q}})=\frac{M_{{B^{+}}}}{2}\left(\varphi_{1}(\stackrel{{\scriptstyle\rightarrow}}{{q}})+\varphi_{2}(\stackrel{{\scriptstyle\rightarrow}}{{q}})\frac{m_{u}+m_{b}}{\omega_{u}+\omega_{b}}\right)$
$\times\left[\frac{\omega_{u}+\omega_{b}}{m_{u}+m_{b}}+{\gamma_{{}_{0}}}-\frac{{\not\\!\vec{q}}(m_{u}-m_{b})}{m_{b}\omega_{u}+m_{u}\omega_{b}}+\frac{{\not\\!\vec{q}}\gamma_{{}_{0}}(\omega_{u}+\omega_{b})}{(m_{b}\omega_{u}+m_{u}\omega_{b})}\right]\gamma_{{}_{5}}\;,$
(6)
where $\omega_{u}=\sqrt{m_{u}^{2}+{\vec{q}}^{2}}$ and
$\omega_{b}=\sqrt{m_{b}^{2}+{\vec{q}}^{2}}$; $\varphi_{1}(\vec{q})$,
$\varphi_{2}(\vec{q})$ are the radial part wave functions, and their numerical
values can be obtained by solving the full Salpter equation of $0^{-}$ state
[22].
The decay width is:
$\Gamma=\frac{1}{8\pi}\frac{|\vec{p}_{{}_{f_{2}}}|}{M^{2}_{B}}|T|^{2}$
$\displaystyle=\frac{1}{8\pi}\frac{|\vec{p}_{{}_{f_{2}}}|}{M^{2}_{B}}\frac{G^{2}_{F}V^{2}_{cs}V^{2}_{bc}}{2}F^{2}_{V}M^{2}_{f_{2}}X,$
(7)
where $\vec{p}_{{}_{f_{2}}}$ and $M_{f_{2}}$ are the three dimensional
momentum and mass of the final new state $D^{*+}_{sJ}$. $M^{2}_{f_{2}}$ come
from the definition of decay constant in Eq.(2), but the square of
polarization vector and transition amplitude of Eq.(5) are symbolized as $X$.
So our result are:
$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))=F^{2}_{V}(2S)\times(3.50\pm
0.38)\times 10^{-14}~{}\rm GeV,$ (8)
$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))=F^{2}_{V}(1D)\times(3.25\pm
0.30)\times 10^{-14}~{}\rm GeV.$ (9)
If we ignore the mass difference between $D^{*+}_{s}(2S)$ and
$D^{*+}_{s}(1D)$, and use $2700$ MeV as input, the results become
$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))=F^{2}_{V}(2S)\times(3.36\pm
0.25)\times 10^{-14}~{}\rm GeV,$ (10)
$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))=F^{2}_{V}(1D)\times(3.36\pm
0.25)\times 10^{-14}~{}\rm GeV.$ (11)
So the difference between this two channel mainly come from the difference of
decay constants. Then our predictions of branching ratios are:
$\displaystyle Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))=(0.72\pm
0.12)\%,$ (12) $\displaystyle
Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))=(0.027\pm 0.007)\%.$ (13)
In Ref. [12], we have calculated the main decay channels of $D^{+}_{sJ}(2S)$
and $D^{+}_{sJ}(1D)$, and we have the following estimates:
$Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})=0.20\pm 0.03$
and
$Br(D^{+}_{sJ}(1D)\rightarrow D^{0}K^{+})=0.32\pm 0.04,$
so we obtain:
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S))\times
Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})=(1.4\pm 0.5)\times 10^{-3}$ (14)
and
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(1D))\times
Br(D^{+}_{sJ}(1D)\rightarrow D^{0}K^{+})=(0.9\pm 0.3)\times 10^{-4}.$ (15)
Our estimate of $Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})\simeq 0.20$ is
larger than than the estimate $0.11$ in Ref. [4] and the estimate $0.05$ in
Ref. [5], if we use their value as input, we will obtain a smaller value of
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S))\times
Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})$. But our estimate of
$Br(D^{+}_{sJ}(1D)\rightarrow D^{0}K^{+})\simeq 0.32$ consist with the value
$0.34$ in Ref. [5]. One can also see that, the final branching ratios depend
strongly on the decay constants, but at this time few papers have calculated
the values of decay constants for $D^{+}_{sJ}(2S)$ and $D^{+}_{sJ}(1D)$. In
Ref. [7], they assumed that the new state $D^{+}_{sJ}(2700)$ is
$D^{+}_{sJ}(2S)$, and they extracted decay constant of $D^{+}_{sJ}(2S)$:
$F_{D^{+}_{sJ}(2S)}=243\pm 41$ MeV.
The Belle Collaboration have the data [3, 13]
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2700))\times
Br(D^{+}_{sJ}(2700)\rightarrow D^{0}K^{+})=(1.13^{~{}+0.26}_{~{}-0.36})\times
10^{-3}.$
Because we only calculated the decay widths of six main channels for
$D^{+}_{sJ}(2S)$ ($D^{+}_{sJ}(1D)$), and based on the summed width of these
six channels, not full width, we give a relative larger branching ratio
$Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})$ and $Br(D^{+}_{sJ}(1D)\rightarrow
D^{0}K^{+})$, the real branching ratios should be smaller than our estimates,
so our estimate of $B^{+}$ decay to $D^{+}_{sJ}(2S)$ is close to the data,
while the estimate of $B^{+}$ decay to $D^{+}_{sJ}(1D)$ is much smaller than
the data, then we can draw a conclusion that the new state $D^{+}_{sJ}(2700)$
is $D^{+}_{sJ}(2S)$.
We have another method to estimate the branching ratio, because the mass of
$B^{+}$ is much heavier than the mass of ${\bar{D}^{0}}$, and the mass of
$D^{+}_{sJ}(2700)$ is close to the mass of $D^{*+}_{s}$, as a rough estimate,
we ignore the mass difference of $D^{+}_{sJ}(2700)$ and $D^{*+}_{s}(2112)$,
from Eq.(1) and Eq.(2), then we have
$\frac{Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S))}{Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1S))}\simeq\frac{F^{2}_{V}(2S)}{F^{2}_{V}(1S)},$
(16)
and from Particle Data Group [13], the branching ratio of
$B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(1S)$:
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1S))=(7.8\pm 1.6)\times 10^{-3}.$
So we have:
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))\simeq\frac{F^{2}_{V}(2S)}{F^{2}_{V}(1S)}\times(7.8\pm
1.6)\times 10^{-3}\simeq(5.4\pm 1.7)\times 10^{-3},$ (17)
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))\simeq\frac{F^{2}_{V}(1D)}{F^{2}_{V}(1S)}\times(7.8\pm
1.6)\times 10^{-3}\simeq(2.0\pm 1.0)\times 10^{-4}.$ (18)
This rough estimate results are very close to our calculations (Eq.(12) and
Eq.(13)), so we have the same conclusion that $D^{+}_{sJ}(2700)$ is
$D^{+}_{sJ}(2S)$.
In Ref. [12], we estimate the full widths of $D^{+}_{sJ}(2S)$ and
$D^{+}_{sJ}(1D)$ by six main decay channels, the estimated full widths are
$46.4\pm 6.2$ MeV for $D^{+}_{sJ}(2S)$, $73.0\pm 10.4$ MeV for
$D^{+}_{sJ}(1D)$, comparing with experimental data, $\Gamma=112\pm 7\pm 36$
MeV for $D^{+}_{sJ}(2690)$ and $\Gamma=108\pm 23^{+36}_{-31}$ MeV for
$D^{+}_{sJ}(2700)$, there is still the possible that there are two states
around $2700$ MeV, one is the $S$ wave dominate $D^{+}_{sJ}(2S)$, the other is
$D$ wave dominate $D^{+}_{sJ}(1D)$.
As summary, from the decays $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
$B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$, we conclude that the new state
$D^{+}_{sJ}(2700)$ is the first radial excited state $D^{+}_{sJ}(2S)$, and
there may exist another state around $2700$, $D^{+}_{sJ}(1D)$, with a mass
around $2718$ MeV, and a width $73.0\pm 10.4$ MeV.
Acknowledgements
This work was supported in part by the National Natural Science Foundation of
China (NSFC) under Grant No. 10875032, and in part by SRF for ROCS, SEM.
## References
* [1] BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 97, (2006) 222001.
* [2] Belle Collaboration, K. Abe et al., e-Print: hep-ex/0608031.
* [3] Belle Collaboration, J. Brodzicka et al., Phys. Rev. Lett. 100, (2008) 092001.
* [4] F. E. Close, C. E. Thomas, Olga Lakhina and Eric S. Swanson, Phys. Lett. B647 (2007) 159.
* [5] Bo Zhang, Xiang Liu, Wei-Zhen Deng and Shi-Lin Zhu, Eur. Phys. J. C50, (2007) 617.
* [6] J. L. Rosner, J. Phys. G34, (2007) S127.
* [7] P. Colangelo, F. De Fazio, S. Nicotri and M. Rizzi, Phys. Rev. D77, (2008) 014012.
* [8] Takayuki Matsuki, Toshiyuki, Kazutaka Sudoh, Eur. Phys. J. A31, (2007) 701.
* [9] De-Min Li, Bing Ma, Yun-Hu Liu, Eur. Phys. J. C51, (2007) 359.
* [10] Xian-Hui Zhong, Qiang Zhao, Phys. Rev. D78, (2008) 014029.
* [11] Xue-Qian Li, Xiang Liu, Zheng-Tao Wei, e-Print: arXiv: 0808.2587.
* [12] Guo-Li Wang, Jin-Mei Zhang and Zhi-Hui Wang, submited.
* [13] Particle Data Group, Phys. Lett. B667 (2008) 1.
* [14] Matthias Neubert, Berthold Stech, Adv. Ser. Direct. High Energy Phys. 15 (1998) 294.
* [15] E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, (1951) 1232.
* [16] E. E. Salpeter, Phys. Rev. 87, (1952) 328.
* [17] Chao-Hsi Chang, Jiao-Kai Chen, Xue-Qian Li and Guo-Li Wang, Commun. Theor. Phys. 43 (2005) 113.
* [18] Guo-Li Wang, Phys. Lett. B633 (2006) 492.
* [19] Ho-Meoyng Choi, Phys. Rev. D75 (2007) 073016.
* [20] S. Mandelstam, Proc. R. Soc. London 233, 248 (1955).
* [21] Chao-Hsi Chang, C. S. Kim, Guo-Li Wang, Phys. Lett. B623 (2005) 218.
* [22] C. S. Kim and Guo-Li Wang, Phys. Lett. B584 (2004) 285.
* [23] F. E. Close and E. S. Swanson, Phys. Rev. D72 (2005) 094004.
|
arxiv-papers
| 2010-01-13T03:11:15 |
2024-09-04T02:49:07.699920
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guo-Li Wang, Jin-Mei Zhang, Zhi-Hui Wang",
"submitter": "Guo-Li Wang",
"url": "https://arxiv.org/abs/1001.2035"
}
|
1001.2063
|
arxiv-papers
| 2010-01-13T02:41:36 |
2024-09-04T02:49:07.707207
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Baishun Lai and Qing Luo",
"submitter": "Baishun Lai",
"url": "https://arxiv.org/abs/1001.2063"
}
|
|
1001.2145
|
1]Institut für Theoretische Physik und Astrophysik
Universität Würzburg
Am Hubland
97074 Würzburg
M. Weidinger
mweidinger@astro.uni-wuerzburg.de
13 January 2010
# Modelling the steady state spectral energy distribution of the BL-Lac Object
PKS 2155-30.4 using a selfconsistent SSC model
M. Weidinger M. Rüger F. Spanier [
(6 April 2009; 19 October 2009; 23 November 2009)
###### Abstract
In this paper we present a fully selfconsistent SSC model with particle
acceleration due to shock and stochastic acceleration (Fermi-I and Fermi-II-
Processes respectively) to model the quiescent spectral energy distribution
(SED) observed from PKS 2155. The simultaneous August/September 2008
multiwavelength data of H.E.S.S., Fermi, RXTE/SWIFT and ATOM give new
constraints to the high-energy peak in the SED concerning its curvature. We
find that, in our model, a monoenergetic injection of electrons at
$\gamma_{0}=910$ into the model region, which are accelerated by Fermi-I- and
Fermi-II-processes while suffering synchrotron and inverse Compton losses,
finally leads to the observed SED of PKS 2155-30.4 shown in H. E. S. S. and
Fermi-LAT collaborations (2009). In contrast to other SSC models our
parameters arise from the jet’s microphysics and the spectrum is evolving
selfconsistently from diffusion and acceleration. The $\gamma_{0}$-factor can
be interpreted as two counterstreaming plasmas due to the motion of the blob
at a bulk factor of $\Gamma=58$ and opposed moving upstream electrons at
moderate Lorentz factors with an average of $\gamma_{u}\approx 8$.
[INTRODUCTION] Among the class of active galactic nuclei (AGN), blazars are
showing a spectral energy distribution (SED) that is strongly dominated by
nonthermal emission across a wide range of wavelengths, from radio waves to
gamma rays, and rapid, large-amplitude variability. Presumably, these
characteristics are due to a highly relativistic jet which covers a small
angle to the line-of-sight, emitting the observable Doppler-boosted
synchrotron and inverse Compton radiation.
In SSC models the characteristic double humped spectra of blazars are
explained by electrons in the jet emitting synchrotron radiation while being
accelerated in a magnetic field, which gives the first peak in the SED. These
high energy electrons upscatter the very same synchrotron photons to TeV
energies due to the inverse Compton effect, resulting in a second peak in the
SED. Another approach to the explanation of the double humped structure are
proton initiated electromagnetic cascades (e.g., Mannheim, 1993) or external
Compton models.
The key issue is to understand which physical mechanisms are leading to such
SEDs. In particular that means to explain the ultrarelativistic electron
spectra within the jet, which are believed to be responsible for the gamma
radiation.
The high peaked BL Lac objects (HBLs) as a subclass of blazars show a peak in
their SED in the X-ray regime, suggesting that an inverse Compton peak should
occur at correspondingly high gamma-ray energies. In fact, a large fraction of
the known nearby HBLs have already been discovered with Cerenkov telescopes,
such as H.E.S.S., MAGIC, and VERITAS. Since 2008 the Fermi satellite measures
at these high gamma-ray energies. The energy range of the Fermi data is
slightly different from the H.E.S.S. and VERITAS-Telescopes which gives new
constraints to the SEDs.
The first Fermi data published is from PKS 2155-30.4, a HBL at redshift
$z=0.117$ (luminosity distance: $d_{\text{L}}=1.67\cdot
10^{27}\leavevmode\nobreak\ \text{cm}$) (H. E. S. S. and Fermi-LAT
collaborations, 2009).
We present a selfconsistent SSC model that is not only able to model the SED
of PKS 2155-30.4 shown in H. E. S. S. and Fermi-LAT collaborations (2009) but
also to partly explain the “ad-hoc” injected particle spectra of many SSC
models. Therefore we introduce and solve the kinetic equation describing the
synchrotron-self-Compton emission numerically in two different zones within
the jet (see section 1). We use the exact Klein-Nishina cross section which is
important at the relevant very high gamma-energies to describe the inverse
Compton radiation and energy losses of the electrons. The emphasis lies on the
accurate treatment of the two possible particle acceleration mechanisms
(Fermi-I- and Fermi-II) which are able to produce high energy electrons as
well as on the selfconsistent treatment of the radiation processes.
## 1 THE MODEL
### 1.1 Model geometry
We extend the well-established SSC model by Fermi-I and Fermi-II acceleration
mechanisms to a selfconsistent SSC model with two zones in a nested setup.
Both regions (the acceleration- and the radiation zone) forming the blob are
assumed to be spherical and homogeneous containing isotropically distributed
non-thermal electrons and a randomly oriented magnetic field. The acceleration
zone is assumed to be spatially significantly smaller than the surrounding
radiation zone. Furthermore every electron leaving the accelerationzone enters
the radiation zone. These assumptions are common place in SSC models (e.g.,
Kirk et al., 1998).
To derive the kinetic equations describing the time evolution of
$n_{e}(\gamma)$, $N_{e}(\gamma)$ ($n_{e}$ in the acceleration zone, $N_{e}$ in
the radiation zone) as the differential electron densities we use the one
dimensional diffusion approximation (eq. (1)) of the relativistic Vlasov
equation (e.g. Schlickeiser, 2002), which is applicable due to the assumptions
made above.
$\displaystyle\frac{\partial}{\partial t}f(p,t)$
$\displaystyle=\frac{1}{p^{2}}\frac{\partial}{\partial
p}\left[F\left(p,f,\frac{\partial}{\partial
p}f\right)\right]+S(p,t)\leavevmode\nobreak\ ,$ (1)
where $f(p,t)$ is a particle distribution function, with the particle’s
momentum value $p$. $F$ describes the contributing processes, such as
synchrotron radiation or acceleration, in momentum space. Catastrophic
particle gains and losses are considered via $S(p,t)$.
Making use of the relativistic approximation $E\approx pc=\gamma mc$ and the
relation $n(p,t)=4\pi p^{2}f(p,t)$ one can derive the kinetic equations
governing the model.
### 1.2 Kinetic equations
#### 1.2.1 Acceleration zone
While the blob propagates through the jet, electrons are continuously injected
into the acceleration zone when considering the blob’s rest frame, leading to
an injection function
$\displaystyle Q_{\text{inj}}(\gamma,t)$
$\displaystyle:=Q_{0}\delta(\gamma-\gamma_{0})\vartheta{(t-t_{0})}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ t_{0}=0$ (2)
which we assume monoenergetic and time independent. These low- to mid-energy
electrons are accelerated systematically and stochastically due to Fermi-I and
Fermi-II processes while suffering synchrotron and inverse Compton losses.
Energy losses due to inverse Compton scattering are calculated using the full
Klein-Nishina cross section, see eq. (1.2.2). This leads to
$P_{\text{IC}}(\gamma)$ given in eq. (8) with the corresponding radiation
field $n_{\text{PH}}$ of the acceleration zone. Due to the non equilibrium of
magnetic and radiative energy in the acceleration zone the energy losses via
inverse Compton scattering can become quite significant and must not be
neglected, also the Thomson limit is not appropriate here. The synchrotron
losses are calculated using eq. (3) from Ginzburg and Syrovatskii (1969) for
isotropic particle distributions
$\displaystyle P_{s}(\gamma)$
$\displaystyle=\frac{1}{6\pi}\frac{\sigma_{\text{T}}B^{2}}{mc}\gamma^{2}=\beta_{s}\gamma^{2}$
(3)
with the Thomson cross section $\sigma_{\text{T}}$. According to Schlickeiser
(1984) particle acceleration via parallel shockfronts and stochastic
acceleration caused by scattering at Alfvén waves leads to
$\displaystyle F\left(p,f,\frac{\partial}{\partial p}f\right)$
$\displaystyle=p^{4}\frac{v_{A}^{2}}{9K_{||}}\frac{\partial f}{\partial
p}+p^{3}\frac{v_{S}^{2}}{4K_{||}}f$ (4)
for the function $F$. With the parallel spatial diffusion coefficient
$K_{||}$, which is momentum independent for hard spheres and the
characteristic speeds $v_{A}$ for the Alfvén mediated stochastic acceleration
and $v_{S}$ for parallel shockfronts. Substituting $p\rightarrow\gamma$ in eq.
(1) and eq. (4) according to the relativistic approximation mentioned above,
one will finally find eq. (1.2.1); the kinetic equation of the acceleration
zone.
$\displaystyle\frac{\partial n_{e}(\gamma,t)}{\partial t}=$
$\displaystyle\frac{\partial}{\partial\gamma}\left[(\beta_{s}\gamma^{2}+P_{\text{IC}}(\gamma)-t_{\text{acc}}^{-1}\gamma)\cdot
n_{e}(\gamma,t)\right]+$
$\displaystyle+\frac{\partial}{\partial\gamma}\left[[(a+2)t_{\text{acc}}]^{-1}\gamma^{2}\frac{\partial
n_{e}(\gamma,t)}{\partial\gamma}\right]+$
$\displaystyle+Q_{0}(\gamma-\gamma_{0})-t_{\text{esc}}^{-1}n_{e}(\gamma,t)\leavevmode\nobreak\
,$ (5)
where the characteristic acceleration timescale $t_{\text{acc}}$ is given by
$\displaystyle t_{\text{acc}}$
$\displaystyle=\left(\frac{v_{s}^{2}}{4K_{||}}+2\frac{v_{A}^{2}}{9K_{||}}\right)^{-1}\leavevmode\nobreak\
.$ (6)
Eq. (6) for $t_{\text{acc}}$ is a direct consequence of the derivation of eq.
(1.2.1) out of eq. (1) using eqn. (4) and (3). The expression in eq. (6)
includes the analytical timescale for non-relativistic shock acceleration.
According to Bednarz and Ostrowski (1996) and especially to Ellison et al.
(1990) the acceleration timescale for parallel relativistic shock waves
decreases approximately by a factor of $3$. We did not take into account this
behavior for it is unclear how the analytical expression looks like in that
case. Secondly we are omitting the energy dependency of $t_{\text{acc}}$ using
hard spheres for the plasma instabilities anyway. This issue is irrelevant for
the modelling (for we are setting numerical values for $t_{\text{acc}}$) and
the type of energy spectrum (a powerlaw) produced by Fermi-I acceleration is
identical in the non-relativistic and relativistic case (Sokolov et al.,
2004). But it has to be kept in mind for the interpretation of the results.
The parameter $a\approx v_{s}^{2}/v_{A}^{2}$ determines the ratio of shock to
stochastic acceleration. $t_{\text{esc}}=\eta R_{\text{acc}}/c$ is the
characteristic timescale for electrons escaping from the acceleration region,
where $\eta$ is an empirical factor set to $\eta=10$ and $R_{\text{acc}}$ the
radius of the acceleration sphere. All escaping electrons enter the radiation
zone downstream the jet. The seperation in two zones can firstly explain the
injected electron spectra and secondly takes account of a much more confined
shock region for Fermi-I acceleration will probably not occur in the whole
blob region when considering physical sources.
Our model can be compared with the model presented by Katarzyński et al.
(2006). The kinetic equation (eq. 3 in their paper) is almost similar to the
kinetic equation in the acceleration zone eq. 1.2.1. One major difference to
our model is their sole use of stochastic acceleration. In fact their model is
the limit of our model for $a\to 0$. Additionally they limit themselves to
radiation in the acceleration zone, which is useful when not taking into
account shock acceleration. Besides that there are number of minor differences
regarding the exact treatment of inverse Compton losses and the derivation of
escape rates.
Due to the small spatial extent the acceleration zone does not contribute to
the SED directly, i.e. $n_{\text{ph}}(\nu)$ is only calculated in order to
determine the inverse Compton loss rate for the electrons in the acceleration
zone.
#### 1.2.2 Radiation zone
The electrons are not accelerated here. Thus the kinetic equation takes the
simple form
$\displaystyle\frac{\partial N_{e}(\gamma,t)}{\partial t}=$
$\displaystyle\frac{\partial}{\partial\gamma}\left[\left(\beta_{s}\gamma^{2}+P_{\text{IC}}(\gamma)\right)\cdot
N_{e}(\gamma,t)\right]-$
$\displaystyle-\frac{N_{e}(\gamma,t)}{t_{\text{rad,esc}}}+\left(\frac{R_{\text{acc}}}{R_{\text{rad}}}\right)^{3}\frac{n_{e}(\gamma,t)}{t_{\text{esc}}}\leavevmode\nobreak\
\text{.}$ (7)
Electrons in the radiation zone suffer synchrotron
($\propto\beta_{s}\gamma^{2}$) and inverse Compton losses (eq. (8)), other
energy losses are irrelevant in jetsystems of such low electron density (see
e.g. Böttcher and Chiang, 2002).
$\displaystyle P_{\text{IC}}(\gamma)$
$\displaystyle=m^{3}c^{7}h\int_{0}^{\alpha_{max}}{d\alpha\alpha\int_{0}^{\infty}{d\alpha_{1}N_{\text{ph}}(\alpha_{1})\frac{dN(\gamma,\alpha_{1})}{dtd\alpha}}}$
(8)
The integrals in eq. (8) are solved numerically using the full Klein-Nishina
cross section for a single electron given in eq. (1.2.2). The photon energies
are rewritten in terms of the electrons rest mass, i.e. $h\nu=\alpha mc^{2}$
for the scattered photons and $h\nu=\alpha_{1}mc^{2}$ for the target photons.
The integration bounds of the outer integral in eq. (8) are a direct
consequence of the kinematics. The non-trivial dependency of
$P_{\text{IC}}(\gamma)$ from $N_{\text{ph}}$ and thus of $N_{e}$ from eq. (8)
makes the numerical treatment of the kinetic equations inevitable leading to a
time resolved model. The loss rates for the electron distribution of PKS 2155
in the steady state are shown in Fig. 1, which indicates that in our case the
inverse Compton losses would be slightly overestimated in the often used
Thomson limit for all Lorentzfactors $\gamma$ because of the dependency on the
photon field $P_{\text{thom}}\propto\int{d\nu\nu N_{\text{ph}}}\gamma^{2}$ and
its special shape due to the modified injection at $\gamma_{0}=910$. This
would not be the case for low energetic injected electrons and the resulting
powerlaw-like photon distribution. For high Lorentzfactors $\gamma$ however
the deviation of the Thompson limit for the inverse Compton scattering to the
real Klein-Nishina treatment becomes more significant in each case.
Figure 1: Lossrates due to the inverse Compton effect for an electron of
Lorentzfactor $\gamma$ in the model photon field of PKS 2155 (Thompson limit,
red and used Klein-Nishina treatment, black) compared to the synchrotron
losses, blue. Other losses like adiabatic cooling, pair production are
irrelevant at typical SSC configurations.
Again electrons escaping the blob are taken into account via
$t_{\text{esc,rad}}=\eta R_{\text{rad}}/c$ with the empirical factor $\eta$
set to $\eta=10$. Both electronic PDEs are connected via the catastrophic
particle loss/gain-term. The factor
$\left(R_{\text{acc}}/R_{\text{rad}}\right)^{3}$ ensures particle
conservation.
To determine the time-dependent spectral energy distribution of blazars we
solve the differential equation for the differential photon number density,
obtained from the radiative transfer equation, including the corresponding
terms with respect to the SSC model,
$\displaystyle\frac{\partial N_{\text{ph}}(\nu,t)}{\partial t}$
$\displaystyle=R_{s}-c\alpha_{\nu}N_{\text{ph}}(\nu,t)+R_{c}-\frac{N_{\text{ph}}(\nu,t)}{t_{\text{ph,esc}}}\leavevmode\nobreak\
\text{.}$ (9)
To describe the synchrotron photon production rate $R_{s}$ in a convenient way
we use the well known Melrose approximation (see e.g., Pohl, 2002)
$\displaystyle R_{s}$ $\displaystyle=1.8\frac{\sqrt{3}e^{3}B_{\bot}}{h\nu
mc^{2}}\int{d\gamma
N_{e}(\gamma,t)\left(\frac{\nu}{\nu_{c}(\gamma)}\right)^{\frac{1}{3}}e^{-\frac{\nu}{\nu_{c}(\gamma)}}}\leavevmode\nobreak\
.$ (10)
with the characteristic frequency eq. (11) for an electron with a Lorentz
factor of $\gamma$.
$\displaystyle\nu_{c}(\gamma)$ $\displaystyle=\frac{3\gamma^{2}eB_{\bot}}{4\pi
mc}$ (11)
In optically thick regimes the emitted synchrotron radiation is absorbed by
the emitting electrons itself. This is described by the synchrotron self
absorption coefficient,
$\displaystyle\alpha_{\nu}$
$\displaystyle=\frac{1}{12}\frac{c}{\nu^{2}eB}P_{\nu}(\gamma_{c})\frac{N_{e}(\gamma_{0},t)}{\gamma_{0}^{2}}\text{.}$
(12)
(with $\gamma_{c}=f(\nu_{c})^{-1}$). Here we made use of the monochromatic
approximation (e.g., Felten and Morrison, 1966) for the synchrotron power:
$\displaystyle P_{\nu}(\nu,\gamma)$
$\displaystyle=\frac{\sqrt{3}e^{2}B_{\bot}}{mc^{2}}\frac{\nu}{\nu_{c}{\gamma}}\int_{0}^{\infty}{d\nu^{\prime}K_{\frac{5}{3}}(\nu^{\prime})}$
(13)
In SSC models the second hump in the SED of a blazar is due to inverse Compton
scattered photons by the synchrotron radiation emitting electrons themselves.
Here the full Klein-Nishina cross section from Blumenthal and Gould (1970) is
used to calculate the inverse Compton photon production rate.
$\displaystyle R_{c}=$ $\displaystyle\int d\gamma\,N_{e}(\gamma)\;\times$
$\displaystyle\times\int
d\alpha_{1}\left[N_{\text{ph}}(\alpha_{1})\frac{dN(\gamma,\alpha_{1})}{dtd\alpha}-N_{\text{ph}}(\alpha)\frac{dN(\gamma,\alpha)}{dtd\alpha_{1}}\right]$
(14)
To fully exploit the Klein-Nishina cross section, eq. (1.2.2), we used the
approximate inverse Compton spectrum of a single electron scattering off a
unit density photon field (e.g., Jones, 1968; Jauch and Rohrlich, 1976):
$\displaystyle\frac{dN(\gamma,\alpha_{1})}{dtd\alpha}=$
$\displaystyle\frac{2\pi r_{0}^{2}c}{\alpha_{1}\gamma^{2}}\bigg{[}2q\ln
q+(1+2q)(1-q)+$ $\displaystyle+\frac{1}{2}\frac{(4\alpha_{1}\gamma
q)^{2}}{(1+4\alpha_{1}\gamma q)}(1-q)\bigg{]}\leavevmode\nobreak\ ,$ (15)
with the electron’s Lorentz radius $r_{0}=e^{2}/(mc^{2})$, the scattering
parameter $q={\alpha}/(4\alpha_{1}\gamma^{2}(1-\alpha/\gamma))$ and $0\approx
1/(4\gamma^{2})<q\leq 1$. Due to momentum and energy conservation this
equation is valid for $\alpha_{1}<\alpha\leq
4\alpha_{1}\gamma^{2}/(1+4\alpha_{1}\gamma)$.
The last catastrophic term in eq. (9) describes photons escaping from the
emitting region, where
$\displaystyle t_{\text{ph, esc}}=\frac{3R_{\text{rad}}}{4c},$ (16)
is the approximate escape time, with $R_{\text{rad}}$ the radius of the
emitting blob. The escape time is chosen to be the light crossing time of the
photons.
The photon lossrate due to the pair production of electrons and positrons is
not taken into account for two reasons. Firstly it is insignificant compared
to the dominating synchrotron and inverse Compton processes. This is a
consequence of the relatively low density (Böttcher and Chiang, 2002).
Secondly it would violate the selfconsistency of our model for positrons are
not treated, hence violating energy conservation.
To compute the SEDs in our model we must shift the frame of reference from the
blob to the observer. For a sphere of radius $R$ the observed flux at distance
$r$ is
$\displaystyle F_{\nu}^{obs}(r)$ $\displaystyle=\pi
I_{\nu}^{obs}\frac{R^{2}}{r^{2}}\leavevmode\nobreak\ \text{.}$ (17)
With the Lorentz boosted intensity $I_{\nu}^{obs}=\delta^{3}I_{\nu}^{blob}$
due to the bulk motion with a doppler factor $\delta$ of the blob and the
Lorentz transformed, red shifted frequency $\nu^{obs}=\delta/(1+z)\nu$. Where
$I_{\nu}^{blob}$ is calculated from the photon unit density
$\displaystyle I_{\nu}^{blob}$ $\displaystyle=\frac{h\nu
c}{4\pi}N_{\text{ph}}(\nu)$ (18)
for homogenous spheres.
## 2 NUMERICS
In our model we numerically solve the kinetic equations forward in time in
order to obtain a model SED. The downstream motion of the electrons induces
the sequence of solving the acceleration zone’s equation before the kinetic
equation of the radiation zone in each time step. The simple Euler scheme was
found adequate to do the time integration.
In the acceleration zone we had to combine the Crank-Nicholson scheme (Press,
2002) with Godunov’s method to provide both correct treatment of the
characteristics and stability for the derivation in $\gamma$. In the radiation
zone the characteristic flows, due to the absence of acceleration, only in one
direction making the Crank-Nicholson scheme sufficient.
With our carefully tested code it is possible to calculate the dynamics of
SEDs in a range of 20 orders of magnitude. The implemented code complies
particle conservation in each zone alone and both together as well as the
conservation of the total energy (i.e. of the electrons and the photons) over
typical simulation times with a maximum error of O(5%). For negligible
stochastic acceleration, i.e. $a\rightarrow\infty$, and without a radiation
field, i.e. no inverse Compton losses, the steady state solution for the
kinetic equation yields
$\displaystyle
n_{\text{e,steady}}(\gamma)=C\frac{1}{\gamma^{2}}\left(\frac{1}{\gamma}-\beta_{\text{s}}t_{\text{acc}}\right)^{\frac{t_{\text{acc}}-t_{\text{esc}}}{t_{\text{esc}}}}$
with $\gamma_{\text{max}}=\left(t_{\text{acc}}\beta_{\text{s}}\right)^{-1}$
and the constant $C$ determined by the injection function $Q$. The implemented
numeric model converges against this solution for sufficient simulation time.
Additionally it was tested against the steady state analytical solution with
Fermi-II processes given in Schlickeiser (1984) with no significant
deviations. Setting $t_{\text{esc,rad}}\rightarrow\infty$ and neglecting
inverse Compton scattering the spectral index of the powerlaw part of the
electron distribution in the radiation zone is (analytically) reduced by one
compared to the one in the acceleration zone, which also was confirmed by the
implemented code. The inverse Compton scattering rate was confirmed against
the approximate analytical results (low energetic Thompson regime and extreme
Klein-Nishina limit) before implementing. Concerning the photon distributions
we validated the expected spectral indices in the steady state solution for
the different frequency regimes, which together with the energy conservation
between electrons and photons approves the integrity of the model.
A detailed description of the used numeric techniques as well as the
implemented model also in context with the variability of the sources will be
given in a paper yet to be published.
## 3 RESULTS
The recent Fermi data give new constraints on the gamma-ray peak of the HBL
PKS 2155-30.4 concerning its curvature. This is leading to a deep dip between
the optical/X-ray and the gamma-ray peak. We are able to model the SED of PKS
2155-30.4 with our model by setting
$\displaystyle\gamma_{0}$ $\displaystyle=910$ (19)
for the monoenergetic injection into the acceleration zone. This is rather
unusual but required to model the SED of PKS 2155. Such moderate but not small
Lorentz factors can be explained e.g. by two counterstreaming plasmas. If the
upstream electrons would be at rest, the bulk doppler factor of $\delta=116$
would automatically lead to $\gamma_{0}=\Gamma\approx 58$. Assuming
speculatively that the upstream electrons moving in the opposite direction of
the blob with a mean velocity of $v_{u}$ hence a upstream Lorentz-Factor
$\gamma_{u}$ the $\gamma_{0}$ factor in the blob’s rest frame must be
calculated according to the relativistic superposition:
$\displaystyle\gamma_{0}$
$\displaystyle=\sqrt{1-\left(\frac{\sqrt{\Gamma^{2}-1}\Gamma+\sqrt{\gamma_{u}^{2}-1}\gamma_{u}}{\Gamma^{2}+\gamma_{u}^{2}-1}\right)^{2}}^{-1}$
(20)
Solving eq. (20) for our setup we find $\gamma_{u}\approx 8$ for the upsteam
electrons which are streaming towards the blob. The numerically solved steady
state electron density in the acceleration zone is shown in Fig. 2. We also
show the time development for a “switched on” injection, i.e.
$\left.n_{\text{e}}(\gamma)\right|_{t<0}=0\leavevmode\nobreak\
\forall_{\gamma}$ and $Q=Q_{0}\delta(\gamma-\gamma_{0})\vartheta(t)$, until
the steady state is reached.
Figure 2: Steady state electron distribution and its time development in the
acceleration zone modelling the SED of PKS 2155-30.4 shown in Fig. 3 as
arising from the injection function $Q_{0}$. The corresponding intrinsic times
are $t=1000\leavevmode\nobreak\ \text{s}$ (dashed black),
$t=5000\leavevmode\nobreak\ \text{s}$ (dashed blue), $t=1\cdot
10^{4}\leavevmode\nobreak\ \text{s}$ (dashed red), $t=2\cdot
10^{4}\leavevmode\nobreak\ \text{s}$ (dashed green). The steady state with its
rising and falling powerlaw and the exponential cutoff at $\gamma\approx
10^{5}$ is reached at about $t=10^{5}\leavevmode\nobreak\ \text{s}$.
In Fig. 2 it can clearly be seen tvhat accelerating electrons using Fermi-I
and Fermi-II processes leads to powerlaw electron distributions with an
exponential cut-off often used as the ad-hoc injection function in onezone-SSC
models (Chiang and Böttcher, 2002) (right side of $\gamma_{0}$ in Fig. 2),
thus explaining them using the diffusion theory derived from plasma physics.
By injecting electrons with eq. (19) and significant stochastic acceleration
(i.e. $a=O(1)$) we are also able to produce rising electron spectra before
decreasing in a power-law and an exponential cut-off, like introduced in
Böttcher and Chiang (2002) (left side of $\gamma_{0}$ in Fig. 2). The Fermi-II
processes are responsible for the rising power-law and exponential cut-off,
whereas the ratio of $t_{\text{acc}}/t_{\text{esc}}$ determines the spectral
index of the power-law at $\gamma>\gamma_{0}$. It can clearly be seen from
Fig. 2 that the convergence against the steady state solution for the electron
density begins relatively rapid while slowing down eventually. The simulation
time when the steady state is reached corresponds to the escape time of the
electrons in the acceleration zone. When concerning variability and time
resolved lightcurves of blazars this is an advantage of the twozone model
because the rising part in such lightcurves corresponds partially to the
escape time of the acceleration zone (while the falling part is connected to
the response time of the system, $t_{\text{rad,esc}}$).
Table 1: Chosen parameters for the model SED shown in Fig. 4 to fit the data
(H. E. S. S. and Fermi-LAT collaborations, 2009) of PKS 2155-30.4.
$Q_{0}(\text{cm}^{-3})$ | $B(\text{G})$ | $R_{\text{acc}}(\text{cm})$ | $R_{\text{rad}}(\text{cm})$ | $t_{\text{acc}}/t_{\text{esc}}$ | $a$ | $\Gamma$
---|---|---|---|---|---|---
$5.25\cdot 10^{4}$ | $0.29$ | $3.0\cdot 10^{13}$ | $6.3\cdot 10^{14}$ | $1.55$ | $1$ | $58$
| | | | | |
An acceleration zone electron density as shown by the solid black curve in
Fig. 2, leads to the desired broken power-law electron spectrum in the
radiation zone which finally is able to model the SED of PKS 2155-30.4 (see
Fig. 3 and Fig. 4). We used the parameters in Table 1 for the model SED in
Fig. 4 (black, solid line). The black dashed curve in Fig. 4 corresponds to a
fit assuming a black body for the thermal contribution of the host galaxy thus
the ATOM optical data is not to be taken into account for the SSC modelling.
The curvature and deep dip in the model SED is a direct consequence of the
rising part in the electron density of the acceleration zone. Thus it can be
modeled by varying the ratio $a$ of shock to stochastic acceleration. All the
parameters in Table 1 are consistent with the limits given via other
observations and statistics, e.g. determination of $\Gamma$ using superluminal
motion of Quasar jets.
The recent Fermi, H.E.S.S. and ATOM data (H. E. S. S. and Fermi-LAT
collaborations, 2009) have been averaged over a period of 14 days and show a
lowstate of the HBL PKS 2155-30.4. This is confirmed by the Aharonian et al.
(2005) data of H.E.S.S. a few years ago which show the same flux level as the
recent data. We used the EBL studies described in Primack et al. (2005) to do
the EBL deabsorption for the H.E.S.S. datapoints, a correction of the Fermi
data is not necessary.
Figure 3: Time evolution due to the switched on injection function until the
steady state of the SED of PKS2155 is reached (see also Fig. 4). The intrinsic
times are $t=1\cdot 10^{4}\leavevmode\nobreak\ \text{s}$ (dashed black),
$t=2\cdot 10^{4}\leavevmode\nobreak\ \text{s}$ (dashed blue), $t=5\cdot
10^{4}\leavevmode\nobreak\ \text{s}$ (dashed red), $t=1\cdot
10^{5}\leavevmode\nobreak\ \text{s}$ (dashed green), the complete steady state
is reached at about $t=2\cdot 10^{6}\leavevmode\nobreak\ \text{s}$, which
correlates to the response time of the system due to $t_{\text{esc,rad}}$.
The time development of the SED due to a switched on injection of electrons
into the acceleration zone at time $t_{0}=0$ is shown in Fig. 3. It can
clearly be seen that the final state of the model SSC correlates with the
response time of the radiation zone $t_{\text{esc,rad}}$ and that the
convergence again begins fast and slows down rapidly at higher simulation
times. [CONCLUSIONS] Our model is able to explain the injection function of
many onezone-SSC models as shock and stochastic acceleration of electrons
upstream the jet entering the blob while continuously suffering synchrotron
losses. By introducing Fermi-II acceleration we get rid of the sharp cut-off
introduced in Kardashev (1962) or Kirk et al. (1998) which probably does not
occur in physical sources. Additionally we are able to model relatively
complex electron densities with increasing and decreasing parts through the
stochastic acceleration of electrons, only by varying the monoenergetic
injection to higher $\gamma_{0}$. In contrast to the ad-hoc injection of some
onezone-SSC models such Lorentz factors have a physically reasonable, but
highly speculative, explanation as upstream previously accelerated but already
partially cooled electrons. These electrons are averagely moving in the
opposite direction of the blob with a mean Lorentz factor of
$\gamma_{u}\approx 8$ resulting, together with the motion of the blob, in
$\gamma_{0}\approx 900$ for the monoenergetic injection function used in the
acceleration zone of our model.
As recent data points out, these complex electron distributions are necessary
to model the new constraints concerning the gamma-ray peak of blazar’s SEDs if
one does not simply shift the synchrotron-peak to achieve the inverse Compton
spectrum (e.g., Kataoka et al., 2000). The curvature of the peak, and thus the
deep dip between the two humps, is a direct consequence of the rising part in
the responsible electron distribution within the blob. This constraint rules
out many SSC models, which are not able to produce such electron spectra.
With our model we are able to form the curvature of the gamma-ray peak and the
dip by varying the influence of the Fermi-II processes. The shape and position
of the synchrotron peak in the model SED is dominated by $t_{\text{acc}}$ and
$R_{\text{acc}},R_{\text{rad}}$ as well as $B$. For the parameters concerning
the acceleration arise from plasmaphysics considerations we gain insight into
the jets microphysics while modelling observed SEDs. We have also shown that
in such environments the Thomson approximation for the inverse Compton effect
can not always be applied, especially when considering time resolution and
hence non equilibria of the energy distribution in the blob.
Here we only introduced steady state solutions of our model, but due to the
spatially relatively small acceleration region, which is at least an oder of
magnitude smaller than the emitting region, this twozone-SSC model is able to
selfconsistently model the rising part in the lightcurves of flaring blazars
which are connected to the behavior in the acceleration zone, especially the
energy transport from low to high energies. This, together with the
consequences of the model geometry on the observable SEDs and lightcurves of
blazars like in Sokolov et al. (2004), will be subject of a following paper.
Figure 4: Lowstate of PKS 2155-30.4 with the simultaneous data of ATOM, SWIFT,
RXTE, Fermi and H.E.S.S of the August/September 2008 campaign from H. E. S. S.
and Fermi-LAT collaborations (2009) (red triangles and circles). The 2003
H.E.S.S. data (blue circles) is also shown, proofing the lowstate of
PKS2155-30.4. The VHE data have been deabsorbed using Primack et al. (2005).
The dashed black curve shows a thermal fit for the contribution of the host-
galaxy. Our model SSC fit, arising from the steady state electron distribution
in the radiation zone is shown in the solid black curve, a moderate energy
injection at $\gamma_{0}\approx 910$ into the acceleration zone together with
stochastic and systematic acceleration is needed to meet the curvature of the
VHE peak given via the Fermi data.
## References
* Aharonian et al. (2005) Aharonian et al.: Multi-wavelength observations of PKS 2155-304 with HESS, 442, 895–907, 10.1051/0004-6361:20053353, 2005.
* Bednarz and Ostrowski (1996) Bednarz, J. and Ostrowski, M.: The acceleration time-scale for first-order Fermi acceleration in relativistic shock waves, 283, 447–456, 1996.
* Blumenthal and Gould (1970) Blumenthal, G. R. and Gould, R. J.: Bremsstrahlung, Synchrotron Radiation, and Compton Scattering of High-Energy Electrons Traversing Dilute Gases, Reviews of Modern Physics, 42, 237–271, 10.1103/RevModPhys.42.237, 1970\.
* Böttcher and Chiang (2002) Böttcher, M. and Chiang, J.: X-Ray Spectral Variability Signatures of Flares in BL Lacertae Objects, Astrophys. J., 581, 127–142, 10.1086/344155, 2002.
* Chiang and Böttcher (2002) Chiang, J. and Böttcher, M.: Synchrotron and Synchrotron Self-Compton Spectral Signatures and Blazar Emission Models, Astrophys. J., 564, 92–96, 10.1086/324294, 2002.
* Ellison et al. (1990) Ellison, D. C., Reynolds, S. P., and Jones, F. C.: First-order Fermi particle acceleration by relativistic shocks, Astrophys. J., 360, 702–714, 10.1086/169156, 1990.
* Felten and Morrison (1966) Felten, J. E. and Morrison, P.: Omnidirectional Inverse Compton and Synchrotron Radiation from Cosmic Distributions of Fast Electrons and Thermal Photons, Astrophys. J., 146, 686–708, 1966.
* Ginzburg and Syrovatskii (1969) Ginzburg, V. L. and Syrovatskii, S. I.: Developments in the Theory of Synchrotron Radiation and its Reabsorption, Ann. Rev. Astron. Astrophys., 7, 375–+, 10.1146/annurev.aa.07.090169.002111, 1969.
* H. E. S. S. and Fermi-LAT collaborations (2009) H. E. S. S. and Fermi-LAT collaborations: Simultaneous observations of PKS 2155-304 with H.E.S.S., Fermi, RXTE and ATOM: spectral energy distributions and variability in a low state, ArXiv e-prints, 2009.
* Jauch and Rohrlich (1976) Jauch, J. M. and Rohrlich, F.: The theory of photons and electrons. The relativistic quantum field theory of charged particles with spin one-half, Texts and Monographs in Physics, New York: Springer, 1976, 2nd ed., 1976.
* Jones (1968) Jones, F. C.: Calculated Spectrum of Inverse-Compton-Scattered Photons, Physical Review, 167, 1159–1169, 10.1103/PhysRev.167.1159, 1968.
* Kardashev (1962) Kardashev, N. S.: Nonstationariness of spectra of young sources of nonthermal radio emission, Soviet Astronomy - AJ, 39, 393–409, 1962.
* Kataoka et al. (2000) Kataoka, J., Takahashi, T., Makino, F., Inoue, S., Madejski, G. M., Tashiro, M., Urry, C. M., and Kubo, H.: Variability Pattern and the Spectral Evolution of the BL Lacertae Object PKS 2155-304, Astrophys. J., 528, 243–253, 10.1086/308154, 2000.
* Katarzyński et al. (2006) Katarzyński, K., Ghisellini, G., Mastichiadis, A., Tavecchio, F., and Maraschi, L.: Stochastic particle acceleration and synchrotron self-Compton radiation in TeV blazars, 453, 47–56, 10.1051/0004-6361:20054176, 2006.
* Kirk et al. (1998) Kirk, J. G., Rieger, F. M., and Mastichiadis, A.: Particle acceleration and synchrotron emission in blazar jets, 333, 452–458, 1998.
* Mannheim (1993) Mannheim, K.: The proton blazar, 269, 67–76, 1993.
* Pohl (2002) Pohl, M.: Einfuehrung in die Hochenergieastrophysik, Shaker Verlag, Aachen, 2002\.
* Press (2002) Press, W. H.: Numerical recipes in C++ : the art of scientific computing, Numerical recipes in C++ : the art of scientific computing by William H. Press. xxviii, 1,002 p. : ill. ; 26 cm. Includes bibliographical references and index. ISBN : 0521750334, 2002.
* Primack et al. (2005) Primack, J. R., Bullock, J. S., and Somerville, R. S.: Observational Gamma-ray Cosmology, in: High Energy Gamma-Ray Astronomy, edited by F. A. Aharonian, H. J. Völk, & D. Horns, vol. 745 of American Institute of Physics Conference Series, pp. 23–33, 10.1063/1.1878394, 2005.
* Schlickeiser (1984) Schlickeiser, R.: An explanation of abrupt cutoffs in the optical-infrared spectra of non-thermal sources. A new pile-up mechanism for relativistic electron spectra, 136, 227–236, 1984.
* Schlickeiser (2002) Schlickeiser, R.: Cosmic ray astrophysics, Astronomy and Astrophysics Library; Physics and Astronomy Online Library. Berlin: Springer. ISBN 3-540-66465-3, 2002, XV + 519 pp., 2002.
* Sokolov et al. (2004) Sokolov, A., Marscher, A. P., and McHardy, I. M.: Synchrotron Self-Compton Model for Rapid Nonthermal Flares in Blazars with Frequency-dependent Time Lags, Astrophys. J., 613, 725–746, 10.1086/423165, 2004.
|
arxiv-papers
| 2010-01-13T13:31:20 |
2024-09-04T02:49:07.712934
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matthias Weidinger, Michael R\\\"uger, Felix Spanier",
"submitter": "Matthias Weidinger",
"url": "https://arxiv.org/abs/1001.2145"
}
|
1001.2159
|
A Riemann theta function formula with its application to double periodic wave
solutions of nonlinear equations
Engui Fana111 E-mail address: faneg@fudan.edu.cn and Kwok Wing Chowb
a. School of Mathematical Sciences and Key Laboratory of Mathematics for
Nonlinear Science,
Fudan University, Shanghai, 200433, P.R. China
b. Department of Mechanical Engineering, University of Hong Kong, Pokfulam,
Hong Kong
Abstract. Based on a Riemann theta function and Hirota’s bilinear form, a
lucid and straightforward way is presented to explicitly construct double
periodic wave solutions for both nonlinear differential and difference
equations. Once such a equation is written in a bilinear form, its periodic
wave solutions can be directly obtained by using an unified theta function
formula. The relations between the periodic wave solutions and soliton
solutions are rigorously established. The efficiency of our proposed method
can be demonstrated on a class variety of nonlinear equations such as those
considered in this paper, shall water wave equation, (2+1)-dimensional
Bogoyavlenskii-Schiff equation and differential-difference KdV equation.
Keywords: Nonlinear equations; Hirota’s bilinear method; Riemann theta
function; double periodic wave solutions; soliton solutions.
PACS numbers: 11\. 30. Pb; 05. 45. Yv; 02. 30. Gp; 45. 10. -b.
1\. Introduction
The bilinear derivative method developed by Hirota is a powerful and direct
approach to construct exact solution of nonlinear equations. Once a nonlinear
equation is written in bilinear forms by a dependent variable transformation,
then multi-soliton solutions are usually obtained [2]-[7]. It was based on
Hirota forms that Nakamura proposed a convenient way to construct a kind of
quasi-periodic solutions of nonlinear equations [8, 9], where the periodic
wave solutions of the KdV equation and the Boussinesq equation were obtained.
Such a method indeed exhibits some advantages. For example, it does not need
any Lax pairs and Riemann surface for the considered equation, allows the
explicit construction of multi-periodic wave solutions, only relies on the
existence of the Hirota’s bilinear form, as well as all parameters appearing
in Riemann matrix are arbitrary. Recently, further development was made to
investigate the discrete Toda lattice, (2+1)-dimensional Kadomtsev-
Petviashvili equation and Bogoyavlenskii’s breaking soliton equation
[10]-[15]. However, where repetitive recursion and computation must be
preformed for each equation [8]-[15].
The motivation of this paper is to considerably improves the key steps of the
above existing methods. To achieve this aim, we devise a theta function
bilinear formula, which actually provides us a lucid and straightforward way
for applying in a class of nonlinear equations. Once a nonlinear equation is
written in bilinear forms, then the double periodic wave solutions of the
nonlinear equation can be obtained directly by using the formula. Moreover, we
propose a simple and effective method to analyze asymptotic properties of the
periodic solutions. As illustrative examples, we shall construct double
periodic wave solutions to the shall water wave equation, (2+1)-dimensional
Bogoyavlenskii-Schiff equation and differential-difference KdV equation.
The organization of this paper is as follows. In section 2, we briefly
introduce a Hirota bilinear operator and a Riemann theta function. In
particular, we provide a key formula for constructing double periodic wave
solutions for both differential and difference equations. As applications of
our method, in sections 3-5, we construct double periodic wave solutions to
the shall water wave equation, (2+1)-dimensional Bogoyavlenskii-Schiff
equation and differential-difference KdV equation, respectively. In addition,
it is rigorously shown that the double periodic wave solutions tend to the
soliton solutions under small amplitude limits.
2\. Hirota bilinear operator and Riemann theta function
To fix the notations we recall briefly some notions that will be used in this
paper. The Hirota bilinear operators $D_{x},D_{t}$ and $D_{n}$ are defined as
follows:
$\displaystyle D_{x}^{m}D_{t}^{k}f(x,t)\cdot
g(x,t)=(\partial_{x}-\partial_{x^{\prime}})^{m}(\partial_{t}-\partial_{t^{\prime}})^{k}f(x,t)g(x^{\prime},t^{\prime})|_{x^{\prime}=x,t^{\prime}=t}$
$\displaystyle e^{\delta D_{n}}f(n)\cdot
g(n)=e^{\delta(\partial_{n}-\partial_{n}^{\prime})}f(n)g(n^{\prime})|_{n^{\prime}=n}=f(n+\delta)g(n-\delta),$
$\displaystyle{\rm cosh}(\delta D_{n})f(n)\cdot g(n)=\frac{1}{2}(e^{\delta
D_{n}}+e^{-\delta D_{n}})f(n)\cdot g(n),$ $\displaystyle{\rm sinh}(\delta
D_{n})f(n)\cdot g(n)=\frac{1}{2}(e^{\delta D_{n}}-e^{-\delta D_{n}})f(n)\cdot
g(n).$
Proposition 1. The Hirota bilinear operators $D_{x},D_{t}$ and $D_{n}$ have
properties [2]-[7]
$\displaystyle D_{x}^{m}D_{t}^{k}e^{\xi_{1}}\cdot
e^{\xi_{2}}=(\alpha_{1}-\alpha_{2})^{m}(\omega_{1}-\omega_{2})^{k}e^{\xi_{1}+\xi_{2}},$
$\displaystyle e^{\delta D_{n}}e^{\xi_{1}}\cdot
e^{\xi_{2}}=e^{\delta(\nu_{1}-\nu_{2})}e^{\xi_{1}+\xi_{2}},$
$\displaystyle{\rm cosh}(\delta D_{n})e^{\xi_{1}}\cdot e^{\xi_{2}}={\rm
cosh}[\delta(\nu_{1}-\nu_{2})]e^{\xi_{1}+\xi_{2}},$ $\displaystyle{\rm
sinh}(\delta D_{n})e^{\xi_{1}}\cdot e^{\xi_{2}}={\rm
sinh}[\delta(\nu_{1}-\nu_{2})]e^{\xi_{1}+\xi_{2}},$
where $\xi_{j}=\alpha_{j}x+\omega_{j}t+\nu_{j}n+\sigma_{j}$, and $\alpha_{j},\
\omega_{j},\nu_{j},\ \sigma_{j}$, $j=1,2$ are parameters and $n\in\mathbb{Z}$
is a discrete variable. More generally, we have
$None$ $\displaystyle F(D_{x},D_{t},D_{n})e^{\xi_{1}}\cdot
e^{\xi_{2}}=F(\alpha_{1}-\alpha_{2},\omega_{1}-\omega_{2},\exp[\delta(\nu_{1}-\nu_{2})])e^{\xi_{1}+\xi_{2}},$
where $F(D_{x},D_{t},D_{n})$ is a polynomial about operators $D_{x},D_{t}$ and
$D_{n}$. This properties are useful in deriving Hirota’s bilinear form and
constructing periodic wave solutions of nonlinear equations.
In the following, we introduce a general Riemann theta function and discuss
its periodicity, which plays a central role in the construction of periodic
solutions of nonlinear equations. The Riemann theta function reads
$None$ $\vartheta\left[\begin{matrix}\varepsilon\\\
s\end{matrix}\right]({\xi},{\tau})=\sum_{{m}\in\mathbb{{Z}}}\exp\\{2\pi
i({\xi}+{\varepsilon})({m}+{s})-\pi{\tau}({m}+{s})^{2}\\}.$
Here the integer value ${m}\in\mathbb{Z}$, complex parameter
${s},{\varepsilon}\in\mathcal{C}$, and complex phase variables
${\xi}\in\mathcal{C}$; The ${\tau}>0$ which is called the period matrix of the
Riemann theta function.
In the definition of the theta function (2.2), for the case
${s}={\varepsilon}={0}$, hereafter we use
$\vartheta({\xi},{{\tau}})=\vartheta\left[\begin{matrix}0\\\
0\end{matrix}\right]({\xi},{\tau})$ for simplicity. Moreover, we have
$\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})=\vartheta({\xi}+{\varepsilon},{\tau})$.
Definition 1. A function $g(t)$ on $\mathbb{C}$ is said to be quasi-periodic
in $t$ with fundamental periods $T_{1},\cdots,T_{k}\in\mathbb{C}$, if
$T_{1},\cdots,T_{k}$ are linearly dependent over $\mathbb{Z}$ and there exists
a function $G(y_{1},\cdots,y_{k})$, such that
$G(y_{1},\cdots,y_{j}+T_{j},\cdots,y_{k})=G(y_{1},\cdots,y_{j},\cdots,y_{k}),\
\ {\rm for\ all}\ y_{j}\in\mathbb{C},\ j=1,\cdots,k.$
$G(t,\cdots,t,\cdots,t)=g(t).$
In particular, $g(t)$ is called double periodic as $k=2$, and it becomes
periodic with $T$ if and only if $T_{j}=m_{j}T,\ j=1,\cdots,k.$ $\square$
Let’s first see the periodicity of the theta function $\vartheta(\xi,\tau)$.
Proposition 2. [16] The theta function $\vartheta({\xi},{\tau})$ has the
periodic properties
$None$ $\displaystyle\vartheta({\xi}+{1}+i{\tau},{\tau})=\exp(-2\pi
i\xi+\pi\tau)\vartheta({\xi},{\tau}).$
We regard the vectors $1$ and $i\tau_{j}$ as periods of the theta function
$\vartheta({\xi},{\tau})$ with multipliers $1$ and $\exp({-2\pi
i\xi+\pi\tau})$, respectively. Here, $i\tau$ is not a period of theta function
$\vartheta({\xi},{\tau})$, but it is the period of the functions
$\partial^{2}_{\xi}\ln\vartheta({\xi},{\tau})$,
$\partial_{\xi}\ln[\vartheta({\xi}+{e},{\tau})/\vartheta({\xi}+{h},{\tau})]$
and
$\vartheta({\xi}+{e},{\tau})\vartheta({\xi}-{e},{\tau})/\vartheta({\xi}+{h},{\tau})^{2}$.
Proposition 3. The meromorphic functions $f({\xi})$ on $\mathbb{C}$ are as
follow
$\displaystyle(i)\ \ \ \ \
f({\xi})=\partial_{\xi}^{2}\ln\vartheta({\xi},{\tau}),\ \ {\xi}\in\mathbb{C},$
$\displaystyle(ii)\ \ \ \
f({\xi})=\partial_{\xi}\ln\frac{\vartheta({\xi}+{e},{\tau})}{\vartheta({\xi}+{h},{\tau})},\
\ {\xi},\ {e},\ {h}\in\mathbb{C}.$ $\displaystyle(ii)\ \ \ \
f({\xi})=\frac{\vartheta({\xi}+{e},{\tau})\vartheta({\xi}-{e},{\tau})}{\vartheta({\xi},{\tau})^{2}},\
\ {\xi},\ {e},\ {h}\in\mathbb{C}.$
then in all three cases (i)–(iii), it holds that
$None$ $\displaystyle f({\xi}+{1}+i{\tau})=f({\xi}),\ \ \ {\xi}\in\mathbb{C},$
that is, $f(\xi)$ is a double periodic function with $1$ and $i\tau$.
Proof. By using (2.3), it is easy to see that
$\displaystyle\frac{\partial_{\xi}\vartheta({\xi}+{1}+i{\tau},{\tau})}{\vartheta({\xi}+{1}+i{\tau},{\tau})}=-2\pi
i+\frac{\partial_{\xi}\vartheta({\xi},{\tau})}{\vartheta({\xi},{\tau})},$
or equivalently
$None$
$\displaystyle\partial_{\xi}\ln\vartheta({\xi}+{1}+i{\tau},{\tau})=-2\pi
i+\partial_{\xi}\ln\vartheta({\xi},{\tau}).$
Differentiating (2.5) with respective to $\xi$ again immediately proves the
formula (2.4) for the case (i). The formula (2.4) can be proved for the cases
(ii) and (iii) in a similar manner. $\square$
Theorem 1. Suppose that $\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})$ and
$\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})$ are two Riemann theta functions, in which
$\xi=\alpha x+\omega t+\nu n+\sigma$. Then Hirota bilinear operators
$D_{x},D_{t}$ and $D_{n}$ exhibit the following perfect properties when they
act on a pair of theta functions
$None$ $\displaystyle
D_{x}\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})\cdot\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})$
$\displaystyle=\left[\sum_{{\mu=0,1}}\partial_{x}\vartheta\left[\begin{matrix}\varepsilon^{\prime}-\varepsilon\\\
-\mu/2\end{matrix}\right](2{\xi},2{\tau})|_{{\xi}={0}}\right]\vartheta\left[\begin{matrix}\varepsilon^{\prime}+\varepsilon\\\
\mu/2\end{matrix}\right](2{\xi},2{\tau}),$ $None$ $\displaystyle\exp(\delta
D_{n})\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})\cdot\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})$
$\displaystyle=\left[\sum_{{\mu=0,1}}\exp(\delta
D_{n})\vartheta\left[\begin{matrix}\varepsilon^{\prime}-\varepsilon\\\
-\mu/2\end{matrix}\right](2{\xi},2{\tau})|_{{\xi}={0}}\right]\vartheta\left[\begin{matrix}\varepsilon^{\prime}+\varepsilon\\\
\mu/2\end{matrix}\right](2{\xi},2{\tau}),$
where the notation $\sum_{{\mu=0,1}}$ represents two different transformations
corresponding to $\mu=0,1$. The bilinear formula for $t$ is the same as (2.6)
by replacing $\partial_{x}$ with $\partial_{t}$.
In general, for a polynomial operator $F(D_{x},D_{t},D_{n})$ with respect to
$D_{x},D_{t}$ and $D_{n}$, we have the following useful formula
$None$ $\displaystyle
F(D_{x},D_{t},D_{n})\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})\cdot\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})=\left[\sum_{{\mu}}C({\varepsilon^{\prime}},{\varepsilon},{\mu})\right]\vartheta\left[\begin{matrix}\varepsilon^{\prime}+\varepsilon\\\
\mu/2\end{matrix}\right](2{\xi},2{\tau}),$
in which, explicitly
$None$ $\displaystyle
C({\varepsilon},{\varepsilon^{\prime}},{\mu})=\sum_{{m}\in\mathbb{Z}^{N}}F({\mathcal{M}})\exp\left[-2\pi{\tau}({m}-{\mu}/2)^{2}-2\pi
i({m}-{\mu}/2)({\varepsilon^{\prime}}-{\varepsilon})\right].$
where we denote vector ${\mathcal{M}}=(4\pi i({m}-{\mu}/2){\alpha},\ 4\pi
i({m}-{\mu}/2){\omega},\exp[4\pi i({m}-{\mu}/2)\delta\nu]).$
Proof. Making use of Proposition 1, we obtain the relation
$\displaystyle D_{x}\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})\cdot\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}{D}_{x}\exp\\{2\pi
im^{\prime}(\xi+\varepsilon^{\prime})-\pi m^{\prime
2}{\tau}\\}\cdot\exp\\{2\pi im(\xi+\varepsilon)-\pi m^{2}{\tau}\\},$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}2\pi
i\alpha(m^{\prime}-m)\exp\left\\{2\pi i(m^{\prime}+m)\xi-2\pi
i(m^{\prime}\varepsilon^{\prime}+m\varepsilon)-\pi{\tau}[m^{\prime
2}+m^{2}]\right\\}$
By shifting sum index as $m=l^{\prime}-m^{\prime}$, then
$\displaystyle\Delta=\sum_{l^{\prime},m^{\prime}\in\mathbb{Z}}2\pi
i\alpha(2m^{\prime}-l^{\prime})\exp\left\\{2\pi il^{\prime}\xi-2\pi
i[m^{\prime}\varepsilon^{\prime}+(l^{\prime}-m^{\prime})\varepsilon]-\pi{\tau}[m^{\prime
2}+(l^{\prime}-m^{\prime})^{2}]\right\\}$
$\displaystyle\stackrel{{\scriptstyle l^{\prime}=2l+\mu}}{{=}}\sum_{\mu=0,1}\
\ \sum_{l,m^{\prime}\in\mathbb{Z}}2\pi i\alpha(2m^{\prime}-2l-\mu)\exp\\{4\pi
i\xi(l+\mu/2)$ $\displaystyle\ \ \ \ \ -2\pi
i[m^{\prime}\varepsilon^{\prime}-(m-2l-\mu)\varepsilon]-\pi[m^{\prime
2}+(m^{\prime}-2l-\mu)^{2}]{\tau}\\}$
Finally letting $m^{\prime}=k+l$, we conclude that
$\displaystyle\Delta=\sum_{\mu=0,1}\left[\sum_{k\in\mathbb{Z}}4\pi
i\alpha[k-\mu/2]\exp\\{-2\pi
i(k-\mu/2)(\varepsilon^{\prime}-\varepsilon)-2\pi{\tau}(k-\mu/2)^{2}\\}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \times\left[\sum_{l\in\mathbb{Z}}\exp\\{2\pi
i(l+\mu/2)(2\xi+\varepsilon^{\prime}+\varepsilon)-2\pi{\tau}(l+\mu/2)^{2}\right]$
$\displaystyle=\left[\sum_{{\mu=0,1}}\partial_{x}\vartheta\left[\begin{matrix}\varepsilon^{\prime}-\varepsilon\\\
-\mu/2\end{matrix}\right](2{\xi},2{\tau})|_{{\xi}={0}}\right]\vartheta\left[\begin{matrix}\varepsilon^{\prime}+\varepsilon\\\
\mu/2\end{matrix}\right](2{\xi},2{\tau}),$
by using the following relations
$\displaystyle k+l=(k-\mu/2)+(l+\mu/2),\ \ k-l-\mu=(k-\mu/2)-(l+\mu/2).$
In a similar way, we can prove the formula (2.7). The formula (2.8) follows
from (2.6) and (2.7). $\Box$
Remark 1. The formulae (2.8) and (2.9) show that if the following equations
are satisfied
$None$ $C({\varepsilon},{\varepsilon^{\prime}},{\mu})=0,$
for $\mu=0,1$, then $\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})$ and
$\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})$ are periodic wave solutions of the
bilinear equation
$F(D_{x},D_{t},D_{n})\vartheta\left[\begin{matrix}\varepsilon^{\prime}\\\
0\end{matrix}\right]({\xi},{\tau})\cdot\vartheta\left[\begin{matrix}\varepsilon\\\
0\end{matrix}\right]({\xi},{\tau})=0.$
The formula (2.10) contains two equations which are called constraint
equations. This formula actually provides us an unified approach to construct
double periodic wave solutions for both differential and difference equations.
Once a equation is written bilinear forms, then its periodic wave solutions
can be directly obtained by solving system (2.10).
Theorem 2. Let $C({\varepsilon},{\varepsilon^{\prime}},{\mu})$ and
$F(D_{x},D_{t},D_{n})$ be given in Theorem 1, and make a choice such that
$\varepsilon^{\prime}-\varepsilon=\pm 1/2$. Then
(i) If $F(D_{x},D_{t},D_{n})$ is an even function in the form
$F(-D_{x},-D_{t},-D_{n})=F(D_{x},D_{t},D_{n}),$
then $C({\varepsilon},{\varepsilon^{\prime}},{\mu})$ vanishes automatically
for the case $\mu=1$, namely
$None$ $C({\varepsilon},{\varepsilon^{\prime}},{\mu})=0,\ \ {\rm for}\ \ \
\mu=1.$
(ii) If $F(D_{x},D_{t},D_{n})$ is an odd function in the form
$F(-D_{x},-D_{t},-D_{n})=-F(D_{x},D_{t},D_{n}),$
then $C({\varepsilon},{\varepsilon^{\prime}},{\mu})$ vanishes automatically
for the case $\mu=0$, namely
$None$ $C({\varepsilon},{\varepsilon^{\prime}},{\mu})=0,\ \ {\rm for}\ \mu=0.$
Proof. We are going to consider the case where $F(D_{x},D_{t},D_{n})$ is an
even function and prove the formula (2.11). The formula (2.12) is analogous.
Making transformation ${m}=-{\bar{m}}+{\mu}$, and noting
$F(D_{x},D_{t},D_{n})$ is even, we then deduce that
$\displaystyle
C({\varepsilon},{\varepsilon^{\prime}},{\mu})=\sum_{{\bar{m}}\in\mathbb{Z}}F(-{\mathcal{M}})\exp\left[-2\pi{\tau}({\bar{m}}-{\mu}/2)^{2}+2\pi
i({\bar{m}}-{\mu}/2)({\varepsilon^{\prime}}-{\varepsilon})\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C({\varepsilon},{\varepsilon^{\prime}},{\mu})\exp\left[4\pi
i({\bar{m}}-{\mu}/2)({\varepsilon^{\prime}}-{\varepsilon})\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C({\varepsilon},{\varepsilon^{\prime}},{\mu})\exp\left(\pm 2\pi
i\bar{m}\right)\exp\left(\pm\pi
i\mu\right)=-C({\varepsilon},{\varepsilon^{\prime}},{\mu}),$
which proves the formula (2.11). $\square$
Corollary 1. Let $\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\
j=1,\cdots,N$. Assume $F(D_{x},D_{t},D_{n})$ is a linear combination of even
and odd functions
$F(D_{x},D_{t},D_{n})=F_{1}(D_{x},D_{t},D_{n})+F_{2}(D_{x},D_{t},D_{n}),$
where $F_{1}(D_{x},D_{t},D_{n})$ is even and $F_{2}(D_{x},D_{t},D_{n})$ is
odd. In addition, $C({\varepsilon},{\varepsilon^{\prime}},{\mu})$
corresponding (2.9) is given by
$C({\varepsilon},{\varepsilon^{\prime}},{\mu})=C_{1}({\varepsilon},{\varepsilon^{\prime}},{\mu})+C_{2}({\varepsilon},{\varepsilon^{\prime}},{\mu}),$
where
$C_{1}({\varepsilon},{\varepsilon^{\prime}},{\mu})=\sum_{{m}\in\mathbb{Z}^{N}}F_{1}({\mathcal{M}})\exp\left[-2\pi{\tau}({m}-{\mu}/2)^{2}-2\pi
i({m}-{\mu}/2)({\varepsilon^{\prime}}-{\varepsilon})\right],$
$C_{2}({\varepsilon},{\varepsilon^{\prime}},{\mu})=\sum_{{m}\in\mathbb{Z}^{N}}F_{2}({\mathcal{M}})\exp\left[-2\pi{\tau}({m}-{\mu}/2)^{2}-2\pi
i({m}-{\mu}/2)({\varepsilon^{\prime}}-{\varepsilon})\right].$
Then
$None$ $\displaystyle
C({\varepsilon},{\varepsilon^{\prime}},{\mu})=C_{2}({\varepsilon},{\varepsilon^{\prime}},{\mu})\
\ {\rm for}\ \ \ \mu=1,$ $None$ $\displaystyle
C({\varepsilon},{\varepsilon^{\prime}},{\mu})=C_{1}({\varepsilon},{\varepsilon^{\prime}},{\mu}),\
\ {\rm for}\ \mu=0.$
Proof. In a similar to the proof of Theorem 2, shifting sum index as
${m}=-{\bar{m}}+{\mu}$, and using $F_{1}(D_{x},D_{t},D_{n})$ even and
$F_{2}(D_{x},D_{t},D_{n})$ odd, we have
$None$ $\displaystyle
C({\varepsilon},{\varepsilon^{\prime}},{\mu})=C_{1}({\varepsilon},{\varepsilon^{\prime}},{\mu})+C_{2}({\varepsilon},{\varepsilon^{\prime}},{\mu})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=\left[C_{1}({\varepsilon},{\varepsilon^{\prime}},{\mu})-C_{2}({\varepsilon},{\varepsilon^{\prime}},{\mu})\right]\exp\left(\pm\pi
i\mu\right).$
Then for $\mu=1$, the equation (2.15) gives
$C_{1}({\varepsilon},{\varepsilon^{\prime}},{\mu})=0,$
which implies the formula (2.13). The formula (2.14) is is analogous.
$\square$
The theorem 2 and corollary 1 are very useful to deal with coupled Hirota’s
bilinear equations, which will be seen in the following section 4.
3\. The shall water wave equation
The shall water wave equation takes the form [17]
$None$ $\displaystyle
u_{t}-u_{xxt}-3uu_{t}+3u_{x}\int_{x}^{\infty}u_{t}dx+u_{x}=0,$
which is like to the KdV equation in the family of shall water wave equations.
Hirota and Satsuma obtained soliton solutions of the equation by means of
bilinear method [18]. Here we construct its a double periodic wave solution
and show that the one-soliton solution can be obtained as limiting case of the
double periodic solution.
To apply the Hirota bilinear method for constructing double periodic wave
solutions of the equation (3.1), we consider a variable transformation
$None$ $u=2\partial_{x}^{2}\ln f(x,t).$
Substituting (3.2) into (3.1) and integrating with respect to $x$, we then get
the following Hirota’s bilinear form
$None$ $\displaystyle F(D_{x},D_{t})f\cdot
f=(D_{x}D_{t}+D_{x}^{2}-D_{t}D_{x}^{3}+c)f\cdot f=0,$
where $c$ is an integration constant. In the special case of $c=0$, starting
from the bilinear equation (3.3), it is easy to find its one-soliton solution
$None$ $u_{1}=2\partial_{x}^{2}\ln(1+e^{\eta}),$
with phase variable $\eta=px+\frac{p}{p^{2}-1}t+\gamma$ for every $p$ and
$\gamma$.
Next, we turn to see the periodicity of the solution (3.2), the function $f$
is chosen to be a Riemann theta function, namely,
$None$ $f(x,t)=\vartheta({\xi},{\tau}),$
where phase variable $\xi=\alpha x+\omega t+\sigma.$ With Proposition 3, we
refer to
$None$
$u=2\partial_{x}^{2}\ln\vartheta({\xi},{\tau})=2\alpha^{2}\partial_{\xi}^{2}\ln\vartheta({\xi},{\tau}),$
which shows that the solution $u$ is a double periodic function with two
fundamental periods $1$ and $i{\tau}$.
We introduce the notations by
$None$
$\displaystyle\lambda=e^{-\pi\tau/2},\quad\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},2\tau)=\sum_{m\in\mathbb{Z}}\lambda^{4m^{2}}\exp(4i\pi
m\xi),$
$\displaystyle\vartheta_{2}(\xi,\lambda)=\vartheta\left[\begin{matrix}0\\\
-1/2\end{matrix}\right](2\mathbf{\xi},2\tau)=\sum_{m\in\mathbb{Z}}\lambda^{(2m-1)^{2}}\exp[2i\pi(2m-1)\xi],$
where the phase variable $\xi=\alpha x+\omega t+\sigma$.
Substituting (3.5) into (3.3), using formula (2.10) and (3.7) leads to a
linear system ( corresponding to $\mu=0$ and $\mu=1$, respectively)
$None$
$\displaystyle[\vartheta_{1}^{\prime\prime}(0,\lambda)\alpha+\vartheta_{1}^{(4)}(0,\lambda)\alpha^{4}]\omega+\vartheta_{1}(0,\lambda)c+\vartheta_{1}^{\prime\prime}(0,\lambda)\alpha^{2}=0,$
$\displaystyle[\vartheta_{2}^{\prime\prime}(0,\lambda)\alpha+\vartheta_{2}^{(4)}(0,\lambda)\alpha^{4}]\omega+\vartheta_{2}(0,\lambda)c+\vartheta_{2}^{\prime\prime}(0,\lambda)\alpha^{2}=0,$
where we have denoted the derivative of $\vartheta_{j}(\xi,\lambda)$ at
$\xi=0$ by notations
$\vartheta_{j}^{(k)}(0,\lambda)=\frac{d^{k}\vartheta_{j}(\xi,\lambda)}{d\xi^{k}}|_{\xi=0},\
\ j=1,2;k=1,2,3,4.$
This system admits an explicit solution $(\omega,c)$. In this way, we obtain
an explicit periodic wave solution (3.6) with parameters $\omega$, $c$ by
(3.8), while other parameters $\alpha,\sigma,\tau,\sigma$ are free.
In summary, double periodic wave (3.6) possesses the following features: (i)
It is is one-dimensional, i.e. there is a single phase variable $\xi$.
Moreover, it has two fundamental periods $1$ and $i\tau$ in phase variable
$\xi$, but it need not to be periodic in $x$ and $t$ . (ii) It can be viewed
as a parallel superposition of overlapping one-soliton waves, placed one
period apart.
In the following, we further consider asymptotic properties of the periodic
wave solution. Interestingly, the relation between the periodic wave solution
(3.6) and the one-soliton solution (3.4) can be established as follows.
Theorem 3. Suppose that the vector $(\omega,c)$ is a solution of the system
(3.8), and for the periodic wave solution (3.6), we let
$None$ $\alpha=\frac{p}{2\pi i},\ \ \sigma=\frac{\gamma+\pi\tau}{2\pi i},$
where the $p$ and $\gamma$ are given in (3.4). Then we have the following
asymptotic properties
$\displaystyle c\longrightarrow 0,\ \ 2\pi
i\xi-\pi\tau\longrightarrow\eta=px+\frac{p}{p^{2}-1}t+\gamma,$
$\displaystyle\vartheta(\xi,\tau)\longrightarrow 1+e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$
In other words, the double periodic solution (3.10) tends to the soliton
solution (3.4) under a small amplitude limit, that is,
$None$ $u\longrightarrow u_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$
Proof. Here we will directly use the system (3.8) to analyze asymptotic
properties of the periodic solution (3.6). Since the coefficients of system
(3.8) are power series about $\lambda$, its solution $(\omega,c)$ also should
be a series about $\lambda$. We explicitly expand the coefficients of system
(3.8) as follows
$None$
$\displaystyle\vartheta_{1}(0,\lambda)=1+2\lambda^{4}+\cdots,\quad\vartheta_{1}^{\prime\prime}(0,\lambda)=-32\pi^{2}\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{1}^{(4)}(0,\lambda)=512\pi^{4}\lambda^{4}+\cdots,\ \
\vartheta_{2}(0,\lambda)=2\lambda+2\lambda^{9}+\cdots$
$\displaystyle\vartheta_{2}^{\prime\prime}(0,\lambda)=-8\pi^{2}\lambda+\cdots,\
\vartheta_{2}^{(4)}(0,\lambda)=32\pi^{4}\lambda+\cdots.$
Let the solution of the system (3.8) be in the form
$None$
$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$\displaystyle c=c_{0}+c_{1}\lambda+c_{2}\lambda^{2}+\cdots=c_{0}+o(\lambda).$
Substituting the expansions (3.11) and (3.12) into the system (3.8) (the
second equation is divided by $\lambda$ ) and letting $\lambda\longrightarrow
0$, we immediately obtain the following relations
$\displaystyle c_{0}=0,\ \
(-8\pi^{2}\alpha+32\pi^{4}\alpha^{3})\omega_{0}-8\pi^{2}\alpha^{2}=0,$
which implies
$None$ $c_{0}=0,\ \ w_{0}=\frac{\alpha}{4\pi^{2}\alpha^{2}-1}.$
Combining (3.12) and (3.13) then yields
$c\longrightarrow 0,\ \ 2\pi i\omega\longrightarrow\frac{2\pi i\alpha}{(2\pi
i\alpha)^{2}-1}=\frac{p}{p^{2}-1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$
Hence we conclude
$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=px+2\pi i\omega t+\gamma$
$\displaystyle\quad\longrightarrow px+\frac{p}{p^{2}-1}t+\gamma=\eta,\ \ {\rm
as}\ \ \lambda\rightarrow 0.$
It remains to consider asymptotic properties of the periodic wave solution
(3.6) under the limit $\lambda\rightarrow 0$. By expanding the Riemann theta
function $\vartheta(\xi,\tau)$ and using (3.14), it follows that
$\displaystyle\vartheta(\xi,\tau)=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots$ $\displaystyle\ \ \ \
\
=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\ \ \ \ \ \quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow
1+e^{\eta},\ \ {\rm as}\ \ \lambda\rightarrow 0,$
which together with (3.6) leads to (3.10). Therefore we conclude that the
double periodic solution (3.6) just goes to the one-soliton solution (3.4) as
the amplitude $\lambda\rightarrow 0$. $\square$
4\. The modified Bogoyavlenskii-Schiff equation
We consider (2+1)-dimensional modified Bogoyavlenskii-Schiff equation [19]
$None$ $\displaystyle
u_{t}-4u^{2}u_{z}-2u_{x}\partial_{x}^{-1}(u^{2})_{z}+u_{xxz}=0,$
which was deduced from the Miura transformation [20]. Equation (4.1) is
reduced to the modified KdV equation in the case of $x=z$.
We shall construct a double periodic wave solution to the equation (4.1) by
using Theorem 1 and 2. The equation (4.1) can be described by a coupled system
$None$ $\displaystyle u=\psi_{x},$ $\displaystyle\rho_{xx}+\psi_{x}^{2}+c=0,$
$\displaystyle\psi_{t}+2\psi_{x}\rho_{xz}+\psi_{z}(\rho_{xx}+\psi_{x}^{2}+c)+\psi_{xxz}=0.$
We perform the dependent variable transformations
$None$ $\displaystyle u=\psi_{x}=\partial_{x}\ln\left(\frac{f}{g}\right),\ \
\rho=\ln(fg),$
then equation (4.2) is reduced to the following bilinear form
$None$ $\displaystyle F(D_{x})f\cdot g=(D_{x}^{2}+c)f\cdot g=0,$
$\displaystyle G(D_{t},D_{x},D_{z})f\cdot
g=(D_{t}+D_{x}^{2}D_{z}+cD_{z})f\cdot g=0,$
where $c$ is a constant. The equation (4.4) is a type of coupled bilinear
equations, which is more difficult to be dealt with than the single bilinear
equation (3.3) due to appearance of two functions and two equations. We will
take full advantages of Theorem 2 to reduce the number of constraint
equations.
Now we take into account the periodicity of the solution (4.3), in which we
take $f$ and $g$ as
$None$ $f=\vartheta({\xi}+e,{\tau}),\ \ g=\vartheta({\xi}+h,{\tau}),\ \
{e},{h}\in\mathbb{C},$
where phase variable $\xi=\alpha x+\beta z+\omega t+\sigma.$ By means of
Proposition 3, we find that the solution
$\displaystyle
u=\alpha\partial_{\xi}\ln\frac{\vartheta({\xi}+e,{\tau})}{\vartheta({\xi}+h,{\tau})}$
is a double periodic function with two fundamental periods $1$ and $i\tau$.
In the special case of $c=0$, the equation (4.2) admits one-soliton solution
$None$ $u_{1}=\partial_{x}\ln\frac{1+e^{\eta}}{1-e^{\eta}},$
where $\eta=px+qy-p^{2}qt+\gamma$ for every $p,q$ and $\gamma$.
We take $e=0,\ h=1/2$ in (4.5), and therefore
$None$ $\displaystyle f=\vartheta(\xi,\tau)=\sum_{m\in\mathbb{Z}}\exp({2\pi
in\xi-\pi m^{2}\tau}),$ $\displaystyle g=\vartheta\left[\begin{matrix}1/2\\\
0\end{matrix}\right](\xi,\tau)=\sum_{m\in\mathbb{Z}}\exp({2\pi im(\xi+1/2)-\pi
m^{2}\tau})$ $\displaystyle\ \ \ =\sum_{m\in\mathbb{Z}}(-1)^{m}\exp({2\pi
im\xi-\pi m^{2}\tau}).$
Due to the fact that $F(D_{x})$ is an even function, its constraint equations
in the formula (2.10) vanish automatically for $\mu=1$. Similarly the
constraint equations associated with $G(D_{t},D_{x},D_{z})$ also vanish
automatically for $\mu=0$. Therefore, the Riemann theta function (4.6) is a
solution of the bilinear equation (4.4), provided the following equations
$None$
$\displaystyle\vartheta_{1}^{\prime\prime}(0,\lambda)\alpha^{2}+\vartheta_{1}(0,\lambda)c=0,$
$\displaystyle\vartheta_{2}^{\prime}(0,\lambda)\omega+\vartheta_{2}^{\prime}(0,\lambda)\beta
c+\vartheta_{2}^{\prime\prime\prime}(0,\lambda)\alpha^{2}\beta=0,$
where we introduce the notations by
$\displaystyle\lambda=e^{-\pi\tau/2},\quad\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},2\tau)=\sum_{m\in\mathbb{Z}}\lambda^{4m^{2}}\exp(4i\pi
m\xi),$
$\displaystyle\vartheta_{2}(\xi,\lambda)=\vartheta\left[\begin{matrix}1/2\\\
-1/2\end{matrix}\right](2\mathbf{\xi},2\tau)=\sum_{m\in\mathbb{Z}}(-1)^{m}\lambda^{(2m-1)^{2}}\exp[2i\pi(2m-1)\xi].$
It is obvious that equation (4.8) admits an explicit solution $\omega$ and
$c$. In this way, a periodic wave solution reads
$None$ $u=\partial_{x}\ln\frac{\vartheta(\xi,\tau)}{\vartheta(\xi+1/2,\tau)},$
where parameters $\omega$ and $c$ are given by (4.11), while other parameters
$\alpha,\beta,\tau,\sigma$ are free. In summary, double periodic wave (4.9)
has the following features: (i) It is one-dimensional and has two fundamental
periods $1$ and $i\tau$ in phase variable $\xi$. (ii) It can be viewed as a
parallel superposition of overlapping one-soliton waves, placed one period
apart.
In the following, we further consider asymptotic properties of the double
periodic wave solution. The relation between the periodic wave solution (4.9)
and the one-soliton solution (4.6) can be established as follows.
Theorem 4. Suppose that the vector $(\omega,c)^{T}$ is a solution of the
system (4.8). In the periodic wave solution (4.9), we choose parameters as
$None$ $\alpha=\frac{p}{2\pi i},\ \ \beta=\frac{q}{2\pi i},\ \
\sigma=\frac{\gamma+\pi\tau}{2\pi i},$
where the $p,q$ and $\gamma$ are the same as those in (4.6). Then we have the
following asymptotic properties
$c\longrightarrow 0,\ \ \xi\longrightarrow\frac{\eta+\pi\tau}{2\pi i},\ \
f\longrightarrow 1+e^{\eta},\ \ g\longrightarrow 1-e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$
In other words, the double periodic solution (4.9) tends to the one-soliton
solution (4.6) under a small amplitude limit , that is,
$None$ $u\longrightarrow u_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$
Proof. Here we will directly use the system (4.8) to analyze asymptotic
properties of periodic solution (4.9). We explicitly expand the coefficients
of system (4.8) as follows
$None$
$\displaystyle\vartheta_{1}(0,\lambda)=1+2\lambda^{4}+\cdots,\quad\vartheta_{1}^{\prime\prime}(0,\lambda)=-32\pi^{2}\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{2}^{\prime}(0,\lambda)=-4\pi i\lambda+12\pi
i\lambda^{9}+\cdots,\ \
\vartheta_{2}^{\prime\prime\prime}(0,\lambda)=16\pi^{3}i\lambda-48\pi^{3}i\lambda^{9}+\cdots,$
Suppose that the solution of the system (4.8) is of the form
$None$
$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$\displaystyle c=c_{0}+c_{1}\lambda+c_{2}\lambda^{2}+\cdots=c_{0}+o(\lambda).$
Substituting the expansions (4.12) and (4.13) into the system (4.8) and
letting $\lambda\longrightarrow 0$, we immediately obtain the following
relations
$\displaystyle c_{0}=0,\ \ -4\pi i\omega_{0}+16\pi^{3}i\alpha^{2}\beta=0,\ \ $
which has a solution
$None$ $c_{0}=0,\ \ w_{0}=4\pi^{2}\alpha^{2}\beta.$
Combining (4.13) and (4.14) leads to
$c\longrightarrow 0,\ \ 2\pi i\omega\longrightarrow
8\pi^{3}i\alpha^{2}\beta=-p^{2}q,\ \ {\rm as}\ \ \lambda\rightarrow 0,$
or equivalently
$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=px+qy+2\pi i\omega t+\gamma$
$\displaystyle\quad\longrightarrow px+qy-p^{2}qt+\gamma=\eta,\ \ {\rm as}\ \
\lambda\rightarrow 0.$
It remains to identify that the periodic wave (4.9) possesses the same form
with the one-soliton solution (4.6) under the limit $\lambda\rightarrow 0$.
For this purpose, we start to expand the functions $f$ and $g$ in the form
$f=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi i\xi})+\lambda^{8}(e^{4\pi
i\xi}+e^{-4\pi i\xi})+\cdots.$ $g=1-\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots.$
By using (4.13)-(4.15), it follows that
$None$ $\displaystyle
f=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow 1+e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0;$ $\displaystyle
g=1-e^{\hat{\xi}}+\lambda^{4}(e^{2\hat{\xi}}-e^{-\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}-e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1-e^{\hat{\xi}}\longrightarrow 1-e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0.$
The expression (4.11) follows from (4.16), and thus we conclude that the
double periodic solution (4.9) just goes to the one-soliton solution (4.6) as
the amplitude $\lambda\rightarrow 0$. $\square$
5\. The differential-difference KdV equation
We consider differential-difference KdV equation
$None$
$\displaystyle\frac{d}{dt}\left(\frac{u(n)}{1+u(n)}\right)=u(n-1/2)-u(n+1/2).$
Hirota and Hu have found its soliton solutions and rational solutions [21,
22], among them one-soliton solution reads
$None$ $u_{1}(n)=\frac{(1+e^{\eta+p/2})(1+e^{\eta-p/2})}{(1+e^{\eta})^{2}}-1,$
where $\eta=pn-\sinh(p/2)t+\gamma$ for every $p$ and $\gamma$.
We shall construct a periodic wave solutions to the equation (5.1) by using
Theorem 1. By means of a variable transformation
$None$ $\displaystyle u(n)=\frac{f({n+1/2})f({n-1/2})}{f(n)^{2}}-1,$
the equation (5.1) is reduced to the bilinear equation
$None$
$\displaystyle\left[\sinh(\frac{1}{4}D_{n})D_{t}+2\sinh(\frac{1}{4}D_{n})\sinh(\frac{1}{2}D_{n})+c\right]f(n)\cdot
f(n)=0,$
where $c$ is a constant.
Now we take into account the periodicity of the solution (5.3), in which we
take $f(n)=\vartheta({\xi},{\tau}),$ where phase variable $\xi=\nu n+\omega
t+\sigma.$ Then solution (5.3) is written as
$None$ $\displaystyle u(\xi)\equiv
u(n)=\frac{\vartheta({\xi}+\frac{1}{2}\nu,{\tau})\vartheta({\xi}-\frac{1}{2}\nu,{\tau})}{\vartheta({\xi},{\tau})^{2}}-1.$
By means of Proposition 2, it is easy to deduce that $u_{n}$ is a double
periodic function with two fundamental periods $1$ and $i\tau$.
Substituting (5.5) into (5.4) and using formula (2.10) leads to a linear
system
$None$
$\displaystyle\sinh(\frac{1}{4}D_{n})\vartheta_{1}^{\prime}(0,\lambda)\omega+\vartheta_{1}(0,\lambda)c+\sinh(\frac{1}{4}D_{n})\sinh(\frac{1}{2}D_{n})\vartheta_{1}(0,\lambda)=0,$
$\displaystyle\sinh(\frac{1}{4}D_{n})\vartheta_{2}^{\prime}(0,\lambda)\omega+\vartheta_{2}(0,\lambda)c+\sinh(\frac{1}{4}D_{n})\sinh(\frac{1}{2}D_{n})\vartheta_{2}(0,\lambda)=0,$
where $\vartheta_{1}(\xi,\lambda)$ and $\vartheta_{2}(\xi,\lambda)$ are the
same as those in (3.7) with $\xi=\nu n+\omega t+\sigma.$ By using the solution
$\omega$ and $c$ of system (5.6), a periodic wave solution is obtained by
(5.5).
In the following, we further consider asymptotic properties of the double
periodic wave solution. The relation between the periodic wave solution (5.5)
and the one-soliton solution (5.2) can be established as follows.
Theorem 5. Suppose that the vector $(\omega,c)^{T}$ is a solution of the
system (5.6). In the periodic wave solution (5.5), we choose parameters as
$None$ $\nu=\frac{p}{2\pi i},\ \ \sigma=\frac{\gamma+\pi\tau}{2\pi i},$
where the $p$ and $\gamma$ are the same as those in (5.2). Then we have the
following asymptotic properties
$c\longrightarrow 0,\ \ \xi\longrightarrow\frac{\eta+\pi\tau}{2\pi i},\ \
\vartheta(\xi,\tau)\longrightarrow 1+e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$
In other words, the periodic solution (5.5) tends to the one-soliton solution
(5.2) under a small amplitude limit , that is,
$None$ $u(n)\longrightarrow u_{1}(n),\ \ {\rm as}\ \ \lambda\rightarrow 0.$
Proof. Here we will directly use the system (5.6) to analyze asymptotic
properties of periodic solution (5.5). We explicitly expand the coefficients
of system (5.6) as follows
$None$
$\displaystyle\vartheta_{1}(0,\lambda)=1+2\lambda^{4}+\cdots,\quad\sinh(\frac{1}{4}D_{n})\vartheta_{1}^{\prime}(0,\lambda)=8\pi
i\sinh(i\pi\nu)\lambda^{4}+\cdots,$
$\displaystyle\sinh(\frac{1}{4}D_{n})\sinh(\frac{1}{2}D_{n})\vartheta_{1}(0,\lambda)=2\sinh(i\pi\nu)\sinh(2i\pi\nu)\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{2}(0,\lambda)=2\lambda+2\lambda^{9}+\cdots,\ \
\sinh(\frac{1}{4}D_{n})\vartheta_{2}^{\prime}(0,\lambda)=4\pi
i\sinh(i\pi\nu/2)\lambda+\cdots,$
$\displaystyle\sinh(\frac{1}{4}D_{n})\sinh(\frac{1}{2}D_{n})\vartheta_{2}(0,\lambda)=2\sinh(i\pi\nu)\sinh(i\pi\nu/2)\lambda+\cdots.$
Suppose that the solution of the system (5.6) is of the form
$None$
$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$\displaystyle c=c_{0}+c_{1}\lambda+c_{2}\lambda^{2}+\cdots=c_{0}+o(\lambda).$
Substituting the expansions (5.9) and (5.10) into the system (5.6) and letting
$\lambda\longrightarrow 0$, we immediately obtain the following relations
$\displaystyle c_{0}=0,\ \ 4\pi
i\sinh(i\pi\nu/2)\omega_{0}+2\sinh(i\pi\nu/2)\sinh(i\pi\nu)=0,\ \ $
which implies
$None$ $c_{0}=0,\ \ w_{0}=-\frac{1}{2\pi i}\sinh(i\pi\nu).$
Combining (5.9) and (5.10) leads to
$c\longrightarrow 0,\ \ 2\pi
i\omega\longrightarrow-\sinh(i\pi\nu)=-\sinh(p/2),\ \ {\rm as}\ \
\lambda\rightarrow 0,$
or equivalently
$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=pn+2\pi i\omega t+\gamma$
$\displaystyle\quad\longrightarrow pn-\sinh(p/2)t+\gamma=\eta,\ \ {\rm as}\ \
\lambda\rightarrow 0.$
It remains to consider asymptotic properties of the periodic wave solution
(5.5) under the limit $\lambda\rightarrow 0$. By expanding the Riemann theta
function $\vartheta(\xi,\tau)$, it follows that
$\displaystyle\vartheta(\xi,\tau)=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\ \ \ \ \ \quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow
1+e^{\eta},\ \ {\rm as}\ \ \lambda\rightarrow 0,$
which together with (5.5) lead to (5.8). Therefore we conclude that the
periodic solution (5.5) just goes to the one-soliton solution (5.2) as the
amplitude $\lambda\rightarrow 0$. $\square$
Acknowledgment
The work described in this paper was supported by grants from Research Grants
Council contracts HKU, the National Science Foundation of China (No.10971031),
Shanghai Shuguang Tracking Project (No.08GG01) and Innovation Program of
Shanghai Municipal Education Commission (No.10ZZ131).
## References
* [1]
* [2] R. Hirota and J. Satsuma: Prog. Theor. Phys. 57 (1977) 797.
* [3] R. Hirota: Direct methods in soliton theory (Springer-verlag, Berlin, 2004).
* [4] X. B. Hu, C X Li, J. J. C. Nimmo and G. F. Yu, J. Phys. A, 38 (2005) 195.
* [5] R Hirota and Y. Ohta: J. Phys. Soc. Jpn. 60 (1991) 798\.
* [6] K W Chow, C K Lam, K Nakkeeran and B Malmed. J. Phys. Soc. Jpn. 77(2008), 054001
* [7] K. Sawada and T Kotera: Prog. Theor. Phys. 51 (1974) 1355.
* [8] A. Nakamura, J. Phys. Soc. Jpn. 47, 1701-1705 (1979).
* [9] A. Nakamura, J. Phys. Soc. Jpn. 48(1980), 1365.
* [10] H. H. Dai, E. G. Fan and X. G. Geng, arxiv.org/pdf/nlin/0602015
* [11] Y. Zhang, L. Y. Ye, Y. N. Lv and H. Q. Zhao, J. Phys A, 40 (2007), 5539.
* [12] Y. C. Hon, E. G. Fan and Z. Y. Qin, Modern Phys Lett B, 22 (2008), 547.
* [13] E. G. Fan and Y. C. Hon, Phys Rev E, 78 (2008), 036607.
* [14] E. G. Fan, J. Phys A, 42 (2009), 095206.
* [15] W. X. Ma, R. G. Zhou, J. Math. Phys, 24 (2009), 1677.
* [16] H. M. Farkas and I. Kra, Riemann Surfaces, New York, Springer-Verlag, 1992.
* [17] P. A. Clarkson and E. L. Mansfield, Nonlinearity, 7 (19994), 975.
* [18] R. Hirota and J. Satsuma, J. Phys. Soc. Jpn. 40 (1976), 611.
* [19] S. J. Yu, K. Toda, N. Sasa and T. Fukuyama, J. Phys A, 31 (1998), 3337.
* [20] O. I. Bogoyavenskii, Math. USSR Izv. 36 (1991), 129.
* [21] R Hirota and J. Satsuma, Progr. Theor. Phys. Suppl. 64 (1976), 64.
* [22] X. B. Hu and P. A. Clarkson, J. Phys. A 28 (1995) 5009.
|
arxiv-papers
| 2010-01-13T13:34:38 |
2024-09-04T02:49:07.721136
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Engui Fan, Kwok Wing Chow",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1001.2159"
}
|
1001.2160
|
2010417-428Nancy, France 417
Serge Grigorieff
Pierre Valarcher
# Evolving MultiAlgebras
unify all usual sequential computation models
S. Grigorieff LIAFA, CNRS & Université Paris Diderot - Paris 7, Case 7014,
75205 Paris Cedex 13 seg@liafa.jussieu.fr http://www.liafa.jussieu.fr and P.
Valarcher LACL, Université de Paris Est, IUT Fontainebleau/Sénart, Route de
Hourtault 77300 Fontainebleau valarcher@univ-paris12.fr http://lacl.univ-
paris12.fr/valarcher/
###### Abstract.
It is well-known that Abstract State Machines (ASMs) can simulate “step-by-
step” any type of machines (Turing machines, RAMs, etc.). We aim to overcome
two facts: 1) simulation is not identification, 2) the ASMs simulating
machines of some type do not constitute a natural class among all ASMs. We
modify Gurevich’s notion of ASM to that of EMA (“Evolving MultiAlgebra”) by
replacing the program (which is a syntactic object) by a semantic object: a
functional which has to be very simply definable over the static part of the
ASM. We prove that very natural classes of EMAs correspond via “literal
identifications” to slight extensions of the usual machine models and also to
grammar models. Though we modify these models, we keep their computation
approach: only some contingencies are modified.
Thus, EMAs appear as the mathematical model unifying all kinds of sequential
computation paradigms.
###### Key words and phrases:
Abstract state machines; Models of machines; Computability; Universality;
Logic in computer science; Theory of algorithms
###### Contents
1. 1 Introduction
2. 2 From ASMs to EMA s: the deterministic case
1. 2.1 How EMAs differ from ASMs
2. 2.2 Deterministic Evolving MultiAlgebras
3. 3 Turing machines
4. 4 Random access machines
5. 5 Other models
6. 6 Uniformly bounded non determinism
7. 7 External non determinism
## 1\. Introduction
What we prove in this paper. The fact that Abstract State Machines (ASMs) can
strict lock-step (i.e. “step-by-step ”) simulate any type of machines (Turing
machines, stack automata, RAM, etc) and grammars was shown long ago by
Gurevich [10, 6]. A systematic study is also done in Börger [2]. A tighter
notion of simulation is also valid as shown in Blass, Dershowitz & Gurevich
[1].
The questions we consider in this paper are the following:
1. (Q1)
Can we replace strict lock-step simulation by literal identity (up to a simple
change of view)?
2. (Q2)
Given a computation model ${\mathcal{C}}$, is it possible to get a natural
characterization of the class of ASMs which are equivalent to machines in
${\mathcal{C}}$?
As far as we know, up to now, there is only one isolated answer which is about
question (Q2): Gurevich & al. [6] proved that Schönhage Storage Modification
Machines correspond exactly (for strict lock-step equivalence) to ASMs with
unary functions only.
We bring positive answers to both questions for the diverse usual computation
models ${\mathcal{C}}$ (Turing machines, stack automata, RAMs, Schönhage
Machines, Chomsky type $0$ grammars, etc.) slightly extended to models
${\mathcal{C}}^{+}$ using a tailored version of ASMs which (resurrecting
Gurevich’s original name for ASMs) we call Evolving Multialgebras (EMAs).
These answers have the following remarkably simple form:
###### Theorem 1.1.
There exists a family of EMA static parts ${\mathcal{M}}$ (fixed semantical
feature) and a family of dynamic signatures ${\mathcal{S}}$ (fixed syntactical
feature) such that, letting ${\mathcal{E}}_{M,S}$ be the family of EMAs with
static part in ${\mathcal{M}}$ and dynamic signature in ${\mathcal{S}}$,
\- any computation device in ${\mathcal{C}}^{+}$ is literally identical to
some EMA in ${\mathcal{E}}_{{\mathcal{M}},{\mathcal{S}}}$,
\- this “literal identity” correspondence is a bijection from
${\mathcal{C}}^{+}$ onto ${\mathcal{E}}_{{\mathcal{M}},{\mathcal{S}}}$.
Of course, literal identity is not a formal notion. What we mean is as
follows: the diverse components of a computation device in ${\mathcal{C}}^{+}$
are in one-one correspondance with the diverse components of the associated
EMA, and this correspondance is an identity up to a change of perspective (for
instance, a “physical” bi-infinite tape will be considered to be identical to
the mathematical set $\mathbb{Z}$ of integers).
###### Remark 1.2.
1\. This theorem is indeed a schema: one theorem per computation model. We
have proved it for a variety of usual sequential computation models (cf. [5]).
2\. As said above, the diverse instances of Theorem 1.1 are proved for slight
extension ${\mathcal{C}}^{+}$ of the usual computation models ${\mathcal{C}}$.
In all cases, ${\mathcal{C}}^{+}$ can be viewed as ${\mathcal{C}}$ considered
with different time units: for any $k\geq 1$, a device ${\mathcal{M}}$ in
${\mathcal{C}}$ is seen as a device ${\mathcal{M}}^{(k)}$ in
${\mathcal{C}}^{+}$ in which one step of ${\mathcal{M}}^{(k)}$ corresponds to
$k$ successive steps of ${\mathcal{M}}$ (or $<k$ successive steps in case the
last of these steps has no successor).
3\. Considering another presentation of ${\mathcal{C}}^{+}$, one can also view
it as ${\mathcal{C}}$ in which some contingencies have been removed (for
instance, the read/write head will be able to scan a window of cells instead
of a single cell) but the computational paradigm has been preserved: local
computation and a particular topology of data storage for Turing machines,
indirect addressing of registers for random access machines, etc. In our
opinion, the classes ${\mathcal{C}}^{+}$ are the right ones to carry the
diverse computation paradigms.
4\. In fact, contingencies can also be captured by families of EMAs with more
technical definitions (cf. [5]): we loose the remarkable simplicity of the
above families ${\mathcal{E}}_{{\mathcal{M}},{\mathcal{S}}}$.
5\. This theorem schema strengthens Gurevich’s claim that ASMs constitute the
natural mathematical modelization of algorithms: EMAs (which are a variant of
ASMs) appear as the computation model which unifies all usual sequential
computation paradigms.
About the proof. No surprise, the proof of Theorem 1.1 for a particular
${\mathcal{C}}$ involves the particular features of the class ${\mathcal{C}}$.
Thus, the claim (point 5 in Remark 1.2) that Theorem 1.1 is true for
extensions ${\mathcal{C}}^{+}$ of every usual sequential computation model
${\mathcal{C}}$ cannot be proved but only be supported by proved instances for
a variety of classes ${\mathcal{C}}$.
As for the common features to all such proofs, they come from an analysis of
what precludes positive solutions to questions (Q1) and (Q2). Let us list some
of the difficulties which are met. Some are easy to solve, other ones force to
adequately tailor the definition of ASMs (as that of EMAs) and those of the
usual computation models.
(1) An ASM program mimicking the transition function $\delta$ of a Turing
machine is a description of $\delta$. Since there are many distinct
descriptions of the same $\delta$, there are many ASMs which tightly simulate
the same Turing machine. Thus, surprisingly as it is, looking at this
component – transition functions –, ASMs are less abstract than Turing
machines. Somehow, there is an extra operational feature in ASMs: the
operational way to use $\delta$ is not part of the formalization of Turing
machines.
This is why we modify ASMs to EMAs: Evolving Multialgebras. The notion of EMA
is that of ASM in which the program (a syntactic object) is replaced by a
semantic object: a (very simply definable) functional operating on the
function sets over the ASM domain. It is then more natural to break the
universe of an ASM into its natural parts: this allows a very useful
rudimentary typing of elements and functions.
(3) Again considering Turing machines, an ASM simulates the tape by the set
$\mathbb{Z}$ of all integers and the moves of the head by the successor and
predecessor operations on $\mathbb{Z}$. Terms in the ASM logical language
allow to name the $i$-th successor and the $i$-th predecessor. Thus, we cannot
avoid the ASM program to move the head more than one cell left or right unless
we constrain terms in ASM programs to be of a simple form (somewhat “flat”).
Which would put technicalities to any positive answer to question (Q2). This
is why we consider slight extensions of the machine models which allow the
read/write head to scan a window of cells rather than only one cell and to
move in a window. This is a kind of extra capability which is much like
allowing several tapes or several heads. Though it does modify the model, it
does preserves its core feature: successive local actions.
(4) For machine models having programs like RAMs and SMM (Schönhage Storage
Modi cation Machines), there are two slight modifications. First, allow
bounded blocks of parallel and/or successive actions. Second, remove the
program and the program counter in favor of a transition function (much in the
vein of Turing machines) which, though operating on an infinite set (the
contents of the accumulator and of the addressed registers in the case of
RAMs) is very simply definable in terms of the original program. Thus, we
replace an operational item (the program) by a denotational one (the
transition function). Again, though it does modify the model, it does
preserves its core feature: indirect addressing (for RAMs), dynamic storage
topology (for Schönhage pointer machines).
EMAs versus ASMs. In our opinion, ASMs and EMAs are complementary models. EMAs
generalize any type of machine: it is the unification model. As for ASMs, they
are closer to programming. Indeed, the functioning of a EMA goes through the
iteration of a functional. To program an EMA, we need to add some operational
information on how to use this functional and this leads back to a program,
hence to an ASM…Thus, ASMs are EMAs plus the instructions for using the
functional: ASMs refine EMAs (in the sense of software engineering) and EMAs
are a (more) abstract version of ASMs.
## 2\. From ASMs to EMA s: the deterministic case
### 2.1. How EMAs differ from ASMs
We detail the diverse features which are peculiar to EMAs.
A functional in place of a program. As said in the introduction, the main
difference between evolving multialgebras and Gurevich’s ASMs is as follows:
the program (i.e. a syntactic object) of an ASM is replaced by a functional
(i.e. a semantic object) which does exactly what the program tells to do.
Thus, an operational feature is removed.
Multi-domains and multialgebras. The above modification leads to another very
minor one, really kind of “semantic sugar”: the universe of an ASM is broken
into its natural constituents and becomes a multi-domain. The reason for such
multialgebras is that they make it possible to type the symbols of the
signature as functions (or elements) between the diverse sets of the multi-
domain.
Multialgebra operations with values in products of domains. Set theoretically,
a map $F:A\to B\times C$ is identified with the pair of its component maps
$(F_{B},F_{C})$ where $F_{B}:A\to B$ and $F_{C}:A\to C$. We do view such an
$F$ as the pair $(F_{B},F_{C})$ plus a correlation condition: one cannot fire
one of these two component maps without firing the other one, and both have to
be fired on the same argument.
We allow operations in the multialgebra to take values in products of domains.
The above condition leads to a notion of multiterms and a constraint in the
definition of formulas associated to the signature of an EMA. It is used in
§LABEL:s:schonhage to deal with Schönhage machines.
Halt/Fail and EMA status. In EMAs, the ASM program is replaced by the
functional which does exactly what the program tells to do. There are still
the questions:
\- is the functional to be applied or not on given arguments?
\- if not, does it “halts and accepts” or “halts and rejects” or “get stuck”?
To deal with the three first alternatives, EMAs have a three valued dynamic
component: the status. Of course, there is no formal component carrying the
information “stuck”.
Inputs and ASMs. In most presentations, Gurevich does not give any formal
status to inputs (his paper [4] with Dershowitz being an exception). When
dealing with question (Q2) it turns out that it is important to give a formal
status to inputs. This is the case for EMA characterizations of machines
having some read-only tapes (e.g., finite automata). We consider that inputs
appear in two ways:
\- as values of some particular static symbols,
\- as initial values of dynamic symbols.
### 2.2. Deterministic Evolving MultiAlgebras
###### Definition 2.1.
Let $n\geq 1$ and ${\mathcal{D}}=(D_{i})_{i=1,\ldots,n}$ be a sequence of $n$
non empty sets (which we call an $n$-multiset). An $n$-sort type is a triple
$(k,\alpha,\ell)$ where $k\in\mathbb{N}$, $\ell\in\\{1,\ldots,n\\}$ and
$\alpha$ is a map $\\{1,\ldots,k\\}\to\\{1,\ldots,n\\}$. Its associated
${\mathcal{D}}$-type $(k,\alpha,\ell)_{{\mathcal{D}}}$ is the family of all
partial functions $D_{\alpha(1)}\times\ldots\times D_{\alpha(k)}\to D_{\ell}$.
A ${\mathcal{D}}$-type is functional if $k\geq 1$. In case $k=0$, the
${\mathcal{D}}$-type $(0,\emptyset,\ell)_{{\mathcal{D}}}$ is the family of
partial functions $\\{\emptyset\\}\to D_{\ell}$, i.e. the set of “partial
elements” of $D_{\ell}$, i.e. $D_{\ell}$ augmented with an “undefined
element”.
Intuition: there are $k$ arguments, $\alpha$ gives their types, and $\ell$ is
the type of the range.
Typed ground terms and their types are defined in the obvious way.
Multialgebras. The notion of multisort algebra is a direct extension to
multiset domains of the usual notion of algebra of partial functions on a
unique domain.
###### Definition 2.2 (Multialgebras).
Let $n\geq 1$ and ${\mathcal{S}}$ be an $n$-sort typed signature containing
function symbols $\varphi_{1},\ldots,\varphi_{p}$. An
${\mathcal{S}}$-multialgebra ${\mathcal{A}}$ is an $n$-multiset
${\mathcal{D}}=(D_{i})_{i=1,\ldots,n}$ endowed with partial functions
$F_{1},\ldots,F_{p}$ which interpret the symbols $\varphi_{i}$’s (Care: arity
$0$ symbols with type $D_{i}$ are interpreted by elements of $D_{i}$ but can
also be undefined).
If defined, the value, relative to ${\mathcal{A}}$, of a ground
${\mathcal{S}}$-term $t$ is denoted by
$\hbox{$[\kern-3.99994pt[\,{t}\,]\kern-3.99994pt]$}_{{\mathcal{A}}}$ (it is an
element of some $D_{i}$).
Semialgebraic functionals. Semialgebraic functionals are those which can be
described by ASM programs. They modify the interpretations in the multialgebra
of constant and functions symbols. For function symbols, this modification
affects the values of only finitely many points in the domain. These points
and the associated new values of the argument are given by ground
${\mathcal{S}}$-terms. As in ASMs programs, there is a disjunction of cases
for the choice of the affected points and their associated new values.
First, a convenient notion.
###### Definition 2.3 (The $\oplus$ operation).
Let $F,G$ be partial functions $X_{1}\times\ldots\times X_{k}\to Y$ and
$Z\subseteq X_{1}\times\ldots\times X_{k}$. We define the partial function
$F\oplus_{Z}G$ as follows:
$\displaystyle\text{\tt Domain}(F\oplus_{Z}G)$ $\displaystyle=$
$\displaystyle(\text{\tt Domain}(F)\setminus Z)\cup(\text{\tt Domain}(G)\cap
Z)$ $\displaystyle(F\oplus_{Z}G)(\vec{x})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}F(\vec{x})&\textit{if\quad$\vec{x}\notin
Z$}\\\ G(\vec{x})&\textit{if\quad$\vec{x}\in Z$}\end{array}\right.$
In case $p=0$, $F,G$ are “partial elements” of $Y$ and
$Z\subseteq\\{\emptyset\\}$ and $F\oplus_{Z}G=F$ if $Z=\emptyset$ and
$F\oplus_{Z}G=G$ if $Z=\\{\emptyset\\}$.
###### Definition 2.4 (Semialgebraic functionals).
Let
• ${\mathcal{D}}=(D_{i})_{i=1,\ldots,n}$ be an $n$-multiset,
• ${\mathcal{S}}$ be an $n$-sort typed signature containing function symbols
$\varphi_{1},\ldots,\varphi_{p}$,
• ${\mathcal{A}}$ be a multialgebra with signature
${\mathcal{S}}\setminus\\{\varphi_{1},\ldots,\varphi_{p}\\}$ on
${\mathcal{D}}$,
• ${\mathcal{F}}_{1},\ldots,{\mathcal{F}}_{p}$ be the ${\mathcal{D}}$-types
associated to $\varphi_{1},\ldots,\varphi_{p}$,
• $m\in\\{1,\ldots,p\\}$ and $(k,\alpha,\ell)$ be the $n$-sort type of
$\varphi_{m}$.
• ${\mathcal{T}}_{i}$ be the family of ground ${\mathcal{S}}$-terms of type
$D_{i}$,
For any $p$-tuple of functions
$\vec{F}=(F_{1},\ldots,F_{p})\in{\mathcal{F}}_{1}\times\ldots\times{\mathcal{F}}_{p}$,
let us denote by ${\mathcal{A}}(\vec{F})$ the multialgebra ${\mathcal{A}}$
expanded to the signature ${\mathcal{S}}$ in which the $\varphi_{i}$’s are
interpreted by the $F_{i}$’s.
A partial functional
$\Phi:\prod_{j=1,\ldots,p}{\mathcal{F}}_{j}\longrightarrow{\mathcal{F}}_{m}$
is $({\mathcal{S}},{\mathcal{A}})$-semialgebraic if there exists a map
$\beta:{\tt
Bool}^{q}\to\mathfrak{P}_{\textit{fin}}({\mathcal{T}}_{\alpha(1)}\times\ldots\times{\mathcal{T}}_{\alpha(k)}\times{\mathcal{T}}_{\ell})$
(where $\mathfrak{P}_{\textit{fin}}(X)$ is the family of finite subsets of
$X$) and ground ${\mathcal{S}}$-terms $t_{1},\ldots,t_{q}$,
$t^{\prime}_{1}\ldots,t^{\prime}_{q}$ such that, for any
$\vec{F}\in{\mathcal{G}}_{1}\times\ldots\times{\mathcal{G}}_{p}$,
$\begin{array}[]{l}\Phi(\vec{F})\mbox{ is defined if and only if}\\\
\quad\left\\{\begin{array}[]{l}(a)\ \ \mbox{all
$\hbox{$[\kern-3.99994pt[\,{t_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}$'s,
$\hbox{$[\kern-3.99994pt[\,{t^{\prime}_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}$'s
are defined}\\\ (b)\ \
\forall(u_{1},\ldots,u_{k},v)\in\beta(\ldots,\hbox{$[\kern-3.99994pt[\,{t_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}=\hbox{$[\kern-3.99994pt[\,{t^{\prime}_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\ldots)\mbox{
all
$\hbox{$[\kern-3.99994pt[\,{u_{j}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}$'s
are defined}\\\ (c)\ \
\forall(\vec{u},v),(\vec{w},z)\in\beta(\ldots,\hbox{$[\kern-3.99994pt[\,{t_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}=\hbox{$[\kern-3.99994pt[\,{t^{\prime}_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\ldots)\quad\mbox{
$\hbox{$[\kern-3.99994pt[\,{u_{j}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}\neq\hbox{$[\kern-3.99994pt[\,{w_{j}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}$
for some $j$}\end{array}\right.\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\Phi(\vec{F})=F_{m}\oplus_{Z}G\mbox{ where }\\\ \quad
Z=\\{(\hbox{$[\kern-3.99994pt[\,{u_{1}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\ldots,\hbox{$[\kern-3.99994pt[\,{u_{k}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})})\mid\exists
v\
(\vec{u},v)\in\beta(\ldots,\hbox{$[\kern-3.99994pt[\,{t_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}=\hbox{$[\kern-3.99994pt[\,{t^{\prime}_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\ldots)\\}\\\
\quad
G=\\{(\hbox{$[\kern-3.99994pt[\,{u_{1}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\ldots,\hbox{$[\kern-3.99994pt[\,{u_{k}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\hbox{$[\kern-3.99994pt[\,{v}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})})\mid(\vec{u},v)\in\beta(\ldots,\hbox{$[\kern-3.99994pt[\,{t_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}=\hbox{$[\kern-3.99994pt[\,{t^{\prime}_{i}}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})},\ldots)\\}\end{array}$
The tuple $(\beta,t_{1},\ldots,t_{q},t^{\prime}_{1}\ldots,t^{\prime}_{q})$ is
called a presentation of $\Phi$.
For $I\subseteq\\{1,\dots,p\\}$, a functional
$\Psi:\prod_{j=1,\ldots,p}{\mathcal{F}}_{j}\longrightarrow\prod_{m\in
I}{\mathcal{F}}_{m}$ is $({\mathcal{S}},{\mathcal{A}})$-semialgebraic if so
are all its components.
###### Remark 2.5.
Condition (a) in Definition 2.4 insures that all equality tests
$t_{i}=t^{\prime}_{i}$ can be achieved. Conditions (b) and (c) insure that, in
equality $\Phi(\vec{F})=F_{m}\oplus_{Z}G$, the finite set $Z$ can be computed
and $G$ is a functional graph.
We do not require the
$\hbox{$[\kern-3.99994pt[\,{v}\,]\kern-3.99994pt]$}_{{\mathcal{A}}(\vec{F})}$’s
to be defined (i.e. $\text{\tt Domain}(G)=Z$): though this is incompatible
with a call by value strategy, it makes sense with a call by name strategy.
###### Definition 2.6 (Deterministic EMAs).
A deterministic evolving multialgebra (EMA) is a tuple ${\mathcal{A}}=(n;\
{\mathcal{S}}_{\text{sta}},{\mathcal{S}}^{\text{sta}}_{\text{input}},{\mathcal{S}}^{\text{dyn}}_{\text{input}},{\mathcal{S}}_{\text{dyn}};\
{\mathcal{D}};\ {\mathcal{M}}_{\text{sta}},{\mathcal{M}}_{\text{ini}};\ \Phi)$
consisting of the following items.
* •
An $n$-multiset ${\mathcal{D}}=(D_{i})_{i=1,\ldots,n}$ such that
$D_{n}=\\{\text{go},\text{acc},\text{rej}\\}$.
Intuition. Sets $D_{1},\ldots,D_{n-1}$ are the $n-1$ different sorts of
objects and $D_{n}=\\{\text{go},\text{acc},\text{rej}\\}$ is the set of
possible statuses of the (evolving) multialgebra during the run: “go on”,
“halt and accept”, “halt and reject”.
* •
Four disjoint $n$-sort typed finite signatures
${\mathcal{S}}_{\text{sta}},{\mathcal{S}}^{\text{sta}}_{\text{input}},{\mathcal{S}}^{\text{dyn}}_{\text{input}},{\mathcal{S}}_{\text{dyn}}$
and two structures ${\mathcal{M}}_{\text{sta}},{\mathcal{M}}_{\text{ini}}$
with respective signatures
${\mathcal{S}}_{\text{sta}},{\mathcal{S}}_{\text{dyn}}$. There is only one
symbol $\mathfrak{s}$ which involves the sort $n$ : it is a constant of type
$D_{n}$ in ${\mathcal{S}}^{\text{dyn}}_{\text{input}}$.
Intuition. ${\mathcal{M}}_{\text{sta}}$ is the static framework on
${\mathcal{D}}$ which remains fixed during any run.
${\mathcal{S}}^{\text{sta}}_{\text{input}}$ is the signature for the static
part of the input: its interpretation remains fixed (hence accessible) during
a run. ${\mathcal{S}}^{\text{dyn}}_{\text{input}}$ is the signature for the
dynamic part of the input: its interpretation can be modified (hence become
inaccessible) during a run. ${\mathcal{M}}_{\text{ini}}$ initializes the part
of the dynamic environment which is not initialized by the input. The
interpretation of $\mathfrak{s}$ represents the status of the multialgebra.
* •
Let
${\mathcal{S}}={\mathcal{S}}_{\text{sta}}\cup{\mathcal{S}}^{\text{sta}}_{\text{input}}\cup{\mathcal{S}}^{\text{dyn}}_{\text{input}}\cup{\mathcal{S}}_{\text{dyn}}$.
$\Phi$ is a $({\mathcal{S}},{\mathcal{M}}_{\text{sta}})$-semialgebraic partial
functional
$\Phi:\left(\prod_{\varphi\in{\mathcal{S}}^{\text{sta}}_{\text{input}}}{\mathcal{F}}_{\varphi}\right)\times\left(\\{\text{go}\\}\times\prod_{\varphi\in({\mathcal{S}}_{\text{dyn}}\cup{\mathcal{S}}^{\text{dyn}}_{\text{input}})\setminus\\{\mathfrak{s}\\}}{\mathcal{F}}_{\varphi}\right)\quad\longrightarrow\quad\prod_{\varphi\in{\mathcal{S}}_{\text{dyn}}\cup{\mathcal{S}}^{\text{dyn}}_{\text{input}}}{\mathcal{F}}_{\varphi}$
where ${\mathcal{F}}_{\varphi}$ denotes the semantic type of the function
symbol $\varphi$. In particular, $\Phi$ rules the evolution of the status. The
sole status which can be an argument of $\Phi$ is “go”: a multialgebra with
status “acc” or “rej” is halted and does not evolve any more. However, in the
image of $\Phi$ the status can take any value.
A state of ${\mathcal{A}}$ is any multialgebra on ${\mathcal{D}}$ with
signature ${\mathcal{S}}$ which expands ${\mathcal{M}}_{\text{sta}}$.
###### Definition 2.7 (Runs of deterministic EMAs).
We keep the notations of Definition 2.6. A run of ${\mathcal{A}}$ is a
sequence $({\mathcal{M}}_{t})_{t\in I}$ of states of ${\mathcal{A}}$ such that
• $I$ is a finite or infinite non empty initial segment of $\mathbb{N}$,
•
$\hbox{$[\kern-3.99994pt[\,{\theta}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{0}}=\hbox{$[\kern-3.99994pt[\,{\theta}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{\text{ini}}}$
for all $\theta\in{\mathcal{S}}_{\text{dyn}}$,
• If $t\in I$ then
$\hbox{$[\kern-3.99994pt[\,{\theta}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{t}}=\hbox{$[\kern-3.99994pt[\,{\theta}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{0}}$
for all $\theta\in{\mathcal{S}}^{\text{sta}}_{\text{input}}$,
• If $t\in I$ then $t+1$ is in $I$ if and only if
$\hbox{$[\kern-3.99994pt[\,{\mathfrak{s}}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{t}}=\text{go}$
and $\Phi$ is defined on
$(\hbox{$[\kern-3.99994pt[\,{\varphi}\,]\kern-3.99994pt]$}_{{\mathcal{M}}}\\})_{\varphi\in{\mathcal{S}}\setminus{\mathcal{S}}_{sta}}$,
• If $t+1\in I$ then
$(\hbox{$[\kern-3.99994pt[\,{\theta}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{t+1}})_{\theta\in{\mathcal{S}}_{\text{dyn}}\cup{\mathcal{S}}^{\text{dyn}}_{\text{input}}}=\Phi((\hbox{$[\kern-3.99994pt[\,{\varphi}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{t}}\\})_{\varphi\in{\mathcal{S}}\setminus{\mathcal{S}}_{sta}})$.
In particular, if
$\hbox{$[\kern-3.99994pt[\,{\mathfrak{s}}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{0}}\neq\text{go}$
then $I=\\{0\\}$. Also, if $t+1\in I$ then
$\hbox{$[\kern-3.99994pt[\,{\mathfrak{s}}\,]\kern-3.99994pt]$}_{{\mathcal{M}}_{t}}=\text{go}$.
## 3\. Turing machines
In order to identify Turing machines with a simple class of EMAs, we introduce
a slight variant of Turing machines, which we call “window Turing machines”:
1) the head is allowed to scan a small window instead of a single cell, and to
move inside a window in a single step, 2) halting (be it accepting or
rejecting) is not related to the current state but to the current local
configuration: the state plus the contents of the scanned window.
###### Definition 3.1.
A deterministic $k$-window $n$-tape (bi-infinite tapes) Turing machine is a
tuple
$(n,k,\Sigma=\\{\sigma_{0},\ldots,\sigma_{s-1}\\},Q=\\{q_{0},\ldots,q_{r-1}\\},F^{+},F^{-},\delta,\omega_{i},\mu_{i})_{i=1,\ldots,n}$
where, for $i=1,\ldots,n$,
* •
$\Sigma$ and $Q$ are finite sets (the alphabet and the set of states),
* •
$F^{+},F^{-}\subseteq Q\times\Sigma^{n(2k+1)}$ (accepting/rejecting final
local configurations),
* •
$\delta:Q\times\Sigma^{n(2k+1)}\to Q$ (state transition),
* •
$\tau_{i}:Q\times\Sigma^{n(2k+1)}\to\Sigma^{n(2k+1)}$ (read/write on tape
$i$),
* •
$\mu_{i}:Q\times\Sigma^{n(2k+1)}\to\\{-k,\ldots,-1,0,1,\ldots,k\\}$ (move on
tape $i$).
On each tape, the head scans the cell on which it is positioned and the $k$
cells to the left and the $k$ cells to the right, a total of $2k+1$ cells. The
argument of type $\Sigma^{n(2k+1)}$ in $\delta,\omega_{i},\mu_{i}$ is the
contents of the $n(2k+1)$ cells scanned on the $n$ tapes. The effect of a
transition is to change the state according to $\delta$, to modify the
contents of the scanned cells of tape $i$ according to $\omega_{i}$ and to
move its head according to $\mu_{i}$.
The notions of run, halt, acceptance and rejection are defined as usual.
###### Remark 3.2.
Usual deterministic $n$-tape Turing machines are the $1$-window ones.
###### Definition 3.3 (The class of EMAs for Turing machines).
We denote by ${\mathcal{C}}^{(n)}_{\textit{wT}}$ the class of EMAs
${\mathcal{A}}=(n+3;\
{\mathcal{S}}_{\text{sta}},{\mathcal{S}}^{\text{sta}}_{\text{input}},{\mathcal{S}}^{\text{dyn}}_{\text{input}},{\mathcal{S}}_{\text{dyn}};\
{\mathcal{D}};\ {\mathcal{M}}_{\text{sta}},{\mathcal{M}}_{\text{ini}};\ \Phi)$
which satisfy the following conditions for some $r,s\in\mathbb{N}$ (for
clarity, we abusively denote by the same letter static constant symbols and
the elements which interpret them in the structure ${\mathcal{D}}$).
(1) The multidomain of ${\mathcal{A}}$ is
${\mathcal{D}}=(\mathbb{Z}^{(1)},\ldots,\mathbb{Z}^{(n)},Q,\Sigma,\mathfrak{S})$
where the $\mathbb{Z}^{(i)}$’s are fixed pairwise disjoint copies of
$\mathbb{Z}$ (for instance, $\mathbb{Z}^{(i)}=\mathbb{Z}\times\\{i\\}$),
$Q,\Sigma$ are finite sets with $r,s$ elements respectively, and
$\mathfrak{S}=\\{\text{go},\text{acc},\text{rej}\\}$.
(2) The static framework signature ${\mathcal{S}}_{\text{sta}}$ contains $r$
constants $q_{0},\ldots,q_{r-1}$ of type $Q$, $s$ constants
$\sigma_{0},\ldots,\sigma_{s-1}$ of type $\Sigma$ and three constants
$\text{go},\text{acc},\text{rej}$ of type $\mathfrak{S}$ which are interpreted
in the obvious way in ${\mathcal{M}}_{\text{sta}}$. It also contains, for each
$i=1,\ldots,n$, two unary functions symbols
$\textit{Succ}^{(i)},\textit{Pred}^{(i)}$ of type
$\mathbb{Z}^{(i)}\to\mathbb{Z}^{(i)}$ which are interpreted in
${\mathcal{M}}_{\text{sta}}$ as the successor and predecessor functions in
$\mathbb{Z}^{(i)}$.
(3) The signature ${\mathcal{S}}^{\text{sta}}_{\text{input}}$ is empty.
(4) The signature ${\mathcal{S}}_{\text{dyn}}$ (for the dynamic environment
non initialized by the input) contains, for each $i=1,\ldots,n$, one constant
$\text{pos}^{(i)}$ of type $\mathbb{Z}^{(i)}$ one constant $q$ of type $Q$,
and one constant $\mathfrak{s}$ of type $\mathfrak{S}$, which are respectively
interpreted in ${\mathcal{M}}_{\text{ini}}$ as $0$, $q_{0}$ and go.
(5) The signature ${\mathcal{S}}^{\text{dyn}}_{\text{input}}$ (for the dynamic
environment initialized by the input) contains, for each $i=1,\ldots,n$, one
unary function $c^{(i)}$ of type $\mathbb{Z}^{(i)}\to\Sigma$.
Thus, the EMAs in ${\mathcal{C}}^{(n)}_{\textit{wT}}$ are defined as those
having particular signature, multidomain, static framework and initialization
of some dynamic symbols with no condition on the functional $\Phi$ (other than
its semialgebraicity).
###### Theorem 3.4 (EMA representation theorem for Turing machines).
Any deterministic $n$-tape window Turing machine is literally identical to
some EMA in the class ${\mathcal{C}}^{(n)}_{\textit{wT}}$. Conversely, any EMA
in ${\mathcal{C}}^{(n)}_{\textit{wT}}$ is literally identical to some
deterministic $n$-tape window Turing machine.
###### Proof 3.5.
The argument is based on the following literal identifications between the
components of a Turing machine (TM) and the interpretations of symbols of the
EMA signature:
1. (1)
(TM) $i$-th tape and the way the read/write head moves on it.
(EMA) the copy $\mathbb{Z}^{(i)}$ of $\mathbb{Z}$ structured as
$\langle\mathbb{Z}^{(i)},\textit{Succ}^{(i)},\textit{Pred}^{(i)}\rangle$.
2. (2)
(TM) diverse states and letters.
(EMA) interpretations of the static symbols $q_{0},\ldots,q_{r-1}$ and
$\sigma_{0},\ldots,\sigma_{s-1}$.
3. (3)
(TM) current state, positions of the $n$ heads and contents of the $n$ tapes.
(EMA) current interpretations of the dynamic symbols $q$, $pos^{(i)}$,
$c^{(i)}$.
4. (4)
(TM) non final or final accepting/rejecting character of the current state.
(EMA) current interpretation of the dynamic symbol $\mathfrak{s}$.
5. (5)
(TM) transition function.
(EMA) semialgebraic functional.
6. (6)
(TM) initial configuration.
(EMA) interpretations of the $c_{i}$’s in the initial multialgebra and of
${\mathcal{S}}^{\text{dyn}}_{\text{dyn}}$ in ${\mathcal{M}}_{\text{ini}}$.
The non trivial identifications are those of points 4 and 5.
Keeping the notations of Definition 2.4, let
$(\beta_{\varphi},t_{1,\varphi},\ldots,t_{q_{\varphi},\varphi},t^{\prime}_{1,\varphi}\ldots,t^{\prime}_{q_{\varphi},\varphi})_{\varphi\in{\mathcal{S}}^{\text{int}}_{\text{dyn}}}$
be a presentation of the semialgebraic functional $\Phi$ of an EMA:
$\beta_{\varphi}:{\tt
Bool}^{q_{\varphi}}\to\mathfrak{P}_{\textit{fin}}({\mathcal{T}}_{\alpha_{\varphi}(1)}\times\ldots\times{\mathcal{T}}_{\alpha_{\varphi}(k_{\varphi})}\times{\mathcal{T}}_{\ell_{\varphi}})$
Observe that terms of type $\mathbb{Z}^{(j)}$ are of the form
$\xi_{1}(\xi_{2}(\ldots))(\text{pos}^{(j)})$ where the $\xi_{k}$’s are
$\textit{Succ}^{(j)}$ or $\textit{Pred}^{(j)}$. Let $k$ be the maximum value
of the $|\xi_{1}(\xi_{2}(\ldots))(0)|$ for all terms of type some
$\mathbb{Z}^{(j)}$ which is among the $t_{i,\varphi},t^{\prime}_{i,\varphi}$
or among the finite sets given by the $\beta_{\varphi}$’s.
First, let us look at the equalities $t_{i,\varphi}=t^{\prime}_{i,\varphi}$
which govern the domain of $\Phi$.
• If $t_{i,\varphi},t^{\prime}_{i,\varphi}$ have type $\mathbb{Z}^{(j)}$ then,
as said above, they are of the form
$\xi_{1}(\xi_{2}(\ldots))(\text{pos}^{(j)})$. Hence any equality
$t_{i,\varphi}=t^{\prime}_{i,\varphi}$ is trivially true or false
independently of the current value of $\text{pos}^{(j)}$.
If $t_{i,\varphi},t^{\prime}_{i,\varphi}$ have type $\mathfrak{S}$ then they
are of the form $\mathfrak{s}$ or $\text{go},\text{acc},\text{rej}$. Since
$\Phi$ and $\beta$ are restricted to values where $\mathfrak{s}=\text{go}$,
all possible equalities are trivial.
Thus, we can suppose that there is no term with type $\mathbb{Z}^{(j)}$ or
$\mathfrak{S}$ among the $t_{i,\varphi},t^{\prime}_{i,\varphi}$’s.
• If $t_{i,\varphi},t^{\prime}_{i,\varphi}$ have type $Q$ then they are of the
form $q$ or $q_{j}$ ($j=0,\ldots,r-1$). Since any equality $q_{j}=q_{k}$ is
trivially true or false, we can suppose that there is at most one equality
between terms of type $Q$ and that it is of the form $q=q_{j}$.
• If $t_{i,\varphi},t^{\prime}_{i,\varphi}$ have type $\Sigma$ then they are
of the form $c^{(j)}(\xi_{1}(\xi_{2}(\ldots))(\text{pos}^{(j)}))$ where the
$\xi_{k}$’s are $\textit{Succ}^{(j)}$ or $\textit{Pred}^{(j)}$. The equalities
between terms of type $\Sigma$ are all comparisons of letters among the values
of $c^{(1)}(-k),\ldots,c^{(1)}(k)$,…, $c^{(n)}(-k),\ldots,c^{(n)}(k)$ where
$k$ is defined above.
This shows that the values of $\Phi$ depend solely on the value of $q$ and
those of the $c^{(j)}(\text{pos}^{(j)}+i)$’s for $j=1,\ldots,n$ and
$i=-k,\ldots,k$). This is exactly to say that what matters is the current
state and the current letters in the $n$ windows of diameter $2k+1$ centered
at the positions of the $n$ heads. Otherwise said, the tuple of arguments of
the functional $\Phi$ is literally identical to the current values of the
state plus the contents of the windows, that is a tuple in
$Q\times\Sigma^{n(2k+1)}$.
Let us look at the image of $\Phi$ which is given through finite families of
tuples of terms given by the $\beta_{\varphi}$’s. Since the only terms of type
$Q$ are $q$ and the $q_{i}$’s. Thus, $\Phi$ can leave the dynamic symbol $q$
unchanged or modify it to any value. The same is valid for the dynamic symbol
$\mathfrak{s}$ (using what is said above about the domain of $\Phi$, this
proves the non easy direction of point 4).
Terms of type $\Sigma$ name the contents of some $c^{(j)}$ at positions which
are at distance $\leq k$ of the position of the $j$-th head. Thus $\Phi$ can
modify the values of the $c^{(j)}$ in the windows around the positions of the
heads.
Terms of type $\mathbb{Z}^{(j)}$ name an integer at distance $k$ of the
position of the $j$-th head. Thus $\Phi$ can move any head left or right of at
most $k$ cells. This proves the non easy direction of point 5. Thus, an EMA in
${\mathcal{C}}^{(n)}_{T}$ is literally identical to some window Turing
machine. The converse is proved in a similar (much easier) way.
###### Remark 3.6.
A slight variation in the EMA model can have strong effect. For instance,
suppose we add a constant $0$ to the static signature and interpret it as $0$
in the structure ${\mathcal{M}}_{\text{sta}}$. Then we get window Turing
machines in which the head can jump to cell $0$.
## 4\. Random access machines
In order to identify RAMs with a simple class of EMAs, we introduce a slight
variant of RAMs, which we call “transition RAM” (TRAM): 1) a bounded number of
registers can be modified in one step, 2) it can test for equality to $0$ and
equality between combinations (via the fixed set of operations on
$\mathbb{N}$) of the contents of the addressed registers, 3) the program is
replaced by a transition function. Though this function operates on an
infinite domain, it is finitarily defined via ground terms.
###### Definition 4.1 ($n$-transition RAMs).
Let $f_{1},\ldots,f_{p}$ operations on non negative integers, A $n$-transition
RAM ($n$-TRAM) with operations $f_{1},\ldots,f_{p}$ is a tuple
$(n,k,Q=\\{q_{0},\ldots,q_{r-1}\\},F^{+},F^{-},\delta,\rho_{i},\tau_{i,j})_{i=1,\ldots,n,\
j=1,\ldots,k}$
where
* •
$n$ is the number of distinguished registers,
* •
$\Sigma$ and $Q$ are finite sets (the alphabet and the set of states),
* •
$F^{+},F^{-}\subseteq Q\times{\tt Bool}^{p}$ (accepting/rejecting final local
configurations),
* •
$\delta:Q\times{\tt Bool}^{p}\to Q$ (state transition),
* •
$\rho_{i}:Q\times{\tt Bool}^{p}\to T$ (modification of register $i$) for
$i=1,\ldots,n$, where $T$ is a finite family of terms built with the
operations $f_{1},\ldots,f_{p}$ and $n(1+k)$ constants (representing the
contents of the addressed registers),
* •
$\tau_{i,j}:Q\times{\tt Bool}^{p}\to T$ (modification of the register
addressed through an iteration of $j$ successive addressing, starting with
register $i$), for $i=1,\ldots,n$, $j=1,\ldots,k$.
At any time the $n$-TRAM accesses registers $1,\ldots,n$ and the registers
addressed addressed through at most $k$ iterated addressing by these
registers. The $p=n(1+k)(1+\frac{n(1+k)-1}{2})$ Boolean arguments in the
$\delta,\rho_{i},\tau_{i}$’s test equalities or equalities to $0$ of the
contents of the $n(1+k)$ adressed registers. Map $\delta$ tells how the state
is modified. Maps $\rho_{i},\tau_{i,j}$’s tell how the contents of the
accessed registers are modified.
The notions of run, halt, acceptance and rejection are defined in the usual
way.
###### Definition 4.2 (The class of EMAs for TRAMS).
Let $f_{1},\ldots,f_{p}$ operations on non negative integers. We denote by
${\mathcal{C}}^{(n)}_{\text{TRAM}}$ the class of EMAs ${\mathcal{A}}$ which
satisfy the following conditions.
(1) ${\mathcal{A}}$ has $4$ sorts and its multidomain is
${\mathcal{D}}=(\mathbb{N},\mathbb{N}^{\text{addr}},Q,\mathfrak{S})$ where
$\mathbb{N}^{\text{addr}}$ is a copy of $\mathbb{N}$, $Q$ is a finite set with
$r$ elements, and $\mathfrak{S}=\\{\text{go},\text{acc},\text{rej}\\}$.
(2) The signature ${\mathcal{S}}_{\text{sta}}$ (for the static framework)
contains $n+r+3$ constants: $1,\dots,n$ of type $\mathbb{N}$,
$q_{0},\ldots,q_{r-1}$ of type $Q$, “go”, “acc”, “rej” of type $\mathfrak{S}$,
and $n+1$ unary function symbols cast of type $\mathbb{N}\to N^{\text{addr}}$,
and, for each $i=1,\ldots,n$, $f_{i}$ of type
$\mathbb{N}^{k_{i}}\to\mathbb{N}$. Their interpretations in
${\mathcal{M}}_{\text{sta}}$ are as follows: i) $f_{i}$ is interpreted as the
given operation on $\mathbb{N}$, ii) the cast function is interpreted as the
identity from $\mathbb{N}$ to its copy $N^{\text{addr}}$, iii) $1,\ldots,n$,
the $q_{i}$’s and “go”, “acc”, “rej” are interpreted in the obvious way.
(3) The signature ${\mathcal{S}}^{\text{sta}}_{\text{input}}$ is empty.
(4) The signature ${\mathcal{S}}_{\text{dyn}}$ contains two constants
$q,\mathfrak{s}$ of types $Q$ and $\mathfrak{S}$. Their interpretations in
${\mathcal{M}}_{\text{ini}}$ are $q_{0}$ and “go”.
(5) The signature ${\mathcal{S}}^{\text{dyn}}_{\text{input}}$ contains one
unary function $c$ of type $\mathbb{N}^{\text{addr}}\to\mathbb{N}$.
Thus, the EMAs in ${\mathcal{C}}^{(n)}_{\text{TRAM}}$ are defined as those
having particular signature, multidomain, static framework and initialization
of some dynamic symbols with no condition on the functional $\Phi$ (other than
its semialgebraicity).
###### Theorem 4.3 (EMA representation theorem for TRAMs).
Any $n$-TRAM is literally identical to some EMA in the class
${\mathcal{C}}^{(n)}_{\text{TRAM}}$. Conversely, any EMA in
${\mathcal{C}}^{(n)}_{\text{TRAM}}$ is literally identical to some $n$-TRAM.
###### Proof 4.4.
Analogous to the proof of Theorem 3.4.
## 5\. Other models
Similar results can be proved with finite atomata, stack automata Schönhage
machines.
Let us mention an interesting feature occurring in the EMA modelization of
Schönhage Storage Modification Machines (SMM) which illustrates what has been
said in §2.1 about operations with values in products of domains. The tape of
an SMM is a dynamic graph which may grow or loose nodes. To manage the current
set of nodes of this graph-tape, it is convenient to introduce the following
items:
• Among the sets of the multi-domain ${\mathcal{D}}$, there is an infinite set
$X$ (where all nodes are taken) and the set $\mathfrak{P}_{\textit{fin}}(X)$
of finite subsets of $X$. There is no structure on $X$ nor on
$\mathfrak{P}_{\textit{fin}}(X)$.
• In the signature ${\mathcal{S}}_{\text{dyn}}$, there is a constant symbol
$U$ of type $\mathfrak{P}_{\textit{fin}}(X)$ (it tells which nodes are in the
current graph-tape).
• In the signature ${\mathcal{S}}_{\text{sta}}$, there is a function symbol
new with type $\mathfrak{P}_{\textit{fin}}(X)\to
X\times\mathfrak{P}_{\textit{fin}}(X)$. It is interpreted as a choice function
$A\mapsto(a,A\cup\\{a\\})$ which picks in $X$ a point outside $A$, i.e. such
that $a\notin A$.
To add a new node to the graph tape, we apply new to $U$. The constraint that
both components of new have to be fired simultaneously and on the same
argument insures that when a new node is picked, it is automatically added to
(the interpretation) of $U$ with no condition on the functional $\Phi$.
## 6\. Uniformly bounded non determinism
Uniformly bounded non determinism allows at each step at most $k$ choices
where $k$ is some fixed constant independent of the step. EMAs with ‘such non
determinism are defined as are deterministic EMAs with the following
modification: replace the semialgebraic functional $\Phi$ by finitely many
such functionals. All litteral identity results mentioned in the previous
sections extend easily to the non deterministic cases.
## 7\. External non determinism
We now deal with a more powerful kind of non determinism: that given by
external choices which may be done during the run. This is the action of
Gurevich’s “Choose” instruction. To deal with such an “external non
determinism”, we enrich EMAs with a fifth signature: the “external dynamic”
signature ${\mathcal{S}}_{\text{ext}}$. We illustrate this notion with the
example of Chomsky type $0$ grammars.
###### Definition 7.1.
A grammar is a finite set of rules $(u_{i},v_{i})_{i=1,\ldots,n}$ where the
$u_{i},v_{i}$’s are words in an alphabet $\Sigma$. The associated relation
$R\subseteq\Sigma^{\star}\times\Sigma^{\star}$ is defined as follows: a pair
$(U,V)$ is in $R$ if and only if there exists a finite sequence
$U=U_{0},\ldots,U_{k}=V$ such that, for all $j<k$ there exists words $P,S$ and
some $i=1,\ldots,n$ such that $U_{j}=Pu_{i}S$ and $U_{j+1}=Pv_{i}S$.
###### Definition 7.2.
We denote by ${\mathcal{C}}_{\text{gra}}$ the class of non deterministic EMAs
${\mathcal{A}}=(3;\
{\mathcal{S}}_{\text{sta}},{\mathcal{S}}^{\text{sta}}_{\text{input}},{\mathcal{S}}^{\text{dyn}}_{\text{input}},{\mathcal{S}}_{\text{dyn}},{\mathcal{S}}_{\text{ext}};\
{\mathcal{D}};\ {\mathcal{M}}_{\text{sta}},{\mathcal{M}}_{\text{ini}};\ \Phi)$
which satisfy the following conditions.
(1) ${\mathcal{A}}$ has $3$ sorts and its multidomain is
${\mathcal{D}}=(\mathbb{N},\Sigma^{*},\mathfrak{S})$ where $\Sigma$ is a
finite set.
(2) The signature ${\mathcal{S}}_{\text{sta}}$ (for the static framework)
contains finitely many binary function symbols $\textit{subst}_{i}$,
$i=1,\ldots,n$ of type $\mathbb{N}\times\Sigma^{*}\to\Sigma^{*}$. There is
some family $(u_{i},v_{i})_{i=1,\ldots,n}$ of pairs of words such that the
interpretation in ${\mathcal{M}}_{\text{sta}}$ (the static framework) of
$\textit{subst}_{i}$ is the function which acts on a pair $(p,U)$ as follows:
if $U$ contains the factor $u_{i}$ in position $p$ then it is replaced by
$v_{i}$, else $U$ is not modified.
(3) The signatures ${\mathcal{S}}^{\text{sta}}_{\text{input}}$ and
${\mathcal{S}}_{\text{dyn}}$ are empty.
(4) The signature ${\mathcal{S}}^{\text{dyn}}_{\text{input}}$ contains one
constant $w$ of type $\Sigma^{*}$.
(5) The signature ${\mathcal{S}}_{\text{ext}}$ (the external dynamic
environment) contains one constant $Choose$ of type $\mathbb{N}$. Its
interpretation during the run is given as an external action: its value
changes at each step.
Thus, the EMAs in ${\mathcal{C}}^{(n)}_{\text{gra}}$ are defined as those
having particular signature, multidomain, static framework and initialization
of some dynamic symbols with no condition on the functional $\Phi$ (other than
its semialgebraicity).
Using the fact that iteration of substitutions is also a substitution, one can
prove :
###### Theorem 7.3.
Any grammar is literally identical to some EMA in the class
${\mathcal{C}}_{\text{gra}}$. Conversely, any EMA in
${\mathcal{C}}_{\text{gra}}$ is literally identical to some grammar.
## References
* [1] Andreas Blass, Nachum Dershowitz and Yuri Gurevich. Exact exploration. Microsoft TechReport MSR-TR-2009-99, 2009.
* [2] Egon Börger. Unifying View of Models of Computation and System Design Frameworks. Annals of Pure and Applied Logic, 133: 149-171, 2005.
* [3] Giuseppe Del Castillo and Yuri Gurevich and Karl Stroetmann. Typed Abstract State Machines. Unfinished manuscript, 25 pages, 1998.
* [4] Nachum Dershowitz and Yuri Gurevich. A natural axiomatization of computability and proof of Church’s Thesis. Bulletin. of Symbolic Logic, 14(3):299–350, 2008.
* [5] Serge Grigorieff and Pierre Valarcher. Evolving MultiAlgebras unify all usual sequential computation models. http://lacl.univ-paris12.fr/valarcher/.
* [6] S. Dexter and P. Boyle and Y. Gurevich. Gurevich Abstract State Machines and Schönhage Storage Modification Machines. JUCS, 3(4): 279–303, 1997.
* [7] Yuri Gurevich. Reconsidering Turing’s Thesis: towards more realistic semantics of programs. Technical Report CRL-TR-38-84, EEC Dept, Univ. Michigan, 1984.
* [8] Yuri Gurevich. A new Thesis. Abstracts, American Math. Soc., 1985.
* [9] Yuri Gurevich. Logic and the Challenge of Computer Science. Current Trends in Theoretical Computer Science, ed. Egon Börger, Computer Sc. Press. 1–57, 1988.
* [10] Yuri Gurevich. Evolving Algebras: An Introductory Tutorial. Bul. EATCS, 43: 264–284, 1991. Reprinted in Current Trends in Theoretical Computer Science, 1993, 266–29, World Scientific, 1993.
* [11] Yuri Gurevich. May 1997 Draft of the ASM Guide. Tech Report CSE-TR-336-97, EECS Dept, University of Michigan, 1997.
* [12] Y Gurevich. The Sequential ASM Thesis. Bul. EATCS, 67: 93–124, 1999. Reprinted in Current Trends in Theoretical Comp. Sc., 2001, 363–392, World Scientific, 2001.
* [13] Yuri Gurevich. Sequential Abstract State Machines capture Sequential Algorithms. ACM Transactions on Computational Logic, 1(1):77–111, July 2000.
|
arxiv-papers
| 2010-01-13T13:35:20 |
2024-09-04T02:49:07.728798
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serge Grigorieff and Pierre Valarcher",
"submitter": "Serge Grigorieff",
"url": "https://arxiv.org/abs/1001.2160"
}
|
1001.2197
|
010007 2009 A. C. Martí 010007
When two immiscible liquids that coexist inside a porous medium are drained
through an opening, a complex flow takes place in which the interface between
the liquids moves, tilts and bends. The interface profiles depend on the
physical properties of the liquids and on the velocity at which they are
extracted. If the drainage flow rate, the liquids volume fraction in the
drainage flow and the physical properties of the liquids are known, the
interface angle in the immediate vicinity of the outlet ($\theta$) can be
determined. In this work, we define four nondimensional parameters that rule
the fluid dynamical problem and, by means of a numerical parametric analysis,
an equation to predict $\theta$ is developed. The equation is verified through
several numerical assessments in which the parameters are modified
simultaneously and arbitrarily. In addition, the qualitative influence of each
nondimensional parameter on the interface shape is reported.
# Parametric study of the interface behavior between two immiscible liquids
flowing through a porous medium
Alejandro David Mariotti [inst1] Elena Brandaleze E-mail:
mariotti.david@gmail.com [inst2] Gustavo C. Buscaglia[inst3] E-mail:
ebrandaleze@frsn.utn.edu.arE-mail: gustavo.buscaglia@icmc.usp.br
(14 October 2009; 28 December 2009)
††volume: 1
99 inst1 Instituto Balseiro, 8400 San Carlos de Bariloche, Argentina. inst2
Departamento de Metalurgia, Universidad Tecnológica Nacional Facultad Regional
San Nicolás, 2900 San Nicolás, Argentina. inst3 Instituto de Ciências
Matemáticas e de Computação, Universidade de São Paulo, 13560-970 São Carlos,
Brasil.
## 1 Introduction
The fluid dynamics of the flow of two immiscible liquids through a porous
medium plays a key role in several engineering processes. Usually, though the
interest is focused on the extraction of one of the liquids, the simultaneous
extraction of both liquids is necessary. This is the case of oil production
and of ironmaking. The water injection method used in oil production consists
of injecting water back into the reservoir, usually to increase pressure and
thereby stimulate production. Normally, just a small percentage of the oil in
a reservoir can be extracted, but water injection increases that percentage
and maintains the production rate of the reservoir over a longer period of
time. The water displaces the oil from the reservoir and pushes it towards an
oil production well [1]. In the steel industry, this multiphase phenomenon
occurs inside the blast furnace hearth, in which the porous medium consists of
coke particles. The slag and pig iron are stratified in the hearth and,
periodically, they are drained through a lateral orifice. The understanding of
this flow is crucial for the proper design and management of the blast furnace
hearth [2]. In both examples above, when the liquids are drained, a complex
flow takes place in which the interface between the liquids moves, tilts and
bends.
Numerical simulation of multiphase flows in porous media is focused mainly in
upscaling methods, aimed at solving for large scale features of interest in
such a way as to model the effect of the small scale features [3–5]. Other
authors [6–8] use the numerical methods to model the complex multiphase flow
that takes place at the pore scale.
In this work, we numerically study the macroscopic behavior of the interface
between two immiscible liquids flowing through a porous medium when they are
drained through an opening. The effect of gravity on this phenomenon is
considered. We define four nondimensional parameters that rule the fluid
dynamical problem and, by means of a numerical parametric analysis, an
equation to predict the interface tilt in the vicinity of the orifice
($\theta$) is developed. The equation is verified through several numerical
cases where the parameters are varied simultaneously and arbitrarily. In
addition, the qualitative influence of each non-dimensional parameter on the
interface shape is reported.
## 2 Parametric Study
The numerical studies in this work were carried out by means of the program
FLUENT 6.3.26. Different models to simulate the two-dimensional parametric
study were used. The volume of fluid (VOF) method was chosen to treat the
interface problem [9]. The drag force in the porous medium was modeled by
means of the source term suggested by Forchheimer [10]. The source term for
the $i^{th}$ momentum equation is:
$\displaystyle S_{i}=-\left(\frac{\mu}{\alpha}V_{i}+\frac{1}{2}\rho
C|\overrightarrow{V}|V_{i}\right).$ (1)
For the constants $\alpha$ and $C$ in Eq. (1), we use the values proposed by
Ergun [10]:
$\displaystyle\alpha=\frac{\varepsilon^{3}d^{2}}{150(1-\varepsilon)^{2}},$ (2)
$\displaystyle C=1.75\frac{(1-\varepsilon)}{\varepsilon^{3}d},$ (3)
where $\varepsilon$ is the porosity, $d$ is the particle equivalent diameter,
$\rho$ is the density, $V$ is the velocity and $\mu$ is the dynamic molecular
viscosity.
Considering that the subscript 1 and 2 represent the fluid 1 and the fluid 2
respectively, three nondimensional parameters were considered in the
parametric study: viscosity ratio, $\mu_{R}=\mu_{1}/\mu_{2}$, density ratio,
$\rho_{R}=\rho_{1}/\rho_{2}$, and nondimensional velocity,
$V_{R}=V_{0}\rho_{2}L/\mu_{2}$; where $V_{0}$ is the outlet velocity and $L$ a
reference length.
### 2.1 Domain description
Figure 1: Sketch of the numerical 2D domain.
The numerical domain considered to carry out the parametric study was a two-
dimensional one composed by the porous medium sub-domain and the outlet sub-
domain. The porous sub-domain is a rectangle $10$m wide and $10$m tall. Inside
of it, a rigid, isotropic and homogeneous porous medium was arranged. We use a
porosity and particle diameter of $0.32$ and $0.006$m, respectively.
For the outlet domain we use a rectangle $0.02m$ wide and with a height
$L=0.01$m divided into two equal parts and located at the center of one of the
lateral edges. The part located at the end of the outlet domain is used to
impose the outlet velocity. Figure 1 shows a complete description of the
domain.
A quadrilateral mesh with $2.2\times 10^{4}$ cells was used, where the outlet
sub-domain mesh consists of $200$ elements in all the cases studied.
As boundary conditions, on edge 1 we define a zero gauge pressure condition
normal to the boundary and impose that only the fluid 1 can enter to the
domain through it. On edge 2 the boundary condition is the same as on edge 1
but the fluid consider in this case is fluid 2. On edge 7 we impose a zero
gauge pressure normal to the boundary but in this case the fluids can only
leave the domain. On the other edges (edges 4, 5, 6, and 8) we impose a wall
condition where the normal and tangential velocity is zero except for edge 3,
at which the tangential velocity is free and the stress tangential to the edge
is zero.
### 2.2 Interface evolution
To illustrate how an interface reaches the stationary position from an
initially horizontal one, three sets of curves were obtained.
Figure 2: Interface evolution when the initial position is below the outlet
level, without gravity.
Figure 2 shows the interface evolution for the case without the gravity effect
and the interface initial position is below the outlet level. The interface
modifies its tilt to reach the exit and it changes its shape to reach the
stationary profile.
Figures 3 and 4 show the interface evolution when the gravity is present but
the interface initial position is below and above the outlet level
respectively.
Figure 3: Interface evolution when the initial position is below the outlet,
with gravity. Figure 4: Interface evolution when the initial position is above
the outlet, with gravity.
### 2.3 Viscosity effect
One of the most important parameters to modify is the dynamical viscosity of
fluid 1. We maintain the properties of the fluid 2 as the properties of water
(density $998$Kg/m3, and dynamical viscosity $0.001$ Pa.s) and the density of
fluid 1 as the density of the oil ($850$ Kg/m3). The dynamical viscosity of
fluid 1 was varied from values smaller than those of fluid 2, to values much
greater.
Figure 5: Stationary interface profiles modifying the fluid 1 viscosity
without the gravity effect.
Two sets of curves were obtained, one considering the effect of gravity and
the other without considering it. Figure 5 shows the stationary interface
profiles for the different values of viscosity, without gravity. A value of
$V_{R}=1.5\times 10^{4}$ and $\rho_{R}=1.17$ were chosen. It is possible to
observe that, when fluid 1 has a viscosity higher than that of fluid 2, the
interface profile is above the outlet and points downwards at the outlet. If
fluid 1 has a lower viscosity, the opposite happens.
Figure 6: Stationary interface profiles modifying the fluid 1 viscosity with
the gravity effect.
When considering gravity, the value of $V_{R}$ was changed to $1\times 10^{5}$
($V_{0}=10$m/s), since for smaller values the interface may not reach the
outlet (this is later studied in Fig. 10). Figure 6 shows the curves obtained
for this situation, where the interface only lies over the outlet level for
the higher $\mu_{R}$ values.
### 2.4 Outlet velocity effect
$V_{0}$ is varied from a small value, similar to the porous medium velocity
($V_{0}=0.2$m/s or $V_{R}=2000$), to a very large one ($V_{0}=50$m/s or
$V_{R}=5\times 105$). Maintaining the properties of fluid 2 similar to those
of water, two sets of curves were obtained ($\mu_{R}>1$ and $\mu_{R}<1$),
shown in Figs. 7 and 8, respectively.
Figure 7: Stationary interface profiles for several values of $V_{R}$, without
gravity, for $\mu_{R}>1$ ($\mu_{R}=35$).
When the effect of gravity was considered, two additional sets of curves
(Figs. 9 and 10) were obtained.
Figure 7 shows the effect of $V_{R}$ when the viscosity of fluid 1 is greater
than that of fluid 2, without gravity. It is possible to see that, as $V_{R}$
increases, the interface tilt at the outlet is maximal for $V_{R}=1\times
10^{5}$.
On the other hand, Fig. 8 shows the interface profiles when the viscosity of
fluid 1 is smaller than that of fluid 2. We observe that as $V_{R}$ increases
the interface tends to the horizontal position.
Figure 8: Stationary interface profiles for several values of $V_{R}$, without
gravity, for $\mu_{R}<1$ ($\mu_{R}=0.01$).
Figure 9 shows the different stationary interface positions when the gravity
effect is present for $\mu_{R}>1$. The effect of gravity is quite significant,
the interface ascends but only for the highest value of $V_{R}$ it lies above
the outlet level.
Figure 9: Stationary interface profiles for several values of $V_{R}$, with
gravity, for $\mu_{R}>1$ ($\mu_{R}=35$).
Figure 10 shows the curves when the viscosity of fluid 1 is lower than that of
fluid 2 ($\mu_{R}<1$). The behavior is different from that without gravity. In
fact, there exists a minimum outlet velocity below which the interface does
not reach the outlet.
Figure 10: Stationary interface profiles for several values of $V_{R}$, with
gravity, for $\mu_{R}<1$ ($\mu_{R}=0.01$).
### 2.5 Density effect
The effect of the density ratio on the interface profile was studied in the
presence of gravity. Keeping fluid 2 with the properties of water and the
viscosity of fluid 1 as $0.035$Pa.s ($\mu_{R}=35$), the density of fluid 1 was
varied from its original value to one three times smaller than that of fluid
2. In Fig. 11 it is seen that as $\rho_{R}$ increases, the interface profile
ascends significantly, with a less significant change in the tilt angle at the
outlet.
Figure 11: Stationary interface profiles for several values of the density of
fluid 1, with $V_{R}=1.5^{4}$ and $\mu_{R}=35$.
## 3 Generic expression
From the study on the influence of each nondimensional parameter on the
interface behavior, an equation that predicts the interface angle at the
immediate vicinity of the outlet ($\theta$) was crafted. For practical
reasons, the cases where gravity is present were considered to develop the
equation.
In Sect. 2.2, it is possible to see that when the nondimensional parameters
$\rho_{R}$, $\mu_{R}$ and $V_{R}$ are constant the interface changes its shape
until it reaches a stationary profile.
For this reason, a fourth nondimensional parameter is considered, the volume
fraction of fluid 1 in the outlet flow (VF).
A generic expression [(Eq. (4)], consisting of three terms and containing 22
constants, was adjusted by trial and error until satisfactory agreement with
the numerical results was found.
$\displaystyle\theta$ $\displaystyle=\alpha\mu_{R}^{b}V_{R}^{c}\rho_{R}^{d}$
$\displaystyle+e\mu_{R}^{f}V_{R}^{g}\rho_{R}^{h}\exp(-i\mu_{R}^{j}V_{R}^{k}\rho_{R}^{l}VF^{m})$
$\displaystyle+n\mu_{R}^{o}V_{R}^{p}\rho_{R}^{q}\exp(-r\mu_{R}^{s}V_{R}^{t}\rho_{R}^{u}VF^{v})$
(4)
Table 1 shows the values of the constants in the generic expression.
a | -29.1 | i | $1.162\times 10^{7}$ | p | 0.1
---|---|---|---|---|---
b | -0.04 | j | -0.18 | q | 0.72
c | 0.1 | k | -1.64 | r | 2763.1
d | 0.45 | l | -1 | s | 0.4
e | 206 | m | 0.63 | t | -0.6
f | 0.035 | n | 12.74 | u | -1.82
g | -0.05 | o | 0.085 | v | 3.2
h | -1.26 | | | |
Table 1: Constant values in the generic expression.
### 3.1 Equation verification
To verify that the generic expression (4) predicts the value of $\theta$
correctly when the parameters are arbitrarily modified, 21 additional
numerical cases were simulated. These cases cover a wide range of physical
properties of the liquids and of the characteristics of the porous medium (by
means of the coefficients $1/\alpha$ and $C$).
The interface angle obtained from the generic expression ($\theta_{GE}$) was
compared with the interface angle obtained from the simulations
($\theta_{S}$). Table 2 shows the five porous medium types (PT) that were
chosen.
PT | $D$ | $\varepsilon$ | $1/\alpha$ | $C$ | Resistance
---|---|---|---|---|---
A | 0.005 | 0.3 | $10.88\times 10^{7}$ | $1.81\times 10^{4}$ | Very high
B | 0.006 | 0.32 | $5.88\times 10^{7}$ | $1.21\times 10^{4}$ | High
C | 0.02 | 0.2 | $3\times 10^{7}$ | $1.75\times 10^{4}$ | Medium
D | 0.02 | 0.25 | $1.35\times 10^{7}$ | 8400 | Low
E | 0.05 | 0.17 | $8.41\times 10^{6}$ | $1.18\times 10^{4}$ | Very low
Table 2: Porous medium types.
Some cases (1, 2, 4, 5, 9, 11, 17-21) were chosen based on the possible
combinations of immiscible liquids that can be manipulated in real situations.
For the remaining cases, the properties of the liquids were fixed at arbitrary
values (fictitious liquids) so that a wide range of the nondimensional
parameters was covered. Table 3 shows the description of each numerical case,
while Table 4 shows the corresponding nondimensional parameter values and PT.
Case | Fluid 1 | Fluid 2 | $\mu_{1}$ | $\mu_{2}$ | $\rho_{1}$ | $\rho_{2}$
---|---|---|---|---|---|---
1 | Heavy oil | Water solution | 0.4 | 0.005 | 850 | 998
2 | Light oil | Water emulsion | 0.012 | 0.06 | 850 | 998
3 | – | – | 0.08 | 0.008 | 4000 | 4680
4 | Kero-sene | Water | $24\times 10^{-4}$ | 0.001 | 780 | 998
5 | Ace-tone | Water | $3.3\times 10^{-4}$ | 0.001 | 791 | 998
6 | – | – | 0.003 | 0.01 | 400 | 720
7 | – | – | 0.2 | 0.01 | 1500 | 2700
8 | – | – | 0.3 | 0.004 | 3300 | 5940
9 | Light slag | Hot pig iron | 0.02 | 0.001 | 2800 | 7000
10 | – | – | 0.5 | 0.013 | 500 | 1250
11 | Medium | pig | 0.4 | 0.005 | 2800 | 7000
| slag | iron | | | |
12-16 | – | – | 0.08 | 0.008 | 4000 | 4680
17-21 | Heavy slag | pig iron | 0.4 | 0.005 | 2800 | 7000
Table 3: Cases description. Case | PT | $\rho_{R}$ | $\mu_{R}$ | $V_{R}$ | VF
---|---|---|---|---|---
1 | B | 1.17 | 80 | $5\times 10^{4}$ | 2.8
2 | B | 1.17 | 0.2 | $5\times 10^{4}$ | 87.2
3 | B | 1.17 | 10 | $10\times 10^{4}$ | 28.4
4 | B | 1.28 | 2.4 | $5\times 10^{4}$ | 96.1
5 | B | 1.26 | 0.33 | $1\times 10^{5}$ | 96.8
6 | B | 1.8 | 0.3 | $1\times 10^{5}$ | 93.2
7 | B | 1.8 | 20 | $1\times 10^{5}$ | 16.1
8 | B | 1.8 | 80 | $8\times 10^{4}$ | 18.4
9 | B | 2.5 | 20 | $1\times 10^{5}$ | 100.0
10 | B | 2.5 | 40 | $1\times 10^{5}$ | 5.5
11 | B | 2.5 | 80 | $5\times 10^{4}$ | 70.8
12 | A | 1.2 | 10.0 | $1\times 10^{5}$ | 23.1
13 | B | 1.2 | 10.0 | $1\times 10^{5}$ | 28.4
14 | C | 1.2 | 10.0 | $1\times 10^{5}$ | 41.5
15 | D | 1.2 | 10.0 | $1\times 10^{5}$ | 53.4
16 | E | 1.2 | 10.0 | $1\times 10^{5}$ | 63.7
17 | A | 2.5 | 80.0 | $5\times 10^{4}$ | 15.8
18 | B | 2.5 | 80.0 | $5\times 10^{4}$ | 24.3
19 | C | 2.5 | 80.0 | $5\times 10^{4}$ | 43.2
20 | D | 2.5 | 80.0 | $5\times 10^{4}$ | 70.8
21 | E | 2.5 | 80.0 | $5\times 10^{4}$ | 94.0
Table 4: Parameter values and PT for all cases described in Table 3.
We define an error ($e=100|\Delta\theta|/180$) as the percentage of the
absolute value of the difference between the interface angles
($\Delta\theta=\theta_{S}-\theta_{GE}$) divided by the interface angle range
($180^{\circ}$). Figure 12 shows the comparison between the generic expression
and the numerical cases. It is seen that the generic expression (4) predicts
the interface angle, for the cases used in this study, with an error smaller
than 10%.
Figure 12: Comparison between the predictions of the generic expression and
the numerical result for the 21 validation cases.
## 4 Conclusions
A numerical study of the macroscopic interface behavior between two immiscible
liquids flowing through a porous medium, when they are drained through an
opening, has been reported. Four nondimensional parameters that rule the
fluid-dynamical problem were identified. Thereby, a numerical parametric
analysis was developed where the qualitative observation of the resulting
interface profiles contributes to the understanding of the effect of each
parameter. In addition, a generic expression to predict the interface angle in
the immediate vicinity of the outlet opening ($\theta$) was developed. To
verify that the generic equation predicts the value of $\theta$ correctly,
$21$ numerical cases with widely different parameters were simulated.
Considering that the cases encompass a large class of liquids and porous
media, the prediction of $\theta$ within an error of 10% is considered
satisfactory.
###### Acknowledgements.
A. D. M. and E. B. are grateful for the support from Metallurgical Department
and DEYTEMA (UTNFSRN). G. C. B. acknowledges partial financial support from
CNPq and FAPESP (Brazil).
## References
* [1] W C Lyons, G J Plisga, Standard Handbook of Petroleum & Natural Gas Engineering \- 2nd ed, Gulf Professional Publishing, Burlington (2005).
* [2] A K Biswas, Principles of blast furnace ironmaking: theory and practice, Cootha Publishing House, Brisbane (1981).
* [3] A Westhead, Upscaling for two-phase flows in porous media, PhD thesis: California Institute of Technology, Pasadena, California (2005).
* [4] R E Ewing, The Mathematics of reservoir simulation, SIAM, Philadelphia (1983).
* [5] M A Cardoso, L J Durlofsky, Linearized reduced-order models for subsurface flow simulation, J. Comput. Phys. 229, 681 (2010).
* [6] M J Blunt, Flow in porous media - pore-network models and multiphase flow, Curr. Opin. Colloid Interface Sci. 6, 197 (2001).
* [7] Z Chen, G Huan, Y Ma, Computational methods for multiphase flows in porous media, SIAM, Philadelphia (2006).
* [8] Y Efendiev, T Houb, Multiscale finite element methods for porous media flows and their applications, Appl. Num. Math. 57, 577 (2007).
* [9] FLUENT 6.3 User’s Guide, Fluent Inc. (2006).
* [10] J Bear, Dynamics of fluids in porous media, Dover Publications Inc., New York (1988).
|
arxiv-papers
| 2010-01-13T15:12:59 |
2024-09-04T02:49:07.737672
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alejandro David Mariotti, Elena Brandaleze, Gustavo C. Buscaglia",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1001.2197"
}
|
1001.2217
|
010008 2009 J. J. Niemela H. Cabrera Morales (Inst. Venezolano Invest.
Científicas, Venezuela) 010008
A nanopatterning scheme is presented by which the structure height can be
controlled in the tens of nanometers range and the lateral resolution is a
factor at least three times better than the point spread function of the
writing beam. The method relies on the initiation of the polymerization
mediated by a very inefficient energy transfer from a fluorescent dye molecule
after single photon absorption. The mechanism has the following distinctive
steps: the dye adsorbs on the substrate surface with a higher concentration
than in the bulk, upon illumination it triggers the polymerization, then
isolated islands develop and merge into a uniform structure (percolation),
which subsequently grows until the illumination is interrupted. This
percolation mechanism has a threshold that introduces the needed nonlinearity
for the fabrication of structures beyond the diffraction limit.
# Surface Percolation and Growth. An alternative scheme for breaking the
diffraction limit in optical patterning
D. Kunik [inst1, inst2] L. I. Pietrasanta E-mail: dkunik@df.uba.ar [inst2,
inst3] O. E. Martínez[inst1, inst2]
(9 November 2009; 29 December 2009)
††volume: 1
99 inst1 Departamento de Física, Universidad de Buenos Aires, Buenos Aires,
Argentina. inst2 Consejo Nacional de Investigaciones Cientí ficas y Técnicas,
Argentina. inst3 Centro de Microscopías Avanzadas Facultad de Ciencias
Exactas, Universidad de Buenos Aires, Buenos Aires, Argentina.
## 1 Introduction
The development of new techniques for the fabrication of smaller and smaller
structures has become an objective of great relevance for many fields in
science and technology [1, 2]. This includes semiconductor industry [2], MEMS
[3, 4], biology [4–7], microfluidics [8], material science and technology,
among the most demanding [2]. Techniques include optical lithography, scanning
electron lithography [1, 2], dip-pen patterning [9], magnetolithograpy [10,
11], ion milling [12], and many others. Among these techniques, optical
lithography is the most widely used due to its inherent simplicity, mature
development, fast production speed and less stringent ambient requirements [1,
2]. The far field optical lithography, both in projection and scanning, has a
first principle limitation in size reduction, the diffraction limit, as light
cannot be focused below a size of about half the wavelength used. The main
approach to circumvent this problem is to reduce the light wavelength and this
is the roadmap traced by the semiconductor industry [13], now using less than
100nm sources mostly from synchrotron radiation but also from other new
developments such as short wavelength lasers [13, 14]. Another approach is to
avoid the far field limit by near field approaches [15], but these techniques
suffer from the same drawbacks of other sophisticated scanning methods, slow
throughput and small scanning areas, unless very sophisticated tricks are
developed [16]. It is now well recognized that the diffraction limit is
derived assuming a linear response of the media to the light and this can be a
way to circumvent the limitation imposed by diffraction [17]. This approach
has been used in many microscopy techniques and also in optical lithography
with the main advantage of allowing the construction of three dimensional
structures, but the increase in resolution was marginal due to the need of
longer optical wavelengths to target the same material transitions with more
photons [17–19]. In the last decade, new microscopic techniques with super-
resolution have been developed, that rely on some complex nonlinearities, such
as stimulated emission depletion (STED) [20]. Recently, several researchers
have reported ways to use these techniques using such nonlinear methods and
switching strategies for photolithography [21–23].
In this work we present a new concept on how to circumvent the diffraction
limit in optical lithography by placing the nonlinearity not in the light-
matter interaction but in the material growth mechanism itself. The technique
uses a dye molecule that initiates the polymerization reaction from an excited
state with a very low efficiency. The mechanism relies on the following
distinctive steps:
* (1)
Mixture of the dye molecules and polymer in adequate proportions.
* (2)
Deposition of a drop of the mixture on a transparent substrate.
* (3)
Adsorption of a fraction of the dye molecules on the substrate material.
* (4)
Illumination of the mixture with a focused beam through the substrate at the
absorption band of the dye.
* (5)
Initiation of the polymerization by due to a very low quantum yield energy
transfer from the dye to the polymer.
* (6)
Percolation of the structure growing at the substrate surface.
* (7)
Deposition of fresh dye molecules from the bulk onto the growing surface.
* (8)
Photoinitiation of the fresh molecules as the structure grows.
We will show how this steps allow the controlled growth of the structures with
10nm resolution in vertical direction (direction of propagation of the light),
and that the lateral resolution can be increased by a factor of at least
three, as compared to the diffraction limit.
## 2 Materials and Methods
The experimental setup is shown in Fig. 1 and it is similar to that used in
previous works [7, 24, 25]. The main difference is that the photo curable
resin was not triggered by exciting the UV initiator by two-photon absorption
directly but instead, the energy transfer to activate the polymerization was
mediated by a dye molecule excited by one photon
Figure 1: Experimental setup.
excitation as described in [24]. Different blends of dyes and resins were
prepared and a drop of the blend was deposited onto a cover slip positioned in
an inverted microscope setup equipped with a motorized stage. The laser beam
was focused onto the sample by means of a high numerical aperture air
objective (UplanSapo 40x, NA=0.9, Olympus, Tokyo, Japan). Control of the focus
was made by means of a CCD camera imaging the back reflection of the laser
beam, and the sample focus was adjusted in order to minimize the image size in
the CCD camera [25]. Because the resin had a refractive index matching that of
the glass cover slip used as substrate, the surface had to be focused before
the blend drop was placed. The choice of the laser to excite the dye that
triggers the polymerization process, depends on the absorption spectrum of the
dye used. The results shown in this work correspond to cyanine (ADS675MT,
American Dye Source, Quebec, Canada) and oxazine dyes (Nile-Blue 690
Perchlorate, Exciton, Ohio, USA) excited with a He-Ne laser emitting at 632.8
nm but similar results were obtained with different infrared cyanines dyes
(ADS740, ADS760 and ADS775PI, American Dye Source, Quebec, Canada and HITC
Iodide and LDS821, Exciton, Ohio, USA) excited with a Ti:Sapphire laser in cw
mode tuned to match the absorption spectrum of these dyes. The laser power was
adjusted by inserting neutral density filters in the beam path, and the
exposure time was controlled with the scanning speed. In order to draw the
desired pattern, a shutter was used to turn on and off the illumination while
the scanning was being made. Tapping-mode AFM was performed in dry nitrogen
using a NanoScope IIIa Multimode-AFM (Digital Instruments-Veeco Metrology,
Santa Barbara, CA, USA) and images were acquired simultaneously with the
height and the phase signals. Images were processed by flattening, using
NanoScope software to remove background slope.
Different polymer-dye blends were prepared changing both the UV curing
adhesive (NOA 60, NOA 63 or NOA65, Norland Products Cranbury, NJ, USA) and the
dyes used. To ensure that the UV photo-sensitizer played no role in the
process, a special batch of NOA60 resin without the UV sensitive additive was
provided by the manufacturer, (Norland Products) which yielded similar
results. The following steps were taken: (1) Select an adequate cosolvent for
both the dye and the adhesive, in most cases ethanol, methanol and acetone
worked well. We used methanol for our reported essays. (2) Make a concentrated
solution of the dye in the solvent, typically 10 mM in methanol. (3) Add the
dye solution to the polymer resin up to the desired concentration. (4) Place a
coverslip on the microscope and set the focus at the surface. (5) Place a drop
of blend on the coverslip. (6) Scan the laser to draw the desired structure.
(7) Rinse the coverslip immersing it in ethanol and acetone. A final rinse of
a few seconds in methylene chloride is helpful to efficiently remove the
remanent unpolymerized resin.
Figure 2: Different stages of the surface percolation process. Upper inset is
an AFM image of a wide line showing the location of the subsequent detailed
images. AFM height images taken from the periphery (a) toward the centre of
the structure (f). At the periphery small clusters are observed. Towards the
center of the structure (higher laser intensity), the size of the clusters
grow until percolation takes place. The size of each image was 500 nm $\times$
500 nm and the displacement between adjacent images was approximately 400 nm.
## 3 Results
In order to study the different stages of the process, we fabricated different
samples under specially designed conditions. After selecting the polymer and
the dye, three parameters remain for the control of the size of the structure.
* (a)
The dye concentration
* (b)
The beam intensity
* (c)
The light exposure time (or scan velocity).
Figure 2 shows AFM images of a wide structure fabricated with a mixture of the
optical adhesive NOA 63 and the laser dye ADS675MT. The dye conentration was 1
mM. The sample was illuminated with a He-Ne laser beam and the power at the
sample was 1 mW. The laser beam was defocused in order to produce a broad line
about 5 $\mu$m wide in such a way that in a cross section the different stages
of the polymerization process can be visualized. The scan speed was 10 $\mu$m
$\rm{s}^{-1}$ and the different degree of coverage can be detected as one
moves towards the line center (maximum intensity). The structure was
detailedly scanned in six different regions going from the periphery of the
structure (a) towards the center of the line (f). The different scans are 500
nm $\times$ 500 nm in size and located at 400 nm steps from each other. The
first scan shows very isolated polymer islands of different sizes with heights
from a few nm to about 7 nm. As the center is approached, the size of the
islands grows, but the height only grows marginally, showing a percolation
process as more dye molecules located at the surface start the polymerization
process at different locations. Towards the center, the percolation phenomenon
becomes evident, with islands merging towards a larger and larger coverage,
but with only a minor increase in the height. With an increase in the
intensity by an improvement in the focusing of the beam, a total coverage is
obtained, as shown in Fig. 3, still maintaining 13 nm
Figure 3: Structure at the percolation threshold showing full coverage. (a) 3D
reconstruction of a AFM topographic image. (b) average cross section.
in height. The laser power was 35 $\mu$W, the scanning velocity was 16 $\mu$m
$\rm{s}^{-1}$ and the blend used was a mix of the optical adhesive NOA60 and
the dye ADS675MT at a concentration of 1 mM.
Once the percolation threshold is reached, the structure grows slowly, as more
dye molecules are adsorbed in the fresh polymer surface. This continues until
the illumination is turned off or the beam walks away from that portion of the
surface. Different concentrations and speeds were used in order to fabricate
lines of different heights and widths. The concentrations used were 1 mM, 500
$\mu$M, 100 $\mu$M, 50 $\mu$M, and 10 $\mu$M of a NOA60-ADS675MT blend. For
each dye concentration, AFM topographies in tapping mode and phase images were
obtained for lines manufactured at different speeds with the same beam
intensity in order to cover a wide range of stages of the percolation and
growth process. The phase images (not shown here) helped to easily distinguish
the polymer from the substrate due to their different hardness. When the
volume dye concentration is very large (1 mM and 500 $\mu$M) the surface
percolation phenomenon can be observed (as shown already for 1 mM) but the
growth was hindered by the fact that at these high concentrations a volume
reaction takes place. This fact is due to the existence of a second threshold
fluency that gives rise to a volume percolation, as will be discussed in the
next section.
In Fig. 4, a plot of the maximum height of the lines as a function of the
exposition time (defined as the ratio between the beam size and the scan
speed) is presented for the samples with dye concentrations 100 $\mu$M
(squares) and 50 $\mu$M (circles). The beam power at the sample was 290 $\mu$W
(100 $\mu$M) and 460 $\mu$W (50 $\mu$M) and the beam diameter, Full Width Half
Maximum (FWHM), was 800 nm. For the sample made with 100 $\mu$M, the speed was
varied from
Figure 4: Height of the lines as a function of the exposition time for samples
with dye concentration of 50 $\mu$M (squares) and 100 $\mu$M (circles).
Examples of structures (topographic AFM images) below, at and above the
percolation threshold are also shown.
line to line in steps of 0.2 $\mu$m $\rm{s}^{-1}$, from 2.2 $\mu$m
$\rm{s}^{-1}$ for the slower scan (higher line) to 3.8 $\mu$m $\rm{s}^{-1}$
for the fastest one (lower line). While for the sample made with 50 $\mu$M the
range of the speed was 1.4 $\mu$m $\rm{s}^{-1}$ to 3 $\mu$m $\rm{s}^{-1}$.
From the AFM topographies, the surface percolation threshold time $t_{thi}$
(threshold time obtained from the image) was determined and is indicated in
the plot. We also show three characteristic lines at both sides of the
percolation threshold of the 100 $\mu$M sample.
As the scanning speed is reduced, the exposure time increases and the
structures generated show a transition from isolated islands to full coverage
and afterwards, a gradual growth in height. A similar experiment was performed
for the five concentrations indicated before. However, for the 10 $\mu$M
sample, the percolation threshold was not reached. For the samples with
concentrations of 1 mM and 500 $\mu$M after the surface percolation was
reached, the structure did not grow gradually but instead grew suddenly from
less than 20 nm to 1 $\mu$m or more. The experimental data presented in Fig. 4
show a clear nonlinear growth mechanism with an apparent linear asymptotic
behavior for large exposure times. It can also be observed that the linear
asymptote crosses the time axis at approximately the threshold exposure time.
Figure 5 shows the lateral reduction below the diffraction limit of structures
fabricated with
Figure 5: Subdriffaction structures made with the blend NOA63-Nile Blue 1 mM.
Normalized PSF of the laser beam (red) and AFM profile of the structure
(blue). Inset: 3D reconstruction of the subdiffraction structures topography.
Nile Blue 690 perchlorate and NOA63. The Point Spread Function (PSF) is
plotted (red) together with a line profile of the structures (blue). A 3D
topographic reconstruction of the structures is also shown as inset. The
height of the structures was 67 nm and the width (FWHM) was 265 nm. The PSF of
the beam was determined by measuring the scattered light from a gold
nanoparticle 80 nm in diameter and the width was 800 nm (FWHM). The lateral
resolution is at least three times better than the beam size, considering that
the tip shape was not deconvolved in the measurement. This reduction below the
diffraction limit is a clear evidence of the nonlinear growth mechanism also
shown in Fig. 4. Similar results were obtained for many combinations of dye
molecules and UV curing resins. The common denominator of all the dyes used,
which included cyanines and oxazines, is that they are all good laser dyes
(meaning high fluorescence efficiency, low triplet formation, low excited
state absorption). Experiments with methylene blue were not successful (no
polymerization was observed) even when the dye was bleached. Another common
aspect is that all the dyes used have the chromophore positively charged and
they were all very poorly soluble in the polymer blend.
## 4 Discussion
As shown in the previous section, the process has two very distinctive stages.
A first stage in which the polymerization process takes place mainly at the
substrate surface with the formation of isolated islands that gradually merge,
and a second stage after the surface percolation in which the structure grows
gradually until the illumination is turned off. As only marginal growth is
detected during the percolation process. We will discuss two models in order
to explain the two stages separately.
### 4.1 Percolation model
As the smallest islands at the very beginning of the process are too small to
be measured with our AFM microscope (24 nm radius tip) we cannot accurately
determine the island size distribution. Hence, we will only model this stage
semi-quantitatively and we will show that the surface percolation process
requires a large adsorption of dye molecules at the substrate surface. If this
is not the case, or else if the volume concentration is too large, a volume
percolation precedes the surface percolation and a thick structure of the size
of the illuminated volume is generated. In order to compare these two
situations, we modeled the percolation process as follows:
* (a)
We assumed that the dye molecules were distributed in an ordered lattice with
a distance between neighbors given by the concentration used.
* (b)
At the substrate surface, a similar lattice was assumed but with a different
neighbor distance kept as a parameter.
Despite the fact that in the experiment the dye molecules are actually
randomly distributed, we found that adding this to the model did not modify
the result. The result being that at equal nearest neighbor distance the
volume percolation precedes the surface percolation. It was assumed that each
dye molecule (lattice site) could initiate the polymerization process at
random with a probability that increases with the illumination time and
intensity. Each polymerization triggered generated a sphere of polymer with a
Gaussian distribution in size. We found no significant differences in the
results by changing the variance of the distribution. After a given time, the
process is stopped, each sphere is linked with any neighboring sphere if their
distance is smaller that the sum of the radii and only the spheres that are
linked (directly or through other neighbors) to the substrate are kept (sample
rinsing). The results of some typical situations are shown in Fig. 6. For very
low concentrations or light doses,
Figure 6: Simulations of the percolation process. Spheres with average radius
r are randomly created. (a), (b) and (c) show structures simulated as the
distance between neighbors $d_{s}$ was diminished. (d) shows the volume
percolation for the case where the nearest neighbor distance in bulk equals
that at the surface.
isolated islands appear at the surface because the structures generated in the
bulk do not touch each other and are washed. As the surface concentration is
increased, a more densely packed structure develops, as shown in the sequence
of Fig. 6 (a) to (c). In this last case, the surface is fully covered by the
polymer spheres. If the surface concentration is kept with a nearest neighbor
distance larger than about one half of the bulk average distance, the surface
percolation never occurs before a massive reaction takes place due to the
percolation of the structure in bulk. This is shown in the simulation of Fig.
6(d) where the same distance between neighbors was used for the surface and
bulk. In brief, the surface percolation process as modeled yields structures
with typical sizes that depend on the nearest neighbor distance, and a much
lower distance (high surface concentration) is required if the surface
phenomenon is to prevail the bulk process. This result, together with the fact
that at the percolation threshold lower lines were obtained with high
concentrations, indicate that the substrate surface was not saturated and
fully covered by the dye molecules. The surface dyes concentration grows with
the volume concentration. This also explains why at higher concentrations (1
mM and 500 $\mu$M) no growth could be observed but instead a sudden bulk
polymerization at the entire illuminated volume. Also that for the low
concentration case (10 $\mu$M) the surface concentration was too low and hence
the large distance between neighbors (larger than the size of the isolated
islands) did not allow the percolation process to take place.
### 4.2 Growth model
Figure 7: Fit of the data for the sample made at 100 $\mu$M dye concentration.
Once the surface is fully covered, the growth of the structure requires the
adsorption of fresh dye molecules at the surface of the growing polymer
structure. We will model this situation by assuming, in a simple manner, that
the growth rate follows the rate equation
$\displaystyle\frac{dh}{dt}=\nu\Phi_{p}I\sigma\rho.$ (1)
Where $I$ is the beam intensity, $\sigma$ is the absorption cross section of
the dye molecule, $\rho$ is the surface dye concentration, $\Phi_{p}$ is the
efficiency of the initiation of the polymerization process (the inverse of the
number of photons required to trigger one polymerization reaction) and $\nu$
is a characteristic volume indicating the size of the structure created once a
polymerization event is triggered. This simple model is consistent with the
assumption that each molecule can trigger only one event and yields a typical
size of the polymer structure or, alternatively, that each molecule can
catalyze more than one event and the size is proportional to the number of
polymer chains initiated. To complete the description, an equation is needed
for the dynamics of the surface density $\rho$. Such equation should take into
account the adsorption-desorption process and the dye consumption due to the
reaction and it can be written as
$\displaystyle{\frac{d\rho}{dt}}=-\kappa\rho+\alpha
C(\rho_{0}-\rho)-\Phi_{p}I\sigma\rho.$ (2)
Where $\kappa$ is the desorption rate, $\alpha$ is the adsorption or sticking
coefficient proportional to the number of collisions one molecule has with the
polymer surface and the probability of sticking to that surface, $C$ is the
volume concentration to be assumed constant, $\rho_{0}$ is the surface density
of sites for the dye molecule (being $\rho_{0}-\rho$ the density of available
sites). The three components in the right hand side of the equation account
for the desorption rate, adsorption rate and dye consumption respectively. Any
gradient in the volume concentration $C$ is neglected for the sake of
simplicity and is a good approximation if the rate is dominated by the dye
consumption term, as the structure would be growing faster than the
development of the concentration gradient.
The solution to Eq. (2) is
$\displaystyle\rho(t)=\rho_{e}\left({1-e^{-\,\frac{t}{\tau}}}\right),$ (3)
were
$\displaystyle\rho_{e}=\frac{{\rho_{0}\,\alpha\,C\,}}{{\alpha\,C+\kappa+\sigma
I\Phi_{p}}}$ (4)
is the equilibrium surface concentration and
$\displaystyle\tau=\frac{1}{{\alpha\,C+\kappa+\sigma I\Phi_{p}}}$ (5)
is a characteristic transition time.
Figure 8: Fit of the data for the sample made at 50 $\mu$M dye concentration.
Inserting Eq. (3) in Eq. (1), the solution for the height evolution is
$\displaystyle
h(t)=m\,(t-t_{0})+h_{0}-m\,\tau\,\left({1-e^{-\,\frac{{t-t_{0}}}{\tau}}}\right);$
(6)
were
$\displaystyle
m=\frac{{v\,\alpha\,C\,\rho_{0}\,\sigma\,I\,\Phi_{p}}}{{\alpha\,C+\kappa+\sigma\,I\,\Phi_{p}}},$
(7)
$t_{0}$ is the percolation threshold time delay (initial time for the growth
equation) and $h_{0}$ is the height of the structure at the percolation
threshold. Equation (6) shows an initial nonlinear growth towards an
asymptotic linear growth with slope $m$. The characteristic transition time
towards the asymptotic behavior is precisely the parameter $\tau$ defined in
Eq. (5). The results shown in Fig. 4 were fit using Eq. (6) and the result is
shown in Fig. 7 for the 100 $\mu$M dye concentration sample and in Fig. 8 for
the 50 $\mu$M dye concentration sample. From these fits, the following
conclusions can be drawn:
* (a)
the linear asymptote is evident in both cases and extrapolates crossing the
time axis at the measured percolation time
* (b)
the slope of the asymptote grows with the dye concentration.
* (c)
The surface percolation height decays with the concentration ranging from
almost 100 nm for the 50 $\mu$M concentration, to 13 nm for the 1 mM case.
The last result is consistent with the fact that the surface is not saturated
with the dye but instead the surface concentration grows with the bulk
concentration. By examining the equations obtained for the growth rate, these
results are consistent with having the surface concentration rate dominated by
the dye consumption
$\displaystyle\sigma I\Phi_{p}\gg\kappa+\alpha C,$ (8)
which yields a slope proportional to the concentration as from Eq. (7) in this
limit
$\displaystyle m=v\,\alpha\,C\,\rho_{0}.$ (9)
And in the same limit Eq. (5) yields
$\displaystyle\tau=\frac{1}{{\sigma\,I\,\Phi_{p}}}.$ (10)
The ratio of the slopes between the 100 $\mu$M and the 50 $\mu$M is
$\frac{m_{100}}{m_{50}}=1.6\pm 0.3$ which is, within the experimental error,
the ratio between concentrations $\frac{C_{100}}{C_{50}}=2\pm 0.4$ . The
experiment for the two concentrations was done at different beam powers
yielding $\frac{\tau_{100}}{\tau_{50}}=1.43\pm 0.9$ and
$\frac{I_{50}}{I_{100}}=1.58\pm 0.0.3$ which show an excellent agreement with
this limiting approximation. From Eq. (10) and the characteristic time
obtained by fitting the experimental results, we found the efficient of initio
of the polymerization process is
$\displaystyle\Phi_{p}=\frac{1}{{\sigma\,I\,\tau}}=(7\pm 2)\,10^{-7}$ (11)
## 5 Conclusions
A technique has been described that allows the fabrication of polymer
structures by scanning a laser beam and yielding features with sizes below the
diffraction limit. The structure height can be controlled with 10 nm
resolution in the range from a few tens of nanometer to hundreds of
nanometers. The width was shown to be reduced by at least a factor of three
times the diffraction limit.
The technique relies on the use of a light sensitive dye that transfers the
absorbed energy to the resin and triggers the polymerization process. The
exact mechanism could not be clearly separated as the reaction efficiency was
shown to be extremely low (below 1ppm). This low yield appears to be crucial
for the success of the technique. Our results show that the surface dye
adsorption and the low dye polymerization efficiency are the keys to allow a
smooth growth from surfaces. The surface dye concentration increases the speed
of the polymerization at the surface with respect to the volume. On the other
hand, the low yield of the triggered polymerization by the excited dye allows
this effect to be observable. If the yield of the dye polymerization was high,
no matter if there are more initiators at the surface than in the polymer
volume, all excited dye would trigger a polymerization event and hence the
whole volume illuminated would be polymerized. If the polymerization speed was
uniform, the surface growth would take place at a lower speed than the volume
growth due to the higher neighbors available in volume, and in this case, the
polymerization would also occur in the whole illuminated volume. As was shown,
two distinct processes take place in an almost sequential manner. The first
one is a surface nucleation of isolated islands that finally percolate into a
uniform surface coverage with a characteristic size that decreases as the dye
concentration increases. This is followed by a structure growth, as fresh dye
molecules are deposited on the growing polymer surface. The process is stopped
when the illumination is interrupted by shutting down the laser beam or moving
it to a different spot where the process starts again. The percolation
mechanism was modeled numerically and the simulations showed that a large
surface adsorption of the dye molecules is needed in order to avoid the volume
percolation and allow a surface growth mechanism. A model developed for the
growth process permitted the fit of the experimental data and the
determination of the extremely low transfer efficiency from photons to the
polymer. How the growth rate increases with the dye concentration after an
incubation time that is inversely proportional to the light intensity, was
also explained.
###### Acknowledgements.
We thank Norland-Products for the sample without photoinitiator. Grants from
Universidad de Buenos Aires, CONICET (Argentina) and ANPCYT (Argentina) are
acknowledged.
## References
* [1] Y Xia, J Rogers, K E Paul, G M Whitesides, Unconventional methods for fabricating and patterning nanostructures, Chem. Rev. 99, 1823 (1999).
* [2] M Geissler, Y Xia, Patterning: Principles and some new developments, Adv. Mater. 16, 1249 (2004).
* [3] J J Yao, RF MEMS from a device perspective, J. Micromech. Microeng. 10, 9 (2000).
* [4] W C Chang, M Kliot, D W Sretavan, Microtechnology and nanotechnology in nerve repair, Neurol. Res. 30, 1053 (2008).
* [5] R Nielson, B Kaehr, J B Shear, Microreplication and design of biological architectures using dynamic-mask multiphoton lithography, Small 5, 120 (2009).
* [6] J M Bélisle, J P Correia, P W Wiseman, T E Kennedy, S Costantino, Patterning protein concentration using laser-assisted adsorption by photobleaching, LAPAP, Lab. Chip 8, 2164 (2008).
* [7] S Costantino, K G Heinze, P De Koninck, P W Wiseman, O E Martínez, Two-photon fluorescent microlithography for live-cell imaging, Microsc. Reser. Tech. 68, 272 (2005).
* [8] J Godin, C Chen, S H Cho, W Qiao, F Tsai, Y H Lo, Microfluidics and photonics for bio-system-on-a-chip: A review of advancements in technology towards a microfluidic flow cytometry chip, J. Biophoton. 1, 355 (2008).
* [9] B Basnar, I Willner, Dip-pen-nanolithographic patterning of metallic, semiconductor, and metal oxide nanostructures on surfaces, Small 5, 28 (2009)
* [10] A. Bardea, R. Naaman, Magnetolithography: From bottom-up route to high throughput, Small 5, 316 (2009)
* [11] E Menéndez, M O Liedke, J Fassbender, T Gemming, A Weber, L J Heyderman, K V Rao, S C Deevi, S Suriñach, M D Baró, J Sort, J Nogués, Direct magnetic patterning due to the generation of ferromagnetism by selective ion irradiation of paramagnetic FeAl alloys, Small 5, 229 (2009).
* [12] M G Ancona, S E Kooi, W Kruppa, Patterning of narrow Au nanocluster lines using V2O5 nanowire masks and ion-beam milling, Nano Lett. 3, 135 (2003).
* [13] B Wu, A Kumar, Extreme ultraviolet lithography: A review J. Vac. Sci. Technol. B 25, 1743 (2007).
* [14] P W Wachulak, M G Capeluto, C S Menoni, J J Rocca, M C Marconi, Nanopatterning in a compact setup using table top extreme ultraviolet lasers, Opto-Electron. Rev. 16, 444 (2008).
* [15] M M Alkaisi, R J Blaikie, S J McNab, Nanolithography in the evanescent near field, Adv. Mater. 13, 877 (2001).
* [16] Y Wang, X Hong, J Zeng, B Liu, B Guo, H Yan, AFM tip hammering nanolithography, Small 5, 477 (2009).
* [17] S Maruo, J T Fourkas, Recent progress in multiphoton microfabrication, Laser Photon. Rev. 2, 100 (2008).
* [18] B H Cumpston, S P Ananthavel, S Barlow, D L Dyer, J E Ehrlich, L L Erskine, A A Heikal, S M Kuebler, I Y S Lee, D McCord-Maughon, J Q Qin, H Rockel, M Rumi, X L Wu, S R Marder, J W Perry, Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication, Nature 398, 51 (1999).
* [19] S Kawata, H B Sun, T Tanaka, K Takada, Finer features for functional microdevices - Micromachines can be created with higher resolution using two-photon absorption, Nature 412, 697 (2001).
* [20] S W Hell, J Wichmann, Breaking the diffraction resolution limit by stimulated-emission - Stimulated-emission-depletion fluorescence microscopy, Opt. Lett. 19, 780 (1994).
* [21] L Li, R R Gattass, E Gershgoren, H Hwang, J T Fourkas, Achieving lambda/20 resolution by one-color initiation and deactivation of polymerization, Science 324, 910 (2009).
* [22] T F Scott, B A Kowalski, A C Sullivan, C N Bowman, R R McLeod, Two-color single-photon photoinitiation and photoinhibition for subdiffraction photolithography, Science 324, 913 (2009).
* [23] T L Andrew, H Tsai, R Menon, Confining light to deep subwavelength dimensions to enable optical nanopatterning, Science 324, 917 (2009).
* [24] D Kunik, P F Aramendia, O E Martínez, Single photon fluorescent microlithography for live-cell imaging, Microsc. Res. Tech. 73, 20 (2010).
* [25] D Kunik, S J Ludueña, S Costantino, O E Martínez, Fluorescent two-photon nanolithography, J. Microsc. (Oxford) 229, 540 (2008).
|
arxiv-papers
| 2010-01-13T16:23:49 |
2024-09-04T02:49:07.744731
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Kunik, L. I. Pietrasanta, O. E. Mart\\'inez",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1001.2217"
}
|
1001.2247
|
On the Goussarov-Polyak-Viro Finite-Type Invariants and the Virtualization Move
Micah W. Chrisman
In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem of [1] which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.
§ INTRODUCTION
[This is a preprint of a paper submitted for consideration for publication to the Journal of Knot Theory and its Ramifications.]In the realm of virtual knots, there are two notions of finite-type invariant. One method, proposed by Kauffman in [4] is closely related to the Vassiliev theory for classical knots. The class of finite-type invariants, due to Goussarov-Polyak-Viro [3] (abbreviated GPV in what follows), are themselves all Kauffman finite-type invariants. However, they are defined by a different filtration in the set of virtual knots.
The two classes of invariants possess similar structures. In the classical case, the Vassiliev finite-type invariants have weight systems arising from semisimple Lie algebras.In the virtual case, the GPV finite-type invariants have weight systems arising from the Lie bialgebras associated to semisimple Lie algebras [5].
On the other hand, not all Kauffman finite-type invariants are of GPV finite-type. This was first observed for small orders by Kauffman in [4]. The result was sharpened in [1] to show that while all of the Birman coefficients $v_n$ are Kauffman finite-type of order $\le n$, none are of GPV finite-type of order $\le m$ for any $m$.
A key player in the proof of this result is the virtualization move. In a small neighborhood of the crossing the virtualization move is given by:
$\begin{array}{c} \scalebox{.25}{\psfig{figure=virtmove1.eps}} \end{array} \leftrightarrow \begin{array}{c} \scalebox{.25}{\psfig{figure=virtmove2.eps}} \end{array}
The Virtualization Move
While this move is not a virtual isotopy move in and of itself, there are numerous virtual knot invariants which are invariant under the virtualization move. The virtualization move was discovered by Kauffman in [4] in his investigation of the Jones polynomial. The Jones-Kauffman polynomial is invariant under the virtualization move. The refined theorem in [1] uses this invariance together with a twist sequence argument to show that the Birman coefficients are not of GPV finite type.
This leads to the question which is the subject of this paper: Are there any nonconstant GPV finite-type invariants which are invariant under the virtualization move? We answer this question in the negative with the following theorem.
If $v$ is a GPV finite-type invariant of virtual knots or long virtual knots which is invariant under the virtualization move, then $v$ is constant. More specifically, if $v$ is a nonconstant GPV finite-type invariant, then there are knots $K$ and $K'$ such that $K$ and $K'$ are obtained from one another by a virtualization move and $v(K) \ne v(K')$.
The proof of this theorem is surprisingly elementary. It uses only a few basic facts about the Polyak algebra, the algebra of arrow diagrams, and module theory. We present here a proof with all details laid bare.
There is a well-known conjecture about the virtualization move called the Virtualization Conjecture [2]. It states that if $K_1$ and $K_2$ are classical knots which are obtained from one another by a sequence of virtualization moves and generalized Reidemeister moves, then $K_1$ and $K_2$ are classically isotopic. The fact that there are many invariants which are unchanged by the virtualization move is positive evidence for this conjecture.
In light of the conjecture, Theorem <ref> is quite curious indeed.
This paper is organized as follows. In the remainder of Section 1, we review the construction of the GPV finite type invariants. The goal of Section 2 is to prove Theorem <ref>.
The author would like to express his deep gratitude to Vassily Manturov who encouraged this investigation and patiently responded to the author's numerous (and characteristically) bad ideas. The author would also like to thank the Monmouth University Mathematics Department for their generous financial support of this research.
§.§ Review of GPV Finite-Type Invariants
In this section, we review the two notions of finite-type invariants for virtual knots. First recall how one obtains a Gauss diagram $D$ of an oriented virtual knot diagram $K$. Traverse the circle in specified direction. Every time one arrives at a classical crossing, mark a corresponding point on a copy of $S^1$ (called the Wilson loop). If two marked points on $S^1$ correspond to the same classical crossing, connect them with an arrow. The arrowhead is incident to the arc on $S^1$ which corresponds to the underpassing arc on $K$. Moreover, we attach a sign to each arrow which gives the orientation of the crossing:
\[
\begin{array}{cc}
\scalebox{.15}{\psfig{figure=orienrightcross.eps}} & \begin{array}{c} \scalebox{.15}{\psfig{figure=orienleftcross.eps}}\end{array} \\
\text{Arrow carries sign: }\oplus & \text{Arrow carries sign: } \ominus
\end{array}
\]
It is known that if two knots have the same Gauss diagram, then they are virtually isotopic via a sequence of virtual moves (see [3]). The same construction works just as well for long virtual knots. Instead of the Wilson loop, we use the Wilson line. It is just a copy of $\mathbb{R}$.
The virtualization move is given in Figure <ref>. Notice that for any of the ways in which the arcs might be directed, the local crossing number remains the same. However, the crossing changes from over to under or vice versa. The affect on the Gauss diagram is easy to describe:
\[
\varepsilon \begin{array}{c} \scalebox{.15}{\psfig{figure=osrneq0.eps}} \end{array} \leftrightarrow \varepsilon \begin{array}{c} \scalebox{.15}{\psfig{figure=osrneq1.eps}} \end{array}
\]
We now proceed to the defintion of Kauffman finite-type invariants. First, the class of virtual knots is extended to the class of four valent graphs modulo rigid vertex isotopy (see [4] for precise definition). Vertices take the place of the singular crossings that appear in the Vassiliev theory of classical knots. Any virtual knot invariant can be extended to these graphs by applying the following relation:
\[
v\left(\begin{array}{c} \scalebox{.15}{\psfig{figure=doublepoint.eps}} \end{array} \right)=v\left( \begin{array}{c} \scalebox{.15}{\psfig{figure=orienrightcross.eps}} \end{array}\right)-v\left(\begin{array}{c} \scalebox{.15}{\psfig{figure=orienleftcross.eps}}\end{array}\right)
\]
A knot invariant is said to be of Kauffman finite type $\le n$ if $v(K_{\dagger})=0$ for all graphs $K_{\dagger}$ with greater than $n$ vertices. The coefficient of $x^n$ in the power series expansion of the Birman substitution (i.e. $A \to e^x$) of the Jones-Kauffman polynomial is known to be of Kauffman finite-type $\le n$ (see [4]).
The second kind of finite-type invariant of virtual knots is due to Goussarov,Polyak, and Viro [3]. Virtual knots are extended to knots having semivirtual crossings. Any virtual knot invariant can be extended to this larger class by applying the relation:
\[
v\left(\begin{array}{c} \scalebox{.15}{\psfig{figure=rightsemivirt.eps}} \end{array} \right)=v\left( \begin{array}{c} \scalebox{.15}{\psfig{figure=rightcross.eps}} \end{array}\right)-v\left(\begin{array}{c} \scalebox{.15}{\psfig{figure=virtcross.eps}}\end{array}\right)
\]
Semivirtual crossings are represented in Gauss diagrams by dashed arrows. Schematically, we have:
\[
\begin{array}{c} \scalebox{.17}{\psfig{figure=dashnbhde.eps}} \\ \end{array} = \begin{array}{c} \scalebox{.17}{\psfig{figure=solidnbhde.eps}} \\ \end{array}-\begin{array}{c} \scalebox{.17}{\psfig{figure=nonbhd.eps}} \\\end{array}
\]
A virtual knot invariant is said to be of GPV finite-type $\le n$ if $v(K_{\circ})=0$ for all diagrams $K_{\circ}$ with greater than $n$ semivirtual crossings. There is a simple and elegant algorithm for constructing all GPV finite-type invariants of order $\le n$ that is bounded only by how much computing power one has readily available . Moreover, every GPV finite-type invariant of order $\le n$ is of Kauffman finite-type $\le n$. However, the aforementioned coefficients obtained from the Jones-Kauffman polynomial are not of GPV finite-type $\le n$ for any $n$ (see [1]).
We now recall the construction of the rational GPV finite-type invariants found in [3]. Let $\mathscr{D}$ denote the set of all Gauss diagrams of virtual knots. A subdiagram of $D \in \mathscr{D}$ is a Gauss diagram consisting of some subset of the edges of $D$. Denote by $\mathscr{A}$ the free abelian group generated by dashed Gauss diagrams. The elements of $A$ are just Gauss diagrams with all arrows drawn dashed. Define a map $i:\mathbb{Z}[\mathscr{D}] \to \mathscr{A}$ to be the map which makes all the arrows of a Gauss diagram dashed. Define $I_{\text{GPV}}:\mathbb{Z}[\mathscr{D}] \to \mathscr{A}$ by:
\[
I_{\text{GPV}}(D)=\sum_{D' \subset D} i(D')
\]
where the sum is over all subdiagrams of $D$. The Polyak algebra is the quotient of $\mathscr{A}$ by the submodule $\Delta\mathscr{P}=\left<\Delta \text{PI},\Delta \text{PII},\Delta \text{PIII} \right>$ generated by the relations in Figure <ref>. The Polyak algebra is denoted $\mathscr{P}$.
\[
\underline{\Delta \text{PI}:}\,\,\, \begin{array}{c}\scalebox{.15}{\psfig{figure=polyak1.eps}} \end{array} =0, \underline{\Delta \text{PII}:}\,\,\,\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak2_1.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak2_2.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak2_3.eps}} \end{array}=0,
\]
\begin{eqnarray*}
\underline{\Delta \text{PIII}:}\,\,\,\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_1.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_2.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_3.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_4.eps}} \end{array} &=& \\ \begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_5.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_6.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_7.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak3_8.eps}} \end{array} & & \\
\end{eqnarray*}
Polyak Relations
Denote by $\mathscr{K}$ the set of virtual knots (where the elements are virtual isotopy classes of knots). The following theorem shows that the theory of virtual knots is entirely encoded in the Polyak algebra.
The map $I_{\text{GPV}}:\mathbb{Z}[\mathscr{D}] \to \mathscr{A}$ is an isomorphism. The inverse can be defined explicitly:
\[
I_{\text{GPV}}^{-1}(A)=\sum_{A'\subset A} (-1)^{|A-A'|} i^{-1}(A)
\]
Here, $|A- A'|$ means the number of arrows in $A$ that are not in $A'$. Furthermore, if $D \in \mathbb{Z}[\mathscr{D}]$ has dashed arrows, then every element in the sum defining $I(D)$ also has every dashed arrow of $D$. Finally, the map extends to an isomorphism of the quotient algebras $I_{\text{GPV}}:\mathbb{Z}[\mathscr{K}] \to \mathscr{P}$.
Let $A_n$ denote submodule of $\mathscr{A}$ generated by those diagrams having more than $n$ arrows. Define $\mathscr{P}_n=\mathscr{A}/(A_n + \Delta \mathscr{P})$. Let $\varphi_n:\mathscr{P} \to \mathscr{P}_n$ denote the natural projection onto the quotient. The following important theorem characterizes all rational valued GPV finite-type invariants.
The map $(I_{\text{GPV}})_n:\mathbb{Z}[\mathscr{K}] \to \mathscr{P} \to \mathscr{P}_n$ is universal in the sense that if $G$ is any abelian group, and $v$ is a GPV finite-type invariant of order $\le n$, then there is a map $v':\mathscr{P}_n \to G$ such that the following diagram commutes:
\[
\xymatrix{\mathbb{Z}[\mathscr{K}] \ar[r]^v \ar[d]_{I_{GPV}} & G \\
\mathscr{P} \ar[r]_{\varphi_n} \ar[ur]_{v I^{-1}_{GPV}} & \mathscr{P}_n \ar@{-->}[u]_{v'} \\
\]
In particular, the vector space of rational valued invariants of type $\le n$ is finite dimensional and can be identified with $\text{Hom}_{\mathbb{Z}}(\mathscr{P}_n,\mathbb{Q})$.
§ PROOF OF MAIN THEOREM
§.§ A model of virtualization invariant knot invariants
Let $\mathscr{C}$ denote the set of signed chord diagrams. These are chord diagrams in the usual sense which have the additional structure of a sign at each chord: $\oplus$ or $\ominus$. Let $C_n$ denote the free abelian group generated by set of signed chord diagrams possessing $>n$ chords. The relations for $\mathbb{Z}[\mathscr{C}]$ are as follows:
\[
\underline{\Delta \text{RI}:}\,\,\, \begin{array}{c}\scalebox{.15}{\psfig{figure=chordR1.eps}} \end{array} =0, \underline{\Delta \text{RII}:}\,\,\, \begin{array}{c}\scalebox{.15}{\psfig{figure=chordR2_3.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR2_1.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR2_2.eps}} \end{array}=0,
\]
\begin{eqnarray*}
\underline{\Delta \text{RIII}:}\,\,\, \begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_1.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_2.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_3.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_4.eps}} \end{array} &=& \\ \begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_5.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_6.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_7.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR3_8.eps}} \end{array} & & \\
\end{eqnarray*}
$\Delta \mathscr{R}$ relations
Let $\Delta \mathscr{R}=\left< \Delta \text{RI}, \Delta \text{RII}, \Delta \text{RIII} \right>$. Define:
\[
\mathscr{V}=\frac{\mathbb{Z}[\mathscr{C}]}{\Delta \mathscr{R}},\,\,\, \mathscr{V}_n=\frac{\mathbb{Z}[\mathscr{C}]}{C_n+\Delta \mathscr{R}}
\]
Let $\mathscr{VK}$ denote the set of virtual knot diagrams and $\mathscr{D}$ the set of Gauss diagrams. Define $\hat{g}:\mathbb{Z}[\mathscr{VK}] \to \mathbb{Z}[\mathscr{D}]$ on generators to be the map which assigns to every virtual knot diagram its Gauss diagram. Define $g:\mathbb{Z}[\mathscr{VK}]/\text{ker}(\hat{g}) \to \mathbb{Z}[\mathscr{D}]$ via Noether's First Isomorphism Theorem. For $D \in \mathscr{D}$, let $\bar{D} \in \mathscr{C}$ denote the chord diagram obtained from $D$ by erasing the arrow head of every arrow of $D$. We will also refer to this operation by the map $\text{Bar}:\mathbb{Z}[\mathscr{D}]\to\mathbb{Z}[\mathscr{C}]$. Define $I:\mathbb{Z}[\mathscr{D}]\to \mathbb{Z}[\mathscr{C}]$ on generators $D \in \mathscr{D}$ by:
\[
I(D)=\sum_{D' \subset D} \overline{D'}=\overline{I_{\text{GPV}}(D)}
\]
where the sum is over all Gauss diagrams $D'$ obtained from $D$ be deleting a subset of its arrows.
For all $v \in \text{Hom}_{\mathbb{Z}}(\mathscr{V},\mathbb{Q})$, $v \circ I$ is a virtual knot invariant that is invariant under the virtualization move.
By the GPV theorem, it is sufficient to show that $v(\overline{\text{PI}})=v(\overline{\text{PII}})=v(\overline{\text{PIII}})=\{0\}$. However, this is clearly true since $\overline{\left< \text{PI}, \text{PII}, \text{PIII} \right>}=\Delta \mathscr{R}$. For the second assertion, note that if $K$ and $K'$ are obtained from one another by a single virtualization move, then their Gauss diagrams differ only in the direction of a single arrow. In that case, $I(g(K))=I(g(K'))$.
§.§ Universality of the model
In this section we establish the universality of $I$ and $I_n$. The following lemma is useful in this regard.
Suppose that $v \in \text{Hom}_{\mathbb{Z}}(\mathscr{P},\mathbb{Q})$ is virtualization invariant. In other words,
\[
v \circ I_{\text{GPV}} \left( \begin{array}{c} \scalebox{.25}{\psfig{figure=virtmove1.eps}} \end{array} \right)=v \circ I_{\text{GPV}} \left( \begin{array}{c} \scalebox{.25}{\psfig{figure=virtmove2.eps}} \end{array}\right)
\]
Then for all signed dashed arrow diagrams $D$, if $D'$ is obtained from $D$ by changing the direction of one arrow, then $v(D)=v(D')$.
The proof is by induction on the number of arrows of the dashed diagram $D$. If $n=1$, the result is obvious in the case of knots. For long knots, we have:
\begin{eqnarray*}
v\circ I_{\text{GPV}}\left( \begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmR1.eps}} \end{array} \right) &=& v \left( \begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmR2.eps}} \end{array}\right)+v\left( \begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmNULL.eps}} \end{array}\right) \\
v\circ I_{\text{GPV}}\left( \begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmL1.eps}} \end{array}\right) &=& v \left(\begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmL2.eps}} \end{array} \right)+v\left(\begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmNULL.eps}} \end{array} \right) \\
\Rightarrow v \left( \begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmL2.eps}} \end{array}\right)&=&v\left(\begin{array}{c} \scalebox{.15}{\psfig{figure=unilemmR2.eps}} \end{array}\right) \\
\end{eqnarray*}
Suppose now that the theorem is true for $n$. Let $D$ be a dashed arrow diagram with $n+1$ arrows, $D'$ a diagram obtained from $D$ by switching the direction of one arrow. Let $B=i^{-1}(D)$ and $B'=i^{-1}(D')$. Let $e$ denote the arrow whose direction is changed from $D$ to $D'$.
\begin{eqnarray*}
v\circ I_{\text{GPV}}(B) &=& v\left( \sum_{R \subset B} i(R) \right) \\
&=& v\left( \sum_{\stackrel{R\subset B}{e \in R}} i(R) \right)+v\left( \sum_{\stackrel{R\subset B}{e \notin R}} i(R)\right)\\
v\circ I_{\text{GPV}}(B') &=& v\left( \sum_{\stackrel{R'\subset B'}{e \in R'}} i(R) \right)+v\left( \sum_{\stackrel{R'\subset B'}{e \notin R'}} i(R')\right)\\
\end{eqnarray*}
Now, for $e \in R$, let $R'$ denote the diagram obtained from $R$ by switching the direction of $e$. By applying linearity of $v$ and subtracting the two equations of interest, we obtain:
\[
\sum_{\stackrel{R\subset B}{e \in R}} v(i(R))-v(i(R'))=0
\]
Now, for $R \subset B$, $R$ has between $1$ and $n+1$ arrows. Since $R$ and $R'$ differ only in the direction of a single arrow the induction hypothesis implies that $v(i(R))=v(i(R'))$ for all $R$ having $\le n$ arrows. Thus,
\[
\]
This establishes the lemma.
The map $I:\mathbb{Z}[\mathscr{D}] \to \mathscr{V}$ is universal in the sense that if $v \in \text{Hom}_{\mathbb{Z}}(\mathscr{P},\mathbb{Q})$ is virtualization invariant, then there is a $v' \in \text{Hom}_{\mathbb{Z}}(\mathscr{V},\mathbb{Q})$ such that the following diagram commutes:
\[
\xymatrix{ \mathbb{Z}[\mathscr{D}] \ar[r]^I \ar[d]_{I_{\text{GPV}}} & \mathscr{V} \ar@{-->}[d]^{v'} \\
\mathscr{P} \ar[r]_v & \mathbb{Q}}
\]
Let $C \in \mathscr{C}$ and let $\vec{C}$ denote the signed arrow diagram obtained from $C$ by directing the chords of $C$. Define $v'(C)=v(\vec{C})$. Note that by Lemma <ref>, if $\vec{C'}$ is an arrow diagram with $\text{Bar}(\vec{C'})=C$, then $v(\vec{C})=v(\vec{C'})$. Thus, $v'$ is well-defined on $\mathbb{Z}[\mathscr{C}]$.
To complete the proof, it is only necessary to show that $v'(r)=0$ for all $r\in \Delta \mathscr{R}$. For each $r\in \Delta \text{RI},\Delta \text{RII}$, or $\Delta \text{RIII}$, there is a $\vec{r} \in \text{Bar}^{-1}(r)$ such that $\vec{r} \in \text{PI},\text{PII}$, or $\text{PIII}$, respectively. For example, we have:
\begin{eqnarray*}
r &=& \begin{array}{c}\scalebox{.15}{\psfig{figure=chordR2_3.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR2_1.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=chordR2_2.eps}} \end{array}\\
\vec{r} &=& \begin{array}{c}\scalebox{.15}{\psfig{figure=polyak2_1.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak2_2.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=polyak2_3.eps}} \end{array} \\
\end{eqnarray*}
Since $v(\text{PI})=v(\text{PII})=v(\text{PIII})=0$, it follows that $v'(r)=0$.
For $v \in \text{Hom}_{\mathbb{Z}}(\mathscr{V}_n,\mathbb{Q})$, $v\circ I$ is a GPV finite-type invariant of order $\le n$.
For $D \in \mathscr{P}$, $v \circ I\circ I_{\text{GPV}}^{-1}(D)=v(\overline{D})$. If $D$ has more than $n$ dashed arrows, then $v(\overline{D})=0$.
Let $p_n:\mathscr{V} \to \mathscr{V}_n$ denote that natural projection. Define $I_n:\mathbb{Z}[D] \to \mathscr{V}_n$ to be the composition: $I_n=p_n \circ I$.
The map $I_n:\mathbb{Z}[\mathscr{D}] \to \mathscr{V}_n$ is universal in the sense that for all $v \in \text{Hom}(\mathscr{P}_n,\mathbb{Q})$ such that $v \circ (I_{\text{GPV}})_n$ is virtualization invariant, then there is a $v'\in \text{Hom}_{\mathbb{Z}}(\mathscr{V}_n,\mathbb{Q})$ such that the following diagram commutes.
\[
\xymatrix{ \mathbb{Z}[\mathscr{D}] \ar[r]^{I_n} \ar[d]_{(I_{\text{GPV}})_n} & \mathscr{V}_n \ar@{-->}[d]^{v'} \\
\mathscr{P}_n \ar[r]_v & \mathbb{Q}}
\]
This follows immediately from Lemma <ref> and Theorem <ref>.
It is obvious that constant invariants are of GPV finite type of every order. For virtual knots, they are generated by the combinatorial formula $\left<\bigcirc,\cdot\right>$. For long virtual knots, they are generated by the combinatorial formula $<\underline{\hspace{.25cm}}\,\,,\cdot>$. Denote both of these generators by $1$.
For small orders, the following computations hold:
* For virtual knots, $\text{Hom}_{\mathbb{Z}}(\mathscr{V}_3,\mathbb{Q})=<1>$
* For long virtual knots, $\text{Hom}_{\mathbb{Z}}(\mathscr{V}_2,\mathbb{Q})=<1>$
For the first assertion, we have from [3] that 1 and the following formula generate the GPV finite-type invariants of order $\le 3$.
\[
\left<3 \cdot \begin{array}{c}\scalebox{.15}{\psfig{figure=casson1.eps}} \end{array}-\begin{array}{c}\scalebox{.15}{\psfig{figure=casson2.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=casson3.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=casson4.eps}} \end{array}-\begin{array}{c}\scalebox{.15}{\psfig{figure=casson5.eps}} \end{array}-\begin{array}{c}\scalebox{.15}{\psfig{figure=casson6.eps}} \end{array}-\begin{array}{c}\scalebox{.15}{\psfig{figure=casson7.eps}} \end{array}+\begin{array}{c}\scalebox{.15}{\psfig{figure=casson8.eps}} \end{array}, \,\,\,\, \cdot \right>
\]
Since the value on diagrams 3 and 5 is different, Lemma <ref> implies our result.
For the second assertion, we have from [3] that 1 and the following two formulas generate the GPV finite-type invariants of order $\le 2$.
\[
\left< \begin{array}{c}\scalebox{.25}{\psfig{figure=longvirt21.eps}} \end{array},\,\,\, \cdot \right>,\,\,\,\left< \begin{array}{c}\scalebox{.25}{\psfig{figure=longvirt22.eps}} \end{array},\,\,\, \cdot \right>
\]
In this case, Lemma <ref> also implies our result.
§.§ Algebraic decomposition of $\text{Hom}_{\mathbb{Z}}(\mathscr{P}_n,\mathbb{Q})$ and $\text{Hom}_{\mathbb{Z}}(\mathscr{V}_n,\mathbb{Q})$
The hero of the decomposition is Polyak's algebra of arrow diagrams. Its precise relationship to the groups $\mathscr{V}_n$ and $\mathscr{P}_n$ is what allows us to obtain Theorem <ref>. While this relationship is already well-known (see [5]), it is prudent to describe it carefully here.
Define $\vec{\mathscr{F}}_n^{\pm}$, $\vec{\mathscr{F}}_n$ to be the free abelian group generated by the set of signed arrow diagrams and unsigned arrow diagrams having exactly $n$ arrows, respectively. Define $\overline{\mathscr{F}}_n^{\pm}$, $\overline{\mathscr{F}}_n$ to be the free abelian group generated by the set of signed chord diagrams and unsigned chord diagrams having exactly $n$ chords, respectively.
The decomposition is the same for the $\mathscr{V}_n$ and $\mathscr{P}_n$. Therefore we define variables which stand in place of either case.
Chord Case Arrow Case
$\mathscr{B}_n$ $\mathscr{V}_n$ $\mathscr{P}_n$
$B_n$ $C_n$ $A_n$
$R$ $\Delta \mathscr{R}$ $\Delta \mathscr{P}$
$\mathscr{F}_n^{\pm}$ $\overline{\mathscr{F}}_n^{\pm}$ $\vec{\mathscr{F}}_n^{\pm}$
$\mathscr{F}_n$ $\overline{\mathscr{F}}_n$ $\vec{\mathscr{F}}_n$
In this section, we investigate the following short exact sequence:
\[
\xymatrix{0 \ar[r] & \frac{B_n+R}{B_{n+1}+R} \ar[r] & \mathscr{B}_{n+1} \ar[r]^{\pi_n} & \mathscr{B}_n \ar[r] & 0}
\]
and (more importantly), its dual:
\[
\xymatrix{0 \ar[r] & \text{Hom}_{\mathbb{Z}}\left(\mathscr{B}_n,\mathbb{Q}\right) \ar[r]^{\pi_n^*} & \text{Hom}_{\mathbb{Z}}\left(\mathscr{B}_{n+1},\mathbb{Q}\right) \ar[r] & \text{Hom}_{\mathbb{Z}}\left(\frac{B_n+R}{B_{n+1}+R},\mathbb{Q}\right)}
\]
In the next section, we will show that the rightmost module in the dual sequence vanishes and hence $\pi_n^*$ is an isomorphism. For this, we use some intermediate groups which are isomorphic to the groups in the grading of Polyak's algebra of arrow diagrams. The relations for the intermediate groups are given below.
\[
\underline{\vec{\text{1T}}_n:} \begin{array}{c} \scalebox{.1}{\psfig{figure=polyak1T.eps}} \end{array}=0 ,\,\, \underline{\vec{\text{NS}}_n:} \begin{array}{c} \scalebox{.1}{\psfig{figure=polyak2_2.eps}} \end{array}+\begin{array}{c} \scalebox{.1}{\psfig{figure=polyak2_3.eps}} \end{array}=0
\]
\begin{eqnarray*}
\underline{\vec{\text{6T}}_n^{\pm}:} \begin{array}{c}\scalebox{.1}{\psfig{figure=polyak3_2.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak3_3.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak3_4.eps}} \end{array} &=& \begin{array}{c}\scalebox{.1}{\psfig{figure=polyak3_6.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak3_7.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak3_8.eps}} \end{array} \\
\end{eqnarray*}
\begin{eqnarray*}
\underline{\vec{\text{6T}}_n:} \begin{array}{c}\scalebox{.1}{\psfig{figure=polyak6T_1.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak6T_2.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak6T_3.eps}} \end{array} &=& \begin{array}{c}\scalebox{.1}{\psfig{figure=polyak6T_4.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak6T_5.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=polyak6T_6.eps}} \end{array} \\
\end{eqnarray*}
Using the same convention as above, define $\overline{\text{1T}}_n^{\pm}=\text{Bar}(\Delta\text{PI}_n)$, $\overline{\text{NS}}_n=\text{Bar}(\vec{\text{NS}}_n)$, $\overline{\text{6T}}_n^{\pm}=\text{Bar}(\vec{\text{6T}}_n^{\pm})$, and $\overline{\text{6T}}_n=\text{Bar}(\vec{\text{6T}}_n)$. Define $\text{6T}_n=\overline{\text{6T}}_n$ or $\vec{\text{6T}}_n$, $\text{6T}_n^{\pm}=\overline{\text{6T}}_n^{\pm}$ or $\vec{\text{6T}}_n^{\pm}$, $\text{1T}_n^{\pm}=\overline{\text{1T}}_n^{\pm}$ or $\Delta\text{PI}_n$, and $\text{NS}_n=\overline{\text{NS}}_n$ or $\vec{\text{NS}}_n$.
There is an injection:
\[
\text{Hom}_{\mathbb{Z}}\left( \frac{B_{n-1}+R}{B_n+R}, \mathbb{Q} \right)\hookrightarrow \text{Hom}_{\mathbb{Z}} \left(\frac{\mathscr{F}_n}{\left<\text{1T}_n,\text{6T}_n\right>} ,\mathbb{Q} \right)
\]
First we rewrite $\frac{B_{n-1}+R}{B_n+R}$ using Noether's Second Isomorphism Theorem.
\begin{eqnarray*}
\frac{B_{n-1}+R}{B_n+R} &=& \frac{\mathscr{F}_n^{\pm}+(B_n+R)}{B_n+R} \\
&\cong& \frac{\mathscr{F}_n^{\pm}}{\mathscr{F}_n^{\pm} \cap (B_n+R)} \\
\end{eqnarray*}
$\left< \text{6T}_n^{\pm}, \text{1T}_n^{\pm}, \text{NS}_n \right> \subset \mathscr{F}_n^{\pm} \cap (B_n+R)$
It is certainly true that $\text{LHS} \subset \mathscr{F}_n^{\pm}$. We will show that $\text{LHS} \subset B_n+R$ for the chord diagram case only:
\begin{eqnarray*}
\underline{\overline{\text{6T}}_n^{\pm}:}\,\,\, \begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_2.eps}} \end{array} &+&\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_3.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_4.eps}} \end{array} - \begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_6.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_7.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_8.eps}} \end{array} \\
&=& \begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_1.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_2.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_3.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_4.eps}} \end{array}\\
&-&\left.\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_5.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_6.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_7.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_8.eps}} \end{array} \right\} \in \Delta \mathscr{R}\\
&+& \left.\left(- \begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_1.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR3_5.eps}} \end{array} \right) \right\} \in C_n
\end{eqnarray*}
\[
\underline{\overline{\text{NS}}_n:}\,\,\, \begin{array}{c} \scalebox{.1}{\psfig{figure=chordR2_1.eps}} \end{array}+\begin{array}{c} \scalebox{.1}{\psfig{figure=chordR2_2.eps}} \end{array} =
\underbrace{\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR2_3.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR2_1.eps}} \end{array}+\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR2_2.eps}} \end{array}}_{ \in \Delta \mathscr{R}}-\underbrace{\begin{array}{c}\scalebox{.1}{\psfig{figure=chordR2_3.eps}} \end{array}}_{\in C_n}
\]
This completes the proof of the lemma.
Therefore there is an exact sequence:
\[
\xymatrix{
\frac{\mathscr{F}_n^{\pm}}{\left< \text{6T}_n^{\pm}, \text{1T}_n^{\pm}, \text{NS}_n \right>} \ar[r] & \frac{\mathscr{F}_n^{\pm}}{\mathscr{F}_n^{\pm} \cap (B_n+R)} \ar[r] & 0}
\]
Define $\Xi:\mathscr{F}_n^{\pm} \to \mathscr{F}_n$ as in [5]. For $F \in \mathscr{F}_n^{\pm}$, let $m(F)$ denote the number of $\ominus$ signs appearing in the diagram. Denote by $|F|\in\mathscr{F}_n$ the diagram obtained by deleting all the signs of $F$. Then $\Xi$ is defined on generators by:
\[
\Xi(F)=(-1)^{m(F)}|F|
\]
Extend $\Xi$ to all of $\mathscr{F}_n^{\pm}$ using linearity. It is clear that:
\begin{eqnarray*}
\Xi(\left<\text{6T}_n^{\pm}\right>) &\subset& \left< \text{6T}_n \right>\\
\Xi(\left<\text{1T}_n^{\pm}\right>) &\subset& \left< \text{1T}_n \right> \\
\Xi(\left<\text{NS}_n \right>) &=& \{0\} \\
\end{eqnarray*}
The surjection $\mathscr{F}_n^{\pm} \to \mathscr{F}_n \to \mathscr{F}_n/\left<\text{1T}_n,\text{6T}_n \right>$ has kernel $\left< \text{6T}_n^{\pm}, \text{1T}_n^{\pm}, \text{NS}_n \right>$. Taking the dual of the above short exact sequence gives the injection:
\[
\text{Hom}_{\mathbb{Z}}\left( \frac{B_{n-1}+R}{B_n+R}, \mathbb{Q} \right)\hookrightarrow \text{Hom}_{\mathbb{Z}}\left(\frac{\mathscr{F}_n^{\pm}}{\left< \text{6T}_n^{\pm}, \text{1T}_n^{\pm}, \text{NS}_n \right>} ,\mathbb{Q}\right) \cong \text{Hom}_{\mathbb{Z}} \left(\frac{\mathscr{F}_n}{\left<\text{1T}_n,\text{6T}_n\right>} ,\mathbb{Q} \right)
\]
As we shall shortly see, it is convenient to change from $\mathbb{Z}$-modules to vector spaces over $\mathbb{Q}$. The following computation shows that this does not affect our result.
\begin{eqnarray*}
\text{Hom}_{\mathbb{Z}} \left(\frac{\mathscr{F}_n}{\left<\text{1T}_n,\text{6T}_n\right>} ,\mathbb{Q} \right) &\cong& \text{Hom}_{\mathbb{Z}} \left(\frac{\mathscr{F}_n}{\left<\text{1T}_n,\text{6T}_n\right>} ,\text{Hom}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Q}) \right)\\
&\cong& \text{Hom}_{\mathbb{Z}} \left(\frac{\mathscr{F}_n}{\left<\text{1T}_n,\text{6T}_n\right>}\otimes \mathbb{Q} ,\mathbb{Q} \right)
\end{eqnarray*}
§.§ Proof of Theorem 1
To prove the theorem, we must also consider the four term relation in $\overline{\mathscr{F}}_n$:
\[
\underline{\overline{\text{4T}}_n:} \,\,\, \begin{array}{c}\scalebox{.15}{\psfig{figure=fourterm1.eps}} \end{array}-\begin{array}{c}\scalebox{.15}{\psfig{figure=fourterm2.eps}} \end{array}=\begin{array}{c}\scalebox{.15}{\psfig{figure=fourterm3.eps}} \end{array}-\begin{array}{c}\scalebox{.15}{\psfig{figure=fourterm4.eps}} \end{array}
\]
Define $\mu_n:\overline{\mathscr{F}}_n \to \vec{\mathscr{F}}_n$ to be:
\[
\mu_n \left( \begin{array}{c}\scalebox{.25}{\psfig{figure=avg1.eps}} \end{array} \right)=\begin{array}{c}\scalebox{.25}{\psfig{figure=avg2.eps}} \end{array}+\begin{array}{c}\scalebox{.25}{\psfig{figure=avg3.eps}} \end{array}
\]
Here, the sum is over all such resolutions. In fact, if $F \in \overline{\mathscr{F}}_n$, then $\mu_n(F)$ is a sum of $2^n$ arrow diagrams. The map $\mu_n$ is also called the average map. The following theorem is due to Polyak.
The average map $\mu_n:\overline{\mathscr{F}}_n \to \vec{\mathscr{F}}_n$ satisfies the following properties.
* $\mu_n\left(\left< \overline{\text{4T}}_n\right>\right)\subset \left< \vec{\text{6T}}_n \right>$
* $\mu_n\left(\left< \overline{\text{1T}}_n\right>\right)\subset \left< \vec{\text{1T}}_n \right>$
* The average map descends to a well-defined map:
\[
\mu_n:\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{4T}}_n \right>}\otimes \mathbb{Q} \to \frac{\vec{\mathscr{F}}_n}{\left<\vec{\text{1T}}_n,\vec{\text{6T}}_n \right>}\otimes \mathbb{Q}
\]
Define $\mu_n'(D)=\frac{1}{2^n} \cdot \overline{\mu_n(D)}$. The following diagrams describe this map.
\[
\mu_n':\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{4T}}_n \right>}\otimes \mathbb{Q} \to \frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{6T}}_n \right>}\otimes \mathbb{Q}
\]
\[
\xymatrix{ \frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{4T}}_n \right>}\otimes \mathbb{Q} \ar[rr]^{2^n \cdot \mu_n'} \ar[dr]_{\mu_n}& & \frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\vec{\text{6T}}_n \right>}\otimes \mathbb{Q} \\ & \frac{\vec{\mathscr{F}}_n}{\left<\vec{\text{1T}}_n,\vec{\text{6T}}_n \right>}\otimes \mathbb{Q} \ar[ur]_{\text{Bar}} & }
\]
In fact, for all $D \in \overline{\mathscr{F}}_n$, $\mu_n'(D)=D$. Hence, $\mu_n'$ is a surjection and $\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{6T}}_n \right>}\otimes \mathbb{Q}$ is a homomorphic image of $\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{4T}}_n \right>}\otimes \mathbb{Q}$. Since $\mu_n'\left(\left<\overline{\text{6T}}_n \right> \right)=\{0\}$, diagrams in $\overline{\mathscr{F}}_n$ satisfy relations $\overline{\text{1T}}_n$, $\overline{\text{6T}}_n$, and $\overline{\text{4T}}_n$ in the homomorphic image $\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{6T}}_n \right>}\otimes \mathbb{Q}$. Now, note that we may write $\overline{\text{6T}}_n$ in the following form:
\[
\underbrace{\begin{array}{c}\scalebox{.1}{\psfig{figure=fourterm1.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=fourterm2.eps}} \end{array}-\left( \begin{array}{c}\scalebox{.1}{\psfig{figure=fourterm3.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=fourterm4.eps}} \end{array}\right)}_{\overline{\text{4T}}_n}+\underbrace{\begin{array}{c}\scalebox{.1}{\psfig{figure=twoterm1.eps}} \end{array}-\begin{array}{c}\scalebox{.1}{\psfig{figure=twoterm2.eps}} \end{array}}_{\overline{\text{2T}}_n}=0
\]
Therefore, the two-term relation (see [6]) is also satisfied in the homomorphic image $\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{6T}}_n \right>}\otimes \mathbb{Q}$.
\[
\underline{\overline{\text{2T}}_n:} \,\,\, \begin{array}{c}\scalebox{.1}{\psfig{figure=twoterm1.eps}} \end{array}=\begin{array}{c}\scalebox{.1}{\psfig{figure=twoterm2.eps}} \end{array}
\]
If the two term relation is satisfied, then all chord diagrams with $n$ chords are equivalent to the diagram given below:
Virtual Knot Case Long Virtual Knot Case
$\underbrace{\begin{array}{c}\scalebox{.25}{\psfig{figure=knotalone.eps}} \end{array}}_{n \text{ chords}} $ $\underbrace{\begin{array}{c}\scalebox{.25}{\psfig{figure=longalone.eps}} \end{array}}_{n \text{ chords}}$
Since these diagrams all vanish in the presence of $\overline{\text{1T}_n}$, it follows that $\frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{6T}}_n \right>}\otimes \mathbb{Q} \cong \{ 0 \}$. Hence the dual $\text{Hom}_{\mathbb{Z}} \left( \frac{\overline{\mathscr{F}}_n}{\left<\overline{\text{1T}}_n,\overline{\text{6T}}_n \right>}\otimes \mathbb{Q},\mathbb{Q}\right) \cong \{ 0 \}$ and we conclude that $\pi_n^*$ is an isomorphism. This implies that $\text{Hom}_{\mathbb{Z}}(\mathscr{V}_n, \mathbb{Q}) \cong \text{Hom}_{\mathbb{Z}}(\mathscr{V}_{n+1}, \mathbb{Q})$ for all $n$. In the case of virtual knots, we have by Lemma <ref> that $\text{Hom}_{\mathbb{Z}}(\mathscr{V}_3,\mathbb{Q})=<1>$. In the case of long virtual knots, $\text{Hom}_{\mathbb{Z}}(\mathscr{V}_2,\mathbb{Q})=<1>$. This completes the proof of Theorem <ref>.
If $v_n$ is the $n$-th Birman coefficient of the Jones-Kauffman polynomial, then $v_n$ is of Kauffman finite-type $\le n$, but not of GPV finite-type $\le m$ for any $m$.
The invariants $v_n$ are invariant under the virtualization move [4].
[1]
Micah Chrisman.
Twist lattices and the jones-kauffman polynomial for long virtual
Journal of Knot Theory and Its Ramifications, to appear.
[2]
Roger Fenn, Louis H. Kauffman, and Vassily O. Manturov.
Virtual knot theory—unsolved problems.
Fund. Math., 188:293–323, 2005.
[3]
Mikhail Goussarov, Michael Polyak, and Oleg Viro.
Finite-type invariants of classical and virtual knots.
Topology, 39(5):1045–1068, 2000.
[4]
Louis H. Kauffman.
Virtual knot theory.
European J. Combin., 20(7):663–690, 1999.
[5]
Michael Polyak.
On the algebra of arrow diagrams.
Lett. Math. Phys., 51(4):275–291, 2000.
[6]
J. Mostovoy S. Chmutov, S. Dushin.
CDBook:Introduction to Vassiliev Knot Invariants.
http://www.math.ohio-state.edu/ chmutov/preprints/.
|
arxiv-papers
| 2010-01-13T18:03:15 |
2024-09-04T02:49:07.753593
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Micah W. Chrisman",
"submitter": "Micah Chrisman",
"url": "https://arxiv.org/abs/1001.2247"
}
|
1001.2332
|
# Superconductivity and Magnetism in BiOCuS
I.I. Mazin Code 6393, Naval Research Laboratory, Washington, D.C. 20375
(Printed on )
###### Abstract
BiOCu1-yS was recently reported to superconduct at $T\approx 5.8$ K at
$y\approx 0.15.$ Band structure calculations indicate that the stoichiometric
BiOCuS is a band insulator. In this paper, I show that the hole-doped (whether
in the virtual crystal approximation or with actual Cu vacancies) BiOCuS is on
the verge of a ferromagnetic instability (cf. Pd metal), and therefore a
conventional superconductivity with $T_{c}\sim 6$ K is quite unlikely.
Presumably, the hole-doped BiOCuS is another example of superconductivity
mediated by spin-fluctuations.
Recently, Ubaldini $et$ $al$ Dirk reported that in the quaternary compound
BiOCu1-yS superconductivity at $T_{c}=6$ K can be induced by a small Cu
deficiency ($y\sim 15\%),$ although not by an electron doping (when
substituting O with F). Although another similar study has not confirmed the
superconductivity Ind , this result has triggered some interest in the
communityInd ; SI . I will show below that this interest is well justified,
for this material is very unlikely to be a conventional s-wave superconductor,
and, if superconducting, is probably another example of a spin-fluctuation
mediated superconductivity.
BiOCuS crystallizes in the well known tetragonal ZrCuSiAs structureInd ; str ,
recently made famous by some Fe-based high-temperature superconductors
(although there is no commonality whatsoever between the latter and the
material in question). The lattice parameters reported in Ref. Dirk, are
3.8726 and 8.5878 Å, in Ref. str, 3.8691 and 8.5602 Å, and in Ref. Ind,
3.868 and 8.557 Å. The internal coordinates of Bi and S, respectively, are
reported as 0.14829 and 0.6710 (Ref. str, ) and 0.151 and 0.648 (Ref. Ind, ).
In my calculations I used the structure reported in Ref. str, . The variation
of crystallographic parameters withing these limits does not affect any
conclusions of this paper.
In agreement with the previous first principles calculationsSI ; str I
foundmethod that in the scalar-relativistic approximation the stoichiometric
compound is a band insulator, with a direct gap underestimated compared to the
measured optical gapstr , as common in the density functional calculations.
Adding spin-orbit interaction on Bi reduces the gap even further, from 0.75 to
0.55 eV, by affecting the unoccupied states. The states right below the gap
are formed predominantly by the Cu $xz$ and $yz$ orbitals, while the empty
states above the gap have quasi-free-electron character (the interstitial
region, being $\approx$60% in volume, contributes up to 50% of the density of
states of these bands). In agreement with, especially, the full potential all-
electron calculations of Ref. SI, , I found a flat band right at the top of
the valence band at the Z point (Fig. 1), essentially not dispersing up to
half-way between Z and Xnote2 . This band results in a small but very sharp
density of states (DOS) peak right below the band gap (Fig. 2), with the total
weight of slightly less than 0.3 electron per copper, and a maximum DOS of 4.1
states/eV. f.u. at 130 meV below the gap (corresponding to 0.10–0.12 hole/Cu).
One can see that this peak indeed originates from the flat band along $Z-A$,
by looking at the Fermi surface for $E_{F}$ set at 120 meV below the band gap,
where the peak just starts (approximately 0.07 hole/Cu doping). This feature
appears as cross-shaped Fermi surface pockets emerging from Z (Fig. 3).
Figure 1: Band structure of BiOCuS near the Fermi level. Note a flat band in
the $Z-A$ direction. Figure 2: Density of states of BiOCuS near the Fermi
level. Note a peak right below the gap, and the leading contribution of the Cu
states below, and the interstitial states above the band gap. (color online)
Figure 3: (color online) Left panel: The Fermi surface of BiOCu1-yS at
$E_{F}$ 120 meV below the gap, where a sharp peak in DOS starts. Note a cross-
shaped pocket emerging along the $Z-A$ direction, responsible for this peak.
For comparison, in the right panel the Fermi surface at $E_{F}=150$ meV
($y\sim 0.15$) is shown, where the peak is already fully developed. Both Fermi
surfaces are colored according the the absolute value of the Fermi velocity
using the same scale (blue is zero).
Given that Cu d orbitals have sizeable Hund rule coupling this result suggests
that Cu ions under 10-20% hole doping should be magnetic. Indeed,
ferromagnetic (antiferromagnetic calculation, however, were invariably
converging to zero magnetizationnote3 ) calculations for $y=0.2$ in the
virtual crystal approximation (0.2 hole per Cu) converge to a stable solution
with the magnetic moment $M\approx 0.12-0.13$ $\mu_{B}$ per Cu, but with
hardly any energy gain. Correspondingly, fixed spin moment calculations (Fig.
4) show that in the GGA approximation the energy, within the computational
accuracy, does not depend on magnetization up to $M=y$ $\mu_{B}$. Note that
this magnetization would correspond to a half-metallic state similar to that
in the Co-doped FeS2CoS2 (in fact the physics of the formation of a magnetic
state is quite similar here). This means that the calculated spin
susceptibility is essentially infinite, a situation extremely close to Pd
metal, where GGA calculations also yield infinite susceptibilityPd . As in Pd,
or, for that matter, in Fe pnictides, fluctuations beyond the mean-field level
should suppress magnetization and reduce the susceptibility to a large, but
finite value. But, again as in Pd, strong spin fluctuations are bound to
destroy any conventional s-wave superconductivity, particularly given that
with this relatively modest carrier density electron-phonon coupling cannot be
too strong.
Figure 4: Fixed spin moment calculations for BiOCu1-yS for $y=0.2$. The self-
consistent solution is indicated by the arrow. Note that the energy is
essentially independent of the magnetic moment up to the half-metallic
magnetization of $y$ $\mu_{B}$/Cu. The inset shows a linear growth of the
total energy when magnetization exceeds the half-metal limit.
This leads us to the conclusion that the superconductivity in the hole-doped
BiOCu1-yS must be unconventional, and probably magnetically mediated, although
it is not clear yet what particular pairing symmetry can be generated in this
material.
One can question the validity of the virtual crystal approximation, given that
holes are introduced not through substitution, but through Cu vacancies. To
verify that, I have performed supercell calculations using four formula units
with one vacancy, that is to say, Bi4O4Cu3S${}_{4},$ corresponding to
$y=0.25.$ The calculations converged to a ferromagnetic state with $M=0.2$
$\mu_{B}$ per formula unit (0.8 $\mu_{B}$ per supercell), again slightly below
the half-metallic limit of 0.25 $\mu_{B},$ thus confirming the validity of the
virtual crystal calculations. Interestingly, the magnetization was quite
delocalized, with Cu ions carrying 0.1 – 0.13 $\mu_{B},$ and the rest of the
magnetic moment distributed among the sulfur ions and in the interstitial
region.
I thank Dirk van der Marel for stimulating discussions related to this work,
and Peter Blaha for a technical consulation. I also acknowledge funding from
the Office of Naval Research.
## References
* (1) A. Ubaldini, E. Giannini, C. Senatore, D. van der Marel, arXiv:0911.5305 (unpublished).
* (2) A. Pal, H. Kishan and V.P.S. Awana, arXiv:0912.0991 (unpublished)
* (3) I. R. Shein and A. L. Ivanovskii, Solid State Communications, 150, 640 (2010)
* (4) H. Hiramatsu, H. Yanagi, T. Kamiya, K. Ueda, M. Hirano and H. Hosono Chem., Mater., 20, 326 (2008)
* (5) The full-potential LAPW method, as implemented in the WIEN2k packege, has been used for all calculations. Up to 840 inequivalent k-points have been utilized to achieved the self-consistency, with the energy convergency better then 0.05 meV.
* (6) In Ref. SI, , unconventional notations for the high symmetry points are used; conventional A ($\pi,0,\pi)$ is called R, and conventional R ($\pi,\pi,\pi)$ is called A. I am using the standard notations.
* (7) I have tried only a checkerboard antiferromagnetic arrangement, since I do not see any a priori reason for any other antiferromagnetic ordering (the fact that the crystal structure coincides with that of pnictide superconductors is not an argument at all that this material would assume the same exotic magnetic structure). Of course, this does not mean that there are no spin fluctuations with a finite wave vector; it only tells us that if such fluctuations occur, they occur at a wave vector, different from ${\pi,\pi}$ (in the unfolded Brillouin zone notation).
* (8) I. I. Mazin, Appl. Phys. Lett. 77, 3000 (2000).
* (9) P. Larson, I.I. Mazin, D.J. Singh, Phys. Rev. B69, 064429 (2004).
|
arxiv-papers
| 2010-01-13T23:22:01 |
2024-09-04T02:49:07.762110
|
{
"license": "Public Domain",
"authors": "I.I. Mazin",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/1001.2332"
}
|
1001.2586
|
# Linear Scaling Solution of the Time-Dependent Self-Consistent-Field
Equations
Matt Challacombe mchalla@lanl.gov Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545
###### Abstract
A new approach to solving the Time-Dependent Self-Consistent-Field equations
is developed based on the double quotient formulation of Tsiper [J. Phys. B,
34 L401 (2001)]. Dual channel, quasi-independent non-linear optimization of
these quotients is found to yield convergence rates approaching those of the
best case (single channel) Tamm-Dancoff approximation. This formulation is
variational with respect to matrix truncation, admitting linear scaling
solution of the matrix-eigenvalue problem, which is demonstrated for bulk
excitons in the polyphenylene vinylene oligomer and the (4,3) carbon nanotube
segment.
Quasi-Independent Optimization, Rayleigh Quotient Iteration, _J_ -Symmetry,
Random Phase Approximation, Time-Dependent Density Functional Theory, Inexact
Linear Algebra.
††preprint: LA-UR 09-06104
The Time-Dependent Self-Consistent-Field equations together with models that
include some portion of the Hartree-Fock (HF) exchange admit control over the
range of self-interaction in the optical response Autschbach (2009); Song _et
al._ (2009); Igumenshchev _et al._ (2007); Magyar and Tretiak (2007), and are
related to new models of electron correlation based on the Random Phase
Approximation (RPA) Lu _et al._ (2009); Furche (2008); Harl and Kresse (2009);
Toulouse _et al._ (2009). Solving the TD-SCF equations is challenging due to
an unconventional J-symmetric structure of the naive molecular orbital (MO)
representation,
$\begin{pmatrix}\bm{\mathbb{A}}&\bm{\mathbf{\mathbb{B}}}\\\
-\mathbb{\bm{B}}&\bm{-\mathbb{A}}\end{pmatrix}\binom{\vec{X}}{\vec{Y}}=\omega\>\binom{\vec{X}}{\vec{Y}}\,,$
(1)
where $\mathbb{A}$ and $\mathbb{B}$ are Hermitian blocks corresponding to
$4^{th}$ order tensors spanning transitions between occupied and virtual sub-
spaces, $\omega$ is the real excitation energy and
$\vec{v}=\binom{\vec{X}}{\vec{Y}}$ is the corresponding transition density. By
construction, the MO representation allows strict separation between the
dyadic particle-hole (_ph_) and hole-particle (_hp_) solutions, $\vec{X}$ and
$\vec{Y}$, for which specialized algorithms exist. Nevertheless, convergence
of the naive $J$-symmetric problem is typically much slower than the
corresponding Hermitian Tamm-Dancoff approximation (TDA),
$\mathbb{A}\vec{X}=\omega\vec{X}$, which is of reduced dimensionality in the
MO representation.
Several TD-SCF eigensolvers are based on the oscillator picture
$\begin{pmatrix}0&\mathbb{\bm{K}}\\\
\boldsymbol{\mathbb{T}}&0\end{pmatrix}\binom{\vec{p}}{\vec{q}}=\omega\>\binom{\vec{p}}{\vec{q}},$
with $\boldsymbol{\mathbb{K}=\mathbb{A}+\mathbb{B}}$ and
$\boldsymbol{\mathbb{T}=\mathbb{A}-\mathbb{B}}$ the Hermitian potential and
kinetic matrices, and the dual
$\left\\{\vec{p},\vec{q\,}\right\\}=\left\\{\vec{X}-\vec{Y},\vec{\,X}+\vec{Y}\right\\}$
corresponding to position and momentum. This picture avoids the imbalance
$\left\|\vec{X}\right\|\gg\left\|\vec{Y}\right\|$ whilst admitting
conventional solutions based on the Hermitian matrix
$\mathbb{G}=\mathbb{K}\cdot\mathbb{T},$ as shown by Tamara and Udagawa Tamura
and Udagawa (1964) and extended by Narita and Shibuya with second order
optimization of the quotient
$\omega^{2}\left[\vec{p},\vec{q}\right]=\vec{q}\cdot\mathbb{G}\cdot\vec{p}/\left|\vec{p}\cdot\vec{q}\right|$
Narita and Shibuya (1992). More recently, Tsiper considered the quotients
${\normalcolor\omega\left[\vec{p},\vec{q}\right]}=\frac{\vec{p}\,\cdotp\boldsymbol{\mathbb{K}}\,\cdotp\vec{p}}{2\,\left|\vec{p}\cdotp\vec{q}\right|}+\frac{q\,\cdotp\boldsymbol{\mathbb{T}}\,\cdotp\vec{q}\>}{2\,\left|\vec{p}\cdotp\vec{q}\right|}\;,$
(2)
and developed a corresponding dual channel Lanczos solver. Subspace solvers in
this dual representation have recently been surveyed by Tretiak, Isborne,
Niklasson and Challacombe (TINC) Tretiak _et al._ (2009), with comparative
results for semi-empirical models.
Another challenge is dimensionality and scaling. Writing Eq. (1) in the
general form $\mathbf{\mathbb{L}}\cdot\vec{v}=\omega\vec{\,v}$, admitting
arbitrary representation, the superoperator matrix $\mathbb{L}$ is a $\sim
N^{2}\times N^{2}$ tetradic, with _N_ the number of basis functions, assumed
proportional to system size. In practice the action of
$\mathbb{\mathbf{\mathbb{L}}}$ onto $\vec{v}$ is carried out implicitly as
$\boldsymbol{L}[\boldsymbol{v}]=[\bm{F},\bm{\,v}]+[\bm{G}[\boldsymbol{v}],\,\bm{P}]\,,$
using an existing framework for construction of the effective Hamiltonian
(Fockian) $\boldsymbol{F}$, where $\bm{P}$ is the one-particle reduced density
matrix, $\bm{G}$ is a screening operator involving Coulomb, exchange and/or
exchange-correlation terms and the correspondence between superoperator and
functional notation is given by a tensorial mapping between diadic and matrix,
$\vec{v}_{{\scriptscriptstyle N^{2}\times 1}}{\bf\Leftrightarrow
v}_{{\scriptscriptstyle N\times N}}$.
Recent efforts have focused on addressing the problem of dimensionality by
employing linear scaling methods that reduce the cost of
$\boldsymbol{L}[\cdot]$ within Density Functional Theory (DFT) to ${\cal
O}(N)$. However, this remains an open problem for the Hartree-Fock (HF)
exchange, an ingredient in models that account for charge transfer in the
dynamic and static response, including the Random Phase Approximation (RPA) at
the pure HF level of theory. Likewise, scaling of the TD-SCF eigenproblem
remains formidable due to associated costs of linear algebra, even when using
powerful Krylov subspace methods. Underscoring this challenge, one of the most
successful approaches to linear scaling TD-DFT avoids the matrix eigenproblem
entirely through explicit time-evolution Yokojima and Chen (1999); Yam _et
al._ (2003).
Linear scaling matrix methods exploit quantum locality, manifest in
approximate exponential decay of matrix elements expressed in a well posed,
local basis; with the dropping of small elements below a threshold,
$\tau_{\mathrm{mtx}}$, this decay leads to sparse matrices and
$\mathcal{O}(N)$ complexity at the forfeit of full precision Challacombe
(1999, 2000); Niklasson _et al._ (2003). Likewise, linear scaling methods for
computing the HF exchange employ an advanced form of direct SCF, exploiting
this decay in the rigorous screening of small exchange interactions bellow the
two-electron integral threshold $\tau_{\mathrm{2e}}$ Schwegler _et al._
(1997). The consequence of these linear scaling approximations is an inexact
linear algebra that challenges Krylov solvers due to nested error
accumulation, a subject of recent formal interest Simoncini (2005); Simoncini
and Szyld (2007). Consistent with this view, TINC found that matrix
perturbation (a truncation proxy) disrupts convergence of Krylov solvers with
slow convergence, _i.e._ Lanczos and Arnoldi for the RPA, but has less impact
on solvers with rapid convergence, _i.e._ generically for the TDA or Davidson
for the RPA. Relative to semi-empirical Hamiltonians, the impact of
incompleteness on subspace iteration may be amplified with first principles
models and large basis sets (ill-conditioning).
An alternative is Rayleigh Quotient Iteration (RQI), which poses the
eigenproblem as non-linear optimization and is variational with respect to
matrix perturbation. Narita and Shibuya Narita and Shibuya (1992) considered
optimization of the quotient $\omega^{2}\left[\vec{p},\vec{q}\right]$ with
second order methods, but these are beyond the capabilities of current linear
scaling technologies and also, convergence is disadvantaged by a power of
$\nicefrac{{1}}{{2}}$. For semi-empirical Hamiltonians, TINC found that
optimization of the Thouless functional
$\omega[\vec{v}]=\vec{v}\cdot\mathbb{L}\cdot\vec{v}/\left|\vec{v}\cdotp\vec{v}\right|,$
corresponding to the solution of Eq. (1), was significantly slower for the RPA
relative to the TDA, and also compared to subspace solvers. For first
principles models and non-trivial basis sets, this naive RQI can become
pathologically slow as shown in Fig. 1. On the other hand, the Tsiper
formulation exposes the underlying pseudo-Hermitian structure of the TD-SCF
equations. Here, this structure is exploited with QUasi-Independent Rayleigh
Quotient Iteration (QUIRQI), involving dual channel optimization of the Tsiper
quotients coupled only weakly through line search.
Figure 1: Convergence of RHF/3-21G TDA and RPA with the RQI and QUIRQI
algorithms for linear decaene (C10H2). Calculations were started from the same
random guess, and tight numerical thresholds were used throughout. In the
representation independent scheme, the cost per iteration is the same for TDA
and RPA.
Our development begins with a brief review of the representation independent
formulation developed by TINC, which avoids the $\mathcal{O}(N^{3})$ cost of
rotating into an explicit _p-h_ , _h-p_ symmetry. Instead, this symmetry is
maintained implicitly via annihilation, $\boldsymbol{x}\leftarrow
f_{a}(\boldsymbol{x})\boldsymbol{=P}\cdotp\boldsymbol{x}\cdotp\boldsymbol{Q}+\boldsymbol{Q}\cdotp\boldsymbol{x}\cdotp\boldsymbol{P}$,
with $\boldsymbol{P}$ the first order reduced density matrix and
$\boldsymbol{Q=I-P}$ its compliment. Likewise, the indefinite metric
associated with the $J$-symmetry of Eq. (1) is carried through the generalized
norm $\left\langle\boldsymbol{x},\boldsymbol{y}\right\rangle={\rm
tr}$$\left\\{\boldsymbol{x}^{{\scriptscriptstyle
T}}\cdotp\left[\boldsymbol{y},\boldsymbol{P}\right]\right\\}$. Introducing the
operator equivalents,
$\boldsymbol{L}[\boldsymbol{p}]\Leftrightarrow\mathbb{K}.\vec{p}$ and
$\boldsymbol{L}[\boldsymbol{q}]\Leftrightarrow\mathbb{T}.\vec{q}$ , the Tsiper
functional becomes
$\omega\left[\boldsymbol{p},\boldsymbol{q}\right]=\frac{\left\langle\boldsymbol{p},\boldsymbol{L\left[p\right]}\right\rangle}{2\left|\left\langle\boldsymbol{p},\boldsymbol{q}\right\rangle\right|}+\frac{\left\langle\boldsymbol{q},\boldsymbol{L}[\boldsymbol{q}]\right\rangle}{2\left|\left\langle\boldsymbol{p},\boldsymbol{q}\right\rangle\right|}.$
Transformations between the transition density and the dual space involves
simple manipulations and minimal cost, allowing Fock builds with the
transition density and optimization in the dual space. The splitting operation
is given by
$\boldsymbol{p}=f_{+}(\boldsymbol{v})=\boldsymbol{P}\cdot\boldsymbol{v}\cdot\boldsymbol{Q}+\left[\boldsymbol{Q}\cdot\boldsymbol{v}\cdot\boldsymbol{P}\right]^{{\scriptscriptstyle
T}}$ and
$\boldsymbol{q}=f_{-}(\boldsymbol{v})=\boldsymbol{P}\cdot\boldsymbol{v}\cdot\boldsymbol{Q}-\left[\boldsymbol{Q}\cdot\boldsymbol{v}\cdot\boldsymbol{P}\right]^{{\scriptscriptstyle
T}}$. Likewise,
$\boldsymbol{L}[\boldsymbol{p}]=f_{-}\left(\boldsymbol{L}[\boldsymbol{v}]\right)$
and
$\boldsymbol{L}[\boldsymbol{q}]=f_{+}\left(\boldsymbol{L}[\boldsymbol{v}]\right)$.
The back transformation (merge) from dual to density is
$\boldsymbol{v}=F(\boldsymbol{p},\boldsymbol{q})=\left(\boldsymbol{p}+\boldsymbol{q}+\left[\boldsymbol{p-q}\right]^{{\scriptscriptstyle
T}}\right)/2$. This framework provides the freedom to work in any orthogonal
representation, and to switch between transition density and oscillator duals
with minimal cost.
QUIRQI is given in Algorithm 1. It begins with a guess for the transition
density, which is then split into its dual (lines 2-3). The choice of initial
guess is discussed later. Lines 4-24 consist of the non-linear conjugate
gradient optimization of the nearly independent channels: In each step, the
flow of information proceeds from optimization of the duals to builds
involving the density and back to the duals in a merge-annihilate-truncate-
build-split-truncate (MATBST) sequence. For the variables $\boldsymbol{v}$,
$\boldsymbol{p}$ and $\boldsymbol{q}$ this sequence is comprised by lines
22-23 and 5-7, and lines 15-19 for the corresponding conjugate gradients
$\boldsymbol{h}_{v}$, $\boldsymbol{h}_{p}$ and $\boldsymbol{h}_{q}$.
Truncation is carried out with the filter operation as described in Ref.
Niklasson _et al._ (2003), with cost and error determined by the matrix
threshold $\tau_{\mathrm{mtx}}$.
The Tsiper functional is the sum of dual quotients $\omega_{p}$ and
$\omega_{q}$, determined at line 8, followed by the gradients
$\boldsymbol{g}_{p}$ and $\boldsymbol{g}_{q}$ computed at line 9. After the
first cycle, the corresponding relative error $e_{\mathrm{rel}}$ (10) and
maximum matrix element of the gradient $g_{\mathrm{max}}$ (11) are computed
and used as an exit criterion at line 4, along with non-variational behavior
$\omega>\omega^{\mathrm{old}}.$
Next, the Polak-Ribiere variant of non-linear conjugate gradients yields the
search direction in each channel, $\boldsymbol{h}_{p}$ and
$\boldsymbol{h}_{q}$ (12-14). The action of $\boldsymbol{L}[\cdot]$ on to
$\boldsymbol{h}_{p}$ and $\boldsymbol{h}_{q}$ is then computed, again with a
MATBST sequence (15-19), followed by a self-consistent dual channel line
search at line 20, as described below. With steps $\lambda_{p}$ and
$\lambda_{q}$ in hand, minimizing updates are taken along each conjugate
direction (22), and the cycle repeats with the MATBST sequence spanning lines
21-23 and 5-7.
Algorithm 1 QUIRQI
1:procedure QUIRQI($\omega,\boldsymbol{v}$)
2: guess $\boldsymbol{v}$
3: $\boldsymbol{p}=f_{+}\left(\boldsymbol{v}\right)$,
$\boldsymbol{q}=f_{-}\left(\boldsymbol{v}\right)$
4: while $e_{\mathrm{rel}}>\epsilon$ and $g_{\mathrm{max}}>\gamma$ and
$\omega<\omega^{\mathrm{old}}$ do
5:
$\boldsymbol{L}[\boldsymbol{v}]=[\bm{F},\bm{\,v}]+[\bm{G}[\boldsymbol{v}],\,\bm{P}]$
6:
$\boldsymbol{L}[\boldsymbol{p}]=f_{-}\left(\boldsymbol{L}[\boldsymbol{v}]\right)$,
$\boldsymbol{L}[\boldsymbol{q}]=f_{+}\left(\boldsymbol{L}[\boldsymbol{v}]\right)$
7:
$\mathtt{\mathtt{\mathrm{\mathtt{filter}}}}\left(\boldsymbol{\boldsymbol{L}[\boldsymbol{p}]},\boldsymbol{\,L}[\boldsymbol{q}],\,\tau_{\mathrm{mtx}}\right)$
8:
$\omega_{p}=\frac{\left\langle\mathbf{p},\boldsymbol{L}[\boldsymbol{p}]\right\rangle}{2\left|\left\langle\boldsymbol{p},\boldsymbol{q}\right\rangle\right|}$,
$\omega_{q}=\frac{\left\langle\mathbf{q},\boldsymbol{L}[\boldsymbol{q}]\right\rangle}{2\left|\left\langle\boldsymbol{p},\boldsymbol{q}\right\rangle\right|}$,
$\omega=\omega_{p}+\omega_{q}$
9:
$\boldsymbol{g}_{p}=\boldsymbol{q\,}\omega_{q}-\boldsymbol{L}[\boldsymbol{p}]$,
$\boldsymbol{g}_{q}=\boldsymbol{p\,}\omega_{p}-\boldsymbol{L}[\boldsymbol{q}]$
10: $e_{\mathrm{rel}}=\left(\omega^{\mathrm{old}}-\omega\right)/\omega$
11:
$g_{\mathrm{max}}=\underset{i,j}{\operatorname{max}}\left\\{\left[\boldsymbol{g}_{p}\right]_{ij},\left[\boldsymbol{g}_{p}\right]_{ij}\right\\}$
12:
$\beta_{p}=\frac{\left\langle\boldsymbol{g}_{p},\,\boldsymbol{g}_{p}-\boldsymbol{g}_{p}^{{\rm
old}}\right\rangle}{\left\langle\boldsymbol{g}_{p}^{{\rm
old}},\boldsymbol{g}_{p}^{{\rm old}}\right\rangle}$,
$\beta_{q}=\frac{\left\langle\boldsymbol{g}_{q},\,\boldsymbol{g}_{q}-\boldsymbol{g}_{q}^{{\rm
old}}\right\rangle}{\left\langle\boldsymbol{g}_{q}^{{\rm
old}},\boldsymbol{g}_{q}^{{\rm old}}\right\rangle}$
13: $\omega^{old}\leftarrow\omega$,
$\boldsymbol{g}_{p}^{old}\leftarrow\boldsymbol{g}_{p}$ ,
$\boldsymbol{g}_{q}^{old}\leftarrow\boldsymbol{g}_{q}$
14:
$\boldsymbol{h}_{p}\leftarrow\boldsymbol{g}_{p}+\beta_{p}\boldsymbol{h}_{p}$,
$\boldsymbol{h}_{q}\leftarrow\boldsymbol{g}_{q}+\beta_{q}\boldsymbol{h}_{q}$
15: $\boldsymbol{h}_{v}=F(\boldsymbol{h}_{p},\boldsymbol{h}_{q})$,
$\boldsymbol{h}_{v}\leftarrow f_{a}\left(\boldsymbol{h}_{v}\right)$
16:
$\mathrm{\mathtt{filter}}\left(\boldsymbol{h}_{p},\boldsymbol{h}_{q},\boldsymbol{h}_{v},\,\tau_{\mathrm{mtx}}\right)$
17:
$\boldsymbol{L}\left[\boldsymbol{h}_{v}\right]=[\bm{F},\,\boldsymbol{h}_{v}]+[\bm{G}[\boldsymbol{h}_{v}],\,\bm{P}]$
18:
$\boldsymbol{L}[\boldsymbol{h}_{p}]=f_{-}\left(\boldsymbol{L}[\boldsymbol{h}_{v}]\right)$,
$\boldsymbol{L}[\boldsymbol{h}_{q}]=f_{+}\left(\boldsymbol{L}[\boldsymbol{h}_{v}]\right)$
19:
$\mathrm{\mathtt{filter}}\left(\boldsymbol{L}[\boldsymbol{h}_{p}],\,\boldsymbol{L}[\boldsymbol{h}_{q}],\,\tau_{\mathrm{mtx}}\right)$
20:
$\left\\{\lambda_{p},\lambda_{q}\right\\}=\underset{\left\\{\lambda_{p},\lambda_{q}\right\\}}{\operatorname{argmin}}\;\omega\left[\boldsymbol{p}+\lambda_{p}\boldsymbol{h}_{p},\boldsymbol{q}+\lambda_{q}\boldsymbol{h}_{q}\right]$
21: $\boldsymbol{p}\leftarrow\boldsymbol{p}+\lambda_{p}\boldsymbol{h}_{p}$,
$\boldsymbol{q}\leftarrow\boldsymbol{q}+\lambda_{q}\boldsymbol{h}_{q}$
22: $\boldsymbol{v}\leftarrow F\left(\boldsymbol{p},\boldsymbol{q}\right)$,
$\boldsymbol{v}\leftarrow f_{a}\left(\boldsymbol{v}\right)$
23:
$\mathrm{\mathtt{filter}}\left(\boldsymbol{p},\boldsymbol{q},\boldsymbol{v},\,\tau_{\mathrm{mtx}}\right)$
24: end while
25:end procedure
Optimization of the Tsipper functional
$\omega\left[\lambda_{p},\lambda_{q}\right]\equiv\omega\left[\boldsymbol{p}+\lambda_{p}\mathbf{h}_{p},\boldsymbol{q}+\lambda_{q}\boldsymbol{h}_{q}\right]$
involves a two dimensional line-search (line 20) corresponding to minimization
of
$\omega\left[\lambda_{p},\lambda_{q}\right]=\frac{A_{p}+\lambda_{p}B_{p}+\lambda_{p}^{2}C_{p}+A_{q}+\lambda_{q}B_{q}+\lambda_{q}^{2}C_{q}}{R_{pq}+\lambda_{p}S_{pq}+\lambda_{q}T_{pq}+\lambda_{p}\lambda_{q}U_{pq}}\>,$
(3)
with coupling entering through terms in the denominator such as
$U_{pq}=\left\langle\boldsymbol{h}_{p},\boldsymbol{h}_{q}\right\rangle$. A
minimum in Eq. (3) can be found quickly to high precision by alternately
substituting one-dimensional solutions one into the other until self-
consistency is reached. This semi-analytic approach starts with a rough guess
at the pair $\left\\{\lambda_{p},\lambda_{q}\right\\}$ (eg. found by a coarse
scan) followed by iterative substitution, where for example the $p$-channel
update is
$\displaystyle\lambda_{p}\leftarrow\left\\{\left[\left(2C_{p}R_{pq}+2C_{p}\lambda_{q}S_{pq}\right)^{2}-4\left(C_{p}T_{pq}+C_{p}\lambda_{q}U_{pq}\right)\right.\right.$
$\displaystyle\left[B_{p}R_{pq}+B_{p}\lambda_{q}S_{pq}-\left(A_{q}+A_{p}+B_{q}\lambda_{q}+C_{q}\lambda_{q}^{2}\right)T_{pq}\right.$
$\displaystyle\left.\vphantom{\left[\left(2C_{q}R_{pq}+2C_{q}\lambda_{p}S_{pq}\right)^{2}-4\left(C_{q}T_{pq}+C_{q}\lambda_{p}U_{pq}\right)\right.}\left.-\left(A_{q}\lambda_{q}+A_{q}\lambda_{q}+B_{q}\lambda_{q}^{2}+C_{q}\lambda_{q}^{3}\right)U_{pq}\right]\right]^{1/2}\qquad$
(4)
$\displaystyle\left.\vphantom{\left\\{\left[\left(2C_{q}R_{pq}+2C_{q}\lambda_{p}S_{pq}\right)^{2}-4\left(C_{q}T_{pq}+C_{q}\lambda_{p}U_{pq}\right)\right.\right.}-2C_{p}R_{pq}-2C_{p}\lambda_{q}S_{pq}\right\\}\big{/}\left[2C_{p}\left(T_{pq}+\lambda_{q}U_{pq}\right)\right]\,,\qquad$
with an analogous update for the $q$-channel obtained by swapping subscripts.
As the solution decouples ($S_{pq},$ $T_{pq}$ and $U_{pq}$ become small) the
steps are found independently.
QUIRQI has been implemented in FreeON Bock _et al._ (2009), which employs the
linear scaling Coulomb and Hartree-Fock exchange kernels QCTC and ONX with
cost and accuracy controlled by the two-electron screening threshold
$\tau_{\mathrm{2e}}$ Schwegler _et al._ (1997). _N_ -scaling solution of the
QUIRQI matrix equations is achieved with the sparse approximate matrix-matrix
multiply (SpAMM), with cost and accuracy determined by the drop tolerance
$\tau_{\mathrm{mtx}}$ Challacombe (1999, 2000); Niklasson _et al._ (2003). All
calculations were carried out with version 4.3 of the gcc/gfortran compiler
under version 8.04 of the Ubuntu Linux distribution and run on a fully loaded
2GHz AMD Quad Opteron 8350.
For systems studied to date, QUIRQI is found to converge monotonically with
rates comparable to the TDA as shown in Fig. 1. Based on the comparative
performance presented by TINC, the TDA rate of convergence appears to be a
lower bound for RPA solvers. In addition to the convergence rate, performance
is strongly determined by the initial guess. The following results have been
obtained using the polarization response density along the polymer axis
Tretiak _et al._ (2009), which can be computed in $\mathcal{O}(N)$ by
Perturbed Projection Weber _et al._ (2004). Also, a relative precision of 4
digits in the excitation energy is targeted with the convergence parameters
$\epsilon=10^{-4}$ and $\gamma=10^{-3}$, with exit from the optimization loop
on violation of monotonic convergence ($\omega>\omega^{\mathrm{old}}$ due to
precision limitations associated with linear scaling approximations).
In Fig. 2, linear scaling and convergence to the bulk limit are demonstrated
for a series of polyphenylene vinylene (PPV) oligomers at the RHF/6-31G**
level of theory for the threshold combinations
$\left\\{\tau{}_{\mathrm{mtx}},\tau_{\mathrm{2e}}\right\\}=\left\\{10^{-4},10^{-5}\right\\}$
and $\left\\{10^{-5},10^{-6}\right\\}$. Significantly more conservative
thresholds have been used for the Coulomb sums, which incur only minor cost.
Convergence is reached in $24-25$ iterations, with the cost of Coulomb
summation via QCTC comparable to the cost of
SpAMM($\tau_{\mathrm{mtx}}=10^{-4}$). In Fig. 3, linear scaling and
convergence to the bulk limit are demonstrated for a series of (4,3) carbon
nanotube segments at the RHF/3-21G level of theory for the same threshold
combinations, again with convergence achieved in about 24-25 cycles. In both
cases, tightening the pair
$\left\\{\tau_{\mathrm{mtx}},\tau_{\mathrm{2e}}\right\\}$ leads to a
systematically improved result. While the $\left\\{10^{-4},10^{-5}\right\\}$
thresholds that work well for PPV lead to a non-monotone behavior with respect
to extent for the nanotube series, dropping one more decade to
$\left\\{10^{-5},10^{-6}\right\\}$ leads to a sharply improved behavior.
Dropping thresholds further to $\left\\{10^{-6},10^{-7}\right\\}$ yields
identical results to within the convergence criteria ($\sim$ four digits)
across the series, also scaling with $N$ but at roughly twice the cost.
These results demonstrate that QUIRQI can achieve both systematic error
control and linear scaling in solution of the RPA eigenproblem for systems
with extended conjugation. Relative to PPV, the greater numerical sensitivity
encountered with the nanotube series is consistent with the ground state
problem, where a smaller band gap and greater atomic connectivity typically
demand tighter thresholds.
QUIRQI exploits decoupling of the Tsipper functional into nearly independent,
pseudo-Hermitian quotients leading to aggressive convergence rates equivalent
to the fully Hermitian TDA, while remaining variational with respect to matrix
truncation ($\tau_{\mathrm{mtx}}$). However, QUIRQI is not variational with
respect to the screening parameter $\tau_{\mathrm{2e}}$. It can be
systematically improved by tightening $\tau_{\mathrm{2e}}$ though, due to
rigorous error bounds based on the Schwartz inequality Schwegler _et al._
(1997). These properties present opportunities for more precise error control
as suggested by Rubensson, Rudberg and Salek Rubensson _et al._ (2008).
Further, these properties are expected to hold even for the most general SCF
models, with the only difference being an increasingly localized transition
density matrix and larger cost prefactor with an increasing DFT component.
Finally, a variational approach allows considerable flexibility in the path to
solution, as errors due to approximation can be overcome by optimization,
offering opportunities for single precision GPU acceleration, variable
thresholding, incremental Fock builds as well as extrapolation techniques.
Figure 2: Approach to the bulk limit of the PPV first excited state at the
6-31G**/RPA level of theory, with inset showing linear scaling cost for HF
exchange (ONX) and sparse linear algebra (SpAMM). The cost of Coulomb sums
with much tighter thresholds are comparable to those for the SpAMM. Figure 3:
Approach to the bulk limit of the first excited state of the (4,3) carbon
nanotube segment at the 3-21G/RPA level of theory, with inset showing linear
scaling cost for HF exchange (ONX), sparse linear algebra (SpAMM) and Coulomb
sums (QCTC).
This work was supported by the U.S. Department of Energy and Los Alamos LDRD
funds. Los Alamos National Laboratory was operated by the Los Alamos National
Security, LLC, for the National Nuclear Security Administration of the U.S.
Department of Energy under Contract No. DE-AC52-06NA25396. Special
acknowledgments go the International Ten Bar Café for scientific vlibations,
and to Sergei Tretiak and Nicolas Bock for valuable input.
## References
* Autschbach (2009) Autschbach, J., 2009, Chem. Phys. Chem. 10, 1757.
* Bock _et al._ (2009) Bock, N., M. Challacombe, C. K. Gan, G. Henkelman, K. Nemeth, A. M. N. Niklasson, A. Odell, E. Schwegler, C. J. Tymczak, and V. Weber, 2009, FreeON ( Sep-18-2009 ), Los Alamos National Laboratory (LA-CC-04-086, Copyright University of California)., URL http://www.nongnu.org/freeon/.
* Challacombe (1999) Challacombe, M., 1999, J. Chem. Phys. 110, 2332.
* Challacombe (2000) Challacombe, M., 2000, Comp. Phys. Comm. 128, 93.
* Furche (2008) Furche, F., 2008, J. Chem. Phys. 129(11), 114105\.
* Harl and Kresse (2009) Harl, J., and G. Kresse, 2009, Phys. Rev. Lett. 103(5), 056401.
* Igumenshchev _et al._ (2007) Igumenshchev, K. I., S. Tretiak, and V. Y. Chernyak, 2007, J. Chem. Phys. 127(11), 114902\.
* Lu _et al._ (2009) Lu, D., Y. Li, D. Rocca, and G. Galli, 2009, Phys. Rev. Lett. 102(20), 206411\.
* Magyar and Tretiak (2007) Magyar, R. J., and S. Tretiak, 2007, J. Chem. Theor. Comp. 3, 976.
* Narita and Shibuya (1992) Narita, S., and T. Shibuya, 1992, Can. J. Chem. 70, 296.
* Niklasson _et al._ (2003) Niklasson, A. M. N., C. J. Tymczak, and M. Challacombe, 2003, J. Chem. Phys. 118, 8611.
* Rubensson _et al._ (2008) Rubensson, E. H., E. Rudberg, and P. Salek, 2008, J. Math. Phys. 49(3), 032103.
* Schwegler _et al._ (1997) Schwegler, E., M. Challacombe, and M. Head-Gordon, 1997, J. Chem. Phys. 106, 9708.
* Simoncini (2005) Simoncini, V., 2005, SIAM J. Num. Anal. 43(3), 1155 .
* Simoncini and Szyld (2007) Simoncini, V., and D. B. Szyld, 2007, Num. Lin. Alg. App. 14(1), 1 .
* Song _et al._ (2009) Song, J.-W., M. A. Watson, and K. Hirao, 2009, J. Chem. Phys. 131(14), 144108\.
* Tamura and Udagawa (1964) Tamura, T., and T. Udagawa, 1964, Nuc. Phys. 53, 33.
* Toulouse _et al._ (2009) Toulouse, J., I. C. Gerber, G. Jansen, A. Savin, and J. G. Ángyán, 2009, Phys. Rev. Lett. 102(9), 096404.
* Tretiak _et al._ (2009) Tretiak, S., C. M. Isborn, A. M. N. Niklasson, and M. Challacombe, 2009, J. Chem. Phys. 130(5), 054111.
* Weber _et al._ (2004) Weber, V., A. M. N. Niklasson, and M. Challacombe, 2004, Phys. Rev. Lett. 92, 193002.
* Yam _et al._ (2003) Yam, C. Y., S. Yokojima, and G. Chen, 2003, J. Chem. Phys. 119(17), 8794.
* Yokojima and Chen (1999) Yokojima, S., and G. H. Chen, 1999, Chem. Phys. Lett. 300(5-6), 540.
|
arxiv-papers
| 2010-01-15T00:13:36 |
2024-09-04T02:49:07.774612
|
{
"license": "Public Domain",
"authors": "Matt Challacombe",
"submitter": "Matt Challacombe",
"url": "https://arxiv.org/abs/1001.2586"
}
|
1001.2667
|
# Isometric group of $(\alpha,\beta)$-type Finsler space and the symmetry of
Very Special Relativity
Xin Li1,4 lixin@itp.ac.cn Zhe Chang2,4 zchang@ihep.ac.cn Xiaohuan Mo3
moxh@pku.edu.cn 1Institute of Theoretical Physics, Chinese Academy of
Sciences, 100190 Beijing, China
2Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
3Key Laboratory of Pure and Applied Mathematics School of Mathematical
Sciences, Peking University, Beijing 100871, China
4Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
###### Abstract
The Killing equation for a general Finsler space is set up. It is showed that
the Killing equation of $(\alpha,\beta)$ space can be divided into two parts.
One is the same with Killing equation of a Riemannian metric, another equation
can be regarded as a constraint. The solutions of Killing equations present
explicitly the isometric symmetry of Finsler space. We find that the isometric
group of a special case of $(\alpha,\beta)$ space is the same with the
symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk
space are given. Unlike Riemannian constant curvature space, the 4 dimensional
Funk space with constant curvature just have 6 independent Killing vectors.
###### pacs:
02.40.-k,11.30.-j
## I Introduction
In the past few years, two interesting theories of investigating the violation
of Lorentz Invariance (LI) are proposed. One is the so called Doubly Special
Relativity (DSR) Amelino1 ; Amelino2 ; Amelino3 ; Smolin1 ; Smolin2 . This
theory takes Planck-scale effects into account by introducing an invariant
Planckian parameter in the theory of special relativity. Another is the so
called Very Special Relativity (VSR) developed by Cohen and Glashow Glashow .
This theory suggested that the exact symmetry group of nature may be
isomorphic to a subgroup SIM(2) of the Poincare group. And the SIM(2) group
semi-direct product with the spacetime translation group gives an
8-dimensional subgroup of the Poincare group called ISIM(2) Kogut . Under the
symmetry of ISIM(2), the CPT symmetry is preserved and many empirical
successes of special relativity are still functioned.
Recently, Physicists found that the two theories mentioned above are related
with Finsler geometry. Girelli, Liberati and Sindoni Girelli showed that the
Modified dispersion relation (MDR) in DSR can be incorporated into the
framework of Finsler geometry. The symmetry of the MDR was described in the
Hamiltonian formalism. Also, Gibbons, Gomis and Pope Gibbons showed that the
Finslerian line element
$ds=(\eta_{\mu\nu}dx^{\mu}dx^{\nu})^{(1-b)/2}(n_{\rho}dx^{\rho})^{b}$ is
invariant under the transformations of the group DISIM${}_{b}(2)$ (1-parameter
family of deformations of ISIM(2)).
Finsler geometry as a natural generation of Riemann geometry could provide new
sight on modern physics. the model based on Finsler geometry could explain the
recent astronomical observations which Einstein’s gravity could not. An
incomplete list includes: the flat rotation curves of spiral galaxies can be
deduced naturally without invoking dark matter Finsler DM ; the anomalous
accelerationAnderson in solar system observed by Pioneer 10 and 11
spacecrafts could account for a special Finsler space-Randers space Finsler PA
; the secular trend in the astronomical unitKrasinsky ; Standish and the
anomalous secular eccentricity variation of the Moon’s orbitWilliams could
account for effect of the length change of unit circle in Finsler
geometryFinsler AU .
Thus, the symmetry of Finslerian spacetime is worth investigating. The way of
describing spacetime symmetry in a covariant language (the symmetry should not
depend on any particular choice of coordinate system) involves the concept of
isometric transformation. In fact, the symmetry of spacetime is described by
the so called isometric group. The generators of isometric group is directly
connected with the Killing vectorsKilling . In this paper, we use solutions of
the Killing equation to establish the symmetry of a class of Finslerian
spacetime. In particular, we show that the isometric group of a special kind
of $(\alpha,\beta)$ space is equivalent to the symmetry of the VSR.
## II Killing vector in Riemann space
In this section, we give a brief review of the Killing vectors in Riemann
space (further material can be found, for example, in Weinberg ). Under a
given coordinate transformation $x\rightarrow\bar{x}$, the Riemannian metric
$g_{\mu\nu}(x)$ transforms as
$\bar{g}_{\mu\nu}(\bar{x})=\frac{\partial
x^{\rho}}{\partial\bar{x}^{\mu}}\frac{\partial
x^{\sigma}}{\partial\bar{x}^{\nu}}g_{\rho\sigma}(x).$ (1)
Any transformation $x\rightarrow\bar{x}$ is called isometry if and only if the
transformation of the metric $g_{\mu\nu}(x)$ satisfies
$g_{\mu\nu}(\bar{x})=\frac{\partial
x^{\rho}}{\partial\bar{x}^{\mu}}\frac{\partial
x^{\sigma}}{\partial\bar{x}^{\nu}}g_{\rho\sigma}(x).$ (2)
One can check that the isometric transformations do form a group. It is
convenient to investigate the isometric transformation under the infinitesimal
coordinate transformation
$\bar{x}^{\mu}=x^{\mu}+\epsilon V^{\mu},$ (3)
where $|\epsilon|\ll 1$. To first order in $|\epsilon|$, the equation (2)
reads
$V^{\kappa}\frac{\partial g_{\mu\nu}}{\partial
x^{\kappa}}+g_{\kappa\mu}\frac{\partial V^{\kappa}}{\partial
x^{\nu}}+g_{\kappa\nu}\frac{\partial V^{\kappa}}{\partial x^{\mu}}=0.$ (4)
By making use of the covariant derivatives with respect to Riemannian
connection, we can write the above equation as
$V_{\mu|\nu}+V_{\nu|\mu}=0,$ (5)
where $``|"$ denotes the covariant derivative. Any vector field $V_{\mu}$
satisfies equation (5) is called Killing vector. Thus, the problem of finding
all isometries of a given metric $g_{\mu\nu}(x)$ is reduced to find the
dimension of the linear space formed by Killing vectors.
In Riemann geometry, by making use of the covariant derivative, one could
obtain the Ricci identities or interchange formula
$V_{\rho|\mu|\nu}-V_{\rho|\nu|\mu}=-V_{\sigma}R^{~{}\sigma}_{\rho~{}\nu\mu},$
(6)
where $R^{~{}\sigma}_{\rho~{}\nu\mu}$ is the Riemannian curvature tensor. And
the first Bianchi identity for the Riemannian curvature tensor gives
$R^{~{}\sigma}_{\rho~{}\nu\mu}+R^{~{}\sigma}_{\nu~{}\mu\rho}+R^{~{}\sigma}_{\mu~{}\rho\nu}=0.$
(7)
Deducing from the equation (6) and (7), we obtain
$V_{\rho|\mu|\nu}=V_{\sigma}R^{~{}\sigma}_{\nu~{}\mu\rho}.$ (8)
Thus, all the derivatives of $V_{\mu}$ will be determined by the linear
combinations of $V_{\mu}$ and $V_{\mu|\nu}$. Once the $V_{\mu}$ and
$V_{\mu|\nu}$ at an arbitrary point of Riemannian space is given, then
$V_{\mu}$ and $V_{\mu|\nu}$ at any other point is determined by integration of
the system of ordinary differential equations. Therefore, the dimension of
linear space formed by Killing vector can be at most $\frac{n(n+1)}{2}$ in $n$
dimensional Riemannian space. If a metric admits that the maximum number
$\frac{n(n+1)}{2}$ of Killing vectors, its Riemann space must homogeneous and
isotropic (or the space is isotropic for every point). Such space is called
maximally symmetry space. In Riemann geometry, the Schur’s lemma tells us that
a Riemannian space with at least 3 dimension is maximally symmetry space if
and only if its sectional curvature is constant. Also, one can check that a 2
dimensional Riemannian space is maximally symmetry space if and only if its
sectional curvature is constant. Thus, the maximal symmetry of a given metric
is an intrinsic property, and not depending on the choice of coordinate
system.
One special maximally symmetry space is the Minkowskian space. The Killing
equation (5) of a given Minkowskian metric $\eta_{\mu\nu}(x)$ reduces to
$\frac{\partial V_{\mu}}{\partial x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}=0.$ (9)
The solution of the above equation is
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu},$ (10)
where $Q_{\mu\nu}=\eta_{\rho\mu}Q^{\rho}_{~{}\nu}$ is an arbitrary constant
skew-symmetric matrix and $C^{\mu}$ is an arbitrary constant vector. Thus,
substituting the solution (10) into the coordinate transformation (3) we
obtain
$\bar{x}^{\mu}=(\delta^{\mu}_{\nu}+\epsilon Q^{\mu}_{~{}\nu})x^{\nu}+\epsilon
C^{\mu}.$ (11)
One should find that the term $\delta^{\mu}_{\nu}+\epsilon Q^{\mu}_{~{}\nu}$
in above equation is just the Lorentz transformation matrix and the term
$\epsilon C^{\mu}$ is related to the spacetime translation. Expanding the
matrix $\delta^{\mu}_{\nu}+\epsilon Q^{\mu}_{~{}\nu}$ and the vector $\epsilon
C^{\mu}$ near identity, one could obtain the famous Poincare algebra.
Other two types of maximally symmetry space are spherical and hyperbolic case.
Without loss of generality, we set its constant sectional curvature to be $\pm
1$ for spherical and hyperbolic case respectively. The length element of
spherical and hyperbolic case is given in a unified form
$ds^{2}=\frac{\sqrt{(1+k(x\cdot x))(dx\cdot dx)-k(x\cdot dx)^{2}}}{1+k(x\cdot
x)},$ (12)
where the $\cdot$ denotes the inner product with respect to Minkowskian metric
and $k=\pm 1$ for spherical and hyperbolic case respectively. The metric is
given as
$g_{\mu\nu}=\left(\frac{\eta_{\mu\nu}}{1+k(x\cdot
x)}-k\frac{x_{\mu}x_{\nu}}{(1+k(x\cdot x))^{2}}\right),$ (13)
where $x_{\mu}\equiv\eta_{\mu\nu}x^{\nu}$. The Christoffel symbols of the
above length element is given as
$\gamma^{\rho}_{\mu\nu}=-k\frac{x_{\mu}\delta^{\rho}_{\nu}+x_{\nu}\delta^{\rho}_{\mu}}{1+k(x\cdot
x)}.$ (14)
Thus, the Killing equation (5) now reads
$\frac{\partial V_{\mu}}{\partial x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}+\frac{2k}{1+k(x\cdot x)}(x_{\mu}V_{\nu}+X_{\nu}V_{\mu})=0.$ (15)
The solution of the above equation is
$V^{\mu}\equiv g^{\mu\nu}V_{\nu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}+k(x\cdot
C)x^{\mu},$ (16)
where the index of $Q$ and $C$ are raise and lower by Minkowskian metric
$\eta^{\mu\nu}$ and its matrix reverse $\eta_{\mu\nu}$.
## III Killing vectors in Finsler space
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is base on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given asBook by Bao
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (17)
Finsler geometry has its genesis in integrals of the form
$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(18)
So the Finsler structure represents the length element of Finsler space.
Like Riemannian case, to investigate the Killing vector we should construct
the isometric transformation of Finsler structure. Let us consider the
coordinate transformation (3) together with the corresponding transformation
for $y$
$\bar{y}^{\mu}=y^{\mu}+\epsilon\frac{\partial V^{\mu}}{\partial
x^{\nu}}y^{\nu}.$ (19)
Under the coordinate transformation (3) and (19), to first order in
$|\epsilon|$, we obtain the expansion of the Finsler structure,
$\bar{F}(\bar{x},\bar{y})=\bar{F}(x,y)+\epsilon V^{\mu}\frac{\partial
F}{\partial x^{\mu}}+\epsilon y^{\nu}\frac{\partial V^{\mu}}{\partial
x^{\nu}}\frac{\partial F}{\partial y^{\mu}},$ (20)
where $\bar{F}(\bar{x},\bar{y})$ should equal $F(x,y)$. Under the
transformation (3) and (19), a Finsler structure is called isometry if and
only if
$F(x,y)=\bar{F}(x,y).$ (21)
Then, deducing from the (20) we obtain the Killing equation $K_{V}(F)$ in
Finsler space
$K_{V}(F)\equiv V^{\mu}\frac{\partial F}{\partial
x^{\mu}}+y^{\nu}\frac{\partial V^{\mu}}{\partial x^{\nu}}\frac{\partial
F}{\partial y^{\mu}}=0.$ (22)
Searching the Killing vectors for general Finsler structure is difficult.
Here, we give the Killing vectors for a class of Finsler
space-$(\alpha,\beta)$ spaceShen with metric defining as
$\displaystyle F=\alpha\phi(s),~{}~{}~{}s=\frac{\beta}{\alpha},$ (23)
$\displaystyle\alpha=\sqrt{g_{\mu\nu}y^{\mu}y^{\nu}}~{}~{}{\rm
and}~{}~{}\beta=b_{\mu}(x)y^{\mu},$ (24)
where $\phi(s)$ is a smooth function, $\alpha$ is a Riemannian metric and
$\beta$ is one form. Then, the Killing equation (22) in $(\alpha,\beta)$ space
reads
$\displaystyle 0$ $\displaystyle=$ $\displaystyle K_{V}(\alpha)\phi(s)+\alpha
K_{V}(\phi(s))$ (25) $\displaystyle=$
$\displaystyle\left(\phi(s)-s\frac{\partial\phi(s)}{\partial
s}\right)K_{V}(\alpha)+\frac{\partial\phi(s)}{\partial s}K_{V}(\beta).$
And by making use of the Killing equation (22), we obtain
$\displaystyle K_{V}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}(V_{\mu|\nu}+V_{\nu|\mu})y^{\mu}y^{\nu},$ (26)
$\displaystyle K_{V}(\beta)$ $\displaystyle=$
$\displaystyle\left(V^{\mu}\frac{\partial b_{\nu}}{\partial
x^{\mu}}+b_{\mu}\frac{\partial V^{\mu}}{\partial x^{\nu}}\right)y^{\nu},$ (27)
where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. The solutions of the Killing equation (22) have three viable
scenarios. The first one is
$\phi(s)-s\frac{\partial\phi(s)}{\partial s}=0~{}~{}{\rm
and}~{}~{}K_{V}(\beta)=0,$ (28)
which implies $F=\lambda\beta$ for all $\lambda\in\mathbb{R}$. The second one
is
$\frac{\partial\phi(s)}{\partial s}=0~{}~{}{\rm and}~{}~{}K_{V}(\alpha)=0,$
(29)
which implies $F=\lambda\alpha$ for all $\lambda\in\mathbb{R}$. The above two
scenarios are just trivial space, we will not consider in this section. Here
we mainly consider the case that $\phi(s)-s\frac{\partial\phi(s)}{\partial
s}\neq 0$ and $\frac{\partial\phi(s)}{\partial s}\neq 0$. This will induce the
last scenario. Apparently, in the last scenario we have such solutions
$\displaystyle V_{\mu|\nu}+V_{\nu|\mu}$ $\displaystyle=$ $\displaystyle 0,$
(30) $\displaystyle V^{\mu}\frac{\partial b_{\nu}}{\partial
x^{\mu}}+b_{\mu}\frac{\partial V^{\mu}}{\partial x^{\nu}}$ $\displaystyle=$
$\displaystyle 0,$ (31)
while $a_{0}\neq 0$. The first equation (30) is no other than the Riemannian
Killing equation (5). The second equation (31) can be regarded as the
constraint for the Killing vectors that satisfy the Killing equation (30).
Therefore, in general the dimension of the linear space formed by Killing
vectors of $(\alpha,\beta)$ space is lower than the Riemannian one. Additional
solutions of (22) should be obtained from (22), it will be discussed in next
section.
## IV Symmetry of VSR
One important physical example of $(\alpha,\beta)$ space is VSR. While we take
$\phi(s)=s^{m}$ and $m$ is an arbitrary constant, the Finsler structure takes
the form proposed by Gibbons et al.Gibbons
$\displaystyle F$ $\displaystyle=$ $\displaystyle\alpha^{1-m}\beta^{m}$ (32)
$\displaystyle=$
$\displaystyle(\eta_{\mu\nu}y^{\mu}y^{\nu})^{(1-m)/2}(b_{\rho}y^{\rho})^{m}.$
We denote it by VSR metric. In this example, $\eta_{\mu\nu}$ is Minkowskian
metric and $b_{\rho}$ is a constant vector. One immediately obtain from the
first Killing equation (30) of $(\alpha,\beta)$ space,
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}.$ (33)
And the second Killing equation (31) gives the constraint for Killing vector
$V^{\mu}$,
$b_{\mu}Q^{\mu}_{~{}\nu}=0.$ (34)
Taking the light cone coordinate Kogut $\alpha=\sqrt{2y^{+}y^{-}-y^{i}y^{i}}$
(with i ranging over the values 1 and 2) and supposing
$b_{\mu}=\\{0,0,0,b_{-}\\}$, we know that in general $Q^{-}_{~{}\mu}\neq 0$.
It means the Killing vectors of VSR metric (32) do not have components
$Q_{+-}$ and $Q_{+i}$. In another word, the generators of the group that
leaves the metric (32) invariant are just four spacetime translation
generators $(P_{+},P_{i},P_{-})$ and three subgenerators of Lorentz algebra
$(M_{-i},M_{ij})$. This result implies that the VSR metric is invariant under
the E(2) group (the group of two-dimensional Euclidean motion).
The above investigation and the Killing equations (30) and (31) obtained in
section 3 are under the premise that the direction of $y^{\mu}$ is arbitrary.
It means that no preferred direction exists in spacetime. If the spacetime
does have a special direction, the Killing equation (25) will have a special
solution. The VSR metric is first suggested by Bogoslovsky Bogoslovsky . He
assumed that the spacetime has a preferred direction. Following the assumption
and taking the null direction to be the preferred direction, we deduce from
Killing equation (25) that
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
s^{m}\left(\frac{1-n}{2\alpha}\left(\frac{\partial V_{\mu}}{\partial
x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}\right)y^{\mu}y^{\nu}+ms^{-1}b_{\rho}\frac{\partial V^{\rho}}{\partial
x^{\kappa}}y^{\kappa}\right)$ (35) $\displaystyle=$ $\displaystyle
s^{m}\frac{1}{\alpha\beta}\Bigg{(}\frac{1-n}{2}\left(\frac{\partial
V_{\mu}}{\partial x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}\right)b_{\kappa}$ $\displaystyle+m\eta_{\mu\nu}b^{\rho}\frac{\partial
V_{\rho}}{\partial x^{\kappa}}\Bigg{)}y^{\mu}y^{\nu}y^{\kappa}.$
The above equations has a special solution
$V_{+}=(Q_{+-}+m\eta_{+-})x^{-}+C_{+},$ (36)
where $Q_{+-}$ is not only an antisymmetrical matrix, but also satisfying the
property
$b_{-}=-b^{+}Q_{+-}.$ (37)
It implies that the Lorentz transformation for $b^{+}$ is
$\left(\delta^{+}_{+}+\epsilon(m\delta^{+}_{+}+Q^{+}_{~{}+})\right)b^{+}=\left(1+\epsilon(n+1)\right)b^{+},$
(38)
which means the null direction $b^{+}$ (or $b_{-}$) is invariant under the
Lorentz transformation. Therefore, if the spacetime has a preferred direction
in null direction, the symmetry corresponded to $Q_{+-}$ is restored. In such
case the VSR metric is invariant under the the transformations of the group
DISIM${}_{b}(2)$ proposed by Gibbons et al.Gibbons .
Another important physical example of $(\alpha,\beta)$ space is Randers
spaceRanders . While we set $\phi(s)=1+s$, the Finsler structure takes the
form
$F=\alpha+\beta.$ (39)
Then, in Randers space the Killing equation (25) reads
$K_{V}(\alpha)+K_{V}(\beta)=0.$ (40)
Since the $K_{V}(\alpha)$ contains irrational term of $y^{\mu}$ and
$K_{V}(\beta)$ only contains rational term of $y^{\mu}$, therefore the
equation (40) satisfies if and only if $K_{V}(\alpha)=0$ and $K_{V}(\beta)=0$.
If Randers space is flat, its Killing vectors satisfy the same Killing
equations with VSR metric. It implies the flat Randers metric is invariant
under the E(2) group semi-direct product with the spacetime translation. Thus,
let us focus on the non-flat Randers space. In section 2, we know that a
Riemannian space is maximally symmetry space if and only if it is a constant
sectional curvature space. Due to the first Killing equation (30) of
Finslerian space is the same with the Riemannian one and the Killing vectors
of Finslerian space must satisfy the constraint (31), we know that a Finsler
space with constant flag curvature (the notion of flag curvature is the
counterpart of sectional curvature) has less independent Killing vectors than
its Riemannian counterpart, in general.
Here, we just investigate a special Randers space with constant flag curvature
$-\frac{1}{4}$. It is called Funk metricFunk . This metric space can be
regarded as a flat Minkowskian space which influenced by a radial “wind”
(vector)Gibbons1 . For Randers space with other types of constant flag
curvature, its $\alpha$ term is not a Riemannian constant sectional curvature
space. Then, its independent Killing vectors should less than Funk one in
general. The Funk metric is given by
$F=\frac{\sqrt{(1-x\cdot x)(y\cdot y)+(x\cdot y)^{2}}}{1-x\cdot
x}+\frac{x\cdot y}{1-x\cdot x},$ (41)
where the $\cdot$ denotes the inner product with respect to Minkowskian
metric. By making use of the result (16) obtained in section 2, we know that
the first Killing equation $K_{V}(\alpha)=0$ (30) implies
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}-(x\cdot C)x^{\mu}.$ (42)
And the Funk metric implies $b_{\mu}(x)=\frac{x_{\mu}}{1-x\cdot x}$. Then, we
obtain the partial derivative for $b_{\mu}(x)$,
$\frac{\partial b_{\nu}}{\partial x^{\mu}}=\frac{\eta_{\mu\nu}}{1-x\cdot
x}+\frac{2x_{\mu}x_{\nu}}{(1-x\cdot x)^{2}},$ (43)
and the partial derivative for Killing vectors $V^{\mu}$,
$\frac{\partial V^{\mu}}{\partial
x^{\nu}}=Q^{\mu}_{~{}\nu}-\delta^{\mu}_{\nu}(x\cdot C)-x^{\mu}C_{\nu}.$ (44)
By making use of the equation (42), (43) and (44), and deducing from the
second Killing equation (31) we obtain
$\frac{2Q_{\nu\mu}x^{\mu}}{1-x\cdot x}+C_{\nu}=0.$ (45)
Substituting the equation (45) into (42), we get
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}\left(\frac{3-x\cdot x}{1-x\cdot x}\right).$
(46)
Therefore, the dimension of the linear space formed by the Killing vectors of
Funk metric is $6$. And the spacetime translation generators corresponded to
$C^{\mu}$ depend on the generators of Lorentz group corresponded to
$Q^{\mu}_{~{}\nu}$.
## V Conclusion
In the past few years, two interesting theories which investigated the
violation of Lorentz Invariance (LI) are proposed. We showed that the Killing
vectors satisfy the same Killing equation (30) of a Riemannian metric, and the
major difference is the Killing vectors of $(\alpha,\beta)$ metric need to
satisfy the constraint (31). We proved that the isometric group of a flat
$(\alpha,\beta)$ space is just the symmetry in VSR proposed by Cohen and
Glashow. If the space do not have a preferred direction, we showed that the
generators induced by such Killing vectors are just isomorphic to the E(2)
group semidirect the spacetime translation. While a preferred direction has
chosen, it is showed that the symmetry of such space isomorphic to the SIM(2)
group semidirect the spacetime translation. For non flat case, we considered
the Funk metric with constant flag curvature. It is showed that the numbers of
independent Killing vectors of Funk metric is just $6$. The determination for
the maximal number of independent Killing vectors of $(\alpha,\beta)$ space
(or general Finsler space) still is a open problem. We hope it could be solved
in the future.
###### Acknowledgements.
We would like to thank Prof. C. J. Zhu for useful discussions. The work was
supported by the NSF of China under Grant No. 10525522, 10875129 and 10771004.
## References
* (1) G. Amelino-Camelia, Phys. Lett. B 510, 255 (2001).
* (2) G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002).
* (3) G. Amelino-Camelia, Nature 418, 34 (2002).
* (4) J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
* (5) J. Magueijo and L. Smolin, Phys. Rev. D 67, 044017 (2003).
* (6) A. G. Cohen and S.L. Glashow, Phys. Rev. Lett. 97, 021601 (2006).
* (7) J. B. Kogut and D. E. Soper, Phys. Rev. D 1, 2901 (1970).
* (8) F. Girelli, S. Liberati and L. Sindoni, Phys. Rev. D 75, 064015 (2007).
* (9) G. W. Gibbons, J. Gomis and C. N. Pope, Phys. Rev. D 76, 081701 (2007).
* (10) Z. Chang and X. Li, Phys. Lett. B 668, 453 (2008).
* (11) J. D. Anderson et al., Phys. Rev. Lett. 81 2858, (1998); Phys. Rev. D 65 082004, (2002); Mod. Phys. Lett. A 17 875, (2002).
* (12) X. Li and Z. Chang, arXiv:gr-qc/0909.3713.
* (13) G. A. Krasinsky and V. A. Brumberg, Celest. Mech. Dyn. Astrn. 90, 267 (2004).
* (14) E. M. Standish, Proc. IAU Colloq. 196, 163 (2005).
* (15) J. G. Williams and D. H. Boggs, in Proceedings of 16th International Workshop on Laser Ranging ed. S. Schillak, (Space Research Centre, Polish Academy of Sciences), 2009.
* (16) X. Li and Z. Chang, arXiv:gr-qc/0911.1890.
* (17) W. Killing, J. f. d. reine u. angew. Math. (Crelle), 109, 121 (1892).
* (18) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, 1972.
* (19) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathmatics 200, Springer, New York, 2000.
* (20) Z. Shen, Some perspectives in Finsler geometry, MSRI Publication Series. Cambridge: Cambridge university press, 2004.
* (21) G. Bogoslovsky, arXiv:gr-qc/0706.2621.
* (22) G. Randers, Phys. Rev. 59, 195 (1941).
* (23) P. Funk, Math. Ann. 101, 226 (1929).
* (24) G. W. Gibbons, et al., Phys. Rev. D 79, 044022 (2009).
|
arxiv-papers
| 2010-01-15T11:42:57 |
2024-09-04T02:49:07.782554
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li, Zhe Chang and Xiaohuan Mo",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1001.2667"
}
|
1001.2806
|
# MIMO Gaussian Broadcast Channels with Confidential and Common Messages
Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz) This research
was supported by the United States National Science Foundation under Grant
CNS-09-05398, CCF-08-45848 and CCF-09-16867, by the Air Force Office of
Scientific Research under Grant FA9550-08-1-0480, by the European Commission
in the framework of the FP7 Network of Excellence in Wireless Communications
NEWCOM++, and by the Israel Science Foundation.Ruoheng Liu and H. Vincent Poor
are with the Department of Electrical Engineering, Princeton University,
Princeton, NJ 08544, USA (e-mail: {rliu,poor}@princeton.edu).Tie Liu is with
the Department of Electrical and Computer Engineering, Texas A&M University,
College Station, TX 77843, USA (e-mail: tieliu@tamu.edu).Shlomo Shamai (Shitz)
is with the Department of Electrical Engineering, Technion-Israel Institute of
Technology, Technion City, Haifa 32000, Israel (e-mail:
sshlomo@ee.technion.ac.il).
###### Abstract
This paper considers the problem of secret communication over a two-receiver
multiple-input multiple-output (MIMO) Gaussian broadcast channel. The
transmitter has two independent, confidential messages and a common message.
Each of the confidential messages is intended for one of the receivers but
needs to be kept perfectly secret from the other, and the common message is
intended for both receivers. It is shown that a natural scheme that combines
secret dirty-paper coding with Gaussian superposition coding achieves the
secrecy capacity region. To prove this result, a channel-enhancement approach
and an extremal entropy inequality of Weingarten et al. are used.
## I Introduction
In this paper, we study the problem of secret communication over a multiple-
input multiple-output (MIMO) Gaussian broadcast channel with two receivers.
The transmitter is equipped with $t$ transmit antennas, and receiver $k$,
$k=1,2$, is equipped with $r_{k}$ receive antennas. A discrete-time sample of
the channel at time $m$ can be written as
$\mathbf{Y}_{k}[m]=\mathbf{H}_{k}\mathbf{X}[m]+\mathbf{Z}_{k}[m],\quad k=1,2$
(1)
where $\mathbf{H}_{k}$ is the (real) channel matrix of size $r_{k}\times t$,
and $\\{\mathbf{Z}_{k}[m]\\}_{m}$ is an independent and identically
distributed (i.i.d.) additive vector Gaussian noise process with zero mean and
identity covariance matrix. The channel input $\\{\mathbf{X}[m]\\}_{m}$ is
subject to the matrix power constraint:
$\frac{1}{n}\sum_{m=1}^{n}\left(\mathbf{X}[m]\mathbf{X}^{\intercal}[m]\right)\preceq\mathbf{S}$
(2)
where $\mathbf{S}$ is a positive semidefinite matrix, and “$\preceq$” denotes
“less than or equal to” in the positive semidefinite partial ordering between
real symmetric matrices. Note that (2) is a rather general power constraint
that subsumes many other important power constraints including the average
total and per-antenna power constraints as special cases.
Figure 1: Channel model
As shown in Fig. 1, we consider the communication scenario in which there is a
common message $W_{0}$ and two independent, confidential messages $W_{1}$ and
$W_{2}$ at the transmitter. Message $W_{0}$ is intended for both receivers.
Message $W_{1}$ is intended for receiver 1 but needs to be kept secret from
receiver 2, and message $W_{2}$ is intended for receiver 1 but needs to be
kept secret from receiver 2. The confidentiality of the messages at the
unintended receivers is measured using the normalized information-theoretic
criteria [1]
$\displaystyle\frac{1}{n}I(W_{1};\mathbf{Y}_{2}^{n})\rightarrow
0\quad\mbox{and}\quad\frac{1}{n}I(W_{2};\mathbf{Y}_{1}^{n})\rightarrow 0$ (3)
where $\mathbf{Y}_{k}^{n}:=(\mathbf{Y}_{k}[1],\ldots,\mathbf{Y}_{k}[n])$,
$k=1,2$, and the limits are taken as the block length $n\rightarrow\infty$.
The goal is to characterize the _entire_ secrecy rate region
${\mathcal{C}}_{s}^{\rm[SBC]}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})=\\{(R_{0},R_{1},R_{2})\\}$
that can be achieved by any coding scheme, where $R_{0}$, $R_{1}$ and $R_{2}$
are the communication rates corresponding to the common message $W_{0}$, the
confidential message $W_{1}$ destined for receiver 1 and the confidential
message $W_{2}$ destined for receiver 2, respectively.
In recent years, MIMO secret communication has been an active area of
research. Several _special_ cases of the communication problem that we
consider here have been studied in the literature. Specifically,
* •
With only one confidential message ($W_{1}$ or $W_{2}$), the problem was
studied as the MIMO Gaussian wiretap channel. The secrecy capacity of the MIMO
Gaussian wiretap channel was characterized in [2] and [3] under the matrix
power constraint (2) and in [4] and [5] under an average total power
constraint.
* •
With both confidential messages $W_{1}$ and $W_{2}$ but _without_ the common
message $W_{0}$, the problem was studied in [6] for the multiple-input single-
output (MISO) case and in [7] for general MIMO case. Rather surprisingly, it
was shown in [7] that, under the matrix power constraint (2), both
confidential messages can be _simultaneously_ communicated at their respected
maximum secrecy rates.
* •
With only one confidential message ($W_{1}$ or $W_{2}$) _and_ the common
message $W_{0}$, the secrecy capacity region of the channel was characterized
in [8] using a channel-enhancement approach [9] and an extremal entropy
inequality of Weingarten et al. [10].
The main contribution of this paper is to provide a precise characterization
of the secrecy capacity region of the MIMO Gaussian broadcast channel with a
more complete message set that includes a common message $W_{0}$ and two
independent, confidential messages $W_{1}$ and $W_{2}$ by generalizing the
channel-enhancement argument of [8].
## II Main Result
The main result of the paper is summarized in the following theorem.
###### Theorem 1 (General MIMO Gaussian broadcast channel)
The secrecy capacity region of the MIMO Gaussian broadcast channel (1) with a
common message $W_{0}$ (intended for both receivers) and confidential messages
$W_{1}$ (intended for receiver 1 but needing to be kept secret from receiver
2) and $W_{2}$ (intended for receiver 2 but needing to be kept secret from
receiver 1) under the matrix power constraint (2) is given by the set of
nonnegative rate triples $(R_{0},R_{1},R_{2})$ such that
$\displaystyle R_{0}$
$\displaystyle\leq\min\left\\{\frac{1}{2}\log\left|\frac{\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}+\mathbf{I}_{r1}}{\mathbf{H}_{1}(\mathbf{S}-\mathbf{B}_{0})\mathbf{H}_{1}^{\intercal}+\mathbf{I}_{r1}}\right|,\right.$
$\displaystyle~{}\qquad\left.\frac{1}{2}\log\left|\frac{\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}+\mathbf{I}_{r2}}{\mathbf{H}_{2}(\mathbf{S}-\mathbf{B}_{0})\mathbf{H}_{2}^{\intercal}+\mathbf{I}_{r2}}\right|\right\\}$
$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}_{1}\mathbf{H}_{1}^{\intercal}\right|$
$\displaystyle~{}\qquad-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}_{1}\mathbf{H}_{2}^{\intercal}\right|$
$\displaystyle\text{and}\qquad R_{2}$
$\displaystyle\leq\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}(\mathbf{S}-\mathbf{B}_{0})\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}_{1}\mathbf{H}_{2}^{\intercal}}\right|$
$\displaystyle~{}\qquad-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}(\mathbf{S}-\mathbf{B}_{0})\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}_{1}\mathbf{H}_{1}^{\intercal}}\right|$
(4)
for some $\mathbf{B}_{0}\succeq 0$, $\mathbf{B}_{1}\succeq 0$ and
$\mathbf{B}_{0}+\mathbf{B}_{1}\preceq\mathbf{S}$. Here, $\mathbf{I}_{r_{k}}$
denotes the identity matrix of size $r_{k}\times r_{k}$ for $k=1,2$.
###### Remark 1
Note that for any given $\mathbf{B}_{0}$, the upper bounds on $R_{1}$ and
$R_{2}$ can be simultaneously maximized by a same $\mathbf{B}_{1}$. In fact,
the upper bounds on $R_{1}$ and $R_{2}$ are fully symmetric with respect to
$\mathbf{H}_{1}$ and $\mathbf{H}_{2}$, even though it is not immediately
evident from the expressions themselves.
To prove Theorem 1, we shall follow [9] and first consider the _canonical_
aligned case. In an _aligned_ MIMO Gaussian broadcast channel [9], the channel
matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ are square and invertible.
Multiplying both sides of (1) by $\mathbf{H}_{k}^{-1}$, the channel can be
equivalently written as
$\mathbf{Y}_{k}[m]=\mathbf{X}_{k}[m]+\mathbf{Z}_{k}[m],\quad k=1,2$ (5)
where $\\{\mathbf{Z}_{k}[m]\\}_{m}$ is an i.i.d. additive vector Gaussian
noise process with zero mean and covariance matrix
$\mathbf{N}_{k}=\mathbf{H}_{k}^{-1}\mathbf{H}_{k}^{-\intercal}$, $k=1,2.$ The
secrecy capacity region of the aligned MIMO Gaussian broadcast channel is
summarized in the following theorem.
###### Theorem 2 (Aligned MIMO Gaussian broadcast channel)
The secrecy capacity region
${\mathcal{C}}_{s}^{\rm[SBC]}(\mathbf{N}_{1},\mathbf{N}_{2},\mathbf{S})$ of
the aligned MIMO Gaussian broadcast channel (5) with a common message $W_{0}$
and confidential messages $W_{1}$ and $W_{2}$ under the matrix power
constraint (2) is given by the set of nonnegative rate triples
$(R_{0},R_{1},R_{2})$ such that
$\displaystyle R_{0}$
$\displaystyle\leq\min\left\\{\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{1}}\right|,\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{2}}\right|\right\\}$
$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{2}\log\left|\frac{\mathbf{B}_{1}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}_{1}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|$
$\displaystyle R_{2}$
$\displaystyle\leq\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{2}}{\mathbf{B}_{1}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{1}}{\mathbf{B}_{1}+\mathbf{N}_{1}}\right|$
(6)
for some $\mathbf{B}_{0}\succeq 0$, $\mathbf{B}_{1}\succeq 0$ and
$\mathbf{B}_{0}+\mathbf{B}_{1}\preceq\mathbf{S}$.
Next, we prove Theorem 2 by generalizing the channel-enhancement argument of
[8]. Extension from the aligned case (6) to the general case (4) follows from
the standard limiting argument [9]; the details are deferred to the extended
version of this work [11].
## III Proof of Theorem 2
### III-A Achievability
The problem of a two-receiver discrete memoryless broadcast channel with a
common message and two confidential common messages was studied in [12], where
an achievable secrecy rate region was given by the set of rate triples
$(R_{0},R_{1},R_{2})$ such that
$\displaystyle R_{0}$
$\displaystyle\leq\min[I(\mathbf{U}_{0};\mathbf{Y}_{1}),~{}I(\mathbf{U}_{0},\mathbf{Y}_{2})]$
$\displaystyle R_{1}$ $\displaystyle\leq
I(\mathbf{V}_{1};\mathbf{Y}_{1}|\mathbf{U}_{0})-I(\mathbf{V}_{1};\mathbf{V}_{2},\mathbf{Y}_{2}|\mathbf{U}_{0})$
$\displaystyle\text{and}\qquad R_{2}$ $\displaystyle\leq
I(\mathbf{V}_{2};\mathbf{Y}_{2}|\mathbf{U}_{0})-I(\mathbf{V}_{2};\mathbf{V}_{1},\mathbf{Y}_{1}|\mathbf{U}_{0})$
(7)
where $\mathbf{U}_{0}$, $\mathbf{V}_{1}$ and $\mathbf{V}_{2}$ are auxiliary
random variables such that
$(\mathbf{U}_{0},\mathbf{V}_{1},\mathbf{V}_{2})\rightarrow\mathbf{X}\rightarrow(\mathbf{Y}_{1},\mathbf{Y}_{2})$
forms a Markov chain. The scheme to achieve this secrecy rate region is a
natural combination of secret dirty-paper coding and superposition coding.
Thus, the achievability of the secrecy rate region (6) follows from that of
(7) by setting
$\displaystyle\mathbf{V}_{1}$
$\displaystyle=\mathbf{U}_{1}+\mathbf{F}\mathbf{U}_{2}$
$\displaystyle\mathbf{V}_{2}$ $\displaystyle=\mathbf{U}_{2}$
$\displaystyle\mbox{and}\quad\mathbf{X}$
$\displaystyle=\mathbf{U}_{0}+\mathbf{U}_{1}+\mathbf{U}_{2}$
where $\mathbf{U}_{0}$, $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are three
independent Gaussian vectors with zero means and covariance matrices
$\mathbf{B}_{0}$, $\mathbf{B}_{1}$ and
$\mathbf{S}-\mathbf{B}_{0}-\mathbf{B}_{1}$, respectively, and
$\mathbf{F}:=\mathbf{B}\mathbf{H}_{1}^{\intercal}(\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal})^{-1}\mathbf{H}_{1}.$
The details of the proof are deferred to the extended version of this work
[11].
### III-B The converse
Next, we prove the converse part of Theorem 2 assuming that $\mathbf{S}\succ
0$. The case where $\mathbf{S}\succeq 0$, $|\mathbf{S}|=0$ can be found in the
extended version of this work [11].
Let
$\displaystyle f_{0}(\mathbf{B}_{0})$
$\displaystyle=\min\left\\{\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{1}}\right|,\right.$
$\displaystyle~{}\quad\left.\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{2}}\right|\right\\}$
$\displaystyle f_{1}(\mathbf{B}_{1})$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{B}_{1}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}_{1}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|$
$\displaystyle\text{and}\qquad f_{2}(\mathbf{B}_{0},\mathbf{B}_{1})$
$\displaystyle=\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{2}}{\mathbf{B}_{1}+\mathbf{N}_{2}}\right|$
$\displaystyle~{}\quad-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0})+\mathbf{N}_{1}}{\mathbf{B}_{1}+\mathbf{N}_{1}}\right|.$
(8)
Then, the secrecy rate region (6) can be rewritten as
$\displaystyle{\mathcal{R}}_{in}:=\bigcup_{\mathbf{B}_{0}\succeq
0,\mathbf{B}_{1}\succeq
0,\mathbf{B}_{0}+\mathbf{B}_{1}\preceq\mathbf{S}}\bigl{\\{}(R_{0},R_{1},R_{2})\bigl{|}$
$\displaystyle\begin{array}[]{l}R_{0}\leq f_{0}(\mathbf{B}_{0}),~{}R_{1}\leq
f_{1}(\mathbf{B}_{1}),~{}R_{2}\leq f_{2}(\mathbf{B}_{0},\mathbf{B}_{1})\\\
\end{array}\bigr{\\}}.$ (10)
To show that ${\mathcal{R}}_{in}$ is indeed the secrecy capacity region of the
aligned MIMO Gaussian broadcast channel (5), we will consider proof by
contradiction. Assume that
$(R_{0}^{{\dagger}},R_{1}^{{\dagger}},R_{2}^{{\dagger}})$ is an achievable
secrecy rate triple that lies _outside_ the region ${\mathcal{R}}_{in}$. Since
$(R_{0}^{{\dagger}},R_{1}^{{\dagger}},R_{2}^{{\dagger}})$ is achievable, we
can bound $R_{0}^{{\dagger}}$ by
$\displaystyle R_{0}^{{\dagger}}$
$\displaystyle\leq\min\left(\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|,\;\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right)=R_{0}^{\rm
max}.$
Moreover, if $R_{1}^{{\dagger}}=R_{2}^{{\dagger}}=0$, then $R_{0}^{\rm max}$
can be achieved by setting $\mathbf{B}_{0}=\mathbf{S}$ and $\mathbf{B}_{1}=0$
in (6). Thus, we can find $\lambda_{1}\geq 0$ and $\lambda_{2}\geq 0$ such
that
$\displaystyle\lambda_{1}R_{1}^{{\dagger}}+\lambda_{2}R_{2}^{{\dagger}}=\lambda_{1}R_{1}^{\star}+\lambda_{2}R_{2}^{\star}+\rho$
(11)
for some $\rho>0$, where $\lambda_{1}R_{1}^{\star}+\lambda_{2}R_{2}^{\star}$
is given by
$\displaystyle\max_{(\mathbf{B}_{0},\;\mathbf{B}_{1})}\qquad$
$\displaystyle\lambda_{1}f_{1}(\mathbf{B}_{1})+\lambda_{2}f_{2}(\mathbf{B}_{0},\mathbf{B}_{1})$
$\displaystyle\text{subject to}\qquad f_{0}(\mathbf{B}_{0})$
$\displaystyle\geq R_{0}^{{\dagger}}$ $\displaystyle\mathbf{B}_{0}$
$\displaystyle\succeq 0$ $\displaystyle\mathbf{B}_{1}$ $\displaystyle\succeq
0$ $\displaystyle\mathbf{B}_{0}+\mathbf{B}_{1}$
$\displaystyle\preceq\mathbf{S}.$ (12)
Let $(\mathbf{B}_{0}^{\star},\mathbf{B}_{1}^{\star})$ be an optimal solution
to the above optimization program (12). Then,
$(\mathbf{B}_{0}^{\star},\mathbf{B}_{1}^{\star})$ must satisfy the following
Karush-Kuhn-Tucker (KKT) conditions:
$\displaystyle(\beta_{1}+\lambda_{2})[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}]^{-1}+\beta_{2}[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}]^{-1}+\mathbf{M}_{0}$
$\displaystyle\qquad\qquad\qquad=\lambda_{2}[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}]^{-1}+\mathbf{M}_{2}$
(13)
$\displaystyle\quad(\lambda_{1}+\lambda_{2})(\mathbf{B}_{1}^{\star}+\mathbf{N}_{1})^{-1}+\mathbf{M}_{1}$
$\displaystyle\qquad\qquad\qquad=(\lambda_{1}+\lambda_{2})(\mathbf{B}_{1}^{\star}+\mathbf{N}_{2})^{-1}+\mathbf{M}_{2}$
(14)
$\displaystyle\mathbf{M}_{0}\mathbf{B}_{0}^{\star}=0,~{}\mathbf{M}_{1}\mathbf{B}_{1}^{\star}=0,~{}\text{and}~{}\mathbf{M}_{2}(\mathbf{S}-\mathbf{B}_{0}^{\star}-\mathbf{B}_{1}^{\star})=0$
(15)
where $\mathbf{M}_{0}$, $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ are positive
semidefinite matrices, and $\beta_{k}$, $k=1,2$, is a nonnegative real scalar
such that $\beta_{k}>0$ if and only if
$\displaystyle\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{k}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{k}}\right|=R_{0}^{{\dagger}}.$
Hence, we have
$\displaystyle(\beta_{1}$
$\displaystyle+\beta_{2})R_{0}^{{\dagger}}+\lambda_{1}R_{1}^{{\dagger}}+\lambda_{2}R_{2}^{{\dagger}}$
$\displaystyle=\frac{\beta_{1}}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}}\right|+\frac{\beta_{2}}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}\right|$
$\displaystyle\quad+\lambda_{1}\left(\frac{1}{2}\log\left|\frac{\mathbf{B}_{1}^{\star}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}_{1}^{\star}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right)$
$\displaystyle\quad+\lambda_{2}\left(\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}{\mathbf{B}_{1}^{\star}+\mathbf{N}_{2}}\right|\right.$
$\displaystyle\qquad\left.-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}}{\mathbf{B}_{1}^{\star}+\mathbf{N}_{1}}\right|\right)+\rho.$
(16)
Next, we shall find a contradiction to (16) by following the following three
steps.
#### III-B1 Step 1–Split Each Receiver into Two Virtual Receivers
Consider the following aligned MIMO Gaussian broadcast channel with four
receivers:
$\displaystyle\mathbf{Y}_{1a}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{1a}[m]$
$\displaystyle\mathbf{Y}_{1b}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{1b}[m]$
$\displaystyle\mathbf{Y}_{2a}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{2a}[m]$
$\displaystyle\text{and}\qquad\mathbf{Y}_{2b}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{2b}[m]$ (17)
where $\\{\mathbf{Z}_{1a}[m]\\}$, $\\{\mathbf{Z}_{1b}[m]\\}$,
$\\{\mathbf{Z}_{2a}[m]\\}$ and $\\{\mathbf{Z}_{2b}[m]\\}$ are i.i.d. additive
vector Gaussian noise processes with zero means and covariance matrices
$\mathbf{N}_{1}$, $\mathbf{N}_{1}$, $\mathbf{N}_{2}$ and $\mathbf{N}_{2}$,
respectively. Suppose that the transmitter has three independent messages
$W_{0}$, $W_{1}$ and $W_{2}$, where $W_{0}$ is intended for both receivers
$1b$ and $2b$, $W_{1}$ is intended for receiver $1a$ but needs to be kept
secret from receiver $2b$, and $W_{2}$ is intended for receiver $2a$ but needs
to be kept secret from receiver $1b$. Note that the channel (17) has the same
secrecy capacity region as the channel (5) under the same power constraints.
#### III-B2 Step 2–Construct an Enhanced Channel
Let $\widetilde{\mathbf{N}}$ be a real symmetric matrix satisfying
$\displaystyle\widetilde{\mathbf{N}}$
$\displaystyle:=\left(\mathbf{N}_{1}^{-1}+\frac{1}{\lambda_{1}+\lambda_{2}}\mathbf{M}_{1}\right)^{-1}.$
(18)
Note that the above definition implies that
$\widetilde{\mathbf{N}}\preceq\mathbf{N}_{1}.$ Since
$\mathbf{M}_{1}\mathbf{B}_{1}^{\star}=0$, following [9, Lemma 11], we have
$\displaystyle(\lambda_{1}+\lambda_{2})(\mathbf{B}_{1}^{\star}+\widetilde{\mathbf{N}})^{-1}$
$\displaystyle=(\lambda_{1}+\lambda_{2})(\mathbf{B}_{1}^{\star}+\mathbf{N}_{1})^{-1}+\mathbf{M}_{1}$
and
$\displaystyle|\mathbf{B}_{1}^{\star}+\widetilde{\mathbf{N}}||{\mathbf{N}_{1}}|$
$\displaystyle=\left|{\mathbf{B}_{1}^{\star}+\mathbf{N}_{1}}\right||{\widetilde{\mathbf{N}}}|.$
(19)
Thus, by (14), we obtain
$\displaystyle(\lambda_{1}+\lambda_{2})(\mathbf{B}_{1}^{\star}+\widetilde{\mathbf{N}})^{-1}$
$\displaystyle=(\lambda_{1}+\lambda_{2})(\mathbf{B}_{1}^{\star}+\mathbf{N}_{2})^{-1}+\mathbf{M}_{2}.$
(20)
This implies that $\widetilde{\mathbf{N}}\preceq\mathbf{N}_{2}.$ Consider the
following enhanced aligned MIMO Gaussian broadcast channel
$\displaystyle\widetilde{\mathbf{Y}}_{1a}[m]$
$\displaystyle=\mathbf{X}[m]+\widetilde{\mathbf{Z}}_{1a}[m]$
$\displaystyle\mathbf{Y}_{1b}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{1b}[m]$
$\displaystyle\widetilde{\mathbf{Y}}_{2a}[m]$
$\displaystyle=\mathbf{X}[m]+\widetilde{\mathbf{Z}}_{2a}[m]$
$\displaystyle\text{and}\qquad\mathbf{Y}_{2b}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{2b}[m]$ (21)
where $\\{\widetilde{\mathbf{Z}}_{1a}[m]\\}$, $\\{\mathbf{Z}_{1b}[m]\\}$,
$\\{\widetilde{\mathbf{Z}}_{2a}[m]\\}$ and $\\{\mathbf{Z}_{2b}[m]\\}$ are
i.i.d. additive vector Gaussian noise processes with zero means and covariance
matrices $\widetilde{\mathbf{N}}$, $\mathbf{N}_{1}$, $\widetilde{\mathbf{N}}$
and $\mathbf{N}_{2}$, respectively. Since
$\widetilde{\mathbf{N}}\preceq\\{\mathbf{N}_{1},\mathbf{N}_{2}\\}$, we
conclude that the secrecy capacity region of the channel (21) is at least as
large as the secrecy capacity region of the channel (17) under the same power
constraints.
Furthermore, based on (20), we have
$\displaystyle[(\mathbf{S}-\mathbf{B}_{0}^{\star})$
$\displaystyle+\widetilde{\mathbf{N}}](\mathbf{B}_{1}^{\star}+\widetilde{\mathbf{N}})^{-1}$
$\displaystyle=[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}](\mathbf{B}_{1}^{\star}+\mathbf{N}_{2})^{-1}$
(22)
and hence,
$\displaystyle\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}}{\mathbf{B}_{1}^{\star}+\widetilde{\mathbf{N}}}\right|$
$\displaystyle=\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}{\mathbf{B}_{1}^{\star}+\mathbf{N}_{2}}\right|.$
(23)
Combining (13) and (20), we may obtain
$\displaystyle(\lambda_{1}$
$\displaystyle+\lambda_{2})[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}]^{-1}$
$\displaystyle=(\lambda_{2}+\beta_{1})[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}]^{-1}$
$\displaystyle\quad+(\lambda_{1}+\beta_{2})[(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}]^{-1}+\mathbf{M}_{0}.$
(24)
Substituting (19) and (23) into (16), we have
$\displaystyle(\beta_{1}+\beta_{2})R_{0}^{{\dagger}}+\lambda_{1}R_{1}^{{\dagger}}+\lambda_{2}R_{2}^{{\dagger}}$
$\displaystyle=\frac{\beta_{1}}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}}\right|+\frac{\beta_{2}}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}\right|$
$\displaystyle\quad+\lambda_{1}\left(\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}}{\widetilde{\mathbf{N}}}\right|-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right)$
$\displaystyle\quad+\lambda_{2}\left(\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}}{\widetilde{\mathbf{N}}}\right|-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|\right)$
$\displaystyle\quad+\rho.$ (25)
#### III-B3 Step 3–Outer Bound for the Enhanced Channel
In the following, we shall consider a four-receiver discrete memoryless
broadcast channel with a common message and two confidential messages and
provide a single-letter outer bound on the secrecy capacity region.
###### Theorem 3 (Discrete memoryless broadcast channel)
Consider a discrete memoryless broadcast channel with transition probability
$p(\widetilde{\mathbf{y}}_{1a},\mathbf{y}_{1b},\widetilde{\mathbf{y}}_{2a},\mathbf{y}_{2b}|\mathbf{x})$
and messages $W_{0}$ (intended for both receivers $1b$ and $2b$), $W_{1}$
(intended for receiver $1a$ but needing to be kept confidential from receiver
$2b$) and $W_{2}$ (intended for receiver $2a$ but needing to be kept
confidential from receiver $1b$). If both
$\displaystyle\mathbf{X}\rightarrow\widetilde{\mathbf{Y}}_{1a}\rightarrow(\mathbf{Y}_{1b},\mathbf{Y}_{2b})\quad\text{and}\quad\mathbf{X}\rightarrow\widetilde{\mathbf{Y}}_{2a}\rightarrow(\mathbf{Y}_{1b},\mathbf{Y}_{2b})$
form Markov chains in their respective order, then the secrecy capacity region
of this channel satisfies
${\mathcal{C}}_{s}^{\rm[DMC]}\subseteq{\mathcal{R}}_{o}$, where
${\mathcal{R}}_{o}$ denotes the set of nonnegative rate triples
$(R_{0},R_{1},R_{2})$ such that
$\displaystyle R_{0}$
$\displaystyle\leq\min[I(\mathbf{U};\mathbf{Y}_{1b}),~{}I(\mathbf{U},\mathbf{Y}_{2b})]$
$\displaystyle R_{1}$ $\displaystyle\leq
I(\mathbf{X};\widetilde{\mathbf{Y}}_{1a}|\mathbf{U})-I(\mathbf{X};\mathbf{Y}_{2b}|\mathbf{U})$
$\displaystyle\text{and}\qquad R_{2}$ $\displaystyle\leq
I(\mathbf{X};\widetilde{\mathbf{Y}}_{2a}|\mathbf{U})-I(\mathbf{X};\mathbf{Y}_{1b}|\mathbf{U})$
(26)
for some $p(\mathbf{u},\mathbf{x})=p(\mathbf{u})p(\mathbf{x}|\mathbf{u})$.
The proof of Theorem 3 can be found in the extended version of this work [11].
Now, we may combine Steps 1, 2 and 3 and consider an upper bound on the
weighted secrecy sum-capacity of the channel (5). By Theorem 3, for any
achievable secrecy rate triple $(R_{0},R_{1},R_{2})$ for the channel (5) we
have
$\displaystyle(\beta_{1}$
$\displaystyle+\beta_{2})R_{0}+\lambda_{1}R_{1}+\lambda_{2}R_{2}$
$\displaystyle\leq\frac{\beta_{1}}{2}\log\left|2\pi
e(\mathbf{S}+\mathbf{N}_{1})\right|+\frac{\beta_{2}}{2}\log\left|2\pi
e(\mathbf{S}+\mathbf{N}_{2})\right|$
$\displaystyle\quad+\frac{\lambda_{1}}{2}\log\left|\frac{\mathbf{N}_{2}}{\widetilde{\mathbf{N}}}\right|+\frac{\lambda_{2}}{2}\log\left|\frac{\mathbf{N}_{1}}{\widetilde{\mathbf{N}}}\right|+\eta(\lambda_{1},\lambda_{2})$
(27)
where
$\displaystyle\eta$
$\displaystyle(\lambda_{1},\lambda_{2}):=\lambda_{1}h(\mathbf{X}+\widetilde{\mathbf{Z}}_{1a}|\mathbf{U})+\lambda_{2}h(\mathbf{X}+\widetilde{\mathbf{Z}}_{2a}|\mathbf{U})$
$\displaystyle\quad-(\lambda_{2}+\beta_{1})h(\mathbf{X}+\mathbf{Z}_{1b}|\mathbf{U})-(\lambda_{1}+\beta_{2})h(\mathbf{X}+\mathbf{Z}_{2b}|\mathbf{U}).$
Note that
$0\prec\widetilde{\mathbf{N}}\preceq\\{\mathbf{N}_{1},\mathbf{N}_{2}\\}$,
$0\prec\mathbf{B}_{0}^{\star}\preceq\mathbf{S}$ and
$\mathbf{B}_{0}^{\star}\mathbf{M}_{0}=0$. Using [10, Corollary 4] and (24), we
may obtain
$\displaystyle\eta(\lambda_{1},\lambda_{2})$
$\displaystyle\leq(\lambda_{1}+\lambda_{2})\log\left|2\pi
e(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}\right|$
$\displaystyle\quad-(\lambda_{2}+\beta_{1})\log\left|2\pi
e(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}\right|$
$\displaystyle\quad-(\lambda_{1}+\beta_{2})\log\left|2\pi
e(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}\right|.$ (28)
Combining (27) and (28), for any achievable secrecy rate triple
$(R_{0},R_{1},R_{2})$ for the channel (5) we have
$\displaystyle($
$\displaystyle\beta_{1}+\beta_{2})R_{0}+\lambda_{1}R_{1}+\lambda_{2}R_{2}$
$\displaystyle\leq\frac{\beta_{1}}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}}\right|+\frac{\beta_{2}}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}\right|$
$\displaystyle\quad+\lambda_{1}\left(\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}}{\widetilde{\mathbf{N}}}\right|-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right)$
$\displaystyle\quad+\lambda_{2}\left(\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\widetilde{\mathbf{N}}}{\widetilde{\mathbf{N}}}\right|-\frac{1}{2}\log\left|\frac{(\mathbf{S}-\mathbf{B}_{0}^{\star})+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|\right)$
$\displaystyle<(\beta_{1}+\beta_{2})R_{0}^{{\dagger}}+\lambda_{1}R_{1}^{{\dagger}}+\lambda_{2}R_{2}^{{\dagger}}.$
(29)
Clearly, this contradicts the assumption that the rate triple
$(R_{0}^{{\dagger}},R_{1}^{{\dagger}},R_{2}^{{\dagger}})$ is achievable.
Therefore, we have proved the desired converse result for Theorem 2.
Figure 2: Secrecy capacity region $\\{(R_{0},R_{1},R_{2})\\}$
## IV Numerical Example
In this section, we provide a numerical example to illustrate the secrecy
capacity region of the MIMO Gaussian broadcast channel with a common message
and two confidential messages. In this example, we assume that both the
transmitter and each of the receivers are equipped with two antennas. The
channel matrices and the matrix power constraint are given by
$\displaystyle\mathbf{H}_{1}$ $\displaystyle=\left(\begin{matrix}1.8&2.0\\\
1.0&3.0\end{matrix}\right),~{}\mathbf{H}_{2}=\left(\begin{matrix}3.3&1.3\\\
2.0&-1.5\end{matrix}\right)$ and
$\displaystyle\qquad\mathbf{S}=\left(\begin{matrix}5.0&1.25\\\
1.25&10.0\end{matrix}\right).$
which yield a _nondegraded_ MIMO Gaussian broadcast channel. The boundary of
the secrecy capacity region
${\mathcal{C}}_{s}^{\rm[SBC]}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ is
plotted in Fig. 2.
In Fig. 3, we have also plotted the boundaries of the secrecy capacity region
$(R_{1},R_{2})$ for some given common rate $R_{0}$. It is particularly worth
mentioning that with $R_{0}=0$, the secrecy capacity region
$\\{(R_{1},R_{2})\\}$ is _rectangular_ , which implies that under the matrix
power constraint, both confidential messages $W_{1}$ and $W_{2}$ can be
simultaneously transmitted at their respective maximum secrecy rates. The
readers are referred to [7] for further discussion of this phenomenon.
## V Conclusion
In this paper, we have considered the problem of communicating two
confidential messages and a common message over a two-receiver MIMO Gaussian
broadcast channel. We have shown that a natural scheme that combines secret
dirty-paper coding and Gaussian superposition coding achieves the entire
secrecy capacity region. To prove the converse result, we have applied a
channel-enhancement argument and an extremal entropy inequality of Weingarten
et al., which generalizes the argument of [8] for the case with a common
message and only one confidential message.
Figure 3: Secrecy rate regions $\\{(R_{1},R_{2})\\}$ for some given $R_{0}$
## References
* [1] I. Csiszár and J. Körner, “Broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 24, no. 3, pp. 339–348, May 1978\.
* [2] T. Liu and S. Shamai (Shitz), “A note on the secrecy capacity of the multiple-antenna wiretap channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 6, pp. 2547–2553, Jun. 2009.
* [3] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “An MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” _EURASIP Journal on Wireless Communications and Networking_ , vol. 2009, Article ID 370970, 8 pages, 2009.
* [4] A. Khisti and G. W. Wornell, “The secrecy capacity of the MIMO wiretap channel,” in _Proc. 45th Annual Allerton Conf. Comm., Contr., Computing_ , Monticello, IL, Sep. 2007.
* [5] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in _Proc. IEEE Int. Symp. Information Theory_ , Toronto, Canada, July 2008, pp. 524–528.
* [6] R. Liu and H. V. Poor, “Secrecy capacity region of a multi-antenna Gaussian broadcast channel with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 3, pp. 1235–1249, Mar. 2009.
* [7] R. Liu, T. Liu, H. V. Poor, and S. Shamai (Shitz), “Multiple-input multiple-output gaussian broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , submitted March 2009. [Online]. Available: http://arxiv.org/abs/0903.3786.
* [8] H. D. Ly, T. Liu, and Y. Liang, “Multiple-input multiple-output Gaussian broadcast channels with common and confidential messages,” _IEEE Trans. Inf. Theory_ , submitted July 2009.
* [9] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 9, pp. 3936–3964, Sep. 2006.
* [10] H. Weingarten, T. Liu, S. Shamai (Shitz), Y. Steinberg, and P. Viswanath, “The capacity region of the degraded multiple-input multiple-output compound broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 11, pp. 5011–5023, Nov. 2009.
* [11] R. Liu, T. Liu, H. V. Poor, and S. Shamai (Shitz), “MIMO Gaussian broadcast channels with confidential and common messages,” _IEEE Trans. Inf. Theory_ , submitted for publication.
* [12] J. Xu, Y. Cao, and B. Chen, “Capacity bounds for broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 10, pp. 4529–4542, Oct. 2009.
|
arxiv-papers
| 2010-01-18T05:44:14 |
2024-09-04T02:49:07.793716
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz)",
"submitter": "Ruoheng Liu",
"url": "https://arxiv.org/abs/1001.2806"
}
|
1001.2831
|
# Quantum Model of Bertrand Duopoly
Salman Khan, M. Ramzan , M. K. Khan
Department of Physics Quaid-i-Azam University
Islamabad 45320, Pakistan sksafi@phys.qau.edu.pk
###### Abstract
We present the quantum model of Bertrand duopoly and study the entanglement
behavior on the profit functions of the firms. Using the concept of optimal
response of each firm to the price of the opponent, we found only one Nash
equilibirum point for maximally entangled initial state. The very presence of
quantum entanglement in the initial state gives payoffs higher to the firms
than the classical payoffs at the Nash equilibrium. As a result the dilemma
like situation in the classical game is resolved.
PACS: 03.65.Ta; 03.65.-w; 03.67.Lx
Keywords: Quantum Bertrand duopoly; profit functions, Nash equilibria.
In economics, oligopoly refers to a market condition in which sellers are so
few that action of each seller has a measurable impact on the price and other
market factors [1]. If the number of firms competing on a commodity in the
market is just two, the oligopoly is termed as duopoly. The competitive
behavior of firms in oligopoly makes it suitable to be analyzed by using the
techniques of game theory. Cournot and Bertrand models are the two oldest and
famous oligopoly models [2, 3]. In Cournot model of oligopoly firms put
certain amount of homogeneous product simultaneously in the market and each
firm tries to maximize its payoff by assuming that the opponent firms will
keep their outputs constant. Later on Stackelberg introduced a modified form
of Cournot oligopoly in which the oligopolistic firms supply their products in
the market one after the other instead of their simultaneous moves. In
Stackelberg duopoly the firm that moves first is called leader and the other
firm is the follower [4]. In Bertrand model the oligopolistic firms compete on
price of the commodity, that is, each firm tries to maximize its payoff by
assuming that the opponent firms will not change the prices of their products.
The output and price are related by the demand curve so the firms choose one
of them to compete on leaving the other free. For a homogeneous product, if
firms choose to compete on price rather than output, the firms reach a state
of Nash equilibrium at which they charge a price equal to marginal cost. This
result is usually termed as Bertrand paradox, because practically it takes
many firms to ensure prices equal to marginal cost. One way to avoid this
situation is to allow the firms to sell differentiated products [1].
For the last one decade quantum game theorists are attempting to study
classical games in the domain of quantum mechanics [$5$-$14$]. Various quantum
protocols have been introduced in this regard and interesting results have
been obtained [15-25]. The first quantization scheme was presented by Meyer
[15] in which he quantized a simple penny flip game and showed that a quantum
player can always win against a classical player by utilizing quantum
superposition.
In this letter, we extend the classical Bertrand duopoly with differentiated
products to quantum domain by using the quantization scheme proposed by
Marinatto and Weber [17]. Our results show that the classical game becomes a
subgame of the quantum version. We found that entanglement in the initial
state of the game makes the players better off. Before presenting the
calculation of quantization scheme, we first review the classical model of the
game.
Consider two firms A and B producing their products at a constant marginal
cost $c$ such that $c<a$, where $a$ is a constant. Let $p_{1}$ and $p_{2}$ be
the prices chosen by each firm for their products, respectively. The
quantities $q_{A}$ and $q_{B}$ that each firm sells is given by the following
key assumption of the classical Bertrand duopoly model
$q_{A}=a-p_{1}+bp_{2}$ $q_{B}=a-p_{2}+bp_{1}$ (1)
where the parameter $0<b<1$ shows the amount of one firm’s product substituted
for the other firm’s product. It can be seen from Eq. (1) that more quantity
of the product is sold by the firm which has low price compare to the price
chosen by his opponent. The profit function of the two firms are given by
$u_{A}\left(p_{1},p_{2},b\right)=q_{A}\left(p_{1}-c\right)=\left(a-p_{1}+bp_{2}\right)\left(p_{1}-c\right)$
$u_{B}\left(p_{1},p_{2},b\right)=q_{B}\left(p_{2}-c\right)=\left(a-p_{2}+bp_{1}\right)\left(p_{2}-c\right)$
(2)
In Bertrand duopoly the firms are allowed to change the quantity of their
product to be put in the market and compete only in price. A firm changes the
price of its product by assuming that the opponent will keep its price
constant. Suppose that firm B has chosen $p_{2}$ as the price of his product,
the optimal response of firm A to $p_{2}$ is obtained by maximizing its profit
function with respect to its own product’s price, that is, $\partial
u_{A}/\partial p_{1}=0$, this leads to
$p_{1}=\frac{1}{2}\left(bp_{2}+a+c\right)$ (3)
Firm B response to a fixed price $p_{1}$ of firm A is obtained in a similar
way and is given by
$p_{2}=\frac{1}{2}\left(bp_{1}+a+c\right)$ (4)
Solution of Eqs.(3 and 4) lead to the following optimal price level that
defines the Nash equilibrium of the game
$p_{1}^{\ast}=p_{2}^{\ast}=\frac{a+c}{2-b}$ (5)
The profit functions of the firms at the Nash equilibrium become
$u_{A}^{\ast}=u_{B}^{\ast}=\left[\frac{a+c}{2-b}-c\right]^{2}$ (6)
From Eq. (6), we see that both firms can be made better off if they choose
higher prices, that is, the Nash equilibrium is Pareto inefficient.
To quantize the game, we consider that the game space of each firm is a two
dimensional Hilbert space of basis vector $|0\rangle$ and $|1\rangle$, that
is, the game consists of two qubits, one for each firm. The composite Hilbert
space $\mathcal{H}$ of the game is a four dimensional space which is formed as
a tensor product of the individual Hilbert spaces of the firms, that is,
$\mathcal{H=H}_{A}\mathcal{\otimes H}_{B}$, where $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$ are the Hilbert spaces of firms A and B, respectively. To
manipulate their respective qubits each firm can have only two strategies $I$
and $C$. Where $I$ is the identity operator and and $C$ is the inversion
operator also called Pauli spin flip operator. If $x$ and $1-x$ stand for the
probabilities of $I$ and $C$ that firm A applies and $y$, $1-y$, are the
probabilities that firm B applies, then the final state $\rho_{f}$ of the game
is given by [17]
$\displaystyle\rho_{f}$ $\displaystyle=$ $\displaystyle xyI_{A}\otimes I_{B}\
\rho_{i}\ I_{A}^{{\dagger}}\otimes
I_{B}^{{\dagger}}+x\left(1-y\right)I_{A}\otimes C_{B}\ \rho_{i}\
I_{A}^{{\dagger}}\otimes C_{B}^{{\dagger}}$ (7)
$\displaystyle+y\left(1-x\right)C_{A}\otimes I_{B}\ \rho_{i}\
C_{A}^{{\dagger}}\otimes I_{B}^{{\dagger}}$
$\displaystyle+\left(1-x\right)\left(1-y\right)C_{A}\otimes C_{B}\ \rho_{i}\
C_{A}^{{\dagger}}\otimes C_{B}^{{\dagger}}$
In Eq. (7) $\rho_{i}=|\psi_{i}\rangle\langle\psi_{i}|$ is the initial density
matrix with initial state $|\Psi_{i}\rangle$, which is given by
$|\psi_{i}\rangle=\cos\gamma|00\rangle+\sin\gamma|11\rangle$ (8)
where $\gamma\in\left[0,\pi\right]$ and represents the degree of entanglement
of the initial state. In Eq. (8) the first qubit corresponds to firm A and the
second qubit corresponds to firm B. The moves (prices) of the firms and the
probabilities $x$, $y$ of using the operators can be related as follows,
$x=\frac{1}{1+p_{1}},\qquad\qquad\qquad y=\frac{1}{1+p_{2}}$ (9)
where the prices $p_{1}$ and $p_{2}\in[0,\infty)$ and the probabilities $x$,
$y\in\left[0,1\right]$. By using Eqs. (7 \- 9), the nonzero elements of the
final density matrix are obtained as
$\displaystyle\rho_{11}$ $\displaystyle=$
$\displaystyle\frac{(\cos^{2}\gamma+p_{1}p_{2}\sin^{2}\gamma)}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$
$\displaystyle\rho_{14}$ $\displaystyle=$
$\displaystyle\rho_{41}=\frac{(1+p_{1}p_{2})\cos\gamma\sin\gamma}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$
$\displaystyle\rho_{22}$ $\displaystyle=$
$\displaystyle\frac{p_{2}\cos^{2}\gamma+p_{1}\sin^{2}\gamma}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$
$\displaystyle\rho_{23}$ $\displaystyle=$
$\displaystyle\rho_{32}=\frac{(p_{1}+p_{2})\cos\gamma\sin\gamma}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$
$\displaystyle\rho_{33}$ $\displaystyle=$
$\displaystyle\frac{p_{1}\cos^{2}\gamma+p_{2}\sin^{2}\gamma}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$
$\displaystyle\rho_{44}$ $\displaystyle=$
$\displaystyle\frac{p_{1}p_{2}\cos^{2}\gamma+\sin^{2}\gamma}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$
(10)
The payoffs of the firms can be found by the following trace operations
$\displaystyle u_{A}\left(p_{1},p_{2},b\right)$ $\displaystyle=$
$\displaystyle\mathrm{Trace}\left(U_{A}^{\mathrm{oper}}\rho_{f}\right)$
$\displaystyle u_{B}\left(p_{1},p_{2},b\right)$ $\displaystyle=$
$\displaystyle\mathrm{Trace}\left(U_{B}^{\mathrm{oper}}\rho_{f}\right)$ (11)
where $U_{A}^{\mathrm{oper}}$ and $U_{B}^{\mathrm{oper}}$ are payoffs
operators of the firms, which we define these as
$\displaystyle U_{A}^{\mathrm{oper}}$ $\displaystyle=$
$\displaystyle\frac{q_{A}}{p_{12}}\left(k_{B}\rho_{11}-\rho_{22}+\rho_{33}\right)$
$\displaystyle U_{B}^{\mathrm{oper}}$ $\displaystyle=$
$\displaystyle\frac{q_{A}}{p_{12}}\left(k_{A}\rho_{11}+\rho_{22}-\rho_{33}\right)$
(12)
where $k_{A}=p_{1}-c$, $k_{B}=p_{2}-c$ and
$p_{12}=\frac{1}{\left(1+p_{1}\right)\left(1+p_{2}\right)}$. By using Eqs. (10
\- 12), the payoffs of the firms are obtained as
$\displaystyle u_{A}\left(p_{1},p_{2},b\right)$ $\displaystyle=$
$\displaystyle(a-p_{1}+bp_{2})[k_{A}\cos^{2}\gamma+\\{p_{2}+p_{1}(-1-cp_{2}+p_{2}^{2})\\}\sin^{2}\gamma]$
$\displaystyle u_{B}\left(p_{1},p_{2},b\right)$ $\displaystyle=$
$\displaystyle(a-p_{2}+bp_{1})[k_{B}\cos^{2}\gamma+\\{p_{1}-p_{2}(1+cp_{1}-p_{1}^{2})\\}\sin^{2}\gamma]$
One can easily see from Eq. (LABEL:E13) that the classical payoffs can be
reproduced by setting $\gamma=0$ in Eq. (LABEL:E13).
We proceed similar to the classical Bertrand duopoly to find the response of
each firm to the price chosen by the opponent firm. For firm B’s price
$p_{2}$, the optimal response of firm A is obtained by maximizing its own
payoff (Eq. (LABEL:E13)) with respect to $p_{1}$. Similarly, the reaction
function of firm B to a known $p_{1}$ is obtained. These reaction functions
can be written as
$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{k_{B}[-1+p_{2}(a+bp_{2})]+[c+p_{2}+2bp_{2}-bp_{2}^{2}k_{B}+a\\{2-p_{2}k_{B}\\}]\cos
2\gamma}{(2-p_{2}k_{B})\cos 2\gamma-2p_{2}k_{B}}$ $\displaystyle p_{2}$
$\displaystyle=$
$\displaystyle\frac{k_{A}[-1+p_{1}(a+bp_{1})]+[c+p_{1}+2bp_{1}+bp_{1}^{2}k_{A}+a\\{2+p_{1}k_{A}\\}]\cos
2\gamma}{(2-p_{1}k_{A})\cos 2\gamma+2p_{1}k_{A}}$
The results of Eq. (LABEL:E14) reduce to the classical results given in Eqs.
(3 and 4) for the initially unentangled state, that leads to the classical
Nash equilibrium. This shows that the classical game is a subgame of the
quantum game.
Now, we discuss the behavior of entanglement in the initial state on the game
dynamics. It can be seen from Eq. (LABEL:E14) that the optimal responses of
the firms to a fixed price of the opponent firm, for a maximally entangled
state, are given by
$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{bp_{2}^{2}+ap_{2}-1}{2p_{2}}$ $\displaystyle p_{2}$
$\displaystyle=$ $\displaystyle\frac{bp_{1}^{2}+ap_{1}-1}{2p_{1}}$ (15)
Solving these equations, we can obtain the optimal price levels and the
corresponding payoffs of each firm. In this case the following four points are
obtained
$\displaystyle p_{1}^{\ast}(1)$ $\displaystyle=$ $\displaystyle
p_{2}^{\ast}(1)=\frac{a+\sqrt{a^{2}+4\beta}}{-2\beta}$ $\displaystyle
p_{1}^{\ast}(2)$ $\displaystyle=$ $\displaystyle
p_{2}^{\ast}(2)=\frac{2}{a+\sqrt{a^{2}+4\beta}}$ $\displaystyle
p_{1}^{\ast}(3,4)$ $\displaystyle=$
$\displaystyle\frac{2b}{a\sqrt{2+b}\left(\sqrt{2+b}\pm\gamma\right)}$
$\displaystyle p_{2}^{\ast}(3,4)$ $\displaystyle=$
$\displaystyle-\frac{1}{2b}\left[a\pm\frac{\gamma}{\sqrt{2+b}}\right]$ (16)
where the numbers in the parentheses correspond to the respective points (the
symbols $\pm$ correspond to points $3$ and $4$ respectively). To verify which
point (points) defines the Nash equilibrium of the game, we use the second
partial derivative condition. That is, for Nash equilibrium, the strategy
(point) must be the global maximum of the payoff function, that is,
$\partial^{2}u_{A(B)}/\partial p_{1(2)}^{2}<0$ and the payoff function at the
point must be higher than the payoff function at the boundary points. It can
easily be verified that this condition is satisfied only at point $1$. Hence
point $1$ defines the Nash equilibrium of the game. The payoffs of the firms
at the Nash equilibrium become
$\displaystyle u_{A}(1)$ $\displaystyle=$ $\displaystyle
u_{B}(1)=\frac{1}{4\beta^{4}}[a^{4}+2\alpha^{2}+2a^{2}b\beta+a^{3}c\beta-a\\{\left(\beta-2\right)\beta-3\\}c\beta^{2}$
(17)
$\displaystyle\qquad+\sqrt{a^{2}+4\beta}(a^{3}+2a\alpha+c\alpha^{2}+a^{2}c\beta)]$
The new parameters introduced in Eqs. (16 and 17) are defined as $\beta=b-2$,
$\alpha=2-3b+b^{2}$. The payoffs of the firms at the Nash equilibrium must be
real and positive for the entire range of substitution parameter $b$. This
condition for marginal cost $c<1.4$ is satisfied when $a\geq 3.5$. The firms’
payoffs at the other three points become
$\displaystyle u_{A}(2)$ $\displaystyle=$ $\displaystyle
u_{B}(2)=-\frac{4}{(a+\sqrt{a^{2}+4\beta})^{4}}[a^{5}c+a(-1+b)\\{(-9+5b)c-2\sqrt{a^{2}+4\beta}\\}$
$\displaystyle\qquad-a^{3}\\{(8-5b)c+\sqrt{a^{2}+4\beta}\\}+(-1+b)^{2}(-2+c\sqrt{a^{2}+4\beta})$
$\displaystyle\qquad+a^{4}(-1+c\sqrt{a^{2}+4\beta})+a^{2}\\{6-4c\sqrt{a^{2}+4\beta}+b(-4+3c\sqrt{a^{2}+4\beta})\\}]$
$\displaystyle u_{A}(3,4)$ $\displaystyle=$
$\displaystyle\frac{(1+b)^{2}(a^{2}(2+b)^{3/2}+a(2+b)(b\sqrt{2+b}c\pm\Gamma)+b(2b\sqrt{2+b}\pm
c\Gamma\left(2+b\right)))}{(2+b)^{3/2}(a(2+b)\pm\sqrt{2+b}\Gamma)^{2}}$
$\displaystyle u_{B}(3,4)$ $\displaystyle=$
$\displaystyle-\frac{(1+b)^{2}\sqrt{2+b}\left[2ac+abc-2b\pm\sqrt{2+b}\Gamma
c\right]}{4b(2+b)^{5/2}}$ (18)
where $\Gamma=\sqrt{4b^{2}+a^{2}(2+b)}$.
(a) | |
---|---|---
Figure 1: The payoffs of the firms at the classical and quantum Nash
equilibria against the substitution parameter $b$. The values of the parameter
$a$ and the marginal cost $c$ are chosen as $3.5$ and $0.1$, respectively. The
superscripts $C$ and $Q$ of $u$ represent the classical and quantum cases,
respectively. The subscripts $A$ stands for firm $A$.
We present a quantization scheme for the Bertrand duopoly with differentiated
products. To analyze the effect of quantum entanglement on the game dynamics,
we plot the payoffs of the firms at the classical and quantum Nash equilibria
against the substitution parameter $b$ in figure ($1$). The values of
parameters $a$ and $c$ are chosen to be $3.5$ and $0.1$, respectively. The
solid line ($u_{A}^{Q}(1)$) represents quantum mechanical payoffs and the
dotted line ($u_{A}^{C}$) represents the classical payoffs of the firms. It is
clear from the figure that quantum payoffs of the firms are higher than the
classical payoffs for the entire range of substitution parameter $b$. The
maximum entanglement in the initial state of the game makes the firms better
off. In figure (2), we plot the payoffs of the firms (Eq. 18) against the
substitution parameter $b$ at the other three points which are not the Nash
equilibria.
(a) | |
---|---|---
Figure 2: The payoffs of the firms at the second and third points as a
function of the substitution parameter $b$. The values of the parameter $a$
and the marginal cost $c$ are chosen as $3.5$ and $0.1$, respectively. The
superscript $Q$ of $u$ represent the classical and quantum cases,
respectively. The subscripts $A$ and $B$ correspond to firms $A$ and $B$
respectively. The numbers in the parentheses represent the corresponding Nash
equilibrium points.
In conclusion, we have used the Marinatto and Weber quantization scheme to
find the quantum version of Bertrand duopoly with differentiated products. We
have studied the entanglement behavior on the payoffs of the firms for a
maximally entangled initial state. We found that for large values of
substitution parameter $b,$ both firms can achieve significantly higher
payoffs as compared to the classical payoffs. Furthermore, for maximally
entangled state the quantum payoffs are higher than the classical payoffs for
the entire range of substitution parameter and is the best situation for both
firms. Thus, the dilemma-like situation in the classical Bertrand duopoly game
is resolved.
Acknowledgment
One of the authors (Salman Khan) is thankful to World Federation of Scientists
for partially supporting this work under the National Scholarship Program for
Pakistan
## References
* [1] Gibbons R Game Theory for applied Economists Princeton Univ. Press Princeton, NJ, 1992: Bierman H.S, Fernandez L Game Theory with Economic Applications, 2nd Edition, Addison - Wesley, Reading MA 1998
* [2] Cournot A 1897 Researches into the Mathematical Principles of the Theory of Wealth, Macmillan Co., New York
* [3] Bertrand J, Savants J 1883 67 499
* [4] Stackelberg H von 1934 Marktform und Gleichgewicht (Julius Springer Vienna)
* [5] Flitney A.P, Ng J, Abbott D 2002 Physica A 314 35, doi:10.1016/S0378-4371(02)01084-1
* [6] Ramzan M et al 2008 J. Phys. A Math. Theor. 41 055307, doi:10.1088/1751-8113/41/5/055307
* [7] D’ Ariano G M 2002 Quantum Information and Computation 2 355
* [8] Flitney A.P, Abbott D 2002 Phys. Rev. 65 062318, doi:10.1103/PhysRevA.65.062318
* [9] Iqbal A, Cheo T, Abbott D 2008 Phys. Lett. A 372 6564, doi:10.1016/j.physleta.2008.09.026
* [10] Zhu X, Kaung L M 2008 Commun. Theor. Phys (china) 49 111
* [11] Ramzan M and Khan M K 2008 J. Phys. A: Math. Theor. 41 435302, doi:10.1088/1751-8113/41/43/435302
* [12] Ramzan M, Khan M.K 2009 J. Phys. A, Math. Theor. 42 025301, doi:10.1088/1751-8113/42/2/025301
* [13] Zhu X, Kaung L M 2007 J. Phys. A 40 7729, doi:10.1088/1751-8113/40/27/021
* [14] Salman Khan et al 2010 Int. J. Theor. Phys.49 31, doi:10.1007/s10773-009-0175-y
* [15] Meyer D A 1999 Phys. Rev. Lett. 82 1052, doi:10.1103/PhysRevLett.82.1052
* [16] Eisert J et al 1999 Phys. Rev. Lett. 83 3077, doi:10.1103/PhysRevLett.83.3077
* [17] Marinatto L and Weber T 2000 Phys. Lett. A 272 291, doi:10.1016/S0375-9601(00)00441-2
* [18] Li H, Du J, Massar S 2002 Phys. Lett. A 306 73, doi:10.1016/S0375-9601(02)01628-6
* [19] Lo C.F, Kiang D 2004 Phys. Lett. A 321 94, doi:10.1016/j.physleta.2003.12.013
* [20] Iqbal A and Abbott D 2009 arXiv:quant-ph/0909.3369
* [21] Iqbal A and Toor A H 2002 Phys. Rev. A 65 052328, doi:10.1103/PhysRevA.65.052328
* [22] Lo C.F, Kiang D 2003 Phys. Lett. 318 333, doi:10.1016/j.physleta.2003.09.047
* [23] Benjamin S.C, Hayden P.M 2001 Phy. Rev. A 64 030301, doi:10.1103/PhysRevA.64.030301
* [24] Lo C.F, Kiang D 2005 Phys. Lett. A 346 65, doi:10.1016/j.physleta.2005.07.055
* [25] Gan Qin et al 2005 J. Phys. A: Math. Gen. 38 4247, doi:10.1088/0305-4470/38/19/013
|
arxiv-papers
| 2010-01-16T14:42:32 |
2024-09-04T02:49:07.800936
|
{
"license": "Public Domain",
"authors": "Salman Khan, M. Ramzan and M. K. Khan",
"submitter": "Salman Khan",
"url": "https://arxiv.org/abs/1001.2831"
}
|
1001.2960
|
# Optimal Dynamical Decoupling Sequence for Ohmic Spectrum
Yu Pan1,2 Zai-Rong Xi1 zrxi@iss.ac.cn Wei Cui1,2 1Key Laboratory of Systems
and Control, Institute of Systems Science, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of
China 2Graduate University of Chinese Academy of Sciences, Beijing 100039,
People’s Republic of China
###### Abstract
We investigate the optimal dynamical decoupling sequence for a qubit coupled
to an ohmic environment. By analytically computing the derivatives of the
decoherence function, the optimal pulse locations are found to satisfy a set
of non-linear equations which can be easily solved. These equations
incorporates the environment information such as high-energy (UV) cutoff
frequency $\omega_{c}$, giving a complete description of the decoupling
process. The solutions explain previous experimental and theoretical results
of locally optimized dynamical decoupling (LODD) sequence in high-frequency-
dominated environment, which were obtained by purely numerical computation and
experimental feedback. As shown in numerical comparison, these solutions
outperform the Uhrig dynamical decoupling (UDD) sequence by one or more orders
of magnitude in the ohmic case.
## I INTRODUCTION
Suppressing decoherence is one of the fundamental issues in the field of
quantum information processing. Decoherence, which has been caused by the
environmental noise, plagues almost all the implementations of quantum bit. To
eliminate the unwanted coupling between a qubit and its environment, several
schemes have been proposed and tested. Among them a promising one is dynamical
decoupling Viola and Lloyd (1998); Ban (1998); Viola et al. (1999); Gordon et
al. (2008), which restores the qubit coherence by applying delicately designed
sequence of control pulses.
For a qubit that can be modeled by a spin-1/2 particle, the oldest dynamical
decoupling sequence is periodic dynamical decoupling (PDD). Originated from
pulse sequences widely used in nuclear magnetic resonance (NMR) Ernst et al.
(1991), the PDD sequence consists of periodic and equidistant $\pi$ pulses. To
achieve better performance, there has been an extensive study in how to
optimize the pulse locations Viola and Knill (2005); Kern and Alber (2005);
Khodjasteh and Lidar (2005, 2007); Uhrig (2007); Yang and Liu (2008); Pryadko
and Quiroz (2008); Pasini and Uhrig (2009); Biercuk et al. (2009a); Uys et al.
(2009). One important progress is the powerful Uhrig DD (UDD) Uhrig (2007),
which employs $n$ pulses located at $t_{j}$ according to the simple rules
$\delta_{j}=\sin^{2}(j\pi/2(n+1)),$
where $\delta_{j}=t_{j}/T$ and $T$ is the total evolution time. UDD is first
derived on spin-boson model and further proved to be universal in the sense
that it can remove the qubit-bath coupling to $n^{th}$ order in generic
environment Yang and Liu (2008).
Beyond UDD, another locally optimized dynamical decoupling (LODD) sequence has
drawn great attention Biercuk et al. (2009a). LODD, along with its simplified
version optimized noise-filtration dynamic decoupling (OFDD) Uys et al.
(2009), generates the decoupling sequence by directly optimizing the
decoherence function using numeric methods as well as experimental feedback.
It has been shown to be able to suppress decoherence effect by orders of
magnitude over UDD for certain noise spectrum, especially for the one with a
high frequency part and sharp high-energy (UV) cutoff.
However, in spite of the great experimental success, analytical results about
the LODD sequence is insufficient. Until recently S. Pasini and G. S. Uhrig
has made an analytical progress in optimizing the decoherence function for
power law spectrum (PLODD) Pasini and Uhrig (2009). The power law spectrum
$\omega^{\alpha}$ for $\alpha<1$ without UV cutoff is considered. They
minimize the decoherence function through expanding the function and
separating, canceling divergences from the relevant terms and solving
variation problems. Inspired by Pasini’s work, we try to analyze the LODD
problem with respect to the ohmic spectrum $S(\omega)\sim\omega$ and a sharp
UV cutoff. Ohmic noise is the major decoherence source often found in a
qubit’s environment, for example, the semiconducting quantum dot Leggett et
al. (1987) and superconducting qubit Weiss (2008). Optimal performance pulse
sequence is found analytically which entirely differs from the UDD sequence in
such environment. We call this kind of optimal sequence HLODD (LODD for ohmic
spectrum) for short.
We organize this paper as follows. In the second section we propose the
optimization problem of the decoherence function. In section III, we derive
the analytical equations for the optimal pulse sequence. In the following
section, we run a simulation to verify our results. Conclusions are put in
section V.
## II OPTIMIZATION OF THE DECOHERENCE FUNCTION
Given a two-level quantum system, when the environmental noise behaves
quantum-mechanically, we use the long-established spin-boson model with pure
dephasing
$H=\sum_{i}\omega_{i}b_{i}^{\dagger}b_{i}+\frac{1}{2}\sigma_{z}\sum_{i}\lambda_{i}(b_{i}^{\dagger}+b_{i}).$
$None$
Here we ignore the qubit free evolution hamiltonian. On the other hand, when
the qubit is subjected to classical noise, the system is modeled as Cywinski
et al. (2008); Kuopanportti et al. (2008)
$H=\frac{1}{2}[\Omega+\beta(t)]\sigma_{z},$ $None$
where the $\Omega$ is the qubit energy splitting and $\beta(t)$ the classical
random noise. Let $t$ be the total evolution time, and $n$ pulses are applied
at $t_{1}<t_{2}<...<t_{n}$ in sequence with negligible pulse durations . We
use the notation $\delta_{j}=\frac{t_{j}}{t}$. This naturally leads to the
definition of $t_{0}=0$ and $t_{n+1}=1$. In either ($1$) or ($2$), the decay
of coherence under the dynamical decoupling sequence can be described by the
decoherence function Uhrig (2007); Cywinski et al. (2008); Uhrig (2008);
Biercuk et al. (2009b) $e^{-2\chi(t)}$ with
$\chi(t)=\int_{0}^{\infty}\frac{S(\omega)}{\omega^{2}}{|y_{n}(\omega{t})|}^{2}d\omega,$
$None$
where $S(\omega)$ is environmental noise spectrum. The filter function
$y_{n}(t)$ is given by
$y_{n}(t)=1+(-1)^{n+1}e^{\mbox{i}\omega{t}}+2\sum_{j=1}^{n}(-1)^{j}e^{\mbox{i}\omega{t}\delta_{j}}.$
$None$
Thus minimization of $\chi(t)$ with respect to $\delta_{j}$ gives the optimal
decoupling sequence.
We now consider the case when the noise spectrum is ohmic with a sharp cutoff
at $\omega_{c}$, i.e. $S(\omega)=S_{0}\omega\Theta(\omega_{c}-\omega)$.
$S_{0}$ is an irrelevant constant factor and $\Theta$ is unit step function.
Then minimization of ($3$) turns to minimization of $I_{n}$ with
$I_{n}=\int_{0}^{z_{c}}\frac{{|y_{n}(z)|}^{2}}{z}dz,$ $None$
where $z_{c}=\omega_{c}{t}$. Since $y_{n}(0)=0$, the IR convergence insures
the integral converges to a finite value Pasini and Uhrig (2009).
## III DERIVATION OF OPTIMAL PULSE SEQUENCE
We follow the approach of Pasini and Uhrig Pasini and Uhrig (2009) to treat
the integral ($5$). Here we use notation
$q_{j}=\left\\{\begin{array}[]{ll}0&\mbox{if $j=0,n+1$,}\\\ 1&\mbox{if
$j\in\\{1,2,...,n\\}$,}\end{array}\right.$
and
$\Delta_{ij}=\mbox{i}(\delta_{i}-\delta_{j}),$
from which we get
$|y_{n}(z)|^{2}=\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}e^{z\Delta_{ij}}.$
Then the integral $I_{n}$ can be expressed as
$I_{n}=\lim_{x\to 0^{+}}I_{n}(x),$
$I_{n}(x)=\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}I_{ij}(x),$ $None$
where the integrals $I_{ij}(x)$ are
$\displaystyle I_{ij}(x)$ $\displaystyle=$
$\displaystyle\int_{x}^{z_{c}}\frac{e^{\Delta_{ij}z}}{z}dz$ (7)
$\displaystyle=$
$\displaystyle\int_{-\Delta_{ij}x}^{-\Delta_{ij}z_{c}}\frac{e^{-z}}{z}dz.$
The limit $x\to 0^{+}$ is carried out because $I_{ij}(0)$ does not exist for
arbitrary $i,j$. Making use of the series representation of exponential
function Abramowitz and Stegun (1964)
$\displaystyle E_{1}(z)$ $\displaystyle=$
$\displaystyle\int_{z}^{\infty}\frac{e^{-t}}{t}dt$ $\displaystyle=$
$\displaystyle-\gamma-\ln{z}+\sum\limits_{k=1}^{\infty}\frac{(-1)^{k+1}}{k!k}z^{k},$
where $\gamma$ is the Euler-Mascheroni constant and the sum converges for all
the complex $z$, $I_{ij}(x)$ can be written as
$\displaystyle I_{ij}(x)$ $\displaystyle=$ $\displaystyle
E_{1}(-\Delta_{ij}x)-E_{1}(-\Delta_{ij}z_{c})$ (8) $\displaystyle=$
$\displaystyle\ln{{(z_{c}}/{x})}+\sum\limits_{k=1}^{\infty}\frac{\Delta_{ij}^{k}}{k!k}(z_{c}^{k}-x^{k}).$
Since we always have $y_{n}(0)=0$ which implies
$|y_{n}(0)|^{2}=\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}=0,$ $None$
we can now proceed by taking the limit $x\to 0+$ in $I_{n}$
$\displaystyle I_{n}$ $\displaystyle=$ $\displaystyle\lim_{x\to
0^{+}}\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}[\ln{{(z_{c}}/{x})}+\sum\limits_{k=1}^{\infty}\frac{\Delta_{ij}^{k}}{k!k}(z_{c}^{k}-x^{k})]$
(10) $\displaystyle=$ $\displaystyle\lim_{x\to
0^{+}}\sum_{i,j=0}^{n+1}2^{q_{i}+q_{j}}(-1)^{i+j}\sum\limits_{k=1}^{\infty}\frac{\Delta_{ij}^{k}}{k!k}(z_{c}^{k}-x^{k})$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{i}+q_{j}}(-1)^{i+j}\frac{\Delta_{ij}^{k}}{k!k}z_{c}^{k}.$
To minimize $I_{n}$, UDD requires the first $n$ derivatives of $y_{n}$ vanish
while OFDD simplifies the optimization process by replacing $S(\omega)$ by a
constant. Here we attempt to minimize $I_{n}$ directly to obtain optimal pulse
sequence. We notice that at the optimal pulse locations $\delta_{j}\
(j=1,2,...,n)$, the gradient of $I_{n}$ vanishes. So we impose the following
conditions $\frac{\partial{I_{n}}}{\partial\delta_{m}}=0,$ for $m$ from $1$ to
$n$. Although ($10$) are complex infinite series, we can still explicitly
compute the derivatives of ($10$) as long as these derivatives converge. For
arbitrary $m$ we have
$\displaystyle\frac{\partial{I_{n}}}{\partial\delta_{m}}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\delta_{m}}\sum_{i,j=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{i}+q_{j}}(-1)^{i+j}\frac{\Delta_{ij}^{k}}{k!k}z_{c}^{k}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\delta_{m}}\\{\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{i}+q_{m}}(-1)^{i+m}\frac{\Delta_{im}^{k}}{k!k}z_{c}^{k}$
$\displaystyle+\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{m}+q_{i}}(-1)^{m+i}\frac{\Delta_{mi}^{k}}{k!k}z_{c}^{k}\\}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{m}+q_{i}}(-1)^{m+i}\frac{z_{c}^{k}}{k!}\mbox{i}[\Delta_{mi}^{k-1}-(-1)^{k-1}\Delta_{mi}^{k-1}].$
The terms with $k$ odd cancel, so the result can be simplified as
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\sum\limits_{k=1}^{\infty}2^{q_{m}+q_{i}+1}(-1)^{m+i}\frac{z_{c}^{2k}}{(2k)!}{\mbox{i}}^{2k}{(\delta_{m}-\delta_{i})}^{2k-1}$
$\displaystyle=$
$\displaystyle\sum_{i\neq{m}}^{n+1}\frac{1}{\delta_{m}-\delta_{i}}2^{q_{m}+q_{i}+1}(-1)^{m+i}\sum\limits_{k=1}^{\infty}\frac{z_{c}^{2k}}{(2k)!}{\mbox{i}}^{2k}{(\delta_{m}-\delta_{i})}^{2k}$
$\displaystyle=$
$\displaystyle\sum_{i\neq{m}}^{n+1}\frac{1}{\delta_{m}-\delta_{i}}2^{q_{m}+q_{i}+1}(-1)^{m+i}\\{\cos[(\delta_{m}-\delta_{i})z_{c}]-1\\}.$
Here we have used the expansion
$\cos{(z)}=\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}z^{2k}$ which
converges on the whole complex plane. From ($12$) we know that the derivatives
of $I_{n}$ indeed converge to a finite value. Thus the optimal pulse locations
$\\{\delta_{1},\delta_{2},...\delta_{n}\\}$ shall satisfy the following non-
linear equations
$\sum_{i\neq{m}}^{n+1}\frac{1}{\delta_{m}-\delta_{i}}2^{q_{m}+q_{i}+1}(-1)^{m+i}\\{\cos[(\delta_{m}-\delta_{i})z_{c}]-1\\}=0.$
$None$
Equations ($13$) are main results of this paper. The optimal sequence obtained
from ($13$) is quite different from the UDD sequence obeying
$\sum\limits_{j=1}^{n+1}2^{q_{j}}(-1)^{j}\delta_{j}^{p}=0$
for $p=\\{1,2,...n\\}$. For the ohmic spectrum, our equations incorporate the
UV cutoff frequency $\omega_{c}$, indicating that the solutions are specially
tailored to combat this kind of noise. Although the UDD sequence is universal
in suppressing decoherence, we believe that the HLODD sequence will outperform
the UDD sequence in the ohmic environment. In the next section, we use numeric
methods to illustrate the performance of HLODD sequence.
Figure 1: Comparison between UDD and HLODD sequence for different UV cutoff
frequency $\omega_{c}$. Pulse sequences $\delta_{i}$ for $n=2$ and $n=5$ are
plotted in one figure under the same $\omega_{c}$.
Figure 2: Semi-log plot of $I_{n}$ versus pulse number $n$. UDD and HLODD
sequences are compared.
## IV NUMERICAL RESULTS
We start our simulation by solving the non-linear equations ($13$) and
evaluate the decoherence function with these solutions. First, we set the
total evolution time $t=1$ without loss of generality. Then $z_{c}=\omega_{c}$
and we can concentrate on analyzing the influence of the cutoff frequency
$\omega_{c}$. Computing solutions to ($13$) for different $\omega_{c}$, we
find that the optimal pulse sequences behave differently. We also evaluate the
UDD sequence for comparison.
As shown in Fig. 1, deviation of the pulse locations $\delta_{i}$ in HLODD
sequence from their UDD counterparts increases with $\omega_{c}$. This agrees
with our intuition since UDD focuses on suppressing decoherence by minimizing
$|y_{n}(z)|$ in the neighborhood of $y_{n}(0)$, weakening its ability to
maintain small $|y_{n}(z)|$ on the other end of the spectrum. For large
$\omega_{c}$, UDD sequence is no longer optimal. In addition, we can see pulse
number $n$ plays an important role. By increasing $n$, UDD can narrow the
difference from HLODD. The difference between the two sequences when $n=2$ is
greatly reduced when we increase $n$ to $5$, see Fig. 1. Especially for the
case $\omega_{c}=1$, the difference is completely removed. However, for larger
$\omega_{c}$ this gap can’t be removed by increasing $n$.
Next, to demonstrate the optimal decoupling ability of HLODD sequence, we
compute $I_{n}$ versus $n$ while $\omega_{c}$ is chosen to be $5$. The results
are depicted in Fig. 2, and again are compared with UDD. The obtained
solutions yield a significant improvement over UDD. For fixed $n$, the HLODD
suppresses decoherence better than UDD by one or two orders of magnitude which
is in agreement with the results in Biercuk et al. (2009a); Uys et al. (2009),
where LODD and OFDD sequences are tested for 9Be+ qubits in a penning ion trap
and various spectrum. The qubit error rates are below $10^{-5}$ when $n>5$,
and we see that HLODD is capable of suppressing the error rates far below the
Fault-Tolerance error threshold Nielsen and Chuang (2000) by increasing $n$.
Furthermore, by inspecting the points on the HLODD curve, we expect the HLODD
sequence suppresses decoherence in power law $n$ as UDD.
At last, we would like to explain the numerical results in another way. If we
fixed UV cutoff frequency $\omega_{c}$ at the beginning, and compare the HLODD
performance for $t=1$, $t=5$, and $t=10$, the numerical results would be the
same since $z_{c}$ did not change. So we can also conclude that for the same
number of pulses $n$, HLODD will beat UDD with increasing total evolution time
$t$.
## V CONCLUSIONS
In this paper we analytically find the optimal pulse locations to decouple a
qubit in an ohmic environment. By deriving the analytical expressions for the
derivatives of decoherence function, we obtain a set of non-linear equations
which the optimal pulse sequence must obey. These equations are completely
different from UDD and are more accurate, because they incorporate the effect
of UV cutoff frequency $\omega_{c}$.
In our numerical simulation, the analytical results provide an improvement
over UDD sequence by an order or two of magnitude, which is consistent with
previous results in LODD and OFDD obtained by purely numerical minimization
and experimental feedback. We have to mention that the pulse performance is
influenced by the sharp UV cutoff frequency $\omega_{c}$ greatly. The larger
the UV cutoff $\omega_{c}$, the more HLODD deviates from UDD. Early work
Pasini and Uhrig (2009); Biercuk et al. (2009a); Cywinski et al. (2008); Uhrig
(2008) has pointed out that for soft large UV cutoff, UDD performs even worse
and LODD is still a better choice. However, the integral ($3$) for $S(\omega)$
with a soft cutoff is hard to analyze.
In conclusion, our work provides an analytical solution to optimal dynamical
decoupling for ohmic case. Our derivation is based on ohmic spectrum, but we
believe it can be extended to super-ohmic case
$S(\omega)\sim\omega^{\alpha}(\alpha>1)$ via slight modification.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
(No. 60774099, No. 60821091), the Chinese Academy of Sciences (KJCX3-SYW-S01),
and by the CAS Special Grant for Postgraduate Research, Innovation and
Practice.
## References
* Viola and Lloyd (1998) L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998).
* Ban (1998) M. Ban, Journal of Modern Optics 45, 2315 (1998).
* Viola et al. (1999) L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
* Gordon et al. (2008) G. Gordon, G. Kurizki, and D. A. Lidar, Phys. Rev. Lett. 101, 010403 (2008).
* Ernst et al. (1991) R. R. Ernst, G. Bodenhausen, and A. Wokaun, _Principles of Nuclear Magnetic Resonance in One and Two Dimensions_ (Clarendon Press, Oxford, 1991).
* Viola and Knill (2005) L. Viola and E. Knill, Phys. Rev. Lett. 94, 060502 (2005).
* Kern and Alber (2005) O. Kern and G. Alber, Phys. Rev. Lett. 95, 250501 (2005).
* Khodjasteh and Lidar (2005) K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95, 180501 (2005).
* Khodjasteh and Lidar (2007) K. Khodjasteh and D. A. Lidar, Phys. Rev. A 75, 062310 (2007).
* Uhrig (2007) G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007).
* Yang and Liu (2008) W. Yang and R.-B. Liu, Phys. Rev. Lett. 101, 180403 (2008).
* Pryadko and Quiroz (2008) L. P. Pryadko and G. Quiroz, Phys. Rev. A 77, 012330 (2008).
* Pasini and Uhrig (2009) S. Pasini and G. S. Uhrig (2009), eprint arXiv/0909.3439.
* Biercuk et al. (2009a) M. J. Biercuk, H. Uys, A. P. Vandevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Nature (London) 458, 996 (2009a).
* Uys et al. (2009) H. Uys, M. J. Biercuk, and J. J. Bollinger, Phys. Rev. Lett. 103, 040501 (2009).
* Leggett et al. (1987) A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).
* Weiss (2008) U. Weiss, _Quantum Dissipative Systems; 3rd ed._ , Series in Modern Condensed Matter Physics (World Scientific, Singapore, 2008).
* Cywinski et al. (2008) L. Cywinski, R. M. Lutchyn, C. P. Nave, and S. D. Sarma, Phys. Rev. B 77, 174509 (2008).
* Kuopanportti et al. (2008) P. Kuopanportti, M. Möttönen, V. Bergholm, O. Saira, J. Zhang, and K. B. Whaley, Phys. Rev. A 77, 032334 (2008).
* Uhrig (2008) G. S. Uhrig, New J. Phys. 10, 083024 (2008).
* Biercuk et al. (2009b) M. J. Biercuk, H. Uys, A. P. Vandevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Phys. Rev. A 79, 062324 (2009b).
* Abramowitz and Stegun (1964) M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables_ (Dover, New York, 1964).
* Nielsen and Chuang (2000) M. Nielsen and I. Chuang, _Quanmtum Computation and Quantum Information_ (Cambridge University Press, Cambridge, 2000).
|
arxiv-papers
| 2010-01-18T07:31:21 |
2024-09-04T02:49:07.813368
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Pan, Zairong Xi, Wei Cui",
"submitter": "Wei Cui",
"url": "https://arxiv.org/abs/1001.2960"
}
|
1001.3251
|
2010585-596Nancy, France 585
George B. Mertzios
Ignasi Sau
Shmuel Zaks
# The Recognition of Tolerance
and Bounded Tolerance Graphs
G.B. Mertzios Department of Computer Science, RWTH Aachen University, Aachen,
Germany mertzios@cs.rwth-aachen.de , I. Sau Department of Computer Science,
Technion, Haifa, Israel ignasi.sau@gmail.com and S. Zaks Department of
Computer Science, Technion, Haifa, Israel zaks@cs.technion.ac.il
###### Abstract.
Tolerance graphs model interval relations in such a way that intervals can
tolerate a certain degree of overlap without being in conflict. This subclass
of perfect graphs has been extensively studied, due to both its interesting
structure and its numerous applications. Several efficient algorithms for
optimization problems that are NP-hard on general graphs have been designed
for tolerance graphs. In spite of this, the recognition of tolerance graphs –
namely, the problem of deciding whether a given graph is a tolerance graph –
as well as the recognition of their main subclass of bounded tolerance graphs,
have been the most fundamental open problems on this class of graphs (cf. the
book on tolerance graphs [14]) since their introduction in 1982 [11]. In this
article we prove that both recognition problems are NP-complete, even in the
case where the input graph is a trapezoid graph. The presented results are
surprising because, on the one hand, most subclasses of perfect graphs admit
polynomial recognition algorithms and, on the other hand, bounded tolerance
graphs were believed to be efficiently recognizable as they are a natural
special case of trapezoid graphs (which can be recognized in polynomial time)
and share a very similar structure with them. For our reduction we extend the
notion of an _acyclic orientation_ of permutation and trapezoid graphs. Our
main tool is a new algorithm that uses _vertex splitting_ to transform a given
trapezoid graph into a permutation graph, while preserving this new acyclic
orientation property. This method of vertex splitting is of independent
interest; very recently, it has been proved a powerful tool also in the design
of efficient recognition algorithms for other classes of graphs [21].
###### Key words and phrases:
Tolerance graphs, bounded tolerance graphs, recognition, vertex splitting, NP-
complete, trapezoid graphs, permutation graphs
###### 1991 Mathematics Subject Classification:
F.2.2 Computations on discrete structures, G.2.2 Graph algorithms
## 1\. Introduction
### 1.1. Tolerance graphs and related graph classes
A simple undirected graph ${G=(V,E)}$ on $n$ vertices is a _tolerance_ graph
if there exists a collection ${I=\\{I_{i}\ |\ i=1,2,\ldots,n\\}}$ of closed
intervals on the real line and a set ${t=\\{t_{i}\ |\ i=1,2,\ldots,n\\}}$ of
positive numbers, such that for any two vertices ${v_{i},v_{j}\in V}$,
${v_{i}v_{j}\in E}$ if and only if ${|I_{i}\cap
I_{j}|\geq\min\\{t_{i},t_{j}\\}}$. The pair ${\langle I,t\rangle}$ is called a
_tolerance representation_ of $G$. If $G$ has a tolerance representation
${\langle I,t\rangle}$, such that ${t_{i}\leq|I_{i}|}$ for every
$i=1,2,\ldots,n$, then $G$ is called a _bounded tolerance_ graph and ${\langle
I,t\rangle}$ a _bounded tolerance representation_ of $G$.
Tolerance graphs were introduced in [11], in order to generalize some of the
well known applications of interval graphs. The main motivation was in the
context of resource allocation and scheduling problems, in which resources,
such as rooms and vehicles, can tolerate sharing among users [14]. If we
replace in the definition of tolerance graphs the operator _min_ by the
operator _max_ , we obtain the class of _max-tolerance_ graphs. Both tolerance
and $\max$-tolerance graphs find in a natural way applications in biology and
bioinformatics, as in the comparison of DNA sequences from different organisms
or individuals [17], by making use of a software tool like BLAST [1].
Tolerance graphs find numerous other applications in constrained-based
temporal reasoning, data transmission through networks to efficiently
scheduling aircraft and crews, as well as contributing to genetic analysis and
studies of the brain [13, 14]. This class of graphs has attracted many
research efforts [14, 12, 13, 15, 8, 4, 18, 2, 24, 22], as it generalizes in a
natural way both interval graphs (when all tolerances are equal) and
permutation graphs (when $t_{i}=|I_{i}|$ for every $i=1,2,\ldots,n$) [11]. For
a detailed survey on tolerance graphs we refer to [14].
A _comparability_ graph is a graph which can be transitively oriented. A _co-
comparability_ graph is a graph whose complement is a comparability graph. A
_trapezoid_ (resp. _parallelogram_ and _permutation_) graph is the
intersection graph of trapezoids (resp. parallelograms and line segments)
between two parallel lines $L_{1}$ and $L_{2}$ [10]. Such a representation
with trapezoids (resp. parallelograms and line segments) is called a
_trapezoid_ (resp. _parallelogram_ and _permutation_) _representation_ of
this graph. A graph is bounded tolerance if and only if it is a parallelogram
graph [19, 2]. Permutation graphs are a strict subset of parallelogram graphs
[3]. Furthermore, parallelogram graphs are a strict subset of trapezoid graphs
[25], and both are subsets of co-comparability graphs [14, 10]. On the
contrary, tolerance graphs are not even co-comparability graphs [14, 10].
Recently, we have presented in [22] a natural intersection model for general
tolerance graphs, given by parallelepipeds in the three-dimensional space.
This representation generalizes the parallelogram representation of bounded
tolerance graphs, and has been used to improve the time complexity of minimum
coloring, maximum clique, and weighted independent set algorithms on tolerance
graphs [22].
Although tolerance and bounded tolerance graphs have been studied extensively,
the recognition problems for both these classes have been the most fundamental
open problems since their introduction in 1982 [14, 10, 5]. Therefore, all
existing algorithms assume that, along with the input tolerance graph, a
tolerance representation of it is given. The only result about the complexity
of recognizing tolerance and bounded tolerance graphs is that they have a
(non-trivial) polynomial sized tolerance representation, hence the problems of
recognizing tolerance and bounded tolerance graphs are in the class NP [15].
Recently, a linear time recognition algorithm for the subclass of _bipartite
tolerance_ graphs has been presented in [5]. Furthermore, the class of
trapezoid graphs (which strictly contains parallelogram, i.e. bounded
tolerance, graphs [25]) can be also recognized in polynomial time [20, 26,
21]. On the other hand, the recognition of $\max$-tolerance graphs is known to
be NP-hard [17]. Unfortunately, the structure of $\max$-tolerance graphs
differs significantly from that of tolerance graphs ($\max$-tolerance graphs
are not even perfect, as they can contain induced $C_{5}$’s [17]), so the
technique used in [17] does not carry over to tolerance graphs.
Since very few subclasses of perfect graphs are known to be NP-hard to
recognize, it was believed that the recognition of tolerance graphs was in P.
Furthermore, as bounded tolerance graphs are equivalent to parallelogram
graphs [19, 2], which constitute a natural subclass of trapezoid graphs and
have a very similar structure, it was plausible that their recognition was
also in P.
### 1.2. Our contribution
In this article, we establish the complexity of recognizing tolerance and
bounded tolerance graphs. Namely, we prove that both problems are surprisingly
NP-complete, by providing a reduction from the monotone-Not-All-Equal-3-SAT
(monotone-NAE-3-SAT) problem. Consider a boolean formula $\phi$ in conjunctive
normal form with three literals in every clause (3-CNF), which is monotone,
i.e. no variable is negated. The formula $\phi$ is called NAE-satisfiable if
there exists a truth assignment of the variables of $\phi$, such that every
clause has at least one true variable and one false variable. Given a monotone
3-CNF formula $\phi$, we construct a trapezoid graph $H_{\phi}$, which is
parallelogram, i.e. bounded tolerance, if and only if $\phi$ is NAE-
satisfiable. Moreover, we prove that the constructed graph $H_{\phi}$ is
tolerance if and only if it is bounded tolerance. Thus, since the recognition
of tolerance and of bounded tolerance graphs are in the class NP [15], it
follows that these problems are both NP-complete. Actually, our results imply
that the recognition problems remain NP-complete even if the given graph is
trapezoid, since the constructed graph $H_{\phi}$ is trapezoid.
For our reduction we extend the notion of an _acyclic orientation_ of
permutation and trapezoid graphs. Our main tool is a new algorithm that
transforms a given trapezoid graph into a permutation graph by _splitting_
some specific vertices, while preserving this new acyclic orientation
property. One of the main advantages of this algorithm is its robustness, in
the sense that the constructed permutation graph does not depend on any
particular trapezoid representation of the input graph $G$. Moreover, besides
its use in the present paper, this approach based on splitting vertices has
been recently proved a powerful tool also in the design of efficient
recognition algorithms for other classes of graphs [21].
Organization of the paper. We first present in Section 2 several properties of
permutation and trapezoid graphs, as well as the algorithm Split-$U$, which
constructs a permutation graph from a trapezoid graph. In Section 3 we present
the reduction of the monotone-NAE-3-SAT problem to the recognition of bounded
tolerance graphs. In Section 4 we prove that this reduction can be extended to
the recognition of general tolerance graphs. Finally, we discuss the presented
results and further research directions in Section 5. Some proofs have been
omitted due to space limitations; a full version can be found in [23].
## 2\. Trapezoid graphs and representations
In this section we first introduce (in Section 2.1) the notion of an _acyclic
representation_ of permutation and of trapezoid graphs. This is followed (in
Section 2.2) by some structural properties of trapezoid graphs, which will be
used in the sequel for the splitting algorithm Split-$U$. Given a trapezoid
graph $G$ and a vertex subset $U$ of $G$ with certain properties, this
algorithm constructs a permutation graph $G^{\\#}(U)$ with $2|U|$ vertices,
which is independent on any particular trapezoid representation of the input
graph $G$.
Notation. We consider in this article simple undirected and directed graphs
with no loops or multiple edges. In an undirected graph $G$, the edge between
vertices $u$ and $v$ is denoted by $uv$, and in this case $u$ and $v$ are said
to be _adjacent_ in $G$. If the graph $G$ is directed, we denote by $uv$ the
arc from $u$ to $v$. Given a graph ${G=(V,E)}$ and a subset ${S\subseteq V}$,
$G[S]$ denotes the induced subgraph of $G$ on the vertices in $S$, and we use
$E[S]$ to denote $E(G[S])$. Whenever we deal with a trapezoid (resp.
permutation and bounded tolerance, i.e. parallelogram) graph, we will consider
w.l.o.g. a trapezoid (resp. permutation and parallelogram) representation, in
which all endpoints of the trapezoids (resp. line segments and parallelograms)
are distinct [9, 14, 16]. Given a permutation graph $P$ along with a
permutation representation $R$, we may not distinguish in the following
between a vertex of $P$ and the corresponding line segment in $R$, whenever it
is clear from the context. Furthermore, with a slight abuse of notation, we
will refer to the line segments of a permutation representation just as
_lines_.
### 2.1. Acyclic permutation and trapezoid representations
Let $P=(V,E)$ be a permutation graph and $R$ be a permutation representation
of $P$. For a vertex $u\in V$, denote by $\theta_{R}(u)$ the angle of the line
of $u$ with $L_{2}$ in $R$. The class of permutation graphs is the
intersection of comparability and co-comparability graphs [10]. Thus, given a
permutation representation $R$ of $P$, we can define two partial orders
$(V,<_{R})$ and $(V,\ll_{R})$ on the vertices of $P$ [10]. Namely, for two
vertices $u$ and $v$ of $G$, $u<_{R}v$ if and only if $uv\in E$ and
$\theta_{R}(u)<\theta_{R}(v)$, while $u\ll_{R}v$ if and only if $uv\notin E$
and $u$ lies to the left of $v$ in $R$. The partial order $(V,<_{R})$ implies
a transitive orientation $\Phi_{R}$ of $P$, such that $uv\in\Phi_{R}$ whenever
$u<_{R}v$.
Let $G=(V,E)$ be a trapezoid graph, and $R$ be a trapezoid representation of
$G$, where for any vertex $u\in V$, the trapezoid corresponding to $u$ in $R$
is denoted by $T_{u}$. Since trapezoid graphs are also co-comparability graphs
[10], we can similarly define the partial order $(V,\ll_{R})$ on the vertices
of $G$, such that $u\ll_{R}v$ if and only if $uv\notin E$ and $T_{u}$ lies
completely to the left of $T_{v}$ in $R$. In this case, we may denote also
$T_{u}\ll_{R}T_{v}$.
In a given trapezoid representation $R$ of a trapezoid graph $G$, we denote by
$l(T_{u})$ and $r(T_{u})$ the left and the right line of $T_{u}$ in $R$,
respectively. Similarly to the case of permutation graphs, we use the relation
$\ll_{R}$ for the lines $l(T_{u})$ and $r(T_{u})$, e.g.
${l(T_{u})\ll_{R}r(T_{v})}$ means that the line $l(T_{u})$ lies to the left of
the line $r(T_{v})$ in $R$. Moreover, if the trapezoids of all vertices of a
subset $S\subseteq V$ lie completely to the left (resp. right) of the
trapezoid $T_{u}$ in $R$, we write $R(S)\ll_{R}T_{u}$ (resp.
$T_{u}\ll_{R}R(S)$). Note that there are several trapezoid representations of
a particular trapezoid graph $G$. Given one such representation $R$, we can
obtain another one $R^{\prime}$ by _vertical axis flipping_ of $R$, i.e.
$R^{\prime}$ is the mirror image of $R$ along an imaginary line perpendicular
to $L_{1}$ and $L_{2}$. Moreover, we can obtain another representation
$R^{\prime\prime}$ of $G$ by _horizontal axis flipping_ of $R$, i.e.
$R^{\prime\prime}$ is the mirror image of $R$ along an imaginary line parallel
to $L_{1}$ and $L_{2}$. We will extensively use these two operations
throughout the article.
###### Definition 2.1.
Let $P$ be a permutation graph with $2n$ vertices
$\\{u_{1}^{1},u_{1}^{2},u_{2}^{1},u_{2}^{2},\ldots,u_{n}^{1},u_{n}^{2}\\}$.
Let $R$ be a permutation representation and $\Phi_{R}$ be the corresponding
transitive orientation of $P$. The simple directed graph $F_{R}$ is obtained
by merging $u_{i}^{1}$ and $u_{i}^{2}$ into a single vertex $u_{i}$, for every
$i=1,2,\ldots,n$, where the arc directions of $F_{R}$ are implied by the
corresponding directions in $\Phi_{R}$. Then,
1. (1)
$R$ is an _acyclic permutation representation with respect to_
$\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{n}$***To simplify the presentation, we use
throughout the paper $\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{n}$ to denote the set
of $n$ unordered pairs
$\\{u_{1}^{1},u_{1}^{2}\\},\\{u_{2}^{1},u_{2}^{2}\\},\ldots,\\{u_{n}^{1},u_{n}^{2}\\}$.
, if $F_{R}$ has no directed cycle,
2. (2)
$P$ is an _acyclic permutation graph with respect to_
$\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{n}$, if $P$ has an acyclic representation
$R$ with respect to $\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{n}$.
###### Definition 2.2.
Let $G$ be a trapezoid graph with $n$ vertices and $R$ be a trapezoid
representation of $G$. Let $P$ be the permutation graph with $2n$ vertices
corresponding to the left and right lines of the trapezoids in $R$, $R_{P}$ be
the permutation representation of $P$ induced by $R$, and
$\\{u_{i}^{1},u_{i}^{2}\\}$ be the vertices of $P$ that correspond to the same
vertex $u_{i}$ of $G$, $i=1,2,\ldots,n$. Then,
1. (1)
$R$ is an _acyclic trapezoid representation_ , if $R_{P}$ is an acyclic
permutation representation with respect to
$\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{n}$,
2. (2)
$G$ is an _acyclic trapezoid graph_ , if it has an acyclic representation $R$.
The following lemma follows easily from Definitions 2.1 and 2.2.
###### Lemma 2.3.
Any parallelogram graph is an acyclic trapezoid graph.
### 2.2. Structural properties of trapezoid graphs
In the following, we state some definitions concerning an arbitrary simple
undirected graph ${G=(V,E)}$, which are useful for our analysis. Although
these definitions apply to any graph, we will use them only for trapezoid
graphs. Similar definitions, for the restricted case where the graph $G$ is
connected, were studied in [6]. For $u\in V$ and $U\subseteq V$, $N(u)=\\{v\in
V\ |\ uv\in E\\}$ is the set of adjacent vertices of $u$ in $G$,
${N[u]=N(u)\cup\\{u\\}}$, and ${N(U)=\bigcup_{u\in U}N(u)\setminus U}$. If
${N(U)\subseteq N(W)}$ for two vertex subsets $U$ and $W$, then $U$ is said to
be _neighborhood dominated_ by $W$. Clearly, the relationship of neighborhood
domination is transitive.
Let ${C_{1},C_{2},\ldots,C_{\omega},\ \omega\geq 1,}$ be the connected
components of ${G\setminus N[u]}$ and ${V_{i}=V(C_{i})}$,
${i=1,2,\ldots,\omega}$. For simplicity of the presentation, we will identify
in the sequel the component $C_{i}$ and its vertex set $V_{i}$,
${i=1,2,\ldots,\omega}$. For ${i=1,2,\ldots,\omega}$, the _neighborhood
domination closure_ of $V_{i}$ with respect to $u$ is the set
${D_{u}(V_{i})=\\{V_{p}\ |\ N(V_{p})\subseteq N(V_{i}),\
p=1,2,\ldots,\omega\\}}$ of connected components of ${G\setminus N[u]}$. A
component $V_{i}$ is called a _master component_ of $u$ if
${|D_{u}(V_{i})|\geq|D_{u}(V_{j})|}$ for all ${j=1,2,\ldots,\omega}$. The
_closure complement_ of the neighborhood domination closure ${D_{u}(V_{i})}$
is the set ${D_{u}^{\ast}(V_{i})=\\{V_{1},V_{2},\ldots,V_{\omega}\\}\setminus
D_{u}(V_{i})}$. Finally, for a subset
${S\subseteq\\{V_{1},V_{2},\ldots,V_{\omega}\\}}$, a component $V_{j}\in S$ is
called _maximal_ if there is no component $V_{k}\in S$ such that
${N(V_{j})\subsetneqq N(V_{k})}$.
For example, consider the trapezoid graph $G$ with vertex set
$\\{u,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3},v_{4}\\}$, which is given by the
trapezoid representation $R$ of Figure 1. The connected components of
${G\setminus N[u]=\\{v_{1},v_{2},v_{3},v_{4}\\}}$ are ${V_{1}=\\{v_{1}\\}}$,
${V_{2}=\\{v_{2}\\}}$, ${V_{3}=\\{v_{3}\\}}$, and ${V_{4}=\\{v_{4}\\}}$. Then,
${N(V_{1})=\\{u_{1}\\}}$, ${N(V_{2})=\\{u_{1},u_{3}\\}}$,
${N(V_{3})=\\{u_{2},u_{3}\\}}$, and ${N(V_{4})=\\{u_{3}\\}}$. Hence,
${D_{u}(V_{1})=\\{V_{1}\\}}$, ${D_{u}(V_{2})=\\{V_{1},V_{2},V_{4}\\}}$,
${D_{u}(V_{3})=\\{V_{3},V_{4}\\}}$, and ${D_{u}(V_{4})=\\{V_{4}\\}}$; thus,
$V_{2}$ is the only master component of $u$. Furthermore,
${D_{u}^{\ast}(V_{1})=\\{V_{2},V_{3},V_{4}\\}}$,
${D_{u}^{\ast}(V_{2})=\\{V_{3}\\}}$,
${D_{u}^{\ast}(V_{3})=\\{V_{1},V_{2}\\}}$, and
${D_{u}^{\ast}(V_{4})=\\{V_{1},V_{2},V_{3}\\}}$.
Figure 1. A trapezoid representation $R$ of a trapezoid graph $G$.
###### Lemma 2.4.
Let $G$ be a simple graph, $u$ be a vertex of $G$, and let
$V_{1},V_{2},\ldots,V_{\omega},\ \omega\geq 1$, be the connected components of
${G\setminus N[u]}$. If $V_{i}$ is a master component of $u$, such that
${D_{u}^{\ast}(V_{i})\neq\emptyset}$, then
${D_{u}^{\ast}(V_{j})\neq\emptyset}$ for every component $V_{j}$ of
$G\setminus N[u]$.
In the following we investigate several properties of trapezoid graphs, in
order to derive the vertex-splitting algorithm Split-$U$ in Section 2.3.
###### Remark 2.5.
Similar properties of trapezoid graphs have been studied in [6], leading to
another vertex-splitting algorithm, called Split-All. However, the algorithm
proposed in [6] is incorrect, since it is based on an incorrect property†††In
Observation 3.1(5) of [6], it is claimed that for an arbitrary trapezoid
representation $R$ of a connected trapezoid graph $G$, where $V_{i}$ is a
master component of $u$ such that ${D_{u}^{\ast}(V_{i})\neq\emptyset}$ and
${R(V_{i})\ll_{R}T_{u}}$, it holds
${R(D_{u}(V_{i}))\ll_{R}T_{u}\ll_{R}R(D_{u}^{\ast}(V_{i}))}$. However, the
first part of the latter inequality is not true. For instance, in the
trapezoid graph $G$ of Figure 1, ${V_{2}=\\{v_{2}\\}}$ is a master component
of $u$, where
$D_{u}^{\ast}(V_{2})=\\{V_{3}\\}=\\{\\{v_{3}\\}\\}{\neq\emptyset}$ and
${R(V_{2})\ll_{R}T_{u}}$. However, $V_{4}=\\{v_{4}\\}\in D_{u}(V_{2})$ and
${T_{u}\ll_{R}}T_{v_{4}}$, and thus, ${R(D_{u}(V_{2}))\not\ll_{R}T}_{u}$. , as
was also verified by [7]. In the sequel of this section, we present new
definitions and properties. In the cases where a similarity arises with those
of [6], we refer to it specifically.
###### Lemma 2.6.
Let $R$ be a trapezoid representation of a trapezoid graph $G$, and $V_{i}$ be
a master component of a vertex $u$ of $G$, such that
$R(V_{i}){\ll}_{R}{T_{u}}$. Then, $T_{u}{\ll}_{R}R(V_{j})$ for every component
$V_{j}\in D_{u}^{\ast}(V_{i})$.
###### Definition 2.7.
Let $G$ be a trapezoid graph, $u$ be a vertex of $G$, and $V_{i}$ be an
arbitrarily chosen master component of $u$. Then, $\delta_{u}=V_{i}$ and
1. (1)
if $D_{u}^{\ast}(V_{i})=\emptyset$, then $\delta_{u}^{\ast}=\emptyset$.
2. (2)
if $D_{u}^{\ast}(V_{i})\neq\emptyset$, then $\delta_{u}^{\ast}={V_{j}}$, for
an arbitrarily chosen maximal component $V_{j}\in{D_{u}^{\ast}(V_{i})}$.
Actually, as we will show in Lemma 2.10, the arbitrary choice of the
components $V_{i}$ and $V_{j}$ in Definition 2.7 does not affect essentially
the structural properties of $G$ that we will investigate in the sequel. From
now on, whenever we speak about $\delta_{u}$ and $\delta_{u}^{\ast}$, we
assume that these arbitrary choices of $V_{i}$ and $V_{j}$ have been already
made.
###### Definition 2.8.
Let $G$ be a trapezoid graph and $u$ be a vertex of $G$. The vertices of
$N(u)$ are partitioned into four possibly empty sets:
1. (1)
$N_{0}(u)$: vertices not adjacent to either ${\delta_{u}}$ or
${\delta_{u}^{\ast}}$.
2. (2)
$N_{1}(u)$: vertices adjacent to ${\delta_{u}}$ but not to
${\delta_{u}^{\ast}}$.
3. (3)
$N_{2}(u)$: vertices adjacent to ${\delta_{u}^{\ast}}$ but not to
${\delta_{u}}$.
4. (4)
$N_{12}(u)$: vertices adjacent to both ${\delta_{u}}$ and
${\delta_{u}^{\ast}}$.
In the following definition we partition the neighbors of a vertex of a
trapezoid graph $G$ into four possibly empty sets. Note that these sets depend
on a given trapezoid representation $R$ of $G$, in contrast to the four sets
of Definition 2.8 that depend only on the graph $G$ itself.
###### Definition 2.9.
Let $G$ be a trapezoid graph, $R$ be a representation of $G$, and $u$ be a
vertex of $G$. Denote by $D_{1}(u,R)$ and $D_{2}(u,R)$ the sets of trapezoids
of $R$ that lie completely to the left and to the right of $T_{u}$ in $R$,
respectively. Then, the vertices of $N(u)$ are partitioned into four possibly
empty sets:
1. (1)
$N_{0}(u,R)$: vertices not adjacent to either $D_{1}(u,R)$ or $D_{2}(u,R)$.
2. (2)
$N_{1}(u,R)$: vertices adjacent to $D_{1}(u,R)$ but not to $D_{2}(u,R)$.
3. (3)
$N_{2}(u,R)$: vertices adjacent to $D_{2}(u,R)$ but not to $D_{1}(u,R)$.
4. (4)
$N_{12}(u,R)$: vertices adjacent to both $D_{1}(u,R)$ and $D_{2}(u,R)$.
Suppose now that $\delta_{u}^{\ast}\neq\emptyset$, and let $V_{i}$ be the
master component of $u$ that corresponds to $\delta_{u}$, cf. Definition 2.7.
Then, given any trapezoid representation $R$ of $G$, we may assume w.l.o.g.
that $R(V_{i}){\ll}_{R}{T_{u}}$, by possibly performing a vertical axis
flipping of $R$. The following lemma connects Definitions 2.8 and 2.9; in
particular, it states that, if $R(V_{i})\ll_{R}T_{u}$, then the partitions of
the set $N(u)$ defined in these definitions coincide. This lemma will enable
us to use in the vertex splitting (cf. Definition 2.11) the partition of the
set $N(u)$ defined in Definition 2.8, independently of any trapezoid
representation $R$ of $G$, and regardless of any particular connected
components $V_{i}$ and $V_{j}$ of $G\setminus N[u]$.
###### Lemma 2.10.
Let $G$ be a trapezoid graph, $R$ be a representation of $G$, and $u$ be a
vertex of $G$ with ${\delta_{u}^{\ast}\neq\emptyset}$. Let $V_{i}$ be the
master component of $u$ that corresponds to $\delta_{u}$. If
${R(V_{i}){\ll}_{R}{T_{u}}}$, then ${N_{X}(u)=N_{X}(u,R)}$ for every
$X\in\\{0,1,2,12\\}$.
### 2.3. A splitting algorithm
We define now the splitting of a vertex $u$ of a trapezoid graph $G$, where
${\delta_{u}^{\ast}\neq\emptyset}$. Note that this splitting operation does
not depend on any trapezoid representation of $G$. Intuitively, if the graph
$G$ was given along with a specific trapezoid representation $R$, this would
have meant that we replace the trapezoid $T_{u}$ in $R$ by its two lines
$l(T_{u})$ and $r(T_{u})$.
###### Definition 2.11.
Let $G$ be a trapezoid graph and $u$ be a vertex of $G$, where
${\delta_{u}^{\ast}\neq\emptyset}$. The graph $G^{\\#}(u)$ obtained by the
_vertex splitting_ of $u$ is defined as follows:
1. (1)
$V(G^{\\#}(u))=V(G)\setminus\\{u\\}\cup\\{u_{1},u_{2}\\}$, where $u_{1}$ and
$u_{2}$ are the two new vertices.
2. (2)
$E(G^{\\#}(u))=E[V(G)\setminus\\{u\\}]\cup\\{u_{1}x\ |\ x\in
N_{1}(u)\\}\cup\\{u_{2}x\ |\ x\in N_{2}(u)\\}\cup\\{u_{1}x,u_{2}x\ |\ x\in
N_{12}(u)\\}$.
The vertices $u_{1}$ and $u_{2}$ are the _derivatives_ of vertex $u$.
We state now the notion of a standard trapezoid representation with respect to
a particular vertex.
###### Definition 2.12.
Let $G$ be a trapezoid graph and $u$ be a vertex of $G$, where
${\delta_{u}^{\ast}\neq\emptyset}$. A trapezoid representation $R$ of $G$ is
_standard with respect to_ $u$, if the following properties are satisfied:
1. (1)
$l(T_{u})\ll_{R}R(N_{0}(u)\cup N_{2}(u))$.
2. (2)
$R(N_{0}(u)\cup N_{1}(u))\ll_{R}r(T_{u})$.
Now, given a trapezoid graph $G$ and a vertex subset
$U=\\{u_{1},u_{2},\ldots,u_{k}\\}$, such that
$\delta_{u_{i}}^{\ast}\neq\emptyset$ for every $i=1,2,\ldots,k$, Algorithm
Split-$U$ returns a graph $G^{\\#}(U)$ by splitting every vertex of $U$
exactly once. At every step, Algorithm Split-$U$ splits a vertex of $U$, and
finally, it removes all vertices of the set $V(G)\setminus U$, which have not
been split.
Algorithm 1 Split-$U$
0: A trapezoid graph $G$ and a vertex subset
$U=\\{u_{1},u_{2},\ldots,u_{k}\\}$, such that
$\delta_{u_{i}}^{*}\neq\emptyset$ for all $i=1,2,\ldots,k$
0: The permutation graph $G^{\\#}(U)$
$\overline{U}\leftarrow V(G)\setminus U$; $H_{0}\leftarrow G$
for $i=1$ to $k$ do
$H_{i}\leftarrow H_{i-1}^{\\#}(u_{i})$ {$H_{i}$ is obtained by the vertex
splitting of $u_{i}$ in $H_{i-1}$}
$G^{\\#}(U)\leftarrow H_{k}[V(H_{k})\setminus\overline{U}]$ {remove from
$H_{k}$ all unsplitted vertices}
return $G^{\\#}(U)$
###### Remark 2.13.
As mentioned in Remark 2.5, a similar algorithm, called Split-All, was
presented in [6]. We would like to emphasize here the following four
differences between the two algorithms. First, that Split-All gets as input a
sibling-free graph $G$ (two vertices $u,v$ of a graph $G$ are called
_siblings_ , if $N[u]=N[v]$; $G$ is called _sibling-free_ if $G$ has no pair
of sibling vertices), while our Algorithm Split-$U$ gets as an input any graph
(though, we will use it only for trapezoid graphs), which may contain also
pairs of sibling vertices. Second, Split-All splits all the vertices of the
input graph, while Split-$U$ splits only a subset of them, which satisfy a
special property. Third, the order of vertices that are split by Split-All
depends on a certain property (inclusion-minimal neighbor set), while
Split-$U$ splits the vertices in an arbitrary order. Last, the main difference
between these two algorithms is that they perform a different vertex splitting
operation at every step, since Definitions 2.7 and 2.8 do not comply with the
corresponding Definitions 4.1 and 4.2 of [6].
###### Theorem 2.14.
Let $G$ be a trapezoid graph and $U=\\{u_{1},u_{2},\ldots,u_{k}\\}$ be a
vertex subset of $G$, such that $\delta_{u_{i}}^{\ast}\neq\emptyset$ for every
$i=1,2,\ldots,k$. Then, the graph $G^{\\#}(U)$ obtained by Algorithm
Split-$U$, is a permutation graph with $2k$ vertices. Furthermore, if $G$ is
acyclic, then $G^{\\#}(U)$ is acyclic with respect to
$\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{k}$, where $u_{i}^{1}$ and $u_{i}^{2}$ are
the derivatives of $u_{i}$, $i=1,2,\ldots,k$.
## 3\. The recognition of bounded tolerance graphs
In this section we provide a reduction from the _monotone-Not-All-Equal-3-SAT
(monotone-NAE-3-SAT)_ problem to the problem of recognizing whether a given
graph is a bounded tolerance graph. The problem of deciding whether a given
monotone 3-CNF formula $\phi$ is NAE-satisfiable is known to be NP-complete.
We can assume w.l.o.g. that each clause has three distinct literals, i.e.
variables. Given a monotone 3-CNF formula $\phi$, we construct in polynomial
time a trapezoid graph $H_{\phi}$, such that $H_{\phi}$ is a bounded tolerance
graph if and only if $\phi$ is NAE-satisfiable. To this end, we construct
first a permutation graph $P_{\phi}$ and a trapezoid graph $G_{\phi}$.
### 3.1. The permutation graph $P_{\phi}$
Consider a monotone 3-CNF formula
${\phi=\alpha_{1}\wedge\alpha_{2}\wedge\ldots\wedge\alpha_{k}}$ with $k$
clauses and $n$ boolean variables ${x_{1},x_{2},\ldots,x_{n}}$, such that
${\alpha_{i}=(x_{r_{i,1}}\vee x_{r_{i,2}}\vee x_{r_{i,3}})}$ for
${i=1,2,\ldots,k}$, where ${1\leq r_{i,1}<r_{i,2}<r_{i,3}\leq n}$. We
construct the permutation graph $P_{\phi}$, along with a permutation
representation $R_{P}$ of $P_{\phi}$, as follows. Let $L_{1}$ and $L_{2}$ be
two parallel lines and let $\theta(\ell)$ denote the angle of the line $\ell$
with $L_{2}$ in $R_{P}$. For every clause $\alpha_{i}$, ${i=1,2,\ldots,k}$, we
correspond to each of the literals, i.e. variables, $x_{r_{i,1}}$,
$x_{r_{i,2}}$, and $x_{r_{i,3}}$ a pair of intersecting lines with endpoints
on $L_{1}$ and $L_{2}$. Namely, we correspond to the variable $x_{r_{i,1}}$
the pair $\\{{a_{i},c_{i}\\}}$, to $x_{r_{i,2}}$ the pair
$\\{{e_{i},b_{i}\\}}$ and to $x_{r_{i,3}}$ the pair $\\{{d_{i},f_{i}\\}}$,
respectively, such that ${\theta(a_{i})>\theta(c_{i})}$,
${\theta(e_{i})>\theta(b_{i})}$, ${\theta(d_{i})>\theta(f_{i})}$, and such
that the lines ${a_{i},c_{i}}$ lie completely to the left of ${e_{i},b_{i}}$
in $R_{P}$, and ${e_{i},b_{i}}$ lie completely to the left of ${d_{i},f_{i}}$
in $R_{P}$, as it is illustrated in Figure 2. Denote the lines that correspond
to the variable $x_{r_{i,j}}$, $j=1,2,3$, by $\ell_{i,j}^{1}$ and
$\ell_{i,j}^{2}$, respectively, such that
${\theta(\ell_{i,j}^{1})>\theta(\ell_{i,j}^{2})}$. That is,
$(\ell_{i,1}^{1},\ell_{i,1}^{2})=(a_{i},c_{i})$,
$(\ell_{i,2}^{1},\ell_{i,2}^{2})=(e_{i},b_{i})$, and
$(\ell_{i,3}^{1},\ell_{i,3}^{2})=(d_{i},f_{i})$. Note that no line of a pair
$\\{\ell_{i,j}^{1},\ell_{i,j}^{2}\\}$ intersects with a line of another pair
$\\{\ell_{i^{\prime},j^{\prime}}^{1},\ell_{i^{\prime},j^{\prime}}^{2}\\}$.
Figure 2. The six lines of the permutation graph $P_{\phi}$, which correspond
to the clause ${\alpha_{i}=(x_{r_{i,1}}\vee x_{r_{i,2}}\vee x_{r_{i,3}})}$ of
the boolean formula $\phi$.
Denote by $S_{p}$, $p=1,2,\ldots,n$, the set of pairs
$\\{\ell_{i,j}^{1},\ell_{i,j}^{2}\\}$ that correspond to the variable $x_{p}$,
i.e. $r_{i,j}=p$. We order the pairs $\\{\ell_{i,j}^{1},\ell_{i,j}^{2}\\}$
such that any pair of $S_{p_{1}}$ lies completely to the left of any pair of
$S_{p_{2}}$, whenever $p_{1}<p_{2}$, while the pairs that belong to the same
set $S_{p}$ are ordered arbitrarily. For two consecutive pairs
$\\{\ell_{i,j}^{1},\ell_{i,j}^{2}\\}$ and
$\\{\ell_{i^{\prime},j^{\prime}}^{1},\ell_{i^{\prime},j^{\prime}}^{2}\\}$ in
$S_{p}$, where $\\{\ell_{i,j}^{1},\ell_{i,j}^{2}\\}$ lies to the left of
$\\{\ell_{i^{\prime},j^{\prime}}^{1},\ell_{i^{\prime},j^{\prime}}^{2}\\}$, we
add a pair
$\\{u_{i,j}^{i^{\prime},j^{\prime}},v_{i,j}^{i^{\prime},j^{\prime}}\\}$ of
parallel lines that intersect both $\ell_{i,j}^{1}$ and
$\ell_{i^{\prime},j^{\prime}}^{1}$, but no other line. Note that
$\theta(\ell_{i,j}^{1})>\theta(u_{i,j}^{i^{\prime},j^{\prime}})$ and
$\theta(\ell_{i^{\prime},j^{\prime}}^{1})>\theta(u_{i,j}^{i^{\prime},j^{\prime}})$,
while
$\theta(u_{i,j}^{i^{\prime},j^{\prime}})=\theta(v_{i,j}^{i^{\prime},j^{\prime}})$.
This completes the construction. Denote the resulting permutation graph by
$P_{\phi}$, and the corresponding permutation representation of $P_{\phi}$ by
$R_{P}$. Observe that $P_{\phi}$ has $n$ connected components, which are
called _blocks_ , one for each variable $x_{1},x_{2},\ldots,x_{n}$.
An example of the construction of $P_{\phi}$ and $R_{P}$ from $\phi$ with
$k=3$ clauses and $n=4$ variables is illustrated in Figure 3. In this figure,
the lines $u_{i,j}^{i^{\prime},j^{\prime}}$ and
$v_{i,j}^{i^{\prime},j^{\prime}}$ are drawn in bold.
The formula $\phi$ has $3k$ literals, and thus the permutation graph
$P_{\phi}$ has $6k$ lines $\ell_{i,j}^{1},\ell_{i,j}^{2}$ in $R_{P}$, one pair
for each literal. Furthermore, two lines
$u_{i,j}^{i^{\prime},j^{\prime}},v_{i,j}^{i^{\prime},j^{\prime}}$ correspond
to each pair of consecutive pairs $\\{\ell_{i,j}^{1},\ell_{i,j}^{2}\\}$ and
$\\{\ell_{i^{\prime},j^{\prime}}^{1},\ell_{i^{\prime},j^{\prime}}^{2}\\}$ in
$R_{P}$, except for the case where these pairs of lines belong to different
variables, i.e. when $r_{i,j}\neq r_{i^{\prime},j^{\prime}}$. Therefore, since
$\phi$ has $n$ variables, there are $2(3k-n)=6k-2n$ lines
$u_{i,j}^{i^{\prime},j^{\prime}},v_{i,j}^{i^{\prime},j^{\prime}}$ in $R_{P}$.
Thus, $R_{P}$ has in total $12k-2n$ lines, i.e. $P_{\phi}$ has $12k-2n$
vertices. In the example of Figure 3, $k=3$, $n=4$, and thus, $P_{\phi}$ has
$28$ vertices.
Figure 3. The permutation representation $R_{P}$ of the permutation graph
$P_{\phi}$ for $\phi=\alpha_{1}\wedge\alpha_{2}\wedge\alpha_{3}=(x_{1}\vee
x_{2}\vee x_{3})\wedge(x_{2}\vee x_{3}\vee x_{4})\wedge(x_{1}\vee x_{2}\vee
x_{4})$.
Let $m=6k-n$, where $2m$ is the number of vertices in $P_{\phi}$. We group the
lines of $R_{P}$, i.e. the vertices of $P_{\phi}$, into pairs
$\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{m}$, as follows. For every clause
$\alpha_{i}$, $i=1,2,\ldots,k$, we group the lines
$a_{i},b_{i},c_{i},d_{i},e_{i},f_{i}$ into the three pairs
$\\{a_{i},b_{i}\\}$, $\\{c_{i},d_{i}\\}$, and $\\{e_{i},f_{i}\\}$. The
remaining lines are grouped naturally according to the construction; namely,
every two lines
$\\{u_{i,j}^{i^{\prime},j^{\prime}},v_{i,j}^{i^{\prime},j^{\prime}}\\}$
constitute a pair.
###### Lemma 3.1.
If the permutation graph $P_{\phi}$ is acyclic with respect to
$\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{m}$ then the formula $\phi$ is NAE-
satisfiable.
The truth assignment $(x_{1},x_{2},x_{3},x_{4})=(1,1,0,0)$ is NAE-satisfying
for the formula $\phi$ of Figure 3. The acyclic permutation representation
$R_{0}$ of $P_{\phi}$ with respect to $\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{m}$,
which corresponds to this assignment, can be obtained from $R_{P}$ by
performing a horizontal axis flipping of the two blocks that correspond to the
variables $x_{3}$ and $x_{4}$, respectively.
### 3.2. The trapezoid graphs $G_{\phi}$ and $H_{\phi}$
Let $\\{u_{i}^{1},u_{i}^{2}\\}_{i=1}^{m}$ be the pairs of vertices in the
permutation graph $P_{\phi}$ and $R_{P}$ be its permutation representation. We
construct now from $P_{\phi}$ the trapezoid graph $G_{\phi}$ with $m$ vertices
$\\{u_{1},u_{2},\ldots,u_{m}\\}$, as follows. We replace in the permutation
representation $R_{P}$ for every $i=1,2,\ldots,m$ the lines $u_{i}^{1}$ and
$u_{i}^{2}$ by the trapezoid $T_{u_{i}}$, which has $u_{i}^{1}$ and
$u_{i}^{2}$ as its left and right lines, respectively. Let $R_{G}$ be the
resulting trapezoid representation of $G_{\phi}$.
Finally, we construct from $G_{\phi}$ the trapezoid graph $H_{\phi}$ with $7m$
vertices, by adding to every trapezoid $T_{u_{i}}$, $i=1,2,\ldots,m$, six
parallelograms $T_{u_{i,1}},T_{u_{i,2}},\ldots,T_{u_{i,6}}$ in the trapezoid
representation $R_{G}$, as follows. Let $\varepsilon$ be the smallest distance
in $R_{G}$ between two different endpoints on $L_{1}$, or on $L_{2}$. The
right (resp. left) line of $T_{u_{1,1}}$ lies to the right (resp. left) of
$u_{1}^{1}$, and it is parallel to it at distance $\frac{\varepsilon}{2}$. The
right (resp. left) line of $T_{u_{1,2}}$ lies to the left of $u_{1}^{1}$, and
it is parallel to it at distance $\frac{\varepsilon}{4}$ (resp.
$\frac{3\varepsilon}{4}$). Moreover, the right (resp. left) line of
$T_{u_{1,3}}$ lies to the left of $u_{1}^{1}$, and it is parallel to it at
distance $\frac{3\varepsilon}{8}$ (resp. $\frac{7\varepsilon}{8}$). Similarly,
the left (resp. right) line of $T_{u_{1,4}}$ lies to the left (resp. right) of
$u_{1}^{2}$, and it is parallel to it at distance $\frac{\varepsilon}{2}$. The
left (resp. right) line of $T_{u_{1,5}}$ lies to the right of $u_{1}^{2}$, and
it is parallel to it at distance $\frac{\varepsilon}{4}$ (resp.
$\frac{3\varepsilon}{4}$). Finally, the right (resp. left) line of
$T_{u_{1,6}}$ lies to the right of $u_{1}^{2}$, and it is parallel to it at
distance $\frac{3\varepsilon}{8}$ (resp. $\frac{7\varepsilon}{8}$), as
illustrated in Figure 4.
After adding the parallelograms $T_{u_{1,1}},T_{u_{1,2}},\ldots,T_{u_{1,6}}$
to a trapezoid $T_{u_{1}}$, we update the smallest distance $\varepsilon$
between two different endpoints on $L_{1}$, or on $L_{2}$ in the resulting
representation, and we continue the construction iteratively for all
$i=2,\ldots,m$. Denote by $H_{\phi}$ the resulting trapezoid graph with $7m$
vertices, and by $R_{H}$ the corresponding trapezoid representation. Note that
in $R_{H}$, between the endpoints of the parallelograms $T_{u_{i,1}}$,
$T_{u_{i,2}}$, and $T_{u_{i,3}}$ (resp. $T_{u_{i,4}}$, $T_{u_{i,5}}$, and
$T_{u_{i,6}}$) on $L_{1}$ and $L_{2}$, there are no other endpoints of
$H_{\phi}$, except those of $u_{i}^{1}$ (resp. $u_{i}^{2}$), for every
$i=1,2,\ldots,m$. Furthermore, note that $R_{H}$ is standard with respect to
$u_{i}$, for every $i=1,2,\ldots,m$.
Figure 4. The addition of the six parallelograms
$T_{u_{i,1}},T_{u_{i,2}},\ldots,T_{u_{i,6}}$ to the trapezoid $T_{u_{i}}$,
${i=1,2,\ldots,m}$, in the construction of the trapezoid graph $H_{\phi}$ from
$G_{\phi}$.
###### Theorem 3.2.
The formula $\phi$ is NAE-satisfiable if and only if the trapezoid graph
$H_{\phi}$ is a bounded tolerance graph.
For the sufficiency part of the proof of Theorem 3.2, the algorithm Split-All
plays a crucial role. Namely, given the parallelogram graph $H_{\phi}$ (which
is acyclic trapezoid by Lemma 2.3), we construct with this algorithm the
acyclic permutation graph $P_{\phi}$ and then a NAE-satisfying assignment of
the formula $\phi$. Since monotone-NAE-3-SAT is NP-complete, the problem of
recognizing bounded tolerance graphs is NP-hard by Theorem 3.2. Moreover,
since this problem lies in NP [15], we summarize our results as follows.
###### Theorem 3.3.
Given a graph $G$, it is NP-complete to decide whether it is a bounded
tolerance graph.
## 4\. The recognition of tolerance graphs
In this section we show that the reduction from the monotone-NAE-3-SAT problem
to the problem of recognizing bounded tolerance graphs presented in Section 3,
can be extended to the problem of recognizing general tolerance graphs. In
particular, we prove that the constructed trapezoid graph $H_{\phi}$ is a
tolerance graph if and only if it is a bounded tolerance graph. Then, the main
result of this section follows.
###### Theorem 4.1.
Given a graph $G$, it is NP-complete to decide whether it is a tolerance
graph. The problem remains NP-complete even if the given graph $G$ is known to
be a trapezoid graph.
## 5\. Concluding remarks
In this article we proved that both tolerance and bounded tolerance graph
recognition problems are NP-complete, by providing a reduction from the
monotone-NAE-3-SAT problem, thus answering a longstanding open question. The
recognition of unit and of proper tolerance graphs, as well as of any other
subclass of tolerance graphs, except bounded tolerance and bipartite tolerance
graphs [5], remain interesting open problems [14].
## References
* [1] S. F. Altschul, W. Gish, W. Miller, E. W. Myers, and D. J. Lipman. Basic local alignment search tool. Journal of molecular biology, 215(3):403–410, 1990.
* [2] K. P. Bogart, P. C. Fishburn, G. Isaak, and L. Langley. Proper and unit tolerance graphs. Discrete Applied Mathematics, 60(1-3):99–117, 1995.
* [3] A. Brandstädt, V. B. Le, and J. P. Spinrad. Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications, 1999.
* [4] A. H. Busch. A characterization of triangle-free tolerance graphs. Discrete Applied Mathematics, 154(3):471–477, 2006.
* [5] A. H. Busch and G. Isaak. Recognizing bipartite tolerance graphs in linear time. In Proceedings of the 33rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 12–20, 2007.
* [6] F. Cheah and D. G. Corneil. On the structure of trapezoid graphs. Discrete Applied Mathematics, 66(2):109–133, 1996.
* [7] F. Cheah and D. G. Corneil, 2009. Personal communication.
* [8] S. Felsner. Tolerance graphs and orders. Journal of Graph Theory, 28:129–140, 1998.
* [9] P. C. Fishburn and W. Trotter. Split semiorders. Discrete Mathematics, 195:111–126, 1999.
* [10] M. C. Golumbic. Algorithmic graph theory and perfect graphs (Annals of Discrete Mathematics, Vol. 57). North-Holland Publishing Co., 2nd edition, 2004.
* [11] M. C. Golumbic and C. L. Monma. A generalization of interval graphs with tolerances. In Proceedings of the 13th Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium 35, pages 321–331, 1982.
* [12] M. C. Golumbic, C. L. Monma, and W. T. Trotter. Tolerance graphs. Discrete Applied Mathematics, 9(2):157–170, 1984.
* [13] M. C. Golumbic and A. Siani. Coloring algorithms for tolerance graphs: reasoning and scheduling with interval constraints. In Proceedings of the Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic Computation (AISC/Calculemus), pages 196–207, 2002.
* [14] M. C. Golumbic and A. N. Trenk. Tolerance graphs. Cambridge Studies in Advanced Mathematics, 2004.
* [15] R. B. Hayward and R. Shamir. A note on tolerance graph recognition. Discrete Applied Mathematics, 143(1-3):307–311, 2004.
* [16] G. Isaak, K. L. Nyman, and A. N. Trenk. A hierarchy of classes of bounded bitolerance orders. Ars Combinatoria, 69, 2003.
* [17] M. Kaufmann, J. Kratochvíl, K. A. Lehmann, and A. R. Subramanian. Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition. In Proceedings of the 17th annual ACM-SIAM symposium on Discrete Algorithms (SODA), pages 832–841, 2006.
* [18] J. M. Keil and P. Belleville. Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs. Discrete Applied Mathematics, 140(1-3):73–89, 2004.
* [19] L. Langley. Interval tolerance orders and dimension. PhD thesis, Dartmouth College, 1993.
* [20] T.-H. Ma and J. P. Spinrad. On the 2-chain subgraph cover and related problems. Journal of Algorithms, 17(2):251–268, 1994.
* [21] G. B. Mertzios and D. G. Corneil. Vertex splitting and the recognition of trapezoid graphs. Technical Report AIB-2009-16, Department of Computer Science, RWTH Aachen University, September 2009.
* [22] G. B. Mertzios, I. Sau, and S. Zaks. A new intersection model and improved algorithms for tolerance graphs. SIAM Journal on Discrete Mathematics, 23(4):1800–1813, 2009.
* [23] G. B. Mertzios, I. Sau, and S. Zaks. The recognition of tolerance and bounded tolerance graphs is NP-complete. Technical Report AIB-2009-06, Department of Computer Science, RWTH Aachen University, April 2009.
* [24] G. Narasimhan and R. Manber. Stability and chromatic number of tolerance graphs. Discrete Applied Mathematics, 36:47–56, 1992.
* [25] S. P. Ryan. Trapezoid order classification. Order, 15:341–354, 1998.
* [26] J. P. Spinrad. Efficient graph representations, volume 19 of Fields Institute Monographs. American Mathematical Society, 2003.
|
arxiv-papers
| 2010-01-19T10:24:34 |
2024-09-04T02:49:07.826683
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "George B. Mertzios, Ignasi Sau, Shmuel Zaks",
"submitter": "George Mertzios",
"url": "https://arxiv.org/abs/1001.3251"
}
|
1001.3283
|
11institutetext: Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425
Jülich, Germany
# Modeling Stop-and-Go Waves in Pedestrian Dynamics
Andrea Portz and Armin Seyfried 1111
###### Abstract
Several spatially continuous pedestrian dynamics models have been validated
against empirical data. We try to reproduce the experimental fundamental
diagram (velocity versus density) with simulations. In addition to this
quantitative criterion, we tried to reproduce stop-and-go waves as a
qualitative criterion. Stop-and-go waves are a characteristic phenomenon for
the single file movement. Only one of three investigated models satisfies both
criteria.
## 1 Introduction
The applications of pedestrians’ dynamics range from the safety of large
events to the planning of towns with a view to pedestrian comfort. Because of
the computational effort involved with an experimental analysis of the complex
collective system of pedestrians’ behavior, computer simulations are run.
Models continuous in space are one possibility to describe this complex
collective system.
In developing a model, we prefer to start with the simplest case: single lane
movement. If the model is able to reproduce reality quantitatively and
qualitatively for that simple case, it is a good candidate for adaption to
complex two-dimensional conditions.
Also in single file movement pedestrians interact in many ways and not all
factors, which have an effect on their behavior, are known. Therefore, we
follow three different modeling approaches in this work. All of them underlie
diverse concepts in the simulation of human behavior.
This study is a continuation and enlargement of the validation introduced in
[7]. For validation, we introduce two criteria: On the one hand, the relation
between velocity and density has to be described correctly. This requirement
is fulfilled, if the modeled data reproduce the fundamental diagram. On the
other hand, we are aiming to reproduce the appearance of collective effects. A
characteristic effect for the single file movement are stop-and-go waves as
they are observed in experiments [5]. We obtained all empirical data from
several experiments of the single file movement. There a corridor with
circular guiding was built, so that it possessed periodic boundary conditions.
The density was varied by increasing the number of the involved pedestrians.
For more information about the experimental set-up, see [5],[6].
## 2 Spatially Continuous Models
The first two models investigated are force based and the dynamics are given
by the following system of coupled differential equations
$m_{i}\frac{d\,v_{i}}{d\,t}=F_{i}\quad\mbox{with}\quad
F_{i}=F_{i}^{drv}+F_{i}^{rep}\quad\mbox{and}\quad\frac{d\,x_{i}}{d\,t}=v_{i}$
(1)
where $F_{i}$ is the force acting on pedestrian $i$. The mass is denoted by
$m_{i}$, the velocity by $v_{i}$ and the current position by $x_{i}$. This
approach derives from sociology [4]. Here psychological forces define the
movement of humans through their living space. This approach is transferred to
pedestrians dynamics and so $F_{i}$ is split to a repulsive force
$F_{i}^{rep}$ and a driven force $F_{i}^{drv}$. In our case the driving force
is defined as
$F_{i}^{drv}=\frac{v_{i}^{0}-v_{i}}{\tau},$ (2)
where $v_{i}^{0}$ is the desired velocity of a pedestrian and $\tau$ their
reaction time.
The other model is event driven. A pedestrian can be in different states. A
change between these states is called event. The calculation of the velocity
of each pedestrian is straightforward and depends on these states.
### 2.1 Social Force Model
The first spatially continuous model was developed by Helbing and Molnár [3]
and has often been modified. According to [2] the repulsive force for
pedestrian $i$ is defined by
$F_{i}^{rep}(t)=\sum_{j\neq
i}f_{ij}(x_{i},x_{j})+\xi_{i}(t)\quad\mbox{with}\quad
f_{ij}(x_{i},x_{j})=-\partial_{x}A(\Delta\,x_{i,j}-D)^{-B},$ (3)
with $A=0.2,B=2,D=1\,[m],\tau=0.2\,[s]$ and $\Delta\,x_{i,j}$ is the distance
between pedestrians $i$ and $j$. The fluctuation parameter $\xi_{i}(t)$
represents the noise of the system. In two-dimensional scenes, this parameter
is used to create jammed states and lane formations [2]. In this study, we are
predominantly interested in the modeled relation between velocity and density
for single file movement. Therefore the fluctuation parameter has no influence
and is ignored.
First tests of this model indicated that the forces are too strong, leading to
unrealistically high velocities. Due to this it is necessary to limit the
velocity to $v_{max}$, as it is done in [3]
$v_{i}(t)=\left\\{\begin{array}[]{cl}v_{i}(t),&\mbox{\quad if }|v_{i}(t)|\leq
v_{max}\\\ v_{max},&\mbox{\quad otherwise}\end{array}\right.\enspace.$ (4)
In our simulation we set $v_{max}=1.34\,[\frac{m}{s}]$.
### 2.2 Model with Foresight
In this model pedestrians possess a degree of foresight, in addition to the
current state of a pedestrian at time step $t$. This approach considers an
extrapolation of the velocity to time step $t+s$. For it [8] employs the
linear relation between the velocity and the distance of a pedestrian $i$ to
the one in front $\Delta x_{i,i+1}(t)$.
$v_{i}(t)=a\,\Delta x_{i,i+1}(t)-b$ (5)
For $a=0.94\,[\frac{m}{s}]$ and $b=0.34\,[\frac{1}{s}]$ this reproduces the
empirical data. So with (5) $v_{i}(t+s)$ can be calculated from $\Delta
x_{i,i+1}(t+s)$ which itself is a result of the extrapolation of the current
state
$\Delta x_{i,i+1}(t)+\Delta v(t)\,s=\Delta x_{i,i+1}(t+s)$ (6)
with $\Delta v(t)=v_{i+1}(t)\,s-v_{i}(t)\,s$. Finally, the repulsive force is
defined as
$F_{i}^{rep}(t)=-\frac{v_{i}^{0}-v_{i}(t+s)}{\tau}\enspace.$ (7)
Obviously the impact of the desired velocity $v_{i}^{0}$ in the driven force
is negated by the one in the repulsive term. After some simulation time, the
system reaches an equilibrium in which all pedestrians walk with the same
velocity. In order to spread the values and keep the right relation between
velocity and density, we added a fluctuation parameter $\zeta_{i}(t)$.
$\zeta_{i}(t)$ uniformly distributed in the interval $[-20,20]$ reproduced the
scatter observed in the empirical data.
### 2.3 Adaptive Velocity Model
In this model pedestrians are treated as hard bodies with a diameter $d_{i}$
[1]. The diameter depends linearly on the current velocity and is equal to the
step length $st$ in addition to the safety distance $\beta$
$d_{i}(t)=e+f\,v_{i}(t)=st_{i}(t)+\beta_{i}(t)\enspace.$ (8)
Based on [9] the step length is a linear function of the current velocity with
following parameters:
$st_{i}(t)=0.235\,[m]+0.302\,[s]\,v_{i}(t)\enspace.$ (9)
$e$ and $f$ can be specified through empirical data and the inverse relation
of (5). Here $e$ is the required space for a stationary pedestrian and $f$
affects the velocity term. For $e=0.36\,[m]$ and $f=1.06\,[s]$ the last
equations (8) and (9) can be summarized to
$\beta_{i}(t)=d_{i}(t)-st_{i}(t)=0.125\,[m]+0.758\,[s]\,v_{i}(t)\enspace.$
(10)
By solving the differential equation
$\frac{dv}{dt}=F^{drv}=\frac{v^{0}-v(t)}{\tau}\quad\Rightarrow\quad
v(t)=v^{0}+c\,\exp\left(-\frac{t}{\tau}\right),\mbox{for }c\in\bbbr,$ (11)
the velocity function is obtained. This is shown in Fig. 1 together with the
parameters of the pedestrians’ movement.
(a) Demonstration of the parameters $d,st$ and $\beta$
(b) Conception of the adaptive velocity
Figure 1: Left: connection between the required space $d$, the step length
$st$ and the safety distance $\beta$. Right: The adaptive velocity with
acceleration until $t_{dec1}$, than deceleration until $t_{acc1}$, again
acceleration until $t_{dec2}$ and so on.
A pedestrian is accelerating to their desired velocity $v_{i}^{0}$ until the
distance to the pedestrian in front is smaller than the safety distance. From
this time on, he/she is decelerating until the distance is larger than the
safety distance and so on. Via $\Delta x_{i,i+1},d_{i}$ and $\beta_{i}$ those
events could be defined: deceleration (12) and acceleration (13).
To ensure good performance for high densities, no events are explicitly
calculated. But in each time step, it is checked whether an event has taken
place and $t_{dec},t_{acc}$ or $t_{coll}$ are set to $t$ accordingly. The time
step, $\Delta t$, of $0.05$ seconds is chosen, so that a reaction time is
automatically included. The discrete time step could lead to configurations
where overlapping occurs. To guarantee volume exclusion, case (14) is
included, in which the pedestrians are too close to each other and have to
stop.
$\displaystyle t$ $\displaystyle=$ $\displaystyle t_{dec},\quad\mbox{if:
}\quad\Delta x_{i,i+1}-0.5*\left(d_{i}(t)+d_{i+1}(t)\right)\leq 0$ (12)
$\displaystyle t$ $\displaystyle=$ $\displaystyle t_{acc},\quad\mbox{if:
}\quad\Delta x_{i,i+1}-0.5*\left(d_{i}(t)+d_{i+1}(t)\right)>0$ (13)
$\displaystyle t$ $\displaystyle=$ $\displaystyle t_{coll},\quad\mbox{if:
}\quad\Delta x_{i,i+1}-0.5*\left(d_{i}(t)+d_{i+1}(t)\right)\leq-\beta_{i}(t)$
(14)
## 3 Validation with Empirical Data
For the comparison of the modeled and experimental data, it is important to
use the same method of measurement. [5] shows that the results from different
measurement methods vary considerably. The velocity $v_{i}$ is calculated by
the entrance and exit times $t_{i}^{in}$ and $t_{i}^{out}$ to two meter
section.
$\displaystyle v_{i}$ $\displaystyle=$
$\displaystyle\frac{2\,[m]}{(t_{i}^{out}-t_{i}^{in})[s]}.$ (15)
To avoid discrete values of the density leading to a large scatter, we define
the density by
$\displaystyle\rho(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{N}\Theta_{i}(t)}{2\,m},$ (16)
where $\Theta_{i}(t)$ gives the fraction to which the space between pedestrian
$i$ and $i+1$ is inside the measured section, see [6]. $\rho_{i}$ is the mean
value of all $\rho(t)$, where $t$ is in the interval
$[t_{i}^{in},t_{i}^{out}]$. We use the same method of measurement for the
modeled and empirical data. The fundamental diagrams are displayed in Fig. 2,
where $N$ is the number of the pedestrians.
(a) Social force model (b) Model with foresight
(c) Adaptive velocity model (d) Empirical data
Figure 2: Validation of the modeled fundamental diagram with the empirical
data (down right) for the single file movement.
The velocities of the social force model are independent of the systems
density and nearly equal to the desired velocity $v_{i}=v_{i}^{0}\sim
1.24\,[\frac{m}{s}]$. Additionally we observe a backward movement of the
pedestrians and pair formation. Because of these unrealistic phenomena are not
observed in the other models, we suggest that this is caused by the
combination of long-range forces and periodic boundary conditions.
In contrast, the model with foresight results in a fundamental diagram in good
agreement with the empirical one. Through the fluctuation parameter, the
values of the velocities and densities vary as in the experimental data.
We are satisfied with the results of the adaptive velocity model. For reducing
computing time, we also tested a linear adaptive velocity function, leading to
a $70\%$ decrease in computing time for $10000$ pedestrians. The fundamental
diagram for this linear adaptive velocity function is not shown, but also
reproduces the empirical one.
## 4 Reproduction of Stop-and-Go Waves
During the experiments of the single file movement, we observed stop-and-go
waves at densities higher than two pedestrians per meter, see Fig. 5 in [5].
Therefore, we compare the experimental trajectories with the modeled ones for
global densities of one, two and three persons per meter. The results are
shown in Fig.3. Since the social force model is not able to satisfy the
criterion for the right relation between velocity and density, we omit this
model in this section. Figure 3 shows the trajectories for global average
densities of one, two and three persons per meter. From left to right the data
of the model with foresight, the adaptive velocity model and the experiment
are shown.
In the experimental data, it is clearly visible that the trajectories get
unsteadier with increasing density. At a density of one person per meter
pedestrians stop for the first time. So a jam is generated. At a density of
two persons per meter stop-and-go waves pass through the whole measurement
range. At densities greater than three persons per meter pedestrians can
hardly move forward.
For the extraction of the empirical trajectories, the pedestrians’ heads were
marked and tracked. Sometimes, there is a backward movement in the empirical
trajectories caused by self-dynamic of the pedestrians’ heads. This dynamic is
not modeled and so the other trajectories have no backward movement. This has
to be accounted for in the comparison.
By adding the fluctuation parameter $\zeta_{i}(t)\in[-20,20]$ to the model
with foresight a good agreement with experiment is obtained for densities of
one and two persons per meter. The irregularities caused by this parameter are
equal to the irregularities of the pedestrians dynamic. Nevertheless, this
does not suffice for stopping so that stop-and-go waves appear, Fig.3(d) and
Fig.3(g).
(a) Model with foresight
(b) Adaptive velocity model
(c) Experimental data
(d) Model with foresight
(e) Adaptive velocity model
(f) Experimental data
(g) Model with foresight
(h) Adaptive velocity model
(i) Experimental data
Figure 3: Comparison of modeled and empirical trajectories for the single lane
movement. The global density of the system is one, two or three persons per
meter (from top to bottom).
With the adaptive velocity model stop-and-go waves already arise at a density
of one pedestrian per meter, something that is not seen in experimental data.
However, this model characterizes higher densities well. So in comparison with
Fig. 3(h) and Fig. 3(i) the stopping-phase of the modeled data seems to last
for the same time as in the empirical data. But there are clearly differences
in the acceleration phase, the adaptive velocity models acceleration is much
lower than seen in experiment.
Finally other studies of stop-and-go waves have to be carried out. The
occurrence of this phenomena has to be clearly understood for further model
modifications. Therefore it is necessarry to measure e. g. the size of the
stop-and-go wave at a fixed position. Unfortunately it is not possible to
measure over a time interval, because the empirical trajectories are only
available in a specific range of 4 meters.
## 5 Conclusion
The well-known and often used social force model is unable to reproduce the
fundamental diagram. The model with foresight provides a good quantitative
reproduction of the fundamental diagram. However, it has to be modified
further, so that stop-and-go waves could be generated as well. The model with
adaptive velocities follows a simple and effective event driven approach. With
the included reaction time, it is possible to create stop-and-go waves without
unrealistic phenomena, like overlapping or interpenetrating pedestrians.
All models are implemented in C and run on a simple PC. They were also tested
for their computing time in case of large system with upto 10000 pedestrians.
The social force model offers a complexity level of $\mathcal{O}(N^{2})$,
whereas the other models only have a level of $\mathcal{O}(N)$. For this
reason the social force model is not qualified for modeling such large
systems. Both other models are able to do this, where the maximal computing
time is one sixth of the simulated time.
In the future, we plan to include steering of pedestrians. For these models
more criteria, like the reproduction of flow characteristics at bottlenecks,
are necessarry. Further we are trying to get a deeper insight into to
occurrence of stop-and-go waves.
## References
* [1] M. Chraibi and A. Seyfried. Pedestrian Dynamics With Event-driven Simulation. In Pedestrian and Evacuation Dynamics 2008, 2009. arXiv:0806.4288, in print.
* [2] D. Helbing, I. J. Farkas, and T. Vicsek. Freezing by Heating in a Driven Mesoscopic System. Phys. Rev. Let., 84:1240–1243, 2000.
* [3] D. Helbing and P. Molnár. Social force model for pedestrian dynamics. Phys. Rev. E, 51:4282–4286, 1995.
* [4] K. Lewin, editor. Field Theory in Social Science. Greenwood Press, Publishers, 1951.
* [5] A. Seyfried, M. Boltes, J. Kähler, W. Klingsch, A. Portz, A. Schadschneider, B. Steffen, and A. Winkens. Enhanced empirical data for the fundamental diagram and the flow through bottlenecks. In Pedestrian and Evacuation Dynamics 2008. Springer, 2009. arXiv:0810.1945, in print.
* [6] A. Seyfried, B. Steffen, W. Klingsch, and M. Boltes. The fundamental diagram of pedestrian movement revisited. J. Stat. Mech., P10002, 2005.
* [7] A. Seyfried, B. Steffen, and T. Lippert. Basics of modelling the pedestrian flow. Physica A, 368:232–238, 2006.
* [8] B. Steffen and A. Seyfried. The repulsive force in continous space models of pedestrian movement. 2008\. arXiv:0803.1319v1.
* [9] U. Weidmann. Transporttechnik der Fussgänger. Technical Report Schriftenreihe des IVT Nr. 90, Institut für Verkehrsplanung,Transporttechnik, Strassen- und Eisenbahnbau, ETH Zürich, ETH Zürich, 1993. Zweite, ergänzte Auflage.
|
arxiv-papers
| 2010-01-19T12:58:56 |
2024-09-04T02:49:07.835785
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrea Portz and Armin Seyfried",
"submitter": "Mohcine Chraibi",
"url": "https://arxiv.org/abs/1001.3283"
}
|
1001.3400
|
A NEW GENERATING FUNCTION OF ($q$-) BERNSTEIN TYPE POLYNOMIALS AND THEIR
INTERPOLATION FUNCTION
∗Yilmaz SIMSEK and ∗∗Mehmet ACIKGOZ
∗University of Akdeniz, Faculty of Arts and Science, Department of
Mathematics, 07058-Antalya, TURKEY, ysimsek@akdeniz.edu.tr
∗∗University of Gaziantep, Faculty of Arts and Science, Department of
Mathematics, 27310 Gaziantep, Turkey, acikgoz@gantep.edu.tr
Abstract
The main object of this paper is to construct a new generating function of the
($q$-) Bernstein type polynomials. We establish elementary properties of this
function. By using this generating function, we derive recurrence relation and
derivative of the ($q$-) Bernstein type polynomials. We also give relations
between the ($q$-) Bernstein type polynomials, Hermite polynomials, Bernoulli
polynomials of higher-order and the second kind Stirling numbers. By applying
Mellin transformation to this generating function, we define interpolation of
the ($q$-) Bernstein type polynomials. Moreover, we give some applications and
questions on approximations of ($q$-) Bernstein type polynomials, moments of
some distributions in Statistics.
2000 Mathematics Subject Classification.11B68, 11M06, 11S40, 11S80, 28B99,
41A10, 41A50, 65D17.
Key Words and Phrases. Generating function, ($q$-) Bernstein polynomials,
Bernoulli polynomials of higher-order, Hermite polynomials, second kind
Stirling numbers, interpolation function, Mellin transformation, moments of
distributions.
CONTENTS
1\. Introduction
2\. Preliminary results related to the classical Bernstein, higher-order
Bernoulli and
Hermit polynomials, the second kind Stirling numbers
3\. Generating function of the ($q$-) Bernstein type polynomials
4\. New identities on ($q$-) Bernstein type polynomials, Hermite polynomials
and
first kind Stirling numbers
5\. Interpolation function of ($q$-) Bernstein type polynomials
6\. Further remarks and observation
## 1\. Introduction
In [4], Bernstein introduced the Bernstein polynomials. Since that time, many
authors have studied on these polynomials and other related subjects (see cf.
[1]-[21]), and see also the references cited in each of these earlier works.
The Bernstein polynomials can also be defined in many different ways. Thus,
recently, many applications of these polynomials have been looked for by many
authors. These polynomials have been used not only for approximations of
functions in various areas in Mathematics but also the other fields such as
smoothing in statistics, numerical analysis and constructing Bezier curve
which have many interesting applications in computer graphics (see cf. [4],
[6], [13], [14], [15], [12], [16], [21]) and see also the references cited in
each of these earlier works.
The ($q$-) Bernstein polynomials have been investigated and studied by many
authors without generating function. So far, we have not found any generating
function of ($q$-) Bernstein polynomials in the literature. Therefore, we will
consider the following question:
How can one construct generating function of ($q$-) Bernstein type
polynomials?
The aim of this paper is to give answer this question and to construct
generating function of the ($q$-) Bernstein type polynomials which is given in
Section 3. By using this generating function, we not only give recurrence
relation and derivative of the ($q$-) Bernstein type polynomials but also find
relations between higher-order Bernoulli polynomials, the second kind Stirling
numbers and the Hermite polynomials. In Section 5, by applying Mellin
transformation to the generating function of the ($q$-) Bernstein type
polynomials, we define interpolation function, which interpolates the ($q$-)
Bernstein type polynomials at negative integers.
## 2\. Preliminary results related to the classical Bernstein, higher-order
Bernoulli and Hermit polynomials, the second kind Stirling numbers
The Bernstein polynomials play a crucial role in approximation theory and the
other branches of Mathematics and Physics. Thus in this section we give
definition and some properties of these polynomials.
Let $f$ be a function on $\left[0,1\right]$. The classical Bernstein
polynomials of degree $n$ are defined by
$\mathbb{B}_{n}f(x)=\sum_{j=0}^{n}f\left(\frac{j}{n}\right)B_{j,n}(x),\text{
}0\leq x\leq 1,$ (2.1)
where $\mathbb{B}_{n}f$ is called the Bernstein operator and
$B_{j,n}(x)=\left(\begin{array}[]{c}n\\\ j\end{array}\right)x^{j}(1-x)^{n-j},$
(2.2)
$j=0$, $1$,$\cdots$,$n$ are called the Bernstein basis polynomials (or the
Bernstein polynomials of degree $n$). There are $n+1$ $n$th degree Bernstein
polynomials. For mathematical convenience, we set $B_{j,n}(x)=0$ if $j<0$ or
$j>n$ cf. ([4], [6], [8]).
If $f:\left[0,1\right]\rightarrow\mathbb{C}$ is a continuous function, the
sequence of Bernstein polynomials $\mathbb{B}_{n}f(x)$ converges uniformly to
$f$ on $\left[0,1\right]$ cf. [9].
A recursive definition of the $k$th $n$th Bernstein polynomials can be written
as
$B_{k,n}(x)=(1-x)B_{k,n-1}(x)+xB_{k-1,n-1}(x).$
For proof of the above relation see [8].
For $0\leq k\leq n$, derivative of the $n$th degree Bernstein polynomials are
polynomials of degree $n-1$:
$\frac{d}{dt}B_{k,n}(t)=n\left(B_{k-1,n-1}(t)-B_{k,n-1}(t)\right),$ (2.3)
cf. ([4], [6], [8]). On the other hand, in Section 3, using our a new
generating function, we give the other proof of (2.3).
Observe that the Bernstein polynomial of degree $n$, $\mathbb{B}_{n}f$, uses
only the sampled values of $f$ at $t_{nj}=\frac{j}{n}$, $j=0$,
$1$,$\cdots$,$n$. For $j=0$, $1$,$\cdots$,$n$,
$\beta_{j,n}(x)\equiv(n+1)B_{j,n}(x),\text{ }0\leq x\leq 1,$
is the density function of beta distribution $beta(j+1,n+1-j)$.
Let $y_{n}(x)$ be a binomial $b(n,x)$ random variable. Then
$E\left\\{y_{n}(x)\right\\}=nt,$
and
$var\left\\{y_{n}(x)\right\\}=E\left\\{y_{n}(x)-nx\right\\}^{2}=nx(1-x),$
and
$\mathbb{B}_{n}f(x)=E\left[f\left\\{\frac{y_{n}(x)}{n}\right\\}\right],$
cf. [6].
The classical higher-order Bernoulli polynomials $\mathcal{B}_{n}^{(v)}(z)$
defined by means of the following generating function
$F^{(v)}(z,t)=e^{tx}\left(\frac{t}{e^{t}-1}\right)^{v}=\sum_{n=0}^{\infty}\mathcal{B}_{n}^{(v)}(z)\frac{t^{n}}{n!}\text{.}$
(2.4)
The higher-order Bernoulli polynomials play an important role in the finite
differences and in (analytic) number theory. So, the coefficients in all the
usual cenral-difference formulae for interpolation, numerical differentiation
and integration, and differences in terms of derivatives can be expressed in
terms of these polynomials cf. ([1], [10], [11], [20]). These polynomials are
related to the many branches of Mathematics. By substituting $v=1$ into the
above, we have
$F(t)=\frac{te^{tx}}{e^{t}-1}=\sum_{n=1}^{\infty}B_{n}\frac{t^{n}}{n!},$
where $B_{n}$ is usual Bernoulli polynomials cf. [18].
The usual second kind Stirling numbers with parameters $(n,k),$ denote by
$S(n,k)$, that is the number of partitions of the set
$\left\\{1,2,\cdots,n\right\\}$ into $k$ non empty set. For any $t$, it is
well known that the second kind Stirling numbers are defined by means of the
generating function cf. ([2], [17], [19])
$F_{S}(t,k)=\frac{(-1)^{k}}{k!}(1-e^{t})^{k}=\sum_{n=0}^{\infty}S(n,k)\frac{t^{n}}{n!}.$
(2.5)
These numbers play an important role in many branches of Mathematics, for
example, combinatorics, number theory, discrete probability distributions for
finding higher order moments. In [7], Joarder and Mahmood demonstrated the
application of Stirling numbers of the second kind in calculating moments of
some discrete distributions, which are binomial distribution, geometric
distribution and negative binomial distribution.
The Hermite polynomials defined by the following generating function:
For $z$, $t\in\mathbb{C}$,
$e^{2zt-t^{2}}=\sum_{n=0}^{\infty}H_{n}(z)\frac{t^{n}}{n!},$ (2.6)
which gives the Cauchy-type integral
$H_{n}(z)=\frac{n!}{2\pi i}\int_{\mathcal{C}}e^{2zt-t^{2}}\frac{dt}{t^{n+1}},$
where $\mathcal{C}$ is a circle around the origin and the integration is in
positive direction cf. [11]. The Hermite polynomials play a crucial role in
certain limits of the classical orthogonal polynomials. These polynomials are
related to the higher-order Bernoulli polynomials, Gegenbauer polynomials,
Laguerre polynomials, the Tricomi-Carlitz polynomials and Buchholz
polynomials, cf. [11]. These polynomials also play a crucial role in not only
in Mathematics but also in Physics and in the other sciences. In section 4 we
give relation between the Hermite polynomials and ($q$-) Bernstein type
polynomials.
## 3\. Generating Function of the Bernstein type polynomials
Let $\left\\{B_{k,n}(x)\right\\}_{0\leq k\leq n}$ be a sequence of Bernstein
polynomials. The aim of this section is to construct generating function of
the sequence $\left\\{B_{k,n}(x)\right\\}_{0\leq k\leq n}$. It is well known
that most of generating functions are obtained from the recurrence formulae.
However, we do not use the recurrence formula of the Bernstein polynomials for
constructing generating function of them.
We now give the following notation:
$[x]=[x:q]=\left\\{\begin{array}[]{c}\frac{1-q^{x}}{1-q}\text{, }q\neq 1\\\
\\\ x\text{, }q=1.\end{array}\right.$
If $q\in\mathbb{C}$, we assume that $\mid q\mid<1$.
We define
$\displaystyle F_{k,q}(t,x)$ $\displaystyle=$
$\displaystyle(-1)^{k}t^{k}\exp\left(\left[1-x\right]t\right)$ (3.4)
$\displaystyle\times\sum_{m,l=0}\left(\begin{array}[]{c}k+l-1\\\
l\end{array}\right)\frac{q^{l}S(m,k)\left(x\log q\right)^{m}}{m!},$
where $\left|q\right|<1$, $\exp(x)=e^{x}$ and $S(m,k)$ denotes the second kind
Stirling numbers and
$\sum_{m,l=0}f(m)g(l)=\sum_{m=0}^{\infty}f(m)\sum_{l=0}^{\infty}g(l).$
By (3.4), we define the following a new generating function of polynomial
$Y_{n}(k;x;q)$ by
$F_{k,q}(t,x)=\sum_{n=k}^{\infty}Y_{n}(k;x;q)\frac{t^{n}}{n!},$ (3.5)
where $t\in\mathbb{C}$.
Observe that if $q\rightarrow 1$ in (3.5), we have
$Y_{n}(k;x;q)\rightarrow B_{k,n}(x),$
hence
$F_{k}(t,x)=\sum_{n=k}^{\infty}B_{k,n}(x)\frac{t^{n}}{n!}.$
From (3.5), we obtain the following theorem.
###### Theorem 1.
Let $n$ be a positive integer with $k\leq n$. Then we have
$\displaystyle Y_{n}(k;x;q)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}n\\\
k\end{array}\right)\frac{(-1)^{k}k!}{(1-q)^{n-k}}$ (3.13)
$\displaystyle\times\sum_{m,l=0}\sum_{j=0}^{n-k}\left(\begin{array}[]{c}k+l-1\\\
l\end{array}\right)\left(\begin{array}[]{c}n-k\\\
k\end{array}\right)\frac{(-1)^{j}q^{l+j(1-x)}S(m,k)\left(x\log
q\right)^{m}}{m!}.$
By using (3.4) and (3.5), we obtain
$\displaystyle F_{k,q}(t,x)$ $\displaystyle=$
$\displaystyle\frac{\left(\left[x\right]t\right)^{k}}{k!}\exp(\left[1-x\right]t)$
$\displaystyle=$
$\displaystyle\sum_{n=k}^{\infty}Y_{n}(k;x;q)\frac{t^{n}}{n!}.$
The generating function $F_{k,q}(t,x)$ depends on integer parameter $k$, real
variable $x$ and complex variable $q$ and $t$. Therefore the proprieties of
this function are closely related to these variables and parameter. By using
this function, we give many properties of the ($q$-) Bernstein type
polynomials and the other well-known special numbers and polynomials. By
applying Mellin transformation to this function, in Section 5, we construct
interpolation function of the ($q$-) Bernstein type polynomials.
By the umbral calculus convention in (3), then we obtain
$\frac{\left(\left[x\right]t\right)^{k}}{k!}\exp(\left[1-x\right]t)=\exp\left(Y(k;x;q)t\right).$
(3.15)
By using the above, we obtain all recurrence formulae of $Y_{n}(k;x;q)$ as
follows:
$\frac{\left(\left[x\right]t\right)^{k}}{k!}=\sum_{n=0}^{\infty}\left(Y(k;x;q)-\left[1-x\right]\right)^{n}\frac{t^{n}}{n!},$
where each occurrence of $Y^{n}(k;x;q)$ by $Y_{n}(k;x;q)$ (symbolically
$Y^{n}(k;x;q)\rightarrow Y_{n}(k;x;q)$).
By (3.15),
$\left[u+v\right]=\left[u\right]+q^{u}\left[v\right]$
and
$\left[-u\right]=-q^{u}\left[u\right],$
we obtain the following corollary:
###### Corollary 1.
Let $n$ be a positive integer with $k\leq n$. Then we have
$Y_{n+k}(k;x;q)=\left(\begin{array}[]{c}n+k\\\
k\end{array}\right)\sum_{j=0}^{n}(-1)^{j}q^{j(1-x)}\left[x\right]^{j+k}.$
###### Remark 1.
By Corollary 1, for all $k$ with $0\leq k\leq n$, we see that
$Y_{n+k}(k;x;q)=\left(\begin{array}[]{c}n+k\\\
k\end{array}\right)\sum_{j=0}^{n}(-1)^{j}q^{j(1-x)}\left[x\right]^{j+k},$
or
$Y_{n+k}(k;x;q)=\left(\begin{array}[]{c}n+k\\\
k\end{array}\right)\left[x\right]^{k}\left[1-x\right]^{n}.$
The polynomials $Y_{n+k}(k;x;q)$ are so-called $q$-Bernstein-type polynomials.
It is easily seen that
$\lim_{q\rightarrow
1}Y_{n+k}(k;x;q)=B_{k,n+k}(x)=\left(\begin{array}[]{c}n+k\\\
k\end{array}\right)x^{k}\left(1-x\right)^{n},$
which give us (2.2).
By using derivative operator
$\frac{d}{dx}\left(\lim_{q\rightarrow 1}Y_{n+k}(k;x;q)\right)$
in (3.4), we obtain
$\displaystyle\sum_{n=k}^{\infty}\frac{d}{dx}\left(Y_{n}(k;x;1)\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\sum_{n=k}^{\infty}nY_{n-1}(k-1;x;1)\frac{t^{n}}{n!}-\sum_{n=k}^{\infty}nY_{n-1}(k;x;1)\frac{t^{n}}{n!}.$
Consequently, we have
$\frac{d}{dx}\left(Y_{n}(k;x;1)\right)=nY_{n-1}(k-1;x;1)-nY_{n-1}(k;x;1),$
or
$\frac{d}{dx}\left(B_{k,n}(x)\right)=nB_{k-1,n-1}(x)-nB_{k,n-1}(x).$
Observe that by using our generating function, we give different proof of
(2.3).
Let $f$ be a function on $\left[0,1\right]$. The ($q$-) Bernstein type
polynomial of degree $n$ is defined by
$\mathbb{Y}_{n}f(x)=\sum_{j=0}^{n}f\left(\frac{\left[j\right]}{\left[n\right]}\right)Y_{n}(j;x;q),$
where $0\leq x\leq 1$. $\mathbb{Y}_{n}$ is called the ($q$-) Bernstein type
operator and $Y_{n}(j;x;q)$, $j=0,\cdots,n$, defined in (3.13), are called the
($q$-) Bernstein type (basis) polynomials.
## 4\. New identities on Bernstein type polynomials, Hermite polynomials and
first kind Stirling numbers
###### Theorem 2.
Let $n$ be a positive integer with $k\leq n$. Then we have
$Y_{n}(k;x;q)=\left[x\right]^{k}\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\mathcal{B}_{j}^{(k)}\left(\left[1-x\right]\right)S(n-j,k),$
where $\mathcal{B}_{j}^{(k)}(x)$ and $S(n,k)$ denote the classical higher-
order Bernoulli polynomials and the second kind Stirling numbers,
respectively.
###### Proof.
By using (2.4), (2.5) and (3.5), we obtain
$\displaystyle\sum_{n=k}^{\infty}Y_{n}(k;x;q)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left[x\right]^{k}\sum_{n=0}^{\infty}S(n,k)\frac{t^{n}}{n!}\sum_{n=0}^{\infty}\mathcal{B}_{j}^{(k)}\left(\left[1-x\right]\right)\frac{t^{n}}{n!}.$
By using Cauchy product in the above, we have
$\displaystyle\sum_{n=k}^{\infty}Y(k,n;x;q)\frac{t^{n}}{n!}$ $\displaystyle=$
$\displaystyle\left[x\right]^{k}\sum_{n=0}^{\infty}\sum_{j=0}^{n}\mathcal{B}_{j}^{(k)}\left(\left[1-x\right]\right)S(n-j,k)\frac{t^{n}}{j!\left(n-j\right)!}.$
From the above, we have
$\displaystyle\sum_{n=k}^{\infty}Y_{n}(k;x;q)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left[x\right]^{k}\sum_{n=0}^{k-1}\sum_{j=0}^{n}\mathcal{B}_{j}^{(k)}\left(\left[1-x\right]\right)S(n-j,k)\frac{t^{n}}{j!\left(n-j\right)!}$
$\displaystyle+\left[x\right]^{k}\sum_{n=k}^{\infty}\sum_{j=0}^{n}\mathcal{B}_{j}^{(k)}\left(\left[1-x\right]\right)S(n-j,k)\frac{t^{n}}{j!\left(n-j\right)!}.$
By comparing coefficients of $t^{n}$ in the both sides of the above equation,
we arrive at the desired result.
###### Remark 2.
In [5], Gould gave a different relation between the Bernstein polynomials,
generalized Bernoulli polynomials and the second kind Stirling numbers. Oruc
and Tuncer [13] gave relation between the $q$-Bernstein polynomials and the
second kind $q$-Stirling numbers. In [12], Nowak studied on approximation
properties for generalized $q$-Bernstein polynomials and also obtained Stancu
operators or Phillips polynomials.
From (4), we get the following corollary:
###### Corollary 2.
Let $n$ be a positive integer with $k\leq n$. Then we have
$\left[x\right]^{k}\sum_{n=0}^{k-1}\sum_{j=0}^{n}\frac{\mathcal{B}_{j}^{(k)}\left(\left[1-x\right]\right)S(n-j,k)}{j!\left(n-j\right)!}=0.$
###### Theorem 3.
Let $n$ be a positive integer with $k\leq n$. Then we have
$H_{n}(1-y)=\frac{k!}{y^{k}}\sum_{n=0}^{\infty}Y_{n+k}(k;y;q)\frac{2^{n}}{\left(n+k\right)!}.$
###### Proof.
By (2.6), we have
$e^{2zt}=\sum_{n=0}^{\infty}\frac{t^{2n}}{n!}\sum_{n=0}^{\infty}H_{n}(z)\frac{t^{n}}{n!}.$
By Cauchy product in the above, we obtain
$e^{2zt}=\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)H_{j}(z)\right)\frac{t^{2n-j}}{n!}.$ (4.2)
By substituting $z=1-y$ into (4.2), we have
$\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)H_{j}(1-y)\right)\frac{t^{2n-j}}{n!}=\frac{k!}{y^{k}}\sum_{n=0}^{\infty}\left(2^{n}Y_{n+k}(k;y;q)\right)\frac{t^{n}}{\left(n+k\right)!}.$
By comparing coefficients of $t^{n}$ in the both sides of the above equation,
we arrive at the desired result.
## 5\. Interpolation Function of the ($q$-) Bernstein type polynomials
The classical Bernoulli numbers interpolate by Riemann zeta function, which
has a profound effect on number theory and complex analysis. Thus, we
construct interpolation function of the ($q$-) Bernstein type polynomials.
For $z\in\mathbb{C}$, and $x\neq 1$, by applying the Mellin transformation to
(3.4), we get
$S_{q}(z,k;x)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{z-k-1}F_{k,q}(-t,x)dt.$
By using the above equation, we defined interpolation function of the
polynomials, $Y_{n}(k;x;q)$ as follows:
###### Definition 1.
Let $z\in\mathbb{C}$ and $x\neq 1$. We define
$S_{q}(z,k;x)=\left(1-q\right)^{z-k}\sum_{m,l=0}\left(\begin{array}[]{c}z+l-1\\\
l\end{array}\right)\frac{q^{l(1-x)}S(m,k)\left(x\log q\right)^{m}}{m!}.$ (5.1)
By using (5.1), we obtain
$S_{q}(z,k;x)=\frac{(-1)^{k}}{k!}\left[x\right]^{k}\left[1-x\right]^{-z},$
where $z\in\mathbb{C}$ and $x\neq 1$.
By (5.1), we have $S_{q}(z,k;x)\rightarrow S(z,k;x)$ as $q\rightarrow 1$. Thus
we have
$S(z,k;x)=\frac{(-1)^{k}}{k!}x^{k}\left(1-x\right)^{-z}.$
By substituting $x=1$ into the above, we have
$S(z,k;1)=\infty.$
We now evaluate the $m$th $z$-derivatives of $S(z,k;x)$ as follows:
$\frac{\partial^{m}}{\partial
z^{m}}S(z,k;x)=\log^{m}\left(\frac{1}{1-x}\right)S(z,k;x),$ (5.2)
where $x\neq 1$.
By substituting $z=-n$ into (5.1), we obtain
$S_{q}(-n,k;x)=\frac{1}{\left(1-q\right)^{n}}\sum_{m,l=0}\left(\begin{array}[]{c}-n+l-1\\\
l\end{array}\right)\frac{q^{l(1-x)}S(m,k)\left(x\log q\right)^{m}}{m!}.$
By substituting (3.13) into the above, we arrive at the following theorem,
which relates the polynomials $Y_{n+k}(k;x;q)$ and the function
$S_{q}(z,k;x)$.
###### Theorem 4.
Let $n$ be a positive integer with $k\leq n$ and $0<x<1$. Then we have
$S_{q}(-n,k;x)=\frac{(-1)^{k}n!}{(n+k)!}Y_{n+k}(k;x;q).$
###### Remark 3.
$\displaystyle\lim_{q\rightarrow 1}S_{q}(-n,k;x)$ $\displaystyle=$
$\displaystyle S(-n,k;x)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{k}n!}{(n+k)!}x^{k}\left(1-x\right)^{n}$
$\displaystyle=$ $\displaystyle\frac{(-1)^{k}n!}{(n+k)!}B_{k,n+k}(x).$
Therefore, for $0<x<1$, the function
$S(z,k;x)=\frac{(-1)^{k}}{k!}x^{k}\left(1-x\right)^{-z}$
interpolates the classical Bernstein polynomials of degree $n$ at negative
integers.
By substituting $z=-n$ into (5.2), we obtain the following corollary.
###### Corollary 3.
Let $n$ be a positive integer with $k\leq n$ and $0<x<1$. Then we have
$\frac{\partial^{m}}{\partial
z^{m}}S(-n,k;x)=\frac{(-1)^{k}n!}{(n+k)!}B_{k,n+k}(x)\log^{m}\left(\frac{1}{1-x}\right).$
## 6\. Further remarks and observation
The Bernstein polynomials are used for important applications in many branches
of Mathematics and the other sciences, for instance, approximation theory,
probability theory, statistic theory, number theory, the solution of the
differential equations, numerical analysis, constructing Bezier curve,
$q$-analysis, operator theory and applications in computer graphics. Thus we
look for the applications of our new functions and the ($q$-) Bernstein type
polynomials. Due to Oruc and Tuncer [13], the $q$-Bernstein polynomials shares
the well-known shape-preserving properties of the classical Bernstein
polynomials. When the function $f$ is convex then
$\beta_{n-1}(f,x)\geq\beta_{n}(f,x)\text{ for }n>1\text{ and }0<q\leq 1,$
where
$\beta_{n}(f,x)=\sum_{r=0}^{n}f_{r}\left[\begin{array}[]{c}n\\\
r\end{array}\right]x^{r}\prod_{s=0}^{n-r-1}\left(1-q^{s}x\right)$
and
$\left[\begin{array}[]{c}n\\\
r\end{array}\right]=\frac{\left[n\right]\cdots\left[n-r+1\right]}{\left[r\right]!}.$
As a consequence of this one can show that the approximation to convex
function by the $q$-Bernstein polynomials is one sided, with $\beta_{n}f\geq
f$ for all $n$. $\beta_{n}f$ behaves is very nice way when one vary the
parameter $q$. In [1], the authors gave some applications on the approximation
theory related to Bernoulli and Euler polynomials.
We conclude this section by the following questions:
1) How can one demonstrate approximation by ($q$-) Bernstein type polynomials,
$Y_{n+k}(k;x;q)$?
2) Is it possible to define uniform expansions of the ($q$-) Bernstein type
polynomials, $Y_{n+k}(k;x;q)$?
3) Is it possible to give applications of the ($q$-) Bernstein type
polynomials in calculating moments of some distributions in Statistics,
$Y_{n+k}(k;x;q)$?
4) How can one give relations between the ($q$-) Bernstein type polynomials,
$Y_{n+k}(k;x;q)$ and the Milnor algebras.
###### Acknowledgement 1.
The first author is supported by the research fund of Akdeniz University.
## References
* [1] M. Acikgoz, Y. Simsek, On multiple interpolation functions of the Nörlund-type $q$-Euler polynomials. Abstr. Appl. Anal. 2009, Art. ID 382574, 14 pp.
* [2] N. Cakic and G. V. Milovanovic, On generalized Stirling numbers and polynomials, preprint.
* [3] S. Berg, Some properties and applications of a ratio of Stirling numbers of the second kind, Scand. J. Statist. 2(2) (1975), 91-94.
* [4] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités, Comm. Soc. Math. Charkow Sér. 2 t. 13 (1912-1913), 1-2.
* [5] H. W. Gould, A theorem concerning the Bernstein polynomials, Math. Magazine 31(5) (1958), 259-264.
* [6] Z. Guan, Iterated Bernstein polynomial approximations, arXiv:0909.0684v3.
* [7] A. H. Joarder and M. Mahmood, An inductive derivation of Stirling numbers of the second kind and their applications in Statistics, J. Appl. Math.&Decision Sciences 1(2) (1997), 151-157.
* [8] K. I. Joy, Bernstein polynomials, On-Line Geometric Modeling Notes, http://en.wikipedia.org/wiki/Bernstein_polynomial.
* [9] E. Kowalski, Bernstein polynomials and Brownian motion, Amer. Math. Monthly 113(10) (2006), 865-886.
* [10] N. E. Nörlund, Vorlesungen über Differenzenrechung, Springer, Berlin, 1924.
* [11] L. Lopez and N. M. Temme, Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel and Buchholz polynomials, Modelling, Analysis and Simulation (MAS), MAS-R9927 September 30, 1999.
* [12] G. Nowak, Approximation properties for generalized $q$-Bernstein polynomials, J. Math. Anal. Appl. 350 (2009), 50-55.
* [13] H. Oruc and N. Tuncer, On the Convergence and Iterates of q-Bernstein Polynomials, J. Approx. Theory 117 (2002), 301-313.
* [14] S. Ostrovska, The approximation by $q$-Bernstein polynomials in the case $q\downarrow 1$, Arch. Math. 86(3) (2006), 282-288.
* [15] S. Ostrovska, On the $q$-Bernstein polynomials, Advan. Stud. Contemp. Math. 11 (2005), 193-204.
* [16] G. M. Phillips, A survey of results on the $q$-Bernstein polynomials, IMA Journal of Numerical Analysis Advance Access published online on June 23, (2009), 1-12, doi:10.1093/imanum/drn088.
* [17] A. Pinter, On a diophantine problem concerning Stirling numbers, Acta Math. Hungar. 65(4) 1994, 361-364.
* [18] Y. Simsek, Twisted $\left(h,q\right)$-Bernoulli numbers and polynomials related to twisted $\left(h,q\right)$-zeta function and $L$-function, J. Math. Anal. Appl. 324 (2006), 790-804.
* [19] Y. Simsek, On q-deformed Stirling numbers, arXiv:0711.0481v1 [math.NT].
* [20] Y. Simsek, V. Kurt and D. Kim, New approach to the complete sum of products of the twisted $(h,q)$-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys. 14(1) (2007), 44-56.
* [21] Z. Wu, The saturation of convergence on the interval $[0,1]$ for the $q$-Bernstein polynomials in the case $q>1$, J. Math. Anal. Appl. 357 (2009), 137-141.
|
arxiv-papers
| 2010-01-19T20:27:18 |
2024-09-04T02:49:07.844244
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yilmaz Simsek and Mehmet Acikgoz",
"submitter": "Yilmaz Simsek",
"url": "https://arxiv.org/abs/1001.3400"
}
|
1001.3429
|
# Alternative solutions of inhomogeneous second–order linear dynamic equations
on time scales
Douglas R. Anderson Department of Mathematics and Computer Science, Concordia
College, Moorhead, MN 56562 USA
visiting the School of Mathematics and Statistics, The University of New South
Wales, Sydney, NSW 2052, Australia andersod@cord.edu
http://www.cord.edu/faculty/andersod/bib.html and Christopher C. Tisdell
School of Mathematics and Statistics, The University of New South Wales,
Sydney, NSW 2052, Australia cct@unsw.edu.au http://web.maths.unsw.edu.au/
cct/
###### Abstract.
We exhibit an alternative method for solving inhomogeneous second–order linear
ordinary dynamic equations on time scales, based on reduction of order rather
than variation of parameters. Our form extends recent (and long-standing)
analysis on $\mathbb{R}$ to a new form for difference equations, quantum
equations, and arbitrary dynamic equations on time scales.
###### Key words and phrases:
Ordinary differential equations; ordinary difference equations; ordinary
quantum equations; ordinary dynamic equations; inhomogeneous equations; time
scales; variation of parameters; exact solutions; nonmultiplicity; reduction
of order
###### 2000 Mathematics Subject Classification:
34N05, 26E70, 39A10
## 1\. Introduction to second-order ordinary dynamic equations
A very common equation in mathematics, mathematical physics, and engineering
is the inhomogeneous second–order linear ordinary differential equation
(1.1) $y^{\prime\prime}(t)+p(t)y^{\prime}(t)+q(t)y(t)=r(t),\quad
t\in\mathbb{R},$
its ordinary difference equation counterparts
(1.2) $\Delta(\Delta y)(t)+p(t)\Delta y(t)+q(t)y(t)=r(t),\quad
t\in\mathbb{Z},\quad\Delta y(t):=y(t+1)-y(t)$
or
(1.3) $y(t+2)+\alpha(t)y(t+1)+\beta(t)y(t)=r(t),\quad t\in\mathbb{Z},$
the related ordinary quantum equation
(1.4) $D_{h}(D_{h}y)(t)+p(t)D_{h}y(t)+q(t)y(t)=r(t),\quad t\in
h^{\mathbb{Z}},\quad h>1,\quad D_{h}y(t):=\frac{y(ht)-y(t)}{ht-t},$
or the recently introduced ordinary dynamic equation on time scales given by
(1.5) $y^{\Delta\Delta}(t)+p(t)y^{\Delta}(t)+q(t)y(t)=r(t),\quad
t\in\mathbb{T}^{\kappa^{2}},$
where the differential/shift operators in (1.1)$-$(1.5) represent
differentiation with respect to $t$ on the corresponding time scales,
respectively. In the general setting represented by (1.5), the functions $p$,
$q$, and $r$ are real-valued right-dense continuous scalar functions of $t$
satisfying the regressivity condition
$1+\mu(t)\left[-p(t)+\mu(t)q(t)\right]\neq 0.$
Recall that on a time scale $\mathbb{T}$, namely any nonempty closed subset of
the real line, the delta derivative is given by
$y^{\Delta}(t):=\lim_{s\rightarrow
t}\frac{y^{\sigma}(t)-y(s)}{\sigma(t)-s},\quad t\in\mathbb{T}^{\kappa},$
provided the limit exists, where $\sigma$ is the forward jump operator
$\sigma(t):=\inf\\{s\in\mathbb{T}:s>t\\}$, and $y^{\sigma}=y\circ\sigma$. The
graininess $\mu$ is simply $\mu(t)=\sigma(t)-t$. For more on time scales see
Hilger [7].
Indeed, extensive analysis of (1.5) and its solution forms can be found in
Bohner and Peterson [3], while the ordinary differential equation (1.1) is
studied by, for example, Blest [2], Boyce and DiPrima [4], Hille [8], Ince
[9], Johnson, Busawon, and Barbot [11], Kelley and Peterson [13], whereas the
difference equation (1.2) appears in Agarwal [1], Elaydi [6], and Kelley and
Peterson [12]. A commonly used technique to solve (1.1)$-$(1.5) is Lagrange’s
variation of parameters method. In this approach, a solution $y$ of
(1.1)$-$(1.5) takes the form $y=y_{u}+y_{d}$, where $y_{u}$ is the
complementary solution of the corresponding homogeneous or undriven ($r\equiv
0$) form of (1.1)$-$(1.5), and $y_{d}$ is any particular solution of the
inhomogeneous or driven equations (1.1)$-$(1.5). If $y_{1}$ and $y_{2}$ are
two linearly independent solutions of the corresponding homogeneous equation,
then it is well known that we may write $y=c_{1}y_{1}+c_{2}y_{2}+y_{d}$ for
arbitrary constants $c_{1}$ and $c_{2}$. For example, using variation of
parameters, the form of a particular solution for (1.5) is [3, Theorem 3.73]
(1.6)
$y_{d}(t)=y_{2}(t)\int_{a}^{t}\frac{y_{1}^{\sigma}(s)r(s)}{W^{\sigma}(y_{1},y_{2})(s)}\Delta
s-y_{1}(t)\int_{a}^{t}\frac{y_{2}^{\sigma}(s)r(s)}{W^{\sigma}(y_{1},y_{2})(s)}\Delta
s,\quad t\in\mathbb{T},$
which reduces to
$y_{d}(t)=y_{2}(t)\int_{a}^{t}\frac{y_{1}(s)r(s)}{W(y_{1},y_{2})(s)}ds-
y_{1}(t)\int_{a}^{t}\frac{y_{2}(s)r(s)}{W(y_{1},y_{2})(s)}ds,\quad
t\in\mathbb{R}$
for (1.1), and to
$y_{d}(t)=y_{2}(t)\sum_{s=a}^{t-1}\frac{y_{1}(s+1)r(s)}{W(y_{1},y_{2})(s+1)}-y_{1}(t)\sum_{s=a}^{t-1}\frac{y_{2}(s+1)r(s)}{W(y_{1},y_{2})(s+1)},\quad
t\in\mathbb{Z}$
for (1.2), where in each case $W(y_{1},y_{2})$ is the Wronskian of $y_{1}$ and
$y_{2}$, defined appropriately for each time scale. Notice that the
integrals/summations always involve both $y_{1}$ and $y_{2}$. Recently,
Johnson, Busawon, and Barbot [11] derived two alternative solution forms for
$y_{d}$ for the ordinary differential equation (1.1), namely
(1.7) $y_{d}(t)=y_{i}(t)\int\frac{e^{-\int p(t)dt}\int r(t)y_{i}(t)e^{\int
p(t)dt}dt}{y_{i}^{2}(t)}dt,\quad t\in\mathbb{R},\quad i=1,2.$
As the authors point out in [11], either $y_{1}$ or $y_{2}$ may be chosen in
(1.7), depending on which one yields an easier integral to compute.
Unfortunately neither the technique nor the results in [11] are new, as they
are discussed in Blest [2] and the undergraduate textbook by Boyce and DiPrima
[4, Exercise 28, p185], to cite just two examples. In fact, Yosida [14, p.29]
calls the technique D’Alembert’s reduction of order for ordinary differential
equations, while according to Jahnke [10, p.332], the method of reduction of
order dates back at least as far as Euler (1750). In addition, Clairaut,
Lagrange and Laplace have been involved with the method and thus it may be
impossible to attribute to one mathematician [10, p.332]. In any case, the
method is at least 250 years old.
Despite the non-novelty of (1.7) for the ode (1.1), such a result would be new
on general time scales for (1.5). Consequently one of our goals in this paper
is to generalize and extend (1.7) to an alternative form for particular
solutions to the ordinary dynamic equation on time scales (1.5), which would
then nevertheless result in new alternative forms for the difference equations
(1.2) or (1.3), and the quantum equation (1.4) as simple corollaries, as well
as contain (1.7) for (1.1). This development is different from that given by
Bohner and Peterson [3, Chapter 3].
An additional goal of this paper is to present some simple results that
guarantee the nonmultiplicity of solutions to initial value problems
associated with (1.5). Our approach is based on simple inequalities and does
not require a knowledge of matrix theory, in contrast to [3, Theorem 3.1;
Corollary 5.90].
In Section 2 we address the question of nonmultiplicity of solutions, while in
Section 3 we develop a method for solving inhomogeneous second–order linear
ordinary dynamic equations on time scales, based on reduction of order rather
than variation of parameters. Section 4 contains some special cases that
illustrate our results.
## 2\. nonmultiplicity
In this section we consider the notion of nonmultiplicity of solutions to the
linear initial value problem (1.5) with initial conditions
(2.1) $y(t_{0})=A,\qquad y^{\Delta}(t_{0})=B,$
where $A,B\in\mathbb{R}$ and $t_{0}\in\mathbb{T}$. Let $I\subseteq\mathbb{R}$
and $t_{0}\in I_{\mathbb{T}}:=I\cap\mathbb{T}$. By “nonmultiplicity of
solutions” we mean that our theorems will present conditions under which
(1.5), (2.1) will have, at most, one solution $y=y(t)$ for $t\geq t_{0}$,
$t\in I_{\mathbb{T}}$. Such information is highly valuable, for example, when
constructing explicit solutions to problems as we can determine when the
constructed solution (or unique linear combination of solutions) will be the
only solution to the problem at hand. Our techniques follow those of
Coddington [5, Ch.2, Sec.3].
In the following, we will make use of the time-scale exponential function
defined in terms of right-dense continuous functions $p$ satisfying the
regressivity condition $1+\mu(t)p(t)\neq 0$ for all $t\in\mathbb{T}^{\kappa}$.
Given such a $p$, the delta exponential function [3, Theorem 2.30] is given by
$e_{p}(t,a)=\begin{cases}\exp\left(\displaystyle\int_{a}^{t}p(\tau)\Delta\tau\right)&:\mu(\tau)=0,\\\
\exp\left(\displaystyle\int_{a}^{t}\frac{1}{\mu(\tau)}\operatorname{Log}(1+p(\tau)\mu(\tau))\Delta\tau\right)&:\mu(\tau)\neq
0,\end{cases}$
where $\operatorname{Log}$ is the principal logarithm. It follows that
$e_{p}(t,a)$ is the unique solution to the initial value problem
$\phi^{\Delta}(t)=p(t)\phi(t)$, $\phi(a)=1$ on $\mathbb{T}$. We will denote
$1/e_{p}(t,a)$ by $e_{\ominus p}(t,a)$.
We will require the following lemma from [3, Theorem 6.1, p.255].
###### Lemma 2.1.
Let $\ell\in\mathcal{R}^{+}$ and $v\in\operatorname{C^{1}_{rd}}(\mathbb{T})$,
and let $t_{0}\in\mathbb{T}$. If $v^{\Delta}(t)\leq\ell(t)v(t)$ for all $t\geq
t_{0}$, then $v(t)\leq v(t_{0})e_{\ell}(t,t_{0})$ for all $t\geq t_{0}$.
Our initial analysis will concern the case of (1.5) with constant
coefficients, namely
(2.2) $y^{\Delta\Delta}(t)+py^{\Delta}(t)+qy(t)=r(t),\quad y(t_{0})=A,\quad
y^{\Delta}(t_{0})=B,$
where $p,q\in\mathbb{R}$ are constants. The following result involves the
homogeneous form of (2.2), that is
(2.3) $y^{\Delta\Delta}(t)+py^{\Delta}(t)+qy(t)=0,$
and provides an estimate on the growth rate of solutions to (2.3). For this
estimate, we define
$\|y(t)\|_{2}:=\left((y(t))^{2}+\left(y^{\Delta}(t)\right)^{2}\right)^{1/2},\quad
t\in I_{\mathbb{T}},\quad t\geq t_{0}.$
###### Theorem 2.2.
Let $k:=1+|p|+|q|$. If $y$ is any solution to (2.3) on $I_{\mathbb{T}}$ then
(2.4) $\|y(t)\|_{2}\leq\|y(t_{0})\|_{2}e_{k}(t,t_{0}),\quad t\in
I_{\mathbb{T}},\quad t\geq t_{0}.$
###### Proof.
Let $u(t)=\|y(t)\|_{2}^{2}$, where $y$ is a solution to (2.3). For all $t\in
I_{\mathbb{T}}^{\kappa}$ we have
$\displaystyle u^{\Delta}(t)$ $\displaystyle=$
$\displaystyle\left(y(t)+y^{\sigma}(t)\right)y^{\Delta}(t)+\left(y^{\Delta}(t)+y^{\Delta\sigma}(t)\right)y^{\Delta\Delta}(t)$
$\displaystyle=$
$\displaystyle\left(2y(t)+\mu(t)y^{\Delta}(t)\right)y^{\Delta}(t)+\left(2y^{\Delta}(t)+\mu(t)y^{\Delta\Delta}(t)\right)y^{\Delta\Delta}(t)$
$\displaystyle=$ $\displaystyle
2y(t)y^{\Delta}(t)+\mu(t)(y^{\Delta}(t))^{2}+2y^{\Delta}(t)y^{\Delta\Delta}(t)+\mu(t)(y^{\Delta\Delta}(t))^{2}.$
Now, apply Young’s inequality $2ab\leq a^{2}+b^{2}$ to the first term and
replace $y^{\Delta\Delta}$ with $-py^{\Delta}-qy$ to obtain
$\displaystyle u^{\Delta}(t)$ $\displaystyle\leq$
$\displaystyle(y(t))^{2}+(y^{\Delta}(t))^{2}+\mu(t)(y^{\Delta}(t))^{2}+2y^{\Delta}(t)[-py^{\Delta}(t)-qy(t)]+\mu(t)[-py^{\Delta}(t)-qy(t)]^{2}$
$\displaystyle\leq$
$\displaystyle(1+2|p|+|q|)\left((y(t))^{2}+(y^{\Delta}(t))^{2}\right)+\mu(t)\left[1+p^{2}+|p||q|+q^{2}\right]\left((y(t))^{2}+(y^{\Delta}(t))^{2}\right)$
$\displaystyle\leq$ $\displaystyle
2(1+|p|+|q|)\left((y(t))^{2}+(y^{\Delta}(t))^{2}\right)+\mu(t)\left[1+2(|p|+|q|)+p^{2}+2|p||q|+q^{2}\right]\left((y(t))^{2}+(y^{\Delta}(t))^{2}\right)$
$\displaystyle=$ $\displaystyle(k\oplus k)u(t).$
Above, $\oplus$ is known as the “circle plus” operator, $z\oplus w:=z+w+\mu
zw$, see [3, p.54].
Thus, the conditions of Lemma 2.1 hold with $v=u$ and $\ell=k\oplus k$.
Consequently,
$u(t)\leq u(t_{0})e_{k\oplus k}(t,t_{0})=u(t_{0})(e_{k}(t,t_{0}))^{2},\quad
t\geq t_{0},\quad t\in I_{\mathbb{T}},$
and therefore
$\|y(t)\|_{2}\leq\|y(t_{0})\|_{2}e_{k}(t,t_{0}),\quad t\geq t_{0},\quad t\in
I_{\mathbb{T}}.$
∎
Theorem 2.2 now leads to the following nonmultiplicity result for solutions to
(2.2), (2.1).
###### Theorem 2.3.
Let $r\in\operatorname{C_{rd}}(I_{\mathbb{T}}^{\kappa^{2}};\mathbb{R})$, and
$t_{0}\in I_{\mathbb{T}}$. The dynamic initial value problem (2.2), (2.1) has,
at most, one solution $y=y(t)$ for $t\geq t_{0}$ with $t\in I_{\mathbb{T}}$.
###### Proof.
Assume there are two solutions $x$ and $y$. Let $z(t):=x(t)-y(t)$ for $t\in
I_{\mathbb{T}}$, and note that $z$ must satisfy the homogeneous equation (2.3)
together with the homogeneous initial conditions $z(t_{0})=0$,
$z^{\Delta}(t_{0})=0$. Now, by Theorem 2.2 we have
$\|z(t)\|_{2}\leq\|z(t_{0})\|_{2}e_{k}(t,t_{0})=0,\quad t\geq t_{0},\quad t\in
I_{\mathbb{T}},$
and thus $z(t)=0$ for all $t\geq t_{0}$, $t\in I_{\mathbb{T}}$. It follows
that $x=y$. ∎
The following result is an extension of Theorem 2.2 and concerns estimates on
solutions to the homogeneous form of (1.5) with variable coefficients, namely
(2.5) $y^{\Delta\Delta}(t)+p(t)y^{\Delta}(t)+q(t)y(t)=0,\quad
t\in\mathbb{T}^{\kappa^{2}}.$
###### Theorem 2.4.
Let $t\in I_{\mathbb{T}}$. Let $p_{1},q_{1}\in\mathbb{R}$ be nonnegative
constants such that
$|p(t)|\leq p_{1},\quad|q(t)|\leq q_{1},\quad t\in
I_{\mathbb{T}}^{\kappa^{2}},\quad t\geq t_{0},$
and let $k_{1}:=1+p_{1}+q_{1}$. If $y$ is any solution to (2.5) on
$I_{\mathbb{T}}$, then
$\|y(t)\|_{2}\leq\|y(t_{0})\|_{2}e_{k_{1}}(t,t_{0}),\quad t\geq t_{0},\quad
t\in I_{\mathbb{T}}.$
###### Proof.
Our proof follows similar lines as that of the proof of Theorem 2.2, and thus
we just sketch the details. Letting $u(t)=\|y(t)\|_{2}^{2}$, we obtain
$\displaystyle u^{\Delta}(t)$ $\displaystyle\leq$
$\displaystyle(y(t))^{2}+(y^{\Delta}(t))^{2}+\mu(t)(y^{\Delta}(t))^{2}+2y^{\Delta}(t)[-p(t)y^{\Delta}(t)-q(t)y(t)]+\mu(t)[-p(t)y^{\Delta}(t)-q(t)y(t)]^{2}$
$\displaystyle\leq$ $\displaystyle
2(1+p_{1}+q_{1})u(t)+\mu(t)\left[1+2(p_{1}+q_{1})+p_{1}^{2}+2p_{1}q_{1}+q_{1}^{2}\right]u(t)$
$\displaystyle\leq$ $\displaystyle(k_{1}\oplus k_{1})u(t).$
Thus, applying Lemma 2.1 we obtain
$u(t)\leq u(t_{0})(e_{k_{1}}(t,t_{0}))^{2},\quad t\geq t_{0},\quad t\in
I_{\mathbb{T}},$
and therefore
$\|y(t)\|_{2}\leq\|y(t_{0})\|_{2}e_{k_{1}}(t,t_{0}),\quad t\geq t_{0},\quad
t\in I_{\mathbb{T}}.$
∎
Theorem 2.4 now leads to the following nonmultiplicity result for solutions to
(1.5), (2.1).
###### Theorem 2.5.
Let $t\in I_{\mathbb{T}}$. The dynamic equation (1.5) with initial conditions
(2.1) has at most one solution $y=y(t)$ for $t\geq t_{0}$ with $t\in
I_{\mathbb{T}}$.
###### Proof.
Assume there are two solutions $x$ and $y$, and let $z(t)=x(t)-y(t)$ for
$t\geq t_{0}$ with $t\in I_{\mathbb{T}}$. Note that $z$ satisfies (2.5) and
the initial homogeneous initial conditions $z(t_{0})=0$,
$z^{\Delta}(t_{0})=0$. Since $I_{\mathbb{T}}$ may be unbounded, the
coefficient functions $p$ and $q$ may not be bounded on $I_{\mathbb{T}}$, and
so Theorem 2.4 may not be directly applied to $z$. We let $t$ be any point in
$I_{\mathbb{T}}$ such that $t>t_{0}$, and let $J_{\mathbb{T}}$ be any closed,
bounded interval of $I_{\mathbb{T}}$ such that $J_{\mathbb{T}}$ has $t_{0}$ as
a left endpoint and $J_{\mathbb{T}}$ contains $t$. On $J_{\mathbb{T}}$, $p$
and $q$ are both bounded, say by $p_{1}$ and $q_{1}$, respectively. We can now
apply Theorem 2.4 to $z$ on $J_{\mathbb{T}}$, and so $z(t)=0$ for all $t\in
J_{\mathbb{T}}$. Now since $t$ was chosen to be any point in $I_{\mathbb{T}}$
with $t>t_{0}$, we have shown that $x(t)=y(t)$ for all $t\in I_{\mathbb{T}}$
with $t\geq t_{0}$. ∎
###### Remark 2.6.
The quest for nonmultiplicity of solutions on intervals to the left of $t_{0}$
is a more delicate affair. The results of this section may be extended to
include nonmultiplicity of solution for $t\leq t_{0}$ by adapting the proofs
and obtaining inequalities like
$u^{\Delta}(t)\geq-2ku(t),\quad\mbox{for all}\quad t\leq t_{0},\ t\in
I_{\mathbb{T}}.$
However, there is a price to pay – the graininess function of the time scale
would need to be bounded above. This is due to regressivity coming into play.
## 3\. alternative solution forms for the general inhomogeneous dynamic
equation
In this section we state and prove the main result, namely a new form for a
solution of (1.5) in the spirit of (1.7). As mentioned previously, this will
then give us a new form for the difference equation (1.2) and quantum equation
(1.4) as well.
###### Theorem 3.1.
Let $\mathbb{T}$ be a time scale and let $p$, $q$ and $r$ be real–valued
right–dense continuous scalar functions of $t\in\mathbb{T}$ with $p$ and $q$
satisfying the regressivity condition
(3.1) $1+\mu(t)\left[-p(t)+\mu(t)q(t)\right]\neq 0,\quad
t\in\mathbb{T}^{\kappa}.$
For all $t\in\mathbb{T}$, let $y_{1}$ and $y_{2}$ satisfy
(3.2) $y_{i}^{\Delta\Delta}(t)+p(t)y_{i}^{\Delta}(t)+q(t)y_{i}(t)=0,\quad
i=1,2,$
and $W(t):=y_{1}(t)y_{2}^{\Delta}(t)-y_{2}(t)y_{1}^{\Delta}(t)\neq 0$. The
general solution of the linear inhomogeneous second–order ordinary dynamic
equation
$y^{\Delta\Delta}(t)+p(t)y^{\Delta}(t)+q(t)y(t)=r(t),\quad
t\in\mathbb{T}^{\kappa^{2}},$
is
(3.3) $y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{1}(t)\int\frac{e_{(-p+\mu
q)}(t,a)\int r(t)y_{1}^{\sigma}(t)e_{\ominus(-p+\mu q)}(\sigma(t),a)\Delta
t}{y_{1}(t)y_{1}^{\sigma}(t)}\Delta t$
or
(3.4) $y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{2}(t)\int\frac{e_{(-p+\mu
q)}(t,a)\int r(t)y_{2}^{\sigma}(t)e_{\ominus(-p+\mu q)}(\sigma(t),a)\Delta
t}{y_{2}(t)y_{2}^{\sigma}(t)}\Delta t,$
where $c_{1}$ and $c_{2}$ are arbitrary constants and $e_{(-p+\mu
q)}(\cdot,a)$ is the time–scale exponential function.
###### Proof.
Clearly (3.3) and (3.4) are of the expected form
$y=c_{1}y_{1}+c_{2}y_{2}+y_{d}$. One could easily verify that a function of
the form
(3.5) $y_{d}(t)=y_{i}(t)\int\frac{e_{(-p+\mu q)}(t,a)\int
r(t)y_{i}^{\sigma}(t)e_{\ominus(-p+\mu q)}(\sigma(t),a)\Delta
t}{y_{i}(t)y_{i}^{\sigma}(t)}\Delta t$
is a particular solution of (1.5) using the time scale calculus, but that
would not give any insight into where (3.5) comes from. Thus, to derive (3.5),
assume we have a particular solution to the inhomogeneous equation (1.5) of
the form $y_{d}(t)=y_{i}(t)v(t)$, where $y_{i}$ solves the homogeneous
equation (3.2) and $v$ is a function to be determined. Then, using the product
rule $(fg)^{\Delta}=gf^{\Delta}+f^{\sigma}g^{\Delta}$, we have
$\displaystyle y_{d}^{\Delta}(t)$ $\displaystyle=$ $\displaystyle
v(t)y_{i}^{\Delta}(t)+y_{i}^{\sigma}(t)v^{\Delta}(t)$ $\displaystyle=$
$\displaystyle
v(t)y_{i}^{\Delta}(t)+y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)/y_{i}(t),$
and using the product rule again with the quotient rule
$\left(\frac{f}{g}\right)^{\Delta}=\frac{gf^{\Delta}-fg^{\Delta}}{gg^{\sigma}}$
we see that
$\displaystyle y_{d}^{\Delta\Delta}(t)$ $\displaystyle=$ $\displaystyle
v(t)y_{i}^{\Delta\Delta}(t)+y_{i}^{\Delta\sigma}(t)v^{\Delta}(t)+\frac{y_{i}(t)\left(y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)\right)^{\Delta}-y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)y_{i}^{\Delta}(t)}{y_{i}(t)y_{i}^{\sigma}(t)}$
$\displaystyle=$ $\displaystyle
v(t)y_{i}^{\Delta\Delta}(t)+y_{i}^{\Delta\sigma}(t)v^{\Delta}(t)+\left(y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)\right)^{\Delta}/y_{i}^{\sigma}(t)-v^{\Delta}(t)y_{i}^{\Delta}(t).$
Since we are assuming $y_{d}$ is a particular solution of the inhomogeneous
equation (1.5), we must have
$\displaystyle r(t)$ $\displaystyle=$ $\displaystyle
y_{d}^{\Delta\Delta}(t)+p(t)y_{d}^{\Delta}(t)+q(t)y_{d}(t)$ $\displaystyle=$
$\displaystyle
v(t)\left(y_{i}^{\Delta\Delta}(t)+p(t)y_{i}^{\Delta}(t)+q(t)y_{i}(t)\right)+p(t)y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)/y_{i}(t)$
$\displaystyle+y_{i}^{\Delta\sigma}(t)v^{\Delta}(t)+\left[y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)\right]^{\Delta}/y_{i}^{\sigma}(t)-v^{\Delta}(t)y_{i}^{\Delta}(t).$
Now $y_{i}$ is a solution of the homogeneous equation (3.2), so after
simplifying we multiply by $y_{i}^{\sigma}$ to get
$r(t)y_{i}^{\sigma}(t)=y_{i}^{\sigma}(t)p(t)\left[y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)\right]/y_{i}(t)+\left[y_{i}(t)y_{i}^{\sigma}(t)v^{\Delta}(t)\right]^{\Delta}+y_{i}^{\sigma}(t)v^{\Delta}(t)\left(y_{i}^{\Delta\sigma}(t)-y_{i}^{\Delta}(t)\right).$
Make the substitution $u=y_{i}y_{i}^{\sigma}v^{\Delta}$, then use the simple
formula $f^{\sigma}-f=\mu f^{\Delta}$ to get
$\displaystyle r(t)y_{i}^{\sigma}(t)$ $\displaystyle=$ $\displaystyle
y_{i}^{\sigma}(t)p(t)u(t)/y_{i}(t)+u^{\Delta}(t)+y_{i}^{\sigma}(t)v^{\Delta}(t)\mu(t)y_{i}^{\Delta\Delta}(t)$
$\displaystyle=$ $\displaystyle
u^{\Delta}(t)+\frac{p(t)y_{i}^{\sigma}(t)+\mu(t)y_{i}^{\Delta\Delta}(t)}{y_{i}(t)}u(t).$
Using the simple formula $f^{\sigma}-f=\mu f^{\Delta}$ again and rearranging,
we see that
$u^{\Delta}(t)+\left(p(t)-\mu(t)q(t)\right)u(t)=r(t)y_{i}^{\sigma}(t).$
Focusing on the coefficient of $u$, we note that
$\displaystyle p(t)-\mu(t)q(t)$ $\displaystyle=$
$\displaystyle-(-p(t)+\mu(t)q(t))$ $\displaystyle=$
$\displaystyle-\ominus\left(\ominus(-p(t)+\mu(t)q(t))\right)$ $\displaystyle=$
$\displaystyle\frac{\ominus(-p(t)+\mu(t)q(t))e_{\ominus(-p+\mu
q)}(t,a)}{\left[1+\mu(t)\left(\ominus(-p(t)+\mu(t)q(t))\right)\right]e_{\ominus(-p+\mu
q)}(t,a)}$ $\displaystyle=$ $\displaystyle\frac{e^{\Delta}_{\ominus(-p+\mu
q)}(t,a)}{e^{\sigma}_{\ominus(-p+\mu q)}(t,a)}.$
Consequently we have that
$\left(e_{\ominus(-p+\mu
q)}(t,a)u(t)\right)^{\Delta}=r(t)y_{i}^{\sigma}(t)e^{\sigma}_{\ominus(-p+\mu
q)}(t,a).$
Solving while recalling that $u=y_{i}y_{i}^{\sigma}v^{\Delta}$, we arrive at
$v(t)=\int\frac{e_{(-p+\mu q)}(t,a)\int r(t)y_{i}^{\sigma}(t)e_{\ominus(-p+\mu
q)}(\sigma(t),a)\Delta t}{y_{i}(t)y_{i}^{\sigma}(t)}\Delta t,$
so that via $y_{d}=y_{i}v$ we obtain (3.5). ∎
###### Remark 3.2.
The regressivity assumption in (3.1) is not at all unusual, as it is automatic
in the case $\mathbb{T}=\mathbb{R}$ since $\mu\equiv 0$, and it is assumed in
the variation of parameters theorem on general time scales; see [3, Definition
3.3]. The following theorem is a simple corollary of Theorem 3.1.
###### Theorem 3.3 (Reduction of Order).
Assume $p$ and $q$ are real–valued right–dense continuous scalar functions of
$t\in\mathbb{T}$ satisfying the regressivity condition
(3.6) $1+\mu(t)\left[-p(t)+\mu(t)q(t)\right]\neq 0,\quad
t\in\mathbb{T}^{\kappa}.$
If $y_{1}$ is a solution of the linear homogeneous second–order ordinary
dynamic equation (2.5), then
(3.7) $y_{2}(t)=y_{1}(t)\int\frac{e_{(-p+\mu
q)}(t,a)}{y_{1}(t)y_{1}^{\sigma}(t)}\Delta t$
is a second linearly independent solution of (2.5). Similarly, if $y_{2}$ is a
solution of (2.5), then
(3.8) $y_{1}(t)=y_{2}(t)\int\frac{e_{(-p+\mu
q)}(t,a)}{y_{2}(t)y_{2}^{\sigma}(t)}\Delta t$
is a second linearly independent solution of (2.5).
###### Proof.
We will prove (3.7), since the proof of (3.8) is similar. Thus, assume $y_{1}$
is a solution of (2.5). Since $r(t)\equiv 0$ in this case, and general
antiderivatives are used in Theorem 3.1, we may choose the constant of
integration in (3.5) in such a way that
$\int r(t)y_{1}^{\sigma}(t)e_{\ominus(-p+\mu q)}(\sigma(t),a)\Delta t=1,$
so that (3.5) becomes
$y_{d}(t)=y_{1}(t)\int\frac{e_{(-p+\mu
q)}(t,a)}{y_{1}(t)y_{1}^{\sigma}(t)}\Delta t=y_{2}(t),$
in other words a particular solution of (2.5). One could also verify (3.7)
directly using the delta derivative rules. To show linear independence, we
calculate the Wronskian of $y_{1}$ and $y_{2}$, namely
(3.9)
$W(y_{1},y_{2})(t)=y_{1}(t)y_{2}^{\Delta}(t)-y_{1}^{\Delta}(t)y_{2}(t)=e_{(-p+\mu
q)}(t,a)\neq 0$
for all $t\in\mathbb{T}$ by the regressivity assumption (3.6); see [3, Theorem
2.48]. ∎
###### Remark 3.4.
It is also possible to show the equivalence of the result shown in (3.5) with
that due to the variation of parameters formula in equation (1.6). Let $y_{1}$
be a solution of the linear homogeneous equation (2.5). Assuming (3.6), and
appropriating the Wronskian of $y_{1}$ and the form of $y_{2}$ given in (3.7)
as calculated in (3.9), the solution given in (1.6) is
(3.10) $\displaystyle y_{d}(t)$ $\displaystyle=$ $\displaystyle
y_{2}(t)\int_{a}^{t}\frac{y_{1}^{\sigma}(s)r(s)}{e_{(-p+\mu
q)}(\sigma(s),a)}\Delta
s-y_{1}(t)\int_{a}^{t}\frac{y_{2}^{\sigma}(s)r(s)}{e_{(-p+\mu
q)}(\sigma(s),a)}\Delta s$ $\displaystyle=$
$\displaystyle\left(y_{1}(t)\int_{a}^{t}\frac{e_{(-p+\mu
q)}(s,a)}{y_{1}(s)y_{1}^{\sigma}(s)}\Delta
s\right)\int_{a}^{t}r(s)y_{1}^{\sigma}(s)e_{\ominus(-p+\mu
q)}(\sigma(s),a)\Delta s$ $\displaystyle-
y_{1}(t)\int_{a}^{t}r(s)y_{1}^{\sigma}(s)e_{\ominus(-p+\mu
q)}(\sigma(s),a)\int_{a}^{\sigma(s)}\frac{e_{(-p+\mu
q)}(\xi,a)}{y_{1}(\xi)y_{1}^{\sigma}(\xi)}\Delta\xi\Delta s.$
If we use the integration by parts formula $\int fg^{\Delta}\Delta
t=f(t)g(t)-\int f^{\Delta}g^{\sigma}\Delta t$, see [3, Theorem 1.77(vi)], on
the integral in (3.5), where we have taken
$f(t)=\int_{a}^{t}r(s)y_{1}^{\sigma}(s)e_{\ominus(-p+\mu
q)}(\sigma(s),a)\Delta s\quad\text{and}\quad g^{\Delta}(t)=\frac{e_{(-p+\mu
q)}(t,a)}{y_{1}(t)y_{1}^{\sigma}(t)},$
we get (3.10).
The next corollary concerns another possible second–order linear dynamic
equation discussed by Bohner and Peterson [3, (3.6)].
###### Corollary 3.5.
Let $\alpha$, $\beta$ and $r$ be real–valued right–dense continuous scalar
functions on $\mathbb{T}$ with $\alpha$ satisfying the regressivity condition
$1+\mu(t)\alpha(t)\neq 0$ for $t\in\mathbb{T}^{\kappa}$. For all
$t\in\mathbb{T}$, let $y_{1}$ and $y_{2}$ satisfy
(3.11)
$y_{i}^{\Delta\Delta}(t)+\alpha(t)y_{i}^{\Delta\sigma}(t)+\beta(t)y_{i}^{\sigma}(t)=0,\quad
i=1,2,$
with $W(t):=y_{1}(t)y_{2}^{\Delta}(t)-y_{2}(t)y_{1}^{\Delta}(t)\neq 0$. The
general solution of the linear inhomogeneous second–order ordinary dynamic
equation
(3.12)
$y^{\Delta\Delta}(t)+\alpha(t)y^{\Delta\sigma}(t)+\beta(t)y^{\sigma}(t)=r(t),\quad
t\in\mathbb{T}^{\kappa^{2}},$
is
(3.13)
$y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{1}(t)\int\frac{e_{\ominus\alpha}(t,a)\int
r(t)y_{1}^{\sigma}(t)e_{\alpha}(\sigma(t),a)\Delta
t}{y_{1}(t)y_{1}^{\sigma}(t)}\Delta t$
or
(3.14)
$y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{2}(t)\int\frac{e_{\ominus\alpha}(t,a)\int
r(t)y_{2}^{\sigma}(t)e_{\alpha}(\sigma(t),a)\Delta
t}{y_{2}(t)y_{2}^{\sigma}(t)}\Delta t,$
where $c_{1}$ and $c_{2}$ are arbitrary constants, and where
$e_{\alpha}(\cdot,a)$ is the time-scale exponential function.
###### Proof.
Rewriting (3.12) using the simple formula $f^{\sigma}=f+\mu f^{\Delta}$, we
see that we arrive at an equation of the form (1.5), where
$p(t)=\frac{\alpha(t)+\mu(t)\beta(t)}{1+\mu(t)\alpha(t)}\quad\text{and}\quad
q(t)=\frac{\beta(t)}{1+\mu(t)\alpha(t)}.$
Then we see that the term $-p+\mu q$ in Theorem 3.1 above is given by
$-p(t)+\mu(t)q(t)=\frac{-\alpha(t)}{1+\mu(t)\alpha(t)}=\ominus\alpha(t),$
and this corollary follows. ∎
## 4\. second–order linear ordinary difference equations
In this section we recount the results of the previous section when
$\mathbb{T}=\mathbb{Z}$, that is to say for second–order linear ordinary
difference equations. The following corollaries to Theorem 3.1 are obtained by
simply taking $\mathbb{T}=\mathbb{Z}$ in Theorem 3.1 above. The first uses the
forward difference operator form of the second–order linear equation (1.2),
and the second uses the shift form (1.3).
###### Theorem 4.1.
Let $p$, $q$ and $r$ be real–valued scalar functions of $t\in\mathbb{Z}$ with
$p$ and $q$ satisfying the regressivity condition
(4.1) $1-p(t)+q(t)\neq 0,\quad t\in\mathbb{Z}.$
Let $y_{1}$ and $y_{2}$ satisfy
$\Delta^{2}y_{i}(t)+p(t)\Delta y_{i}(t)+q(t)y_{i}(t)=0,\quad i=1,2,$
and $W(t):=y_{1}(t)\Delta y_{2}(t)-y_{2}(t)\Delta y_{1}(t)\neq 0$. The general
solution of the linear inhomogeneous second–order ordinary difference equation
$\Delta^{2}y(t)+p(t)\Delta y(t)+q(t)y(t)=r(t),\quad t\in\mathbb{Z},$
is given by
(4.2)
$y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{1}(t)\sum\frac{\prod_{j=a}^{t-1}\left(1-p(j)+q(j)\right)\sum\frac{r(t)y_{1}(t+1)}{\prod_{j=a}^{t}\left(1-p(j)+q(j)\right)}}{y_{1}(t)y_{1}(t+1)}$
or
(4.3)
$y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{2}(t)\sum\frac{\prod_{j=a}^{t-1}\left(1-p(j)+q(j)\right)\sum\frac{r(t)y_{2}(t+1)}{\prod_{j=a}^{t}\left(1-p(j)+q(j)\right)}}{y_{2}(t)y_{2}(t+1)},$
and where $c_{1}$ and $c_{2}$ are arbitrary constants.
###### Theorem 4.2.
Let $\alpha$, $\beta$ and $r$ be real–valued scalar functions of
$t\in\mathbb{Z}$ with regressivity condition $\beta(t)\neq 0$ for
$t\in\mathbb{Z}$. Let $y_{1}$ and $y_{2}$ satisfy
$y_{i}(t+2)+\alpha(t)y_{i}(t+1)+\beta(t)y_{i}(t)=0,\quad i=1,2,$
with $W(t):=y_{1}(t)y_{2}(t+1)-y_{2}(t)y_{1}(t+1)\neq 0$. The general solution
of the linear inhomogeneous second–order ordinary difference equation
(4.4) $y(t+2)+\alpha(t)y(t+1)+\beta(t)y(t)=r(t),\quad t\in\mathbb{Z},$
is given by
(4.5)
$y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{1}(t)\sum\frac{\prod_{j=a}^{t-1}\beta(j)\sum\frac{r(t)y_{1}(t+1)}{\prod_{j=a}^{t}\beta(j)}}{y_{1}(t)y_{1}(t+1)}$
or
(4.6)
$y(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{2}(t)\sum\frac{\prod_{j=a}^{t-1}\beta(j)\sum\frac{r(t)y_{2}(t+1)}{\prod_{j=a}^{t}\beta(j)}}{y_{2}(t)y_{2}(t+1)},$
where $c_{1}$ and $c_{2}$ are arbitrary constants.
In [11, Section 3] the authors applied their alternative formula (1.7) to give
a general treatment of (1.1) in the case of constant coefficient functions
$p(t)\equiv 2P$ and $q(t)\equiv Q$ with $P$ and $Q$ real constants with $Q\neq
0$ and $P^{2}\neq Q$. We follow suit by giving a general treatment of (4.4)
with constant coefficients using the results above from Theorem 4.2. To wit,
we analyze the ordinary difference equation
(4.7) $y(t+2)+2\alpha y(t+1)+\beta y(t)=r(t),\quad t\in\mathbb{Z},$
where $\alpha$ and $\beta$ are real constants with $\beta\neq 0$ and
$\alpha^{2}\neq\beta$. The solution of (4.7) can be written as
$y(t)=c_{1}\left(-\alpha-\sqrt{\alpha^{2}-\beta}\right)^{t}+c_{2}\left(-\alpha+\sqrt{\alpha^{2}-\beta}\right)^{t}+y_{d}.$
From Theorem 4.2 we know that a particular solution to the inhomogeneous
equation (4.7) has the form
$y_{d}(t)=y_{i}(t)\sum_{x=a}^{t-1}\frac{\prod_{j=a}^{x-1}\beta\sum_{s=a}^{x-1}\frac{r(s)y_{i}(s+1)}{\prod_{j=a}^{s}\beta}}{y_{i}(x)y_{i}(x+1)},\quad
i=1,2,$
which simplifies to
$y_{d}(t)=\sum_{x=a}^{t-1}\sum_{s=a}^{x-1}r(s)\left(-\alpha\pm\sqrt{\alpha^{2}-\beta}\right)^{t+s-2x}\beta^{x-1-s}.$
Thus, complete expressions for the solution of (4.7) are given by
$y(t)=c_{1}\lambda_{1}^{t}+c_{2}\lambda_{2}^{t}+\sum_{x=a}^{t-1}\sum_{s=a}^{x-1}r(s)\lambda_{1}^{t+s-2x}\beta^{x-1-s}$
or
$y(t)=c_{1}\lambda_{1}^{t}+c_{2}\lambda_{2}^{t}+\sum_{x=a}^{t-1}\sum_{s=a}^{x-1}r(s)\lambda_{2}^{t+s-2x}\beta^{x-1-s},$
where for simplicity we have taken
$\lambda_{1}=-\alpha-\sqrt{\alpha^{2}-\beta}$ and
$\lambda_{2}=-\alpha+\sqrt{\alpha^{2}-\beta}$.
###### Remark 4.3.
A general treatment of (1.5) with constant coefficients on arbitrary time
scales and even quantum equations is made difficult by the fact that, as seen
in (3.5), even with $p$ and $q$ taken to be constant functions, the term
$-p+\mu q$ is not constant except for the very special cases of
$\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=h\mathbb{Z}$.
## References
* [1] R. P. Agarwal, _Difference Equations and Inequalities: Theory, Methods, and Applications_ , Second Edition, Revised and Expanded, Marcel Dekker, New York, 2000.
* [2] D. Blest, Reduction of order as a neglected method for the solution of inhomogeneous linear ordinary differential equations, _Int. J. Math. Educ. Sci. Technol._ , 21(3) (1990) 393–401.
* [3] M. Bohner and A. Peterson, _Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2001.
* [4] W. E. Boyce and R. C. DiPrima, _Elementary Differential Equations_ , Seventh Edition, John Wiley & Sons, New York, 2001.
* [5] E. A. Coddington, _An Introduction to Ordinary Differential Equations_ , Prenctice–Hall Inc., Englewood Cliffs, 1961.
* [6] S. N. Elaydi, _An Introduction to Difference Equations_ , Springer, New York, 1996.
* [7] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, _Results Math._ , 18 (1990) 18–56.
* [8] E. Hille, _Ordinary Differential Equations in the Complex Domain_ , Dover, New York, 1997.
* [9] E. L. Ince, _Ordinary Differential Equations_ , Dover, New York, 1956.
* [10] _A History of Analysis_ , translated from the German, edited by Hans Niels Jahnke, _History of Mathematics_ , 24, American Mathematical Society, Providence, RI, London Mathematical Society, London, 2003.
* [11] P. Johnson, K. Busawon, and J. P. Barbot, Alternative solution of the inhomogeneous linear differential equation of order two, _J. Math. Anal. Appl._ , 339 (2008) 582–589.
* [12] W. Kelley and A. Peterson, _Difference Equations: An Introduction with Applications_ , Second Edition, Academic Press, San Diego, 2000.
* [13] W. Kelley and A. Peterson, _The Theory of Differential Equations: Classical and Qualitative_ , Pearson Prentice Hall, Upper Saddle River, NJ, 2004.
* [14] K. Yosida, _Lectures on Differential and Integral Equations_ , Interscience Publishers, New York, 1960.
|
arxiv-papers
| 2010-01-19T22:24:21 |
2024-09-04T02:49:07.850996
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson and Christopher C. Tisdell",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1001.3429"
}
|
1001.3463
|
# Relating diameter and mean curvature for
Riemannian submanifolds
Jia-Yong Wu Department of Mathematics, Shanghai Maritime University, Haigang
Avenue 1550, Shanghai 201306, P. R. China jywu81@yahoo.com and Yu Zheng
Department of Mathematics, East China Normal University, Dong Chuan Road 500,
Shanghai 200241, P. R. China zhyu@math.ecnu.edu.cn
(Date: December 20, 2009.)
###### Abstract.
Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly
isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we
estimate the diameter of $M$ in terms of its mean curvature field integral
under some geometric restrictions, and therefore generalize a recent work of
Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539–546).
###### Key words and phrases:
Mean curvature; Riemannian submanifolds; Geometric inequalities; Diameter
estimate.
###### 2000 Mathematics Subject Classification:
Primary 53C40; Secondly 57R42.
This work is partially supported by the NSFC10871069.
## 1\. Introduction
Let $M\rightarrow N$ be an isometric immersion of Riemannian manifolds of
dimension $m$ and $n$, respectively. In this paper, we estimate the intrinsic
diameter of the closed submanifold $M$ in terms of its mean curvature vector
integral, under some geometric restrictions involving the volume of $M$, the
sectional curvatures of $N$ and the injectivity radius of $N$. In particular,
we can estimate the intrinsic diameter of the closed submanifold $M$ in terms
of its mean curvature vector integral without any geometric restriction,
provided the sectional curvatures of the ambient manifold $N$ is non-positive.
Our work was inspired by the following result of P. Topping [11] who treated
the case $N=\mathbb{R}^{n}$.
Theorem A. (P. Topping [11]) _For $m\geq 1$, suppose that $M$ is an
$m$-dimensional closed (compact, no boundary) connected manifold smoothly
immersed in $\mathbb{R}^{n}$. Then there exists a constant $C(m)$ dependent
only on $m$ such that its intrinsic diameter $d_{int}$ and mean curvature $H$
are related by_
(1.1) $d_{int}\leq C(m)\int_{M}|H|^{m-1}d\mu,$
_where $d_{int}:=\max_{x,y\in M}dist_{M}(x,y)$ and $\mu$ is the measure on $M$
induced by the ambient space. In particular, we can take
$C(2)=\frac{32}{\pi}$._
Prior to the Topping’s work, L. Simon in [6] (see also [8]) derived an
interesting estimate of the external diameter $d_{ext}:=\max_{x,y\in
M^{2}\hookrightarrow\mathbb{R}^{3}}|x-y|_{\mathbb{R}^{3}}$ of a closed
connected surface $M^{2}$ immersed in $\mathbb{R}^{3}$ in terms of its area
and Willmore energy:
(1.2)
$d_{ext}<\frac{2}{\pi}\left(Area(M^{2})\cdot\int_{M^{2}}|H|^{2}d\mu\right)^{\frac{1}{2}}.$
At the core of the proof of (1.2) is the following assertion that one cannot
simultaneously have small area and small mean curvature in a ball within the
surface. In other words, for all $r>0$, we have
(1.3)
$\pi\leq\frac{A_{ext}(x,r)}{r^{2}}+\frac{1}{4}\int_{B_{ext}(x,r)}|H|^{2}d\mu,$
where $B_{ext}(x,r)$ and $A_{ext}(x,r)$ denote the subset of $M^{2}$ immersed
inside the open extrinsic ball in $\mathbb{R}^{3}$ centred at $x$ of radius
$r>0$ and its area, respectively. This type of estimate is from [6], and with
these sharp constants from [8]. Combining this fact with a simple covering
argument, one can derive (1.2). Note that if $M^{2}$ is a surface of constant
mean curvature $H$ immersed in $\mathbb{R}^{3}$, P. Topping in [7] used a
different method and established the following inequality
$d_{ext}\leq\frac{A|H|}{2\pi}.$
Equality is achieved when $M^{2}$ is a sphere.
Following the idea of L. Simon’s proof, P. Topping in [11] proved Theorem A by
considering a refined version of (1.3) for any dimensional manifold immersed
in $\mathbb{R}^{n}$. Roughly speaking, P. Topping asserted that the maximal
function and volume ratio (see their definitions in Section 2) cannot be
simultaneously smaller than a fixed dimensional constant. This assertion can
be confirmed by means of the Michael-Simon Sobolev inequality for submanifolds
of Euclidean space [4]. Then using this assertion and a covering lemma, one
can derive (1.1) immediately. As an application, H.-Z. Li in a recent paper
[3] used Theorem A to discuss the convergence of the volume-preserving mean
curvature flow in Euclidean space under some initial integral pinching
conditions.
On the other hand, as we all known, D. Hoffman and J. Spruck in [2] extended
the Michael-Simon result [4] to a general Sobolev inequality for submanifolds
of a Riemannian manifold under some geometric restrictions. To formulate their
result, we need some notations in [2]. Let $M\rightarrow N$ be an isometric
immersion of Riemannian manifolds of dimension $m$ and $n$, respectively. We
denote the sectional curvatures of $N$ by $K_{N}$. The mean curvature vector
field of the immersion is given by $H$. We write $\bar{R}(M)$ for the
injectivity radius of $N$ restricted to $M$ (or minimum distance to the cut
locus in $N$ for all points in $M$). Let us denote by $\omega_{m}$ the volume
of the unit ball in $\mathbb{R}^{m}$ and let $b$ be a positive real number or
a pure imaginary one.
Theorem B. (D. Hoffman and J. Spruck [2]) _Let $M\rightarrow N$ be an
isometric immersion of Riemannian manifolds of dimension $m$ and $n$,
respectively. Some notations are adopted as above. Assume $K_{N}\leq b^{2}$
and let $h$ be a non-negative $C^{1}$ function on $M$ vanishing on $\partial
M$. Then_
(1.4) $\left(\int_{M}h^{m/(m-1)}d\mu\right)^{(m-1)/m}\leq
c(m)\int_{M}\left[|\nabla h|+h|H|\right]d\mu,$
_provided_
(1.5)
$b^{2}(1-\beta)^{-2/m}\left(\omega_{m}^{-1}Vol(\mathrm{supp}h)\right)^{2/m}\leq
1$
_and_
(1.6) $2\rho_{0}\leq\bar{R}(M),$
_where_
$\rho_{0}=\left\\{\begin{aligned}
b^{-1}\sin^{-1}\left[b(1-\beta)^{-1/m}\left(\omega_{m}^{-1}Vol(\mathrm{supp}h)\right)^{1/m}\right]\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{real},\\\
(1-\beta)^{-1/m}\left(\omega_{m}^{-1}Vol(\mathrm{supp}h)\right)^{1/m}\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{imaginary}.\end{aligned}\right.$
_Here $\beta$ is a free parameter, $0<\beta<1$, and_
(1.7) $c(m):=c(m,\beta)=\pi\cdot
2^{m-1}\beta^{-1}(1-\beta)^{-1/m}\frac{m}{m-1}\omega_{m}^{-1/m}.$
###### Remark 1.1.
In Theorem B, we may replace the assumption $h\in C^{1}(M)$ by $h\in
W^{1,1}(M)$. As the mentioned remark in [2], the optimal choice of $\beta$ to
minimize $c$ is $\beta=m/(m+1)$. When $b$ is real we may replace condition
(1.6) by the stronger condition $\bar{R}\geq\pi b^{-1}$. When $b$ is a pure
imaginary number and the Riemannian manifold $N$ is simply connected and
complete, $\bar{R}(M)=+\infty$. Hence conditions (1.5) and (1.6) are
automatically satisfied.
Motivated by the work of P. Topping, it is natural to expect that there exists
a general geometric inequality for submanifolds of a Riemannian manifold,
which is similar to Theorem A. Fortunately, following closely the lines of the
Topping’s proof of Theorem A in [11], we can employ the Hoffman-Spruck Sobolev
inequality for submanifolds of a Riemannian manifold together with a covering
lemma to derive the desired results.
###### Theorem 1.2.
For $m\geq 1$, suppose that $M$ is an $m$-dimensional closed connected
Riemannian manifold smoothly isometrically immersed in an $n$-dimensional
complete Riemannian manifold $N$ with $K_{N}\leq b^{2}$. For any $0<\alpha<1$,
if
(1.8) $b^{2}(1-\alpha)^{-2/m}\left(\omega_{m}^{-1}Vol(M)\right)^{2/m}\leq 1$
and
(1.9) $2\rho_{0}\leq\bar{R}(M),$
where
$\rho_{0}=\left\\{\begin{aligned}
b^{-1}\sin^{-1}\left[b(1-\alpha)^{-1/m}\left(\omega_{m}^{-1}Vol(M)\right)^{1/m}\right]\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{real},\\\
(1-\alpha)^{-1/m}\left(\omega_{m}^{-1}Vol(M)\right)^{1/m}\quad\quad&\mathrm{for}\,\,\,b\,\,\,\mathrm{imaginary},\end{aligned}\right.$
then there exists a constant $C(m,\alpha)$ dependent only on $m$ and $\alpha$
such that
$d_{int}\leq C(m,\alpha)\int_{M}|H|^{m-1}d\mu.$
In particular, we can take $C(2,\alpha)=\frac{576\pi}{\alpha^{2}(1-\alpha)}$.
###### Remark 1.3.
In Theorem 1.2, the coefficients $C(m,\alpha)$ are not identical to (but
strongly dependent on) the coefficients $c(m)$ in Theorem B. From (2.7) and
(3.3) we can find that $C(m,\alpha)$ can still arrive at the minimum, when
$\alpha=\frac{m}{m+1}$. The conditions of (1.8) and (1.9) are similar to the
restrictions of (1.5) and (1.6) in Theorem B, and they guarantee that the
Hoffman-spruck Sobolev for submanifolds of a Riemannian manifold can be
applied in the proof of our theorem. When $b$ is real we may replace condition
(1.9) by the stronger condition $\bar{R}\geq\pi b^{-1}$. When $b$ is a pure
imaginary number and the Riemannian manifold $N$ is simply connected and
complete, $\bar{R}(M)=+\infty$, and hence conditions (1.8) and (1.9) are
automatically satisfied.
In particular, when $N=\mathbb{R}^{n}$, $K_{N}\equiv 0$ and
$\bar{R}(M)=+\infty$, and hence there are also no volume restrictions on $M$.
Combining this with Remark 1.3, if $b$ is pure imaginary or zero, then we see
that conditions (1.8) and (1.9) are automatically satisfied, and hence we
conclude that
###### Corollary 1.4.
For $m\geq 1$, suppose that $M$ is an $m$-dimensional closed connected
Riemannian manifold smoothly isometrically immersed in an $n$-dimensional
simply connected, complete, nonpositively curved Riemannian manifold $N$
($K_{N}\leq 0$). For any $0<\alpha<1$, then there exists a constant
$C(m,\alpha)$ dependent only on $m$ and $\alpha$ such that
$d_{int}\leq C(m,\alpha)\int_{M}|H|^{m-1}d\mu,$
where $\min_{0<\alpha<1}C(m,\alpha)=C(m,\frac{m}{m+1})$. In particular, we can
take $\min_{0<\alpha<1}C(2,\alpha)=C(2,\frac{2}{3})=3888\pi$.
We remark that the constants $C(2,\alpha)$ in Theorem 1.2 and Corollary 1.4
are not optimal in general. The proof of Theorem 1.2 follows the proof in the
Euclidean case [11]. Theorem 1.2 and Corollary 1.4 may have many interesting
applications which we have not discussed here. For example, we may borrow Li’s
idea of [3] and apply our Theorem 1.2 to study the convergence problem of the
volume-preserving mean curvature flow in Riemannian manifolds. We will explore
this aspect in the future.
Besides the above works, the closest precedent for our theorem is another P.
Topping’s work on diameter estimates for intrinsic manifolds evolving under
the Ricci flow [9]. In the Ricci flow case, P. Topping explored a log-Sobolev
inequality of the Ricci flow (see Theorem 3.4 in [9]), which can be derived by
the monotonicity of Perelman’s $\mathcal{W}$-functional (see [1], [5], [10]).
However a core tool of proving Theorem 1.2 is the Hoffman-Spruck Sobolev
inequality.
The rest of this paper is organized as follows. In Section 2, we will prove
Lemma 2.1. The proof needs the key Hoffman-Spruck Sobolev inequality. In
Section 3, we will finish the proof of Theorem 1.2 using Lemma 2.1 of Section
2 and a covering lemma.
## 2\. Estimates for maximal function and volume ratio
In this section we first introduce two useful geometric quantities: the
maximal function and the volume ratio. Then we apply the Hoffman-Spruck
Sobolev inequality to prove the following important Lemma 2.1, which is
essential in the proof of Theorem 1.2.
Given $x\in M^{m}$, with respect to a given metric, we denote the open
geodesic ball in $M^{m}$ centred at $x$ and of intrinsic radius $r>0$ by
$B(x,r)$, and its volume by
$V(x,r):=Vol(B(x,r)).$
Following Topping’s definitions in [11], when $m\geq 2$, we introduce the
_maximal function_
(2.1)
$M(x,R):=\sup_{r\in(0,R]}r^{-\frac{1}{m-1}}[V(x,r)]^{-\frac{m-2}{m-1}}\int_{B(x,r)}|H|d\mu$
and the _volume ratio_
(2.2) $\kappa(x,R):=\inf_{r\in(0,R]}\frac{V(x,r)}{r^{m}}$
for any $R>0$.
Similar to Lemma 1.2 in [11], we have the following general result.
###### Lemma 2.1.
For $m\geq 2$, suppose that $M$ is an $m$-dimensional Riemannian manifold
smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$
with $K_{N}\leq b^{2}$, which is complete with respect to the induced metric.
For any $0<\alpha<1$, if conditions (1.8) and (1.9) are satisfied, then there
exists a constant $\delta>0$ dependent only on $m$ and $\alpha$ such that for
any $x\in M$ and $R>0$, at least one of the following is true:
1. (1)
$M(x,R)\geq\delta$;
2. (2)
$\kappa(x,R)>\delta$.
In the case of closed surfaces ($m=2$) in $N$, we can choose
$\delta=\frac{\alpha^{2}(1-\alpha)}{144\pi}$.
###### Remark 2.2.
In Lemma 2.1, when $b$ is a pure imaginary number and the Riemannian manifold
$N$ is simply connected and complete, $\bar{R}(M)=+\infty$. Hence conditions
(1.8) and (1.9) are automatically satisfied.
Now we will finish the proof of Lemma 2.1.
###### Proof of Lemma 2.1.
We follow the ideas of the proof of Lemma 1.2 in [11]. Suppose that
$M(x,R)<\delta$ for some constant $\delta>0$, which will be chosen later.
According to the definition of the maximal function $M(x,R)$, we know that for
all $r\in(0,R]$
(2.3) $\int_{B(x,r)}|H|d\mu<\delta
r^{\frac{1}{m-1}}[V(x,r)]^{\frac{m-2}{m-1}}.$
Note that for fixed $x$, $V(r):=V(x,r)$ is differentiable for almost all
$r>0$. For such $r\in(0,R]$, and any $s>0$, we define a Lipschitz cut-off
function $h$ on $M$ by
(2.4) $h(y)=\left\\{\begin{aligned} 1\quad\quad&y\in B(x,r)\\\
1-\frac{1}{s}(dist_{M}(x,y)-r)\quad\quad&y\in B(x,r+s)\setminus B(x,r)\\\
0\quad\quad&y\not\in B(x,r+s).\end{aligned}\right.$
Since function $\sin^{-1}x$ is increasing on $[0,1]$ and
$Vol(\mathrm{supp}h)\leq Vol(M)$, we easily see that conditions (1.8) and
(1.9) guarantee the function $h$ of (2.4) to satisfy conditions (1.5) and
(1.6), where $\beta=\alpha$. Substituting this function to the Hoffman-Spruck
Sobolev inequality from Theorem B, we derive that
$\displaystyle V(r)^{(m-1)/m}$
$\displaystyle\leq\left(\int_{M}h^{m/(m-1)}d\mu\right)^{(m-1)/m}$
$\displaystyle\leq
c(m)\left[\frac{V(r+s)-V(r)}{s}+\int_{B(x,r+s)}|H|d\mu\right],$
where $c(m):=c(m,\alpha)=\pi\cdot
2^{m-1}\alpha^{-1}(1-\alpha)^{-1/m}\frac{m}{m-1}\omega_{m}^{-1/m}$. Letting
$s\downarrow 0$, we conclude that
$\displaystyle V(r)^{(m-1)/m}\leq
c(m)\left[\frac{dV}{dr}+\int_{B(x,r)}|H|d\mu\right].$
Combining this with (2.3), we have
(2.5) $\frac{dV}{dr}+\delta
r^{\frac{1}{m-1}}V(r)^{\frac{m-2}{m-1}}-c(m)^{-1}V(r)^{\frac{m-1}{m}}>0.$
Now we assume that $\delta>0$ is sufficiently small so that
$\delta<\omega_{m}$, and define another smooth function
$v(r):=\delta r^{m}.$
Then a straightforward computation yields
(2.6) $\frac{dv}{dr}+\delta
r^{\frac{1}{m-1}}v(r)^{\frac{m-2}{m-1}}-c(m)^{-1}v(r)^{\frac{m-1}{m}}=\left(m\delta+\delta^{\frac{2m-3}{m-1}}-c(m)^{-1}\delta^{\frac{m-1}{m}}\right)r^{m-1}.$
We can see that
(2.7) $\frac{dv}{dr}+\delta
r^{\frac{1}{m-1}}v(r)^{\frac{m-2}{m-1}}-c(m)^{-1}v(r)^{\frac{m-1}{m}}\leq 0$
as long as $\delta>0$ is sufficiently small, depending only on $m$ and
$\alpha$.
Notice the fact that $V(r)/r^{m}\rightarrow\omega_{m}$ as $r\downarrow 0$,
while $v(r)/r^{m}=\delta<\omega_{m}$. And combining inequalities (2.5) and
(2.7), we conclude that
$V(r)>v(r)$
for all $r\in(0,R]$. Otherwise, there exists a fixed $r_{0}$ such that
$V(r_{0})=v(r_{0})$ and $V(r)>v(r)$ for all $r\in(0,r_{0})$. Then from (2.5)
and (2.7), we can derive
$\frac{dV}{dr}\Big{|}_{r=r_{0}}>\frac{dv}{dr}\Big{|}_{r=r_{0}}.$
Namely,
$\frac{dV}{dr}>\frac{dv}{dr}$
in any sufficiently small neighborhood of $r_{0}$, which is impossible since
$V(r_{0})=v(r_{0})$ and $V(r)>v(r)$ for all $r\in(0,r_{0})$.
Therefore
$\kappa(x,R):=\inf_{r\in(0,R]}\frac{V(x,r)}{r^{m}}>\delta,$
which completes the proof of Lemma 2.1.
In the case of closed surfaces ($m=2$) in $N$, we can choose
$\delta=\frac{c(2,\alpha)^{-2}}{9}=\frac{\alpha^{2}(1-\alpha)}{144\pi}$ to
satisfy (2.7) and the constraint condition $\delta<\omega_{2}=\pi$. ∎
## 3\. Diameter Control
In this section we can follow the lines of [11] or [9], and easily prove
Theorem 1.2 by using Lemma 2.1 and a covering lemma. For the completeness of
this paper, here we still give the detailed proof of Theorem 1.2.
###### Proof of Theorem 1.2.
We may assume $m\geq 2$ since the case $m=1$ is trivial. Now we choose $R>0$
sufficiently large so that the total volume of the closed manifold $M$ is less
than $\delta R^{m}$, where $\delta$ is given by Lemma 2.1 (Notice that
$\delta$ does not depend on $R$). In particular, for all $z\in M$, we must
have
$\kappa(z,R)\leq\frac{V(z,R)}{R^{m}}\leq\delta.$
Hence by Lemma 2.1, as long as conditions (1.8) and (1.9) are satisfied, we
must have the maximal function $M(x,R)\geq\delta$. Namely, there exists
$r=r(z)$ such that
(3.1) $\displaystyle\delta$ $\displaystyle\leq
r^{-\frac{1}{m-1}}V(z,r)^{-\frac{m-2}{m-1}}\int_{B(z,r)}|H|d\mu$
$\displaystyle\leq
r^{-\frac{1}{m-1}}\left(\int_{B(z,r)}|H|^{m-1}d\mu\right)^{\frac{1}{m-1}},$
where we used the Hölder inequality for the second inequality above. Hence
(3.2) $r(z)\leq\delta^{1-m}\int_{B(z,r(z))}|H|^{m-1}d\mu.$
Now we have to pick appropriate points $z$ at which to apply (3.2). Let
$x_{1},x_{2}\in M$ be extremal points in $M$. This means that
$d_{int}=dist_{M}(x_{1},x_{2})$. Let $\Sigma$ be a shortest geodesic
connecting $x_{1}$ and $x_{2}$. Obviously, $\Sigma$ is covered by the balls
$\\{B(z,r(z)):z\in\Sigma\\}$. By a modification of the covering lemma (see
Lemma 5.2 in [9]), there exists a countable (possibly finite) set of points
$\\{z_{i}\in\Sigma\\}$ such that the balls $\\{B(z_{i},r(z_{i}))\\}$ are
disjoint, and cover at least a fraction $\rho$, where $\rho\in(0,\frac{1}{2})$
of $\Sigma$:
$\rho d_{int}\leq\sum_{i}2r(z_{i}).$
Combining this with (3.2), we have
$\displaystyle d_{int}$ $\displaystyle\leq\frac{2}{\rho}\sum_{i}r(z_{i})$
$\displaystyle\leq\frac{2}{\rho}\delta^{1-m}\sum_{i}\int_{B(z_{i},r(z_{i}))}|H|^{m-1}d\mu$
$\displaystyle\leq\frac{2}{\rho}\delta^{1-m}\int_{M}|H|^{m-1}d\mu,$
where $\delta>0$ is sufficiently small, depending only on $m$ and $\alpha$.
Letting $\rho\to\frac{1}{2}$, we arrived at
(3.3) $d_{int}\leq 4\delta^{1-m}\int_{M}|H|^{m-1}d\mu.$
Hence the desired theorem follows. If $m=2$, we can choose
$4\delta^{1-m}=\frac{576\pi}{\alpha^{2}(1-\alpha)}$, since
$\delta=\frac{\alpha^{2}(1-\alpha)}{144\pi}$. ∎
## Acknowledgment
The authors thank the referee for various comments and suggestions that helped
improve this paper.
## References
* [1] B. Chow, P. Lu, L. Ni, Hamilton’s Ricci flow, Lectures in Contemporary Mathematics 3, Science Press and American Mathematical Society, 2006.
* [2] D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure. Appl. Math., 27 (1974), 715–727. Erratum, Comm. Pure. Appl. Math., 28 (1975), 765–766.
* [3] H.-Z. Li, The volume-preserving mean curvature flow in Euclidean space, Pacific of Math., 243 (2009), 331–355.
* [4] J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $\mathbb{R}^{n}$, Comm. Pure Appl. Math., 26 (1973), 361–379.
* [5] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159.
* [6] L. M. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281–326.
* [7] P. M. Topping, The optimal constant in Wente’s $L^{\infty}$ estimate, Comment. Math. Helv., 72 (1997), 316–328.
* [8] P. M. Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math., 503 (1998), 47–61.
* [9] P. Topping, Diameter control under Ricci flow, Comm. Anal. Geom., 13 (2005), 1039–1055.
* [10] P. M. Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series 325, Cambridge University Press, 2006.
* [11] P. M. Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv., 83 (2008), 539–546.
|
arxiv-papers
| 2010-01-20T05:58:26 |
2024-09-04T02:49:07.858237
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jia-Yong Wu, Yu Zheng",
"submitter": "Jia-Yong Wu",
"url": "https://arxiv.org/abs/1001.3463"
}
|
1001.3470
|
KU-TP 041
Friedmann Equations from Entropic Force
Rong-Gen Caia, ***e-mail: cairg@itp.ac.cn, Li-Ming Caob,†††e-mail:
caolm@phys.kindai.ac.jp, and Nobuyoshi Ohtab,‡‡‡e-mail:
ohtan@phys.kindai.ac.jp
a Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical
Physics,
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
b Department of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502,
Japan
Abstract
In this note by use of the holographic principle together with the
equipartition law of energy and the Unruh temperature, we derive the Friedmann
equations of a Friedmann-Robertson-Walker universe.
It has a long history that gravity is not regarded as a fundamental
interaction in Nature. The earliest idea on this was proposed by Sakharov in
1967 [1]. In this so-called induced gravity, spacetime background emerges as a
mean field approximation of underlying microscopic degrees of freedom, similar
to hydrodynamics or continuum elasticity theory from molecular physics [2].
This idea has been further developed since the discovery of the thermodynamic
properties of black hole in 1970s. Black hole thermodynamics tells us that a
black hole has an entropy proportional to its horizon area and a temperature
proportional to its surface gravity at the black hole horizon, and the entropy
and temperature together with the mass of the black hole satisfy the first law
of thermodynamics [3, 4, 5].
The geometric feature of thermodynamic quantities of black hole leads Jacobson
to ask an interesting question whether it is possible to derive Einstein’s
equations of gravitational field from a point of view of thermodynamics [6].
It turns out that it is indeed possible. Jacobson derived Einstein’s equations
by employing the fundamental Clausius relation $\delta Q=TdS$ together with
the equivalence principle. Here the key idea is to demand that this relation
holds for all the local Rindler causal horizon through each spacetime point,
with $\delta Q$ and $T$ interpreted as the energy flux and Unruh temperature
seen by an accelerated observer just inside the horizon. In this way,
Einstein’s equation is nothing but an equation of state of spacetime.
Further, assuming the apparent horizon of a Friedmann-Robertson-Walker (FRW)
universe has temperature $T$ and entropy $S$ satisfying
$T=1/2\pi\tilde{r}_{A}$ and $S=A/4G$, where $\tilde{r}_{A}$ is the radius of
the apparent horizon and $A$ is the area of the apparent horizon, one is able
to derive Friedmann equations of the FRW universe with any spatial curvature
by applying the Clausius relation to apparent horizon [7]. This works not only
in Einstein’s gravitational theory, but also in Gauss-Bonnet and Lovelock
gravity theories. Here a key ingredient is to replace the entropy area formula
in Einstein’s theory by using entropy expressions of black hole horizon in
those higher order curvature theories. Recently the Hawking temperature
associated with the apparent horizon of FRW universe has been shown [8]. There
exist a lot of papers investigating the relation between the first law of
thermodynamics and the Friedmann equations of FRW universe in various gravity
theories. For more references see, for example [9, 10] and references therein.
Another hint appears on the relation between thermodynamics and gravitational
dynamics by investigating the relation between the first law of thermodynamics
and gravitational field equation in the setup of black hole spacetime.
Padmanabhan [11] first noticed that the gravitational field equation in a
static, spherically symmetric spacetime can be rewritten as a form of the
ordinary first law of thermodynamics at a black hole horizon. This indicates
that Einstein’s equation is nothing but a thermodynamic identity. This
observation was then extended to the cases of stationary axisymmetric horizons
and evolving spherically symmetric horizons in Einstein’s gravity [12], static
spherically symmetric horizons [13] and dynamical apparent horizons [14] in
Lovelock gravity, and three dimensional BTZ black hole horizons [15]. Very
recently it has been shown it also holds in Horava-Lifshitz gravity [16]. For
a recent review on this topic and some relevant issues, see [17].
In a very recent paper by Verlinde [18], the viewpoint of gravity being not a
fundamental interaction has been further advocated. Gravity is explained as an
entropic force caused by changes in the information associated with the
positions of material bodies. Among various interesting observations made by
Verlinde, here we mention two of them. One is that with the assumption of the
entropic force together with the Unruh temperature [19], Verlinde is able to
derive the second law of Newton. The other is that the assumption of the
entropic force together with the holographic principle and the equipartition
law of energy leads to Newton’s law of gravitation. Similar observations are
also made by Padmanabhan [20]. He observed that the equipartition law of
energy for the horizon degrees of freedom combing with the thermodynamic
relation $S=E/2T$, also leads to Newton’s law of gravity, here $S$ and $T$ are
thermodynamic entropy and temperature associated with the horizon and $E$ is
the active gravitational mass producing the gravitational acceleration in the
spacetime [21]. Finally we mention that there exist some earlier attempts to
build microscopic models of spacetime, for example, see [22, 23, 24].
In this short note we are going to derive the Friedmann equations governing
the dynamical evolution of the FRW universe from the viewpoint of entropic
force together with the equipartition law of energy and the Unruh temperature
by generalizing some arguments of Verlinde to dynamical spacetimes.
Consider the FRW Universe with metric
$ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+r^{2}d\Omega^{2}),$ (1)
where $a(t)$ is the scale factor of the universe. Following [18], consider a
compact spatial region ${\cal V}$ with a compact boundary ${\partial\cal V}$,
which is a sphere with physical radius $\tilde{r}=ar$. The compact boundary
$\partial\cal V$ acts as the holographic screen. The number of bits on the
screen is assumed as
$N=\frac{Ac^{3}}{G\hbar},$ (2)
where $A$ is the area of the screen (note that there is a factor difference
$1/4$ from the Bekenstein-Hawking area entropy formula of black hole).
Assuming the temperature on the screen is $T$, and then according to the
equipartition law of energy, the total energy on the screen is
$E=\frac{1}{2}Nk_{B}T.$ (3)
Further just as in [18], we need the relation
$E=Mc^{2},$ (4)
where $M$ represents the mass that would emerge in the compact spatial region
${\cal V}$ enclosed by the boundary screen $\partial{\cal V}$.
Suppose the matter source in the FRW universe is a perfect fluid with stress-
energy tensor
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (5)
Due to the pressure, the total mass $M=\rho V$ in the region enclosed by the
boundary $\partial{\cal V}$ is no longer conserved, the change in the total
mass is equal to the work made by the pressure $dM=-pdV$, which leads to the
well-known continuity equation
$\dot{\rho}+3H(\rho+p)=0,$ (6)
where $H=\dot{a}/a$ is the Hubble parameter.
The total mass in the spatial region ${\cal V}$ can be expressed as
$M=\int_{\cal V}dV(T_{\mu\nu}u^{\mu}u^{\nu}),$ (7)
where $T_{\mu\nu}u^{\mu}u^{\nu}$ is the energy density measured by a comoving
observer. On the other hand, the acceleration for a radial comoving observer
at $r$, namely at the place of the screen, is
$a_{r}=-d^{2}\tilde{r}/dt^{2}=-\ddot{a}\ r,$ (8)
where the negative sign arises because we consider the acceleration is caused
by the matter in the spatial region enclosed by the boundary ${\partial\cal
V}$. Note that the proper acceleration vanishes for a comoving observer.
However, the acceleration (8) is crucial in the following discussions.
According to the Unruh formula, we assume that the acceleration corresponds to
a temperature
$T=\frac{1}{2\pi k_{B}c}\hbar a_{r}.$ (9)
Now it is straightforward to derive the following equation from eqs. (2), (3),
(4), (7) and (9):
$\ddot{a}=-\frac{4\pi G}{3}\rho a.$ (10)
This is nothing but the dynamical equation for Newtonian cosmology (Page 10 in
[25]). Note that the reference [25] derives (10) from the Newtonian gravity
law, while we obtain (10) by using the holographic principle and the
equipartition law of energy in statistical physics. To produce the Friedmann
equations of FRW universe in general relativity, let us notice that producing
the acceleration is the so-called active gravitational mass ${\cal M}$ [21],
rather than the total mass $M$ in the spatial region ${\cal V}$. The active
gravitational mass is also called Tolman-Komar mass, defined as
${\cal M}=2\int_{\cal
V}dV\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}.$ (11)
Replacing $M$ by ${\cal M}$, we have in this case
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+3p\right).$ (12)
This is just the acceleration equation for the dynamical evolution of the FRW
universe. Multiplying $\dot{a}a$ on both sides of eq. (12), and using the
continuity equation (6), we integrate the resulting equation and obtain
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho.$ (13)
Note that $k$ appears in (13) as an integration constant, but it is clear that
the constant $k$ has the interpretation as the spatial curvature of the region
${\cal V}$ in the Einstein theory of gravity. $k=1$, $0$ and $-1$ correspond
to a close, flat and open FRW universe, respectively.
The above discussion can be extended to any spacetime dimension $d\geq 4$. In
that case, the number of bits on the screen is changed to [18]
$N=\frac{1}{2}\frac{d-2}{d-3}\frac{Ac^{3}}{G\hbar},$ (14)
the continuity equation becomes $\dot{\rho}+(d-1)H(\rho+p)=0$, and the active
mass ${\cal M}$ is defined as
${\cal M}=\frac{d-2}{d-3}\int_{\cal
V}dV\left(T_{\mu\nu}-\frac{1}{d-2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}.$ (15)
The acceleration equation (12) is changed to
$\frac{\ddot{a}}{a}=-\frac{8\pi G}{(d-1)(d-2)}\left((d-3)\rho+(d-1)p\right).$
(16)
Integrating (16) we then have
$H^{2}+\frac{k}{a^{2}}=\frac{16\pi G}{(d-1)(d-2)}\rho.$ (17)
This is just the Friedmann equation of the FRW universe in $d$ dimensions.
Thus we have derived the Friedmann equations of a FRW universe starting from
the holographic principle and the equipartition law of energy by using
Verlinde’s argument that gravity appears as an entropic force. Before we close
this note, however, some remarks are in order. First it is claimed that
Verlinde’s arguments are applicable to any spacetime, but Verlinde mainly
discusses the cases of static and/or stationary spacetimes. In particular,
when he derives Einstein’s equation, a time-like Killing vector is employed.
The time-like Killing vector exists for static or stationary spacetimes, and
it does not for a dynamical spacetime. Here we have applied his arguments to
the FRW universe, a special dynamical spacetime, and obtained the dynamical
equations governing the evolution of the FRW universe. Second, in deriving
Newton’s law of gravity, Verlinde considers a spherical surface with a fixed
radius as the holographic screen, and does not take into account the evolution
of the background spacetime itself. This is right since in Newton’s gravity,
the background spacetime is a fixed one. In our case, the holographic screen
is a dynamical one, in some sense, so it can be viewed as a surface of the
spherical symmetric dust matter [25]. The surface evolves due to the
selfgravity. Thus, an observer (or a test particle) on the screen will feel a
force which leads to an acceleration (8). The final comment is concerned with
the assumed relation (9). According to Unruh, the acceleration could
correspond to a local Unruh temperature on the screen (9). Note that the
acceleration (8) is not a proper acceleration; the proper acceleration
vanishes for a comoving observer in the FRW universe. In fact, $a_{r}$ is just
the acceleration of geodesic deviation vector [28]. Let us recall that
Verlinde arrives at the second law of Newton starting from entropic force
together with the Unruh relation (Eq. (3.8) in [18]), which relates the
temperature on the screen to an acceleration. Note that the second law of
Newton is a nonrelativistic form, where the acceleration has a form
$\ddot{x}$. The situation is the same as the case of the discussions in the
present paper. Indeed Eq. (10) has a nonrelativistic origin. It is argued by
Verlinde that here the Unruh relation should be read as a formula for the
temperature on the screen that is required to cause the acceleration, not as
usual, as the temperature caused by an acceleration. Therefore the relation
(9) may be regarded as a working ansatz here. Thus it is a very interesting
issue to see whether there exists such a relation between the Unruh
temperature and the acceleration. To this aim, some useful references are
already available [26]: It is known that Hawking temperature in de Sitter
space and in some black hole spacetimes can be viewed as a Unruh temperature
for a Rindler observer in higher-dimensional flat spacetimes in which the de
Sitter spacetime and black holes spacetime can be embedded.
Note added: When we are in the final stage of writing the manuscript, two
papers appear [27, 28] in the preprint archive, which discuss some relevant
issues and have some overlap with our discussions in this paper.
## Acknowledgments
RGC thanks his colleagues and students at ITP/CAS and T. Padmanabhan for
various discussions on the entropic force. RGC is supported partially by
grants from NSFC, China (No. 10821504 and No. 10975168) and a grant from the
Ministry of Science and Technology of China national basic research Program
(973 Program) (No. 2010CB833004). LMC and NO were supported in part by the
Grants-in-Aid for Scientific Research Fund of the JSPS Nos. 20540283 and
21$\cdot$ 09225, and also by the Japan-U.K. Research Cooperative Program.
## References
* [1] A. D. Sakharov, “Vacuum quantum fluctuations in curved space and the theory of gravitation,” Sov. Phys. Dokl. 12, 1040 (1968) [Dokl. Akad. Nauk Ser. Fiz. 177, 70 (1967 SOPUA,34,394.1991 GRGVA,32,365-367.2000)].
* [2] M. Visser, “Sakharov’s induced gravity: A modern perspective,” Mod. Phys. Lett. A 17, 977 (2002) [arXiv:gr-qc/0204062].
* [3] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333 (1973).
* [4] S. W. Hawking, “Particle Creation By Black Holes,” Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)].
* [5] J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161 (1973).
* [6] T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. 75, 1260 (1995) [arXiv:gr-qc/9504004].
* [7] R. G. Cai and S. P. Kim, “First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe,” JHEP 0502, 050 (2005) [arXiv:hep-th/0501055].
* [8] R. G. Cai, L. M. Cao and Y. P. Hu, “Hawking Radiation of Apparent Horizon in a FRW Universe,” Class. Quant. Grav. 26, 155018 (2009) [arXiv:0809.1554 [hep-th]].
* [9] R. G. Cai, L. M. Cao and Y. P. Hu, “Corrected Entropy-Area Relation and Modified Friedmann Equations,” JHEP0808:090,2008 [arXiv:0807.1232[hep-th]].
* [10] R. G. Cai, “Thermodynamics of Apparent Horizon in Brane World Scenarios,” Prog.Theor.Phys.Suppl.172:100-109,2008 [arXiv:0712.2142[hep-th]].
* [11] T. Padmanabhan, “Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes,” Class. Quant. Grav. 19, 5387 (2002) [arXiv:gr-qc/0204019].
* [12] D. Kothawala, S. Sarkar and T. Padmanabhan, “Einstein’s equations as a thermodynamic identity: The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons,” Phys. Lett. B 652, 338 (2007) [arXiv:gr-qc/0701002].
* [13] A. Paranjape, S. Sarkar and T. Padmanabhan, “Thermodynamic route to field equations in Lancos-Lovelock gravity,” Phys. Rev. D 74, 104015 (2006) [arXiv:hep-th/0607240].
* [14] R. G. Cai, L. M. Cao, Y. P. Hu and S. P. Kim, “Generalized Vaidya Spacetime in Lovelock Gravity and Thermodynamics on Apparent Horizon,” Phys. Rev. D 78, 124012 (2008) [arXiv:0810.2610 [hep-th]].
* [15] M. Akbar, “Thermodynamic Interpretation of Field Equations at Horizon of BTZ Black Hole,” Chin. Phys. Lett. 24, 1158 (2007) [arXiv:hep-th/0702029]; M. Akbar and A. A. Siddiqui, ‘Charged rotating BTZ black hole and thermodynamic behavior of field equations at its horizon,” Phys. Lett. B 656, 217 (2007).
* [16] R. G. Cai and N. Ohta, “Horizon Thermodynamics and Gravitational Field Equations in Horava-Lifshitz Gravity,” arXiv:0910.2307 [hep-th].
* [17] T. Padmanabhan, “Thermodynamical Aspects of Gravity: New insights,” arXiv:0911.5004 [gr-qc].
* [18] E. P. Verlinde, “On the Origin of Gravity and the Laws of Newton,” arXiv:1001.0785 [hep-th].
* [19] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D 14, 870 (1976).
* [20] T. Padmanabhan, “Equipartition of energy in the horizon degrees of freedom and the emergence of gravity,” arXiv:0912.3165 [gr-qc].
* [21] T. Padmanabhan, “Gravitational entropy of static spacetimes and microscopic density of states,” Class. Quant. Grav. 21, 4485 (2004) [arXiv:gr-qc/0308070].
* [22] F. Piazza, “Glimmers of a pre-geometric perspective,” arXiv:hep-th/0506124; for a review, F. Piazza, “New views on the low-energy side of gravity,” arXiv:0910.4677 [gr-qc].
* [23] O. Dreyer, “Why things fall,” PoS QG-PH, 016 (2007) [arXiv:0710.4350 [gr-qc]].
* [24] J. Makea, “Notes Concerning ’On the Origin of Gravity and the Laws of Newton’ by E. Verlinde (arXiv:1001.0785),” arXiv:1001.3808 [gr-qc].
* [25] V. Mukhanov, “Physical foundations of cosmology,” Cambridge, UK: Univ. Pr. (2005) 421 p.
* [26] S. Deser and O. Levin, “Accelerated detectors and temperature in (anti) de Sitter spaces,” Class. Quant. Grav. 14, L163 (1997) [arXiv:gr-qc/9706018]; S. Deser and O. Levin, “Equivalence of Hawking and Unruh temperatures through flat space embeddings,” Class. Quant. Grav. 15, L85 (1998) [arXiv:hep-th/9806223]; S. Deser and O. Levin, “Mapping Hawking into Unruh thermal properties,” Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159].
* [27] F. W. Shu and Y. Gong, “Equipartition of energy and the first law of thermodynamics at the apparent horizon,”[arXiv:1001.3237[gr-qc]].
* [28] T. Padmanabhan, “Why Does the Universe Expand ?” [arXiv:1001.3380[gr-qc]].
|
arxiv-papers
| 2010-01-20T08:30:46 |
2024-09-04T02:49:07.863410
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Gen Cai, Li-Ming Cao and Nobuyoshi Ohta",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/1001.3470"
}
|
1001.3646
|
###### Abstract
The asymptotic expansion of $n$-dimensional cyclic integrals was expressed as
a series of functionals acting on the symmetric function involved in the
cyclic integral. In this article, we give an explicit formula for the action
of these functionals on a specific class of symmetric functions. These results
are necessary for the computation of the $\mathrm{O}\left(1\right)$ part in
the long-distance asymptotic behavior of correlation functions in integrable
models.
LPENSL-TH-03/09
Fine structure of the asymptotic expansion of cyclic integrals
K. K. Kozlowski111 Laboratoire de Physique, ENS Lyon et CNRS, France,
karol.kozlowski@ens-lyon.fr
## 1 Introduction
There is a great interest in obtaining exact representations for the
correlation functions of integrable models in one-dimensional quantum
mechanics[7, 2, 11, 12]. This is especially due to the recent development of
experimental techniques allowing one to directly observe such models. Among
experimentally interesting correlation functions, one could mention the
dynamical ones that are for instance measured through neutron scattering on
anti-ferromagnets. It was the determinant representation for the form factors
of the spin-$1/2$ XXZ chain [12] or the non-linear Schrodinger model [13] that
opened the possibility for an effective numerical study of these dynamical
correlation functions in relatively long chains [5] and diffuse lattice Bose
gases [4] (this should be put in contrast with numerical diagonalization that
is limited to twenty or so sites). One had to resort to numerics so as to
obtain a satisfactory answer as, in general, the exact representations for the
correlation functions in integrable models are involved and often not
computable in a closed form. Even if simple closed expressions for the
correlators do not exist, it is sometimes possible to built on these explicit
answers so as to derive the asymptotic behavior of certain correlation
functions. This, in turn, allows one to test the conformal field theoretic
predictions [3] for the power-law decay with the distance of two-point
functions in gapless one-dimensional quantum Hamiltonians [1, 6]. This last
program has been carried recently by the author and his collaborators [9].
There, it was shown that it is possible to extract, starting from first-
principle based computations on the XXZ spin-$1/2$ chain, the long-distance
$x$ power-law decay of the longitudinal spin-spin
$\langle\sigma^{z}_{1}\sigma^{z}_{x}\rangle$ correlation function. As
expected, these asymptotics were found to match the conformal field theoretic
predictions for this quantity.
We now briefly outline the method used for deriving these asymptotics. This
will settle the context of the results presented in this paper. The method for
extracting the asymptotic behavior of spin-spin correlation functions builds
on the master equation [8, 11]. It is an $N$-fold contour integral
representation for the generating function
$\langle\mathrm{e}^{\beta\mathcal{Q}_{x}}\rangle$ of two-point functions.
Starting from this master equation, one should first reach the conformal
regime of the model. The latter is supposed to manifest itself in the case of
a gapless spectrum above the ground state. For integrable models, this
corresponds to sending the size of the model to infinity and restricting
oneself to a given range of the coupling constants. In particular, to reach
this limit, one should send the number of integrals $N$ to infinity. However,
the latter limit is impossible directly in the aforementioned integral
representation. To do so, one has first to expand the $N$-fold contour
integral into a series, whose summand $a_{n}$ of order $n$ is a linear
combination of $2n$-fold integrals. Each of these integrals depends on the
large parameter $x$ and is expressed as products of the so-called cyclic
integrals:
$\mathcal{I}_{n}\left[\mathcal{F}_{n}\right]=\int\limits_{\Gamma\left(\left[\,-q\,;q\,\right]\right)}{\mathchoice{\dfrac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}{\dfrac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}}\int\limits_{-q}^{q}\\!{\mathchoice{\dfrac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}{\dfrac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}}\,\mathcal{F}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)\prod\limits_{k=1}^{n}{\mathchoice{\dfrac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}{\dfrac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}{\frac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}{\frac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}}\;.$
(1.1)
The functions $\mathcal{F}_{n}$ and $p$ are specific of the considered model.
The asymptotic behavior of (1.1) drives the one of $a_{n}$. It was shown in
[10] that the asymptotic expansion (AE) of cyclic integrals can be deduced
from the asymptotic solution of a Riemann–Hilbert problem related to the
generalized sine kernel. This AE is written as a series of functionals
$I^{\left(N,m\right)}_{n}$ acting on the function $\mathcal{F}_{n}$. In order
to deduce the AE of $\langle\mathrm{e}^{\beta\mathcal{Q}_{x}}\rangle$, one
should replace every cyclic integral in $a_{n}$ by its AE and then re-order an
re-sum the resulting terms. Due to the product structure of $a_{n}$ such a
procedure involves summing-up series of products of functionals
$I^{\left(N,m\right)}_{n}$. In the intermediate steps of this summation
procedure, it happens that the $x$-independent function one started with,
acquires an $x$-dependence. It thus follows that one has to know how to
evaluate the action of the functionals $I_{n}^{\left(N,m\right)}$ on certain
classes of $x$-dependent functions. This last topic constitutes the subject of
this article. It seems rather difficult to give an explicit formula for
$I_{n}^{\left(N,m\right)}\left[\mathcal{F}_{n}\right]$ when $\mathcal{F}_{n}$
is a generic symmetric function. Indeed, the functionals
$I_{n}^{\left(N,m\right)}$ have been built rather implicitly through a
recursive procedure. However, we found that it is possible to write a simple
explicit formula for the leading order in $x$ part of the action of
$I_{n}^{\left(N,m\right)}$ when focusing on a specific class of $x$-dependent
functions. Such a result should not be understood as some part of the
asymptotics of $2n$-fold cyclic integrals versus some $x$-dependent functions
but rather as the action of given functionals (defined as the ones appearing
in the AE of cyclic integrals) on a particular kind of functions. In this
paper, we only discuss the leading order in $x$ part of the action although it
is quite clear how to apply the method we present so as to obtain further,
sub-leading, corrections. Of course, the size and complexity of the formulae
will grow very quickly with the precision of the corrections
The article is organized as follows. In section 2 we remind some properties of
the AE of the generalized sine kernel which is a building block of cyclic
integrals (1.1). We prove some results on the structure of these asymptotics.
In section 3, we recall the link between the generalized sine kernel and
cyclic integrals. We also remind the overall properties of their AE and derive
some sum-rules. Finally, in section 4, we apply these sum rules so as to
determine explicitly the leading $\mathrm{O}\left(1\right)$ part of the action
of these functionals on a certain class of $x$-dependent functions.
## 2 The generalized sine kernel
The generalized sine kernel is an integral operator $I+V$ on
$L^{2}\left(\left[\,-q\,;q\,\right]\right)$. Its kernel $V$ reads
$V\left(\lambda,\mu\right)=\gamma\sqrt{F\left(\lambda\right)F\left(\mu\right)}{\mathchoice{\dfrac{e\left(\lambda\right)/e\left(\mu\right)-e\left(\mu\right)/e\left(\lambda\right)}{2i\pi\left(\lambda-\mu\right)}}{\dfrac{e\left(\lambda\right)/e\left(\mu\right)-e\left(\mu\right)/e\left(\lambda\right)}{2i\pi\left(\lambda-\mu\right)}}{\frac{e\left(\lambda\right)/e\left(\mu\right)-e\left(\mu\right)/e\left(\lambda\right)}{2i\pi\left(\lambda-\mu\right)}}{\frac{e\left(\lambda\right)/e\left(\mu\right)-e\left(\mu\right)/e\left(\lambda\right)}{2i\pi\left(\lambda-\mu\right)}}}\;\;,\quad
e\left(\lambda\right)=\mathrm{e}^{ixp\left(\lambda\right)/2+g\left(\lambda\right)/2}\;.$
(2.1)
The asymptotic behavior of the Fredholm determinant of this kernel was derived
in [10] under the assumption of $p,F,g$ holomorphic in a neighborhood of
$\left[\,-q\,;q\,\right]$, $p$ injective on this neighborhood and
$\left|\gamma\right|$ small enough. We remind that the AE of its logarithm has
the structure
$\ln\operatorname{det}\left[I+V\right]=\sum\limits_{N=-1}^{+\infty}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}\mathcal{R}^{\left(N,0\right)}\left[p,g,\nu\right]+\sum\limits_{m\in\mathbb{Z}^{*}}\sum\limits_{N\geq
2\left|m\right|}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}\mathcal{R}^{\left(N,m\right)}\left[p,g,\nu\right]\mathrm{e}^{imx\left(p_{+}-p_{-}\right)}x^{m\left(\nu_{+}+\nu_{-}\right)}\;.$
(2.2)
Here and in the following, we use the notation $H_{\pm}=H\left(\pm q\right)$
for the boundary values of any function $H$. The function $\nu$ is related to
$F$ by
$\nu\left(\lambda\right)={\mathchoice{\dfrac{i}{2\pi}}{\dfrac{i}{2\pi}}{\frac{i}{2\pi}}{\frac{i}{2\pi}}}\ln\left(1+\gamma
F\left(\lambda\right)\right)\;.$ (2.3)
and $\mathcal{R}^{\left(N,m\right)}\left[p,g,\nu\right]$, $m\in\mathbb{Z}$,
are functional in $p,g$ and $\nu$. They are also polynomials in $\ln x$ of
degree $N$ (we decided not to insist explicitly on this dependence in order to
keep the notation light). $\mathcal{R}^{\left(N\right)}$ is expressible in
terms of integrals over $\left[\,-q\,;q\,\right]$ involving $\nu$ and products
of functions in $p_{\pm},g_{\pm},\nu_{\pm}$ or of their higher order
derivatives.
Actually, it is easy to see from the details of the recursive construction of
$\mathcal{R}^{\left(N,m\right)}$ given in [10] that these functionals only
produce algebraic expressions in $p$ or its derivatives. Moreover, only
$p^{\prime}_{\pm}$ or $p_{+}-p_{-}$ appear in the denominator of such
expressions222 Another argument to convince oneself of this fact is that,
otherwise, the AE would be ill-defined on the functions fulfilling the
assumptions of [10]. Indeed, the AE is obtained to any order in $x$ only by
assuming holomorphy of the involved functions, smallness of
$\left|\gamma\right|$ and injectivity of $p$. These conditions can only
guarantee that $p_{+}-p_{-}$ and $p^{\prime}_{\pm}$ do not vanish..
Note that one can represent the operator $I+V$ in several ways. Namely, the
kernels $V_{p,F,g}$ and $V_{p-ig/x,F,0}$ parameterize the same generalized
sine kernel333Here we have explicitly stressed out the dependence of $V$ on
the functions $p,F,g$. However, these two different parameterizations lead to,
a priori distinct, asymptotic series444We stress that one can also apply the
results of [10] in the case of an $x$-dependent $p$ function, as long as
$p=p_{0}+\mathrm{o}\left(1\right)$ and $p_{0}$ is holomorphic in a
neighborhood of $\left[\,-q\,;q\,\right]$ and injective on this neighborhood.
This stems from the fact that when $x\rightarrow+\infty$, only $p_{0}$ plays a
role in the details of the analysis carried in [10]. (2.2). In virtue of the
uniqueness of an asymptotic expansions, these have to coincide
$\sum\limits_{N=-1}^{+\infty}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}\mathcal{R}^{\left(N,0\right)}\left[p,g,\nu\right]+\sum\limits_{m\in\mathbb{Z}^{*}}\sum\limits_{N\geq
2\left|m\right|}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}\mathcal{R}^{\left(N,m\right)}\left[p,g,\nu\right]\mathrm{e}^{imx\left(p_{+}-p_{-}\right)}x^{m\left(\nu_{+}+\nu_{-}\right)}\;\sim\\\
\;\sum\limits_{N=-1}^{+\infty}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}\mathcal{R}^{\left(N,0\right)}\left[p-ig/x,0,\nu\right]+\sum\limits_{m\in\mathbb{Z}^{*}}\sum\limits_{N\geq
2\left|m\right|}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}\mathcal{R}^{\left(N,m\right)}\left[p-ig/x,0,\nu\right]\mathrm{e}^{imx\left(p_{+}-p_{-}\right)}\mathrm{e}^{m\left(g_{+}-g_{-}\right)}x^{m\left(\nu_{+}+\nu_{-}\right)}\;.$
(2.4)
Due to the aforementioned properties of the functionals
$\mathcal{R}^{\left(N,m\right)}$, the dependence in the RHS of (2.4) is
algebraic in $p-ig/x$ and the only terms appear in the denominator have a well
defined Taylor series ($ie$ one that only produces corrections in respect to
the $x\rightarrow+\infty$ limit). Hence,
$\mathcal{R}^{\left(N\right)}\left[p-ig/x,0,\nu\right]$ can be expanded in a
Taylor series of inverse powers of $x$. Such an expansion yields
$\mathcal{R}^{\left(N\right)}\left[p,0,\nu\right]$ and terms that involve $g$
but are subdominant with respect to it. In other words, the structure in $g$
of the RHS of (2.4) follows by reordering and gathering the powers in $x$
issued from the various Taylor series expansions in the LHS of (2.4). The
mechanism of this effect can be seen explicitly on the leading asymptotics of
$\ln\operatorname{det}\left[I+V_{p,g,F}\right]$:
$x\mathcal{R}^{\left(-1,0\right)}\left[p,0,\nu\right]=x\int\limits_{-q}^{q}p^{\prime}\left(\lambda\right)\nu\left(\lambda\right)\mathrm{d}\lambda\;.$
(2.5)
The substitution $p\mapsto p-ig/x$ produces
$x\int\limits_{-q}^{q}p^{\prime}\left(\lambda\right)\nu\left(\lambda\right)\mathrm{d}\lambda-i\int\limits_{-q}^{q}g^{\prime}\left(\lambda\right)\nu\left(\lambda\right)\mathrm{d}\lambda\;.$
(2.6)
This additional $g$-dependent term is precisely the one appearing in the
$\mathrm{O}\left(1\right)$ asymptotics of
$\ln\operatorname{det}\left[I+V_{p,F,g}\right]$, cf [10].
### 2.1 Some results on the structure of the asymptotic series
One can build on the two ways of presenting the AE so as to deduce some
properties on the structure of the coefficients involved in the AE (2.2). Such
results can hardly by inferred from a direct computation based on the
techniques presented in [10]. We prove the results that are important for the
purpose of the further analysis in the two lemmas below.
###### Lemma 2.1
The $1/x^{N}$ order in the AE of $\ln\operatorname{det}\left[I+V\right]$
contains the term
$\sum\limits_{\sigma=\pm}{\mathchoice{\dfrac{1}{N}}{\dfrac{1}{N}}{\frac{1}{N}}{\frac{1}{N}}}\left({\mathchoice{\dfrac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\dfrac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}}\right)^{N}\nu^{2}_{\sigma}\;.$
(2.7)
This is the highest possible power of $g$ that can appear in the non-
oscillating part of the $x^{-N}$ order in the AE of
$\ln\operatorname{det}\left[I+V\right]$ (2.2).
Proof — As it was discussed previously, it is possible to deduce the whole
$g$-dependence of $\mathcal{R}^{\left(N,0\right)}\left[p,g,\nu\right]$ by
expanding
$\mathcal{R}^{\left(0,0\right)}\left[p-ig/x,0,\nu\right],\dots,x^{1-N}\mathcal{R}^{\left(N-1,0\right)}\left[p-ig/x,0,\nu\right]$
in to Taylor series of inverse powers of $x$. Note that the $g$-dependent part
of $x^{-k}\mathcal{R}^{\left(k,0\right)}\left[p-ig/x,0,\nu\right]$, $k\geq N$
only produces subdominant contribution with respect to
$x^{-N}\mathcal{R}^{\left(N\right)}$. Moreover, we did not include
$\mathcal{R}^{\left(-1,0\right)}$ in the list, as we have already seen (2.6)
that it only generates the $g$ dependent part of
$\mathcal{R}^{\left(0\right)}\left[p,g,\nu\right]$.
The Taylor expansion into inverse powers of $g$ of the functional
$R^{\left(\ell,0\right)}\left[p-ig/x,0,\nu\right]$ is of the type
$R^{\left(\ell,0\right)}\left[p,0,\nu\right]+\sum\limits_{n\geq
1}x^{-n}F_{n}^{\left(\ell\right)}\left[p,g,\nu\right]\;.$ (2.8)
The functional $F^{\left(\ell\right)}_{n}\left[p,g,\nu\right]$ is homogeneous
in $g$ of order $n$, ie
$F^{\left(p\right)}_{n}\left[p,tg,\nu\right]=t^{n}F^{\left(p\right)}_{n}\left[p,g,\nu\right]$.
In other words it only contains powers of $g$ whose total degree in $g$ is
$n$, for instance terms of the type
$\left(g_{+}\right)^{n_{+}^{\left(0\right)}}\dots\left(g^{\left(k\right)}_{+}\right)^{n_{+}^{\left(k\right)}}\left(g_{-}\right)^{n_{-}^{\left(0\right)}}\dots\left(g^{\left(k\right)}_{-}\right)^{n_{-}^{\left(k\right)}}\;,\qquad\mathrm{where}\quad\sum\limits_{p=1}^{k}n_{+}^{\left(p\right)}+n_{-}^{\left(p\right)}=n.$
(2.9)
Hence, the highest possible degree in $g$ of
$\mathcal{R}^{\left(N,0\right)}\left[p,g,\nu\right]$ can only be issued from
the $N^{th}$ term of the Taylor series in $x^{-1}$ of
$\mathcal{R}^{\left(0\right)}\left[p-ig/x,0,\nu\right]$. As
$-\Sigma_{\sigma=\pm}\nu_{\sigma}^{2}\ln p^{\prime}_{\sigma}$ represents the
$p$ dependent part of $\mathcal{R}^{\left(0\right)}\left[p,0,\nu\right]$, the
substitution $p\mapsto p-ig/x$ generates the series of inverse powers of $x$:
$-\sum\limits_{\sigma=\pm}\nu^{2}_{\sigma}\ln
p_{\sigma}^{\prime}+\sum\limits_{\sigma=\pm}\nu^{2}_{\sigma}\sum\limits_{\ell\geq
1}{\mathchoice{\dfrac{1}{\ell\,x^{\ell}}}{\dfrac{1}{\ell\,x^{\ell}}}{\frac{1}{\ell\,x^{\ell}}}{\frac{1}{\ell\,x^{\ell}}}}\left({\mathchoice{\dfrac{i\,g^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\dfrac{i\,g^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{i\,g^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{i\,g^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}}\right)^{\ell}\;.$
(2.10)
Thus, the highest possible degree in $g$ contained in
$\mathcal{R}^{\left(N,0\right)}\left[p,g,\nu\right]$ is $N$ and the
corresponding contribution is
$N^{-1}\cdot\left[\left(ig^{\prime}_{+}/p^{\prime}_{+}\right)^{N}+\left(ig^{\prime}_{-}/p^{\prime}_{-}\right)^{N}\right]$.
□
One can also prove an analogous lemma in respect to the oscillating part.
###### Lemma 2.2
The $1/x^{N+2}$ part of the oscillating corrections with period
$\mathrm{e}^{i\sigma x\left(p_{+}-p_{-}\right)}$, $\sigma=\pm$, contains the
series
$\mathrm{e}^{\sigma\left(g_{+}-g_{-}\right)}\sum\limits_{k=0}^{n}\left({\mathchoice{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}}\right)^{k}\left({\mathchoice{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}}\right)^{n-k}\widetilde{O}_{n,\,k}\left[\sigma\,\nu\right]\;.$
(2.11)
We have introduced the functional
$\displaystyle\widetilde{O}_{n,\,k}\left[\nu\right]$ $\displaystyle=$
$\displaystyle{\mathchoice{\dfrac{\left[2qx\right]^{2\left(\nu_{+}+\nu_{-}\right)}}{\left(2q\right)^{2}}}{\dfrac{\left[2qx\right]^{2\left(\nu_{+}+\nu_{-}\right)}}{\left(2q\right)^{2}}}{\frac{\left[2qx\right]^{2\left(\nu_{+}+\nu_{-}\right)}}{\left(2q\right)^{2}}}{\frac{\left[2qx\right]^{2\left(\nu_{+}+\nu_{-}\right)}}{\left(2q\right)^{2}}}}{p^{\prime}_{+}}^{2\nu_{+}-1}{p^{\prime}_{-}}^{2\nu_{-}-1}{\mathchoice{\dfrac{\Gamma\left(1-\nu_{-}\right)\Gamma\left(1-\nu_{+}\right)\kappa^{2}_{-}}{\Gamma\left(\nu_{-}\right)\Gamma\left(\nu_{+}\right)\kappa_{+}^{2}}}{\dfrac{\Gamma\left(1-\nu_{-}\right)\Gamma\left(1-\nu_{+}\right)\kappa^{2}_{-}}{\Gamma\left(\nu_{-}\right)\Gamma\left(\nu_{+}\right)\kappa_{+}^{2}}}{\frac{\Gamma\left(1-\nu_{-}\right)\Gamma\left(1-\nu_{+}\right)\kappa^{2}_{-}}{\Gamma\left(\nu_{-}\right)\Gamma\left(\nu_{+}\right)\kappa_{+}^{2}}}{\frac{\Gamma\left(1-\nu_{-}\right)\Gamma\left(1-\nu_{+}\right)\kappa^{2}_{-}}{\Gamma\left(\nu_{-}\right)\Gamma\left(\nu_{+}\right)\kappa_{+}^{2}}}}{\mathchoice{\dfrac{\left(2\nu_{+}-1\right)_{k}\left(2\nu_{-}-1\right)_{n-k}}{k!\left(n-k\right)!}}{\dfrac{\left(2\nu_{+}-1\right)_{k}\left(2\nu_{-}-1\right)_{n-k}}{k!\left(n-k\right)!}}{\frac{\left(2\nu_{+}-1\right)_{k}\left(2\nu_{-}-1\right)_{n-k}}{k!\left(n-k\right)!}}{\frac{\left(2\nu_{+}-1\right)_{k}\left(2\nu_{-}-1\right)_{n-k}}{k!\left(n-k\right)!}}}\;\;,$
(2.12) $\displaystyle\mathrm{with}\;\;\ln\kappa\left(\lambda\right)$
$\displaystyle=$
$\displaystyle\int\limits_{-q}^{q}{\mathchoice{\dfrac{\nu\left(\lambda\right)-\nu\left(\mu\right)}{\lambda-\mu}}{\dfrac{\nu\left(\lambda\right)-\nu\left(\mu\right)}{\lambda-\mu}}{\frac{\nu\left(\lambda\right)-\nu\left(\mu\right)}{\lambda-\mu}}{\frac{\nu\left(\lambda\right)-\nu\left(\mu\right)}{\lambda-\mu}}}\mathrm{d}\mu\;.$
(2.13)
Proof — The first oscillating term appearing in the asymptotic expansion of
$\ln\operatorname{det}\left[I+V\right]_{\mid_{g=0}}$ reads [10]:
${\mathchoice{\dfrac{1}{x^{2}}}{\dfrac{1}{x^{2}}}{\frac{1}{x^{2}}}{\frac{1}{x^{2}}}}\sum\limits_{\sigma=\pm}\widetilde{O}_{0,0}\left[\sigma\nu\right]\mathrm{e}^{ix\sigma\left(p_{+}-p_{-}\right)}\;\quad\mathrm{so}\,\mathrm{that}\quad
R^{\left(2,\sigma\right)}\left[p,g,\nu\right]=x^{-2\sigma\left(\nu_{+}+\nu_{-}\right)}\widetilde{O}_{0,0}\left[\sigma\nu\right]$
(2.14)
Just as for the non-oscillating corrections discussed above, it is the only
one responsible for producing the highest degree in $g$ of the oscillating
corrections subordinate to the frequency
$\mathrm{e}^{ix\sigma\left(p_{+}-p_{-}\right)}$, $\sigma=\pm$. To deduce the
structure of these terms, we set $p\mapsto p-ig/x$ in the above functional and
perform a Taylor series expansion. For $x$ large enough:
${\mathchoice{\dfrac{1}{\left(1-\epsilon_{1}/x\right)^{\alpha}\left(1-\epsilon_{2}/x\right)^{\beta}}}{\dfrac{1}{\left(1-\epsilon_{1}/x\right)^{\alpha}\left(1-\epsilon_{2}/x\right)^{\beta}}}{\frac{1}{\left(1-\epsilon_{1}/x\right)^{\alpha}\left(1-\epsilon_{2}/x\right)^{\beta}}}{\frac{1}{\left(1-\epsilon_{1}/x\right)^{\alpha}\left(1-\epsilon_{2}/x\right)^{\beta}}}}=\sum\limits_{n\geq
0}{\mathchoice{\dfrac{1}{x^{n}}}{\dfrac{1}{x^{n}}}{\frac{1}{x^{n}}}{\frac{1}{x^{n}}}}\sum\limits_{k=0}^{n}\epsilon_{2}^{n-k}\epsilon_{1}^{k}{\mathchoice{\dfrac{\left(\alpha\right)_{k}\left(\beta\right)_{n-k}}{k!\left(n-k\right)!}}{\dfrac{\left(\alpha\right)_{k}\left(\beta\right)_{n-k}}{k!\left(n-k\right)!}}{\frac{\left(\alpha\right)_{k}\left(\beta\right)_{n-k}}{k!\left(n-k\right)!}}{\frac{\left(\alpha\right)_{k}\left(\beta\right)_{n-k}}{k!\left(n-k\right)!}}}\;,\qquad\mathrm{where}\quad\left(\alpha\right)_{k}={\mathchoice{\dfrac{\Gamma\left(\alpha+k\right)}{\Gamma\left(\alpha\right)}}{\dfrac{\Gamma\left(\alpha+k\right)}{\Gamma\left(\alpha\right)}}{\frac{\Gamma\left(\alpha+k\right)}{\Gamma\left(\alpha\right)}}{\frac{\Gamma\left(\alpha+k\right)}{\Gamma\left(\alpha\right)}}}\;.$
(2.15)
We get that the highest degree in $g$ appearing in the $x^{-2-N}$ part of the
oscillating $\mathrm{e}^{i\sigma\left(p_{+}-p_{-}\right)}$ corrections is
given by
$\sum\limits_{\sigma=\pm}\sum\limits_{k=0}^{N}\widetilde{O}_{N,k}\left[\sigma\nu\right]\mathrm{e}^{\sigma\left(g_{+}-g_{-}\right)}\left({\mathchoice{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}}\right)^{k}\left({\mathchoice{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}}\right)^{N-k}\;.$
(2.16)
□
It is clear that one can look at lower order terms like
$\mathcal{R}^{\left(1,0\right)}\left[p,0,\nu\right]$ so as to obtain the
coefficient in front of the terms of total degree $N-1$ in $g$ at any order in
$x^{-N}$. In the next sections, we will actually need the highest degree in
$g$ dependence in the AE of
$J_{n}\left[p,g,F\right]={\mathchoice{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}}\partial_{\gamma}^{n}\ln\operatorname{det}\left[I+V\right]_{\mid_{\gamma=0}}\;.$
(2.17)
The latter is easily deduced from the previous lemmas due to the fact that the
AE of $\ln\operatorname{det}\left[I+V\right]$ in uniform to any finite order
derivative in $\gamma$. We gather these results in the
###### Corollary 2.1
The highest degree of $g$ in the non-oscillating $x^{-N}$ order of the
asymptotic expansion of $J_{n}\left[p,g,F\right]$ reads
${\mathchoice{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}}\sum\limits_{\sigma=\pm}{\mathchoice{\dfrac{1}{N}}{\dfrac{1}{N}}{\frac{1}{N}}{\frac{1}{N}}}\left({\mathchoice{\dfrac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\dfrac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}}\right)^{N}\left(F_{\sigma}\right)^{n}{\partial_{\gamma}^{n}\left(\nu^{2}_{0}\right)}_{\mid_{\gamma=0}}\quad\mathrm{where}\quad\nu_{0}={\mathchoice{\dfrac{i}{2\pi}}{\dfrac{i}{2\pi}}{\frac{i}{2\pi}}{\frac{i}{2\pi}}}\ln\left(1+\gamma\right)\;.$
(2.18)
Similarly, the highest degree in $g$ present in the $x^{-N-2}$ oscillating
part at frequency $\mathrm{e}^{ix\sigma\left(p_{+}-p_{-}\right)}$,
$\sigma=\pm$, in the AE of $J_{n}\left[p,g,F\right]$ is given by the series
${\mathchoice{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}}\mathrm{e}^{\sigma\left(g_{+}-g_{-}\right)}\sum\limits_{k=0}^{N}\left({\mathchoice{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}}\right)^{k}\left({\mathchoice{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}}\right)^{N-k}\partial_{\gamma}^{n}\widetilde{O}_{N,\,k}\left[\sigma\nu\right]_{\mid_{\gamma=0}}\;.$
(2.19)
## 3 Cyclic multiple integrals
One of the main results of [10], was the proof of the asymptotic expansion for
cyclic integrals:
$\mathcal{I}_{n}\left[\mathcal{F}_{n}\right]=\int\limits_{\Gamma\left(\left[\,-q\,;q\,\right]\right)}{\mathchoice{\dfrac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}{\dfrac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}z}{\left(2i\pi\right)^{n}}}}\int\limits_{-q}^{q}\\!{\mathchoice{\dfrac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}{\dfrac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}{\frac{\mathrm{d}^{n}\lambda}{\left(2i\pi\right)^{n}}}}\,\mathcal{F}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)\prod\limits_{k=1}^{n}{\mathchoice{\dfrac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}{\dfrac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}{\frac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}{\frac{\mathrm{e}^{ix\left[p\left(z_{k}\right)-p\left(\lambda_{k}\right)\right]}}{\left(\lambda_{k}-z_{k}\right)\left(\lambda_{k}-z_{k+1}\right)}}}\;.$
(3.1)
There, $\mathcal{F}_{n}$ is a holomorphic function of the $n$ variables
$\left\\{z\right\\}$ and the $n$ variables $\left\\{\lambda\right\\}$ that is
symmetric separately in respect to variables belonging to each of these two
sets. $\Gamma\left(\left[\,-q\,;q\,\right]\right)$ is a sufficiently small
loop of index $1$ around $\left[\,-q\,;q\,\right]$. Lastly, we agree upon
$z_{n+1}\equiv z_{1}$. It was shown in [10], that there exists functionals
$I_{n}^{\left(N,m\right)}$ such that the asymptotic expansion of
$\mathcal{I}_{n}\left[\mathcal{F}_{n}\right]$ reads
$\mathcal{I}_{n}\left[\mathcal{F}_{n}\right]=\sum\limits_{m\in\mathbb{Z}}\sum\limits_{N\geq
2\left|m\right|-\delta_{m,0}}\mathrm{e}^{imx\left(p_{+}-p_{-}\right)}{\mathchoice{\dfrac{1}{x^{N}}}{\dfrac{1}{x^{N}}}{\frac{1}{x^{N}}}{\frac{1}{x^{N}}}}I_{n}^{\left(N,m\right)}\left[\mathcal{F}_{n}\right]\;.$
(3.2)
This equality is to be understood in the sense of an asymptotic expansion. In
particular, it excludes any $\mathrm{O}\left(x^{-\infty}\right)$ terms. For
the purpose of our study, we now recall a couple of general properties of the
functionals appearing in (3.2). First, all the oscillating terms
$\mathrm{e}^{imx\left(p_{+}-p_{-}\right)}$ are factored out from
$I_{n}^{\left(N,m\right)}$ and these functionals are polynomial in $\ln x$ of
at most degree $n+N$. Their action involves the finite-difference operator
$\eth^{\left(t\right)}_{z_{\ell}}\left(\mu\right)\cdot\mathcal{G}_{n}\left[\left\\{\left\\{z_{i}\right\\}^{\bar{p}_{i}}\right\\}_{i=1,\ldots,r}\right]=\mathcal{G}_{n}\left[\left\\{\left\\{z_{i}\right\\}^{\bar{p}_{i}}\right\\}_{i=1,\ldots,r}\right]-\mathcal{G}_{n}\left[\left\\{\left\\{z_{k}\right\\}^{\bar{p}_{k}}\right\\}_{k\not=\ell},\left\\{\mu\right\\}^{t},\left\\{z_{\ell}\right\\}^{\bar{p}_{\ell}-m}\right]\,.$
(3.3)
The function $\mathcal{G}_{n}$ appearing in (3.3) is assumed symmetric in its
variables. Moreover the notation, $\left\\{z_{k}\right\\}^{\overline{p}_{k}}$
means that the variable $z_{k}$ is repeated $\overline{p}_{k}$ times
$\left\\{z_{k}\right\\}^{\overline{p}_{k}}=(\underbrace{z_{1},\dots,z_{1}}_{\overline{p}_{1}\,\mathrm{times}},\dots,\underbrace{z_{r},\dots
z_{r}}_{\overline{p}_{r}})\;.$ (3.4)
Hence, $\eth^{\left(t\right)}_{z_{\ell}}\left(\mu\right)$ acts by replacing
$t\leq\overline{p}_{\ell}$ variables $z_{\ell}$ by $\mu$ and then performing
the difference with the function one started with. The action of
$I_{n}^{\left(N,m\right)}$ takes the form
$I_{n}^{\left(N,m\right)}\left[\mathcal{F}_{n}\right]=\mathfrak{S}^{\left(m\right)}_{N}\,*\\!\\!\sum\limits_{\Sigma
k_{\ell}\leq
N-2\left|m\right|}G^{\mathfrak{S}^{\left(m\right)}_{N}}_{\left\\{k_{\ell}\right\\}}\cdot\left\\{\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}\prod\limits_{\ell=1}^{r}\prod\limits_{t=1}^{n}\prod\limits_{j=1}^{p_{\ell\,,t}}\left\\{\int\limits_{-q}^{q}{\mathchoice{\dfrac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\dfrac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\frac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\frac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}}\eth^{\left(t\right)}_{z_{\ell}}\left(\mu_{\ell,\,t,\,j}\right)\right\\}\cdot\mathcal{F}_{n}\left(\begin{array}[]{c}\left\\{z_{\ell}\right\\}^{\overline{p}_{\ell}}\\\
\left\\{z_{\ell}\right\\}^{\overline{p}_{\ell}+\epsilon_{\ell}}\end{array}\right)\right\\}_{z_{\ell}=\sigma_{\ell}q}.$
(3.5)
Here $\mathfrak{S}^{\left(m\right)}_{N}*$ is a shorthand notation for the
multiple summation symbol
$\mathfrak{S}^{\left(m\right)}_{N}*=\sum\limits_{r=1+\left|m\right|}^{n}\sum\limits_{\sigma_{\ell}=\pm}\sum\limits_{\begin{subarray}{c}\epsilon_{\ell}=\pm
1,0\\\
\Sigma\epsilon_{\ell}=0\end{subarray}}\sum\limits_{\begin{subarray}{c}\Sigma\overline{p}_{\ell}=n\\\
p_{\ell,\,0}\geq 0\end{subarray}}$ (3.6)
over the points $\sigma_{\ell}q=$, $\sigma_{\ell}=\pm$, where the variables
$z_{\ell}$ are evaluated, the integers $\epsilon_{\ell}=\pm 1,\,0$,
$\ell=1,\dots r$ parameterizing the sets of unequal variables and integers
$p_{\ell,t}$ labeling the number of finite difference operators involved.
These integers are subject to the constraints $p_{\ell,t}\geq 0$,
$p_{\ell,0}\geq 1$ and $\overline{p}_{\ell}=\Sigma_{t=1}^{n}tp_{\ell,t}\;$,
$\;\Sigma_{\ell=1}^{r}\epsilon_{\ell}=0$ and
$\Sigma_{\ell=1}^{r}\epsilon_{\ell}p_{\sigma_{\ell}}=m\left(p_{+}-p_{-}\right)$.
The coefficients
$G^{\mathfrak{S}^{\left(m\right)}_{N}}_{\left\\{k_{\ell}\right\\}}$ depend on
all of these additional summation indices and are polynomials in $\ln x$ of
degree $n+N$. We chose to keep this dependence implicit, so as to have as
light notations as possible.
$I_{n}^{\left(N;m\right)}$ is thus written as a sum of terms. Each of these
terms corresponds to several operation performed on $\mathcal{F}_{n}$. The set
the initial $\lambda$-variables of $\mathcal{F}_{n}$ is set to
$\left\\{z_{\ell}\right\\}^{\overline{p}_{\ell}}$ and the $z$-variables to
$\left\\{z_{\ell}\right\\}^{\overline{p}_{\ell}+\epsilon_{\ell}}$. It thus
follows that most of the variables in the upper and lower hypergeometric type
notation for $\mathcal{F}_{n}$ are set equal. The variables
$\left\\{z_{\ell}\right\\}$ are acted upon by the finite-difference operators,
the result of the action being integrated versus a $z_{\ell}$ dependent weight
along $\left[\,-q\,;q\,\right]$. After the action of the finite-difference
operators, one acts with the derivative operators
$\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}$ with
$0\leq\Sigma_{s=1}^{r}k_{s}\leq N-2\left|m\right|$ and sets
$z_{\ell}=\sigma_{\ell}q$, $\sigma_{\ell}=\pm$. One can be more specific about
the structure of the part of $I_{n}^{\left(N,m\right)}$ containing only the
highest order $N-2\left|m\right|$ of derivatives. There, the variables
$z_{1},\dots,z_{r}$ are set equal to $\sigma_{s}q=z_{s}$, where the sequence
$\sigma_{s}\in\left\\{\pm 1\right\\}$ and has precisely $\left|m\right|$
jumps. For instance, when $m=0$, only two sequences $\sigma_{\ell}$ are
possible, $\sigma_{s}=\sigma\,\forall s$, $\sigma=\pm$.
There exist a relation between the Fredholm determinant
$\ln\operatorname{det}\left[I+V\right]$ and cyclic integrals involving a
special class of of symmetric functions in $n$ variables
$\left\\{\lambda\right\\}$ and $n$ variables $\left\\{z\right\\}$ that we call
pure product functions:
$\mathcal{F}^{\left(\varphi,g\right)}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)=\prod\limits_{k=1}^{n}\varphi\left(\lambda_{k}\right)\mathrm{e}^{g\left(z_{k}\right)}\;.$
(3.7)
Indeed, on the one hand by looking at the $n^{\mathrm{th}}$ term in the series
in $\gamma$ of $\ln\operatorname{det}\left[I+V\right]$ and, on the other hand,
by computing the residues at $z_{k}=\lambda_{k}$ or $z_{k}=\lambda_{k+1}$ in
(1.1), one shows that
$J_{n}\left[p,g,\varphi\mathrm{e}^{g}\right]=\mathcal{I}_{n}\left[\mathcal{F}_{n}^{\left(\varphi,g\right)}\right]\;.$
(3.8)
This observation was used in [10] to build on the AE of
$\ln\operatorname{det}\left[I+V\right]$ so as to deduce the one of general
cyclic integrals (1.1). Due to (2.2) and (3.8) the $g$-dependent part of the
AE of $\mathcal{I}_{n}\left[\mathcal{F}_{n}^{\left(\varphi,g\right)}\right]$
can be deduced from the one of
$\mathcal{R}^{\left(N,m\right)}\left(x\right)\left[p,g,\nu^{\left(\varphi\mathrm{e}^{g}\right)}\right]$,
where $\nu^{\left(\varphi\mathrm{e}^{g}\right)}$ is obtained from $\nu$ by
setting $F=\varphi\mathrm{e}^{g}$ in (2.3). Yet, this $g$-dependence has also
to be reproduced if one chooses to act with the functionals
$I_{n}^{\left(N,m\right)}$ (3.2) on $\mathcal{F}_{n}^{\left(\varphi,g\right)}$
instead. This fact allows us to derive certain sum-rules for part of the terms
appearing in (3.5). Such identities will be used in the next section so as to
obtain the leading $\mathrm{O}\left(1\right)$ order in $x$ of the action of
$x^{-N}I_{n}^{\left(N,m\right)}$ on $x$-dependent functions.
### 3.1 Non-oscillating sum-rules
We identify the part of
$I_{n}^{\left(N,0\right)}\left[\mathcal{F}_{n}^{\left(\varphi,g\right)}\right]$
producing the highest order in $g$ of
$J_{n}\left[p,g,\varphi\mathrm{e}^{g}\right]$. Due to the pure-product
structure of $\mathcal{F}_{n}^{\left(\varphi,g\right)}$ we have
$\mathcal{F}^{\left(\varphi,g\right)}_{n}\left(\begin{array}[]{c}\left\\{z_{\ell}\right\\}^{\overline{p}_{\ell}}\\\
\left\\{z_{\ell}\right\\}^{\overline{p}_{\ell}+\epsilon_{\ell}}\end{array}\right)=\prod\limits_{s=1}^{r}F\left(z_{s}\right)^{\overline{p}_{s}}\mathrm{e}^{\epsilon_{s}g\left(z_{s}\right)}\;\;,\quad
F=\varphi\mathrm{e}^{g}\;.$ (3.9)
This function admits a simple representation for the action of
$\eth^{\left(t\right)}_{z}\left(\mu_{\ell,\,t,\,j}\right)$:
$\prod\limits_{\ell=1}^{r}\prod\limits_{t=1}^{n}\prod\limits_{j=1}^{p_{\ell\,,t}}\left\\{\int\limits_{-q}^{q}{\mathchoice{\dfrac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\dfrac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\frac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\frac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}}\eth^{\left(t\right)}_{z_{\ell}}\left(\mu_{\ell,\,t,\,j}\right)\right\\}\cdot\prod\limits_{s=1}^{r}\left\\{F\left(z_{s}\right)^{\overline{p}_{s}}\mathrm{e}^{\epsilon_{s}g\left(z_{s}\right)}\right\\}=\\\
\prod\limits_{s=1}^{r}\left\\{F\left(z_{s}\right)^{p_{s,0}}\mathrm{e}^{\epsilon_{s}g\left(z_{s}\right)}\prod\limits_{t=1}^{n}\left[\int\limits_{-q}^{q}{\mathchoice{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}}\mathrm{d}\mu\right]^{p_{s\,t}}\right\\}\;.$
(3.10)
Thence,
$I_{n}^{\left(N,0\right)}\left[\mathcal{F}_{n}\right]=\mathfrak{S}^{\left(0\right)}_{N}\,*\\!\\!\sum\limits_{\Sigma
k_{\ell}\leq
N}G^{\mathfrak{S}^{\left(0\right)}_{N}}_{\left\\{k_{\ell}\right\\}}\left\\{\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}\prod\limits_{s=1}^{r}\left\\{F\left(z_{s}\right)^{p_{s,0}}\mathrm{e}^{\epsilon_{s}g\left(z_{s}\right)}\right\\}\prod\limits_{t=1}^{n}\left\\{\int\limits_{-q}^{q}{\mathchoice{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}}\mathrm{d}\mu\right\\}^{p_{s\,t}}\right\\}$
(3.11)
The maximal degree in $g$ can only stem from the part of the sum involving the
highest possible number of derivatives. Indeed, it is obtained by hitting with
all of the derivatives $\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}$
on the exponent $\prod_{k=1}^{r}\mathrm{e}^{\epsilon_{k}g\left(z_{k}\right)}$.
Any other type of action produces a lower degree in $g$. More explicitly, for
an holomorphic function $H$
$\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}\left\\{\prod\limits_{s=1}^{r}\mathrm{e}^{\epsilon_{s}g\left(z_{s}\right)}\cdot
H\left(\left\\{z\right\\}\right)\right\\}=\prod\limits_{s=1}^{r}\left(\epsilon_{s}g^{\prime}\left(z_{s}\right)\right)^{k_{s}}\mathrm{e}^{\epsilon_{s}g\left(z_{s}\right)}\cdot
H\left(\left\\{z\right\\}\right)+\mathrm{LD}\left(g\right)\;\;.$ (3.12)
$\mathrm{LD}\left(g\right)$ stands for the terms that are of a lower degree in
$g$. Once the derivatives have been acted with, one should set
$z_{1}=\dots=z_{r}=\sigma q$ as there are no jumps in the sequence
$\sigma_{\ell}$ when $m=0$ and $\Sigma k_{\ell}=N$. In particular, this allows
us to simplify the products of the exponents
$\prod\limits_{s=1}^{r}\mathrm{e}^{\epsilon_{s}g\left(\sigma
q\right)}=\exp\left\\{g\left(\sigma
q\right)\sum\limits_{s=1}^{r}\epsilon_{s}\right\\}=1\quad\mathrm{due}\,\mathrm{to}\,\mathrm{the}\,\mathrm{constraint}\,\quad\sum\limits_{s=1}^{r}\epsilon_{s}=0\;\;.$
(3.13)
By lemma 2.1, the highest possible degree in $g$ appearing in the non-
oscillating $x^{-N}$ order of the AE of
$\ln\operatorname{det}\left[I+V\right]$ is given by (2.7). Thus we obtain the
sum-rule:
$I_{n}^{\left(N,0\right)}\left[\mathcal{F}^{\left(\varphi,g\right)}_{n}\right]=\mathfrak{S}^{\left(0\right)}_{N}*\sum\limits_{\Sigma
k_{\ell}=N}G^{\mathfrak{S}^{\left(0\right)}_{N}}_{\left\\{k_{\ell}\right\\}}\prod\limits_{s=1}^{r}\left\\{\left[\epsilon_{s}g^{\prime}\left(z_{s}\right)\right]^{k_{s}}\left[F\left(z_{s}\right)\right]^{p_{s,0}}\prod\limits_{t=1}^{n}\left\\{\int\limits_{-q}^{q}{\mathchoice{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}}\mathrm{d}\mu\right\\}^{p_{s\,t}}\right\\}\\\
=\sum\limits_{\sigma={\pm}}{\mathchoice{\dfrac{1}{N}}{\dfrac{1}{N}}{\frac{1}{N}}{\frac{1}{N}}}\left({\mathchoice{\dfrac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\dfrac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}{\frac{ig^{\prime}_{\sigma}}{p^{\prime}_{\sigma}}}}\right)^{N}\left\\{{\mathchoice{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}}\partial_{\gamma}^{n}\left(\nu_{0}\right)_{\mid_{\gamma=0}}\right\\}F^{n}_{\sigma}\;\;.$
(3.14)
### 3.2 Oscillating sum-rules
One can repeat word for word the above analysis in the case of the action of
$I_{n}^{\left(N,m\right)}$, $m=\pm$, on pure product functions. One gets that
$\mathfrak{S}^{\left(m\right)}_{N}\,*\\!\\!\sum\limits_{\Sigma
k_{\ell}=N-2}G^{\mathfrak{S}^{\left(m\right)}_{N}}_{\left\\{k_{\ell}\right\\}}\prod\limits_{s=1}^{r}\left\\{\left[\epsilon_{s}g^{\prime}\left(z_{s}\right)\right]^{k_{s}}\left[F\left(z_{s}\right)\right]^{p_{s,0}}\prod\limits_{t=1}^{n}\left\\{\int\limits_{-q}^{q}{\mathchoice{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\dfrac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[F\left(z_{s}\right)\right]^{t}-\left[F\left(\mu\right)\right]^{t}}{z_{s}-\mu}}}\mathrm{d}\mu\right\\}^{p_{s\,t}}\right\\}_{z_{\ell}=\sigma_{\ell}q}\\\
=\sum\limits_{k=0}^{N-2}\left({\mathchoice{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\dfrac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}{\frac{ig_{+}^{\prime}}{p^{\prime}_{+}}}}\right)^{k}\left({\mathchoice{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\dfrac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}{\frac{ig_{-}^{\prime}}{p^{\prime}_{-}}}}\right)^{N-k-2}\partial_{\gamma}^{n}\widetilde{O}_{N,\,k}\left[\sigma\nu\right]_{\mid_{\gamma=0}}\;.$
(3.15)
The only difference in the proof of the oscillating sum-rule lies in the
specialization of the variables $z_{\ell}$ to $\sigma_{\ell}q$. For a maximal
number $N-2$ of derivatives in $I_{n}^{\left(N,m\right)}$ there can be only
one jump in the sequence $\sigma_{\ell}$ (3.6). Moreover, the sequence
$\epsilon_{s}$ has to fulfill
$\Sigma\epsilon_{s}p_{\sigma_{s}}=m\left(p_{+}-p_{-}\right)$. It thus follows
that $\Sigma\epsilon_{s}p_{\sigma_{s}}=m\left(g_{+}-g_{-}\right)$ and the
resulting prefactor cancels with the one appearing (2.11).
## 4 The action on $x$-dependent functions
We will now apply the sum-rules derived in the previous section so as to
obtain the $\mathrm{O}\left(1\right)$ part of the action of the functionals
$I_{n}^{\left(N;m\right)}$, $m\in\left\\{0,\pm 1\right\\}$, on a certain class
of $x$-dependent functions. We however first define the class of functions
that we consider in the following.
### 4.1 The type of $x$-dependent function
In the rest of this article, we assume that $\mathcal{F}_{n}$ has the typical
structure of functions involved in the asymptotic analysis of correlation
functions of integrable models [9]. Such functions $\mathcal{F}_{n}$ are
written as
$\mathcal{F}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)=\exp\left\\{x\mathcal{G}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)+\ln
x\,\mathcal{H}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)\right\\}\mathcal{W}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right)\prod\limits_{i=1}^{n}\mathcal{V}_{n}\left(\lambda_{i}\left|\begin{array}[]{c}\left\\{\lambda\right\\}\\\
\left\\{z\right\\}\end{array}\right.\right)\;.$ (4.1)
All of the function in (4.1) are symmetric in the $\left\\{\lambda\right\\}$
and $\left\\{z\right\\}$ variables. Moreover, they satisfy reduction
properties in respect to variables appearing in hypergeometric notation, eg:
$\mathcal{W}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}_{1}^{n}\\\
\left\\{z\right\\}_{1}^{n}\end{array}\right)_{\mid_{z_{s}=\lambda_{k}}}=\mathcal{W}_{n}\left(\begin{array}[]{c}\left\\{\lambda\right\\}_{1,\,\not=k}^{n}\\\
\left\\{z\right\\}_{1,\,\not=s}^{n}\end{array}\right)\;.$ (4.2)
In the above equation, we have indicated the label dependence of the variables
so as to make clear what we mean by reduction property. Yet, as long as it
does not lead to confusion, we will omit to write this dependence explicitly
later on.
### 4.2 The action of $I_{n}^{\left(N,0\right)}$
###### Proposition 4.1
Let $\mathcal{F}_{n}$ be as in (4.1), then
$x^{-N}I_{n}^{\left(N;0\right)}\left[\mathcal{F}_{n}\right]=\mathrm{e}^{x\mathcal{G}+\ln
x\mathcal{H}}W\left\\{{\mathchoice{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\dfrac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}{\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}}}\partial_{\gamma}^{n}\left(\nu_{0}\right)_{\mid_{\gamma=0}}\right\\}\sum\limits_{\sigma=\pm}{\mathchoice{\dfrac{1}{N}}{\dfrac{1}{N}}{\frac{1}{N}}{\frac{1}{N}}}\left({\mathchoice{\dfrac{i\mathcal{G}^{\prime}\left(\sigma
q\right)}{p^{\prime}\left(\sigma
q\right)}}{\dfrac{i\mathcal{G}^{\prime}\left(\sigma
q\right)}{p^{\prime}\left(\sigma
q\right)}}{\frac{i\mathcal{G}^{\prime}\left(\sigma
q\right)}{p^{\prime}\left(\sigma
q\right)}}{\frac{i\mathcal{G}^{\prime}\left(\sigma
q\right)}{p^{\prime}\left(\sigma
q\right)}}}\right)^{N}\mathcal{V}^{N}\left(\sigma
q\right)\;+\mathrm{o}\left(1\right)\;.$ (4.3)
We have introduced shorthand notations for the fully reduced functions
$\displaystyle\mathcal{G}=\mathcal{G}_{0}\left(\begin{array}[]{c}\cdot\\\
\cdot\end{array}\right)=\mathcal{G}_{1}\left(\begin{array}[]{c}z\\\
z\end{array}\right)\;,\quad\mathcal{H}=\mathcal{H}_{0}\left(\begin{array}[]{c}\cdot\\\
\cdot\end{array}\right)\;,\quad\mathcal{W}=\mathcal{W}_{0}\left(\begin{array}[]{c}\cdot\\\
\cdot\end{array}\right)\;,$ (4.12)
$\displaystyle\quad\mathcal{V}\left(\lambda\right)=\mathcal{V}\left(\lambda\left|\begin{array}[]{c}\cdot\\\
\cdot\end{array}\right.\right)\;\;\;,\qquad\quad\mathcal{G}^{\prime}\left(\lambda\right)=\partial_{\epsilon}\mathcal{G}_{1}\left(\begin{array}[]{c}\lambda\\\
\lambda+\epsilon\end{array}\right)_{\mid_{\epsilon=0}}\;.$ (4.17)
Proof —
The key observation is that for functions satisfying reduction properties, may
terms in the action of $I_{n}^{\left(N,m\right)}$ decouple and one is almost
reduced to the pure-product case. Due to the reduction properties of
$\mathcal{F}_{n}$, the action of the finite-difference operators simplifies:
$\eth^{\left(t\right)}_{z}\left(\mu_{\ell,\,t,\,j}\right)\cdot\mathcal{F}_{n}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{\overline{p}_{s}}\\\
\left\\{z_{s}\right\\}^{\overline{p}_{s}+\epsilon_{s}}\end{array}\right)=\mathcal{F}_{n-t}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{\overline{p}_{s}-t\delta_{s,\,\ell}}\\\
\left\\{z_{s}\right\\}^{\overline{p}_{s}+\epsilon_{s}-t\delta_{s,\,\ell}}\end{array}\right)\left\\{\left[\mathcal{V}_{n-t}\right]^{t}\left(z_{\ell}\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{\overline{p}_{s}-t\delta_{s,\,\ell}}\\\
\left\\{z_{s}\right\\}^{\overline{p}_{s}+\epsilon_{s}-t\delta_{s,\,\ell}}\end{array}\right.\right)\;\right.\\\
\left.-\left[\mathcal{V}_{n-t}\right]^{t}\left(\mu_{\ell,\,m,\,j}\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{\overline{p}_{s}-t\delta_{s,\,\ell}}\\\
\left\\{z_{s}\right\\}^{\overline{p}_{s}+\epsilon_{s}-t\delta_{s,\,\ell}}\end{array}\right.\right)\right\\}\;.$
(4.18)
There, $\delta_{s,\ell}$ is the Kronecker $\delta$ symbol. Thus,
$\prod\limits_{\ell=1}^{r}\prod\limits_{t=1}^{n}\prod\limits_{j=1}^{p_{\ell\,,t}}\left\\{\int\limits_{-q}^{q}{\mathchoice{\dfrac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\dfrac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\frac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}{\frac{\mathrm{d}\mu_{\ell,\,t,\,j}}{z_{\ell}-\mu_{\ell,\,t,\,j}}}}\eth^{\left(t\right)}_{z}\left(\mu_{\ell,\,t,\,j}\right)\right\\}\cdot\mathcal{F}_{n}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{\overline{p}_{s}}\\\
\left\\{z_{s}\right\\}^{\overline{p}_{s}+\epsilon_{s}}\end{array}\right)\\\
=\mathcal{F}_{\left|p\right|}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right)\prod\limits_{\ell=1}^{r}\prod\limits_{t=1}^{n}\left\\{\int\limits_{-q}^{q}\mathrm{d}\mu{\mathchoice{\dfrac{\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(z_{\ell}\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)-\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(\mu\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)}{z_{\ell}-\mu}}{\dfrac{\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(z_{\ell}\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)-\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(\mu\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)}{z_{\ell}-\mu}}{\frac{\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(z_{\ell}\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)-\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(\mu\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)}{z_{\ell}-\mu}}{\frac{\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(z_{\ell}\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)-\left[\mathcal{V}_{\left|p\right|}\right]^{t}\left(\mu\left|\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right.\right)}{z_{\ell}-\mu}}}\right\\}^{p_{\ell\,t}}\;.$
(4.19)
$\left|p\right|=\Sigma_{\ell}p_{\ell\,0}$ is the number of variables remaining
after computing the action the finite-difference operators. Indeed, all the
dependence on the variables $\left\\{z_{s}\right\\}$ that are repeated
$p_{s,\,\ell}$ times, $\ell=1,\dots,n$, disappears from the function. To
evaluate the action of $I_{n}^{\left(N,0\right)}$, one should hit with the
product of $z_{s}$ derivatives on the RHS of (4.19) just as it was done in
(3.12). Clearly, such an operation generates positive powers of $x$ (by
differentiating the exponent $\mathrm{e}^{x\mathcal{G}_{n}}$) or of $\ln x$
(by differentiating $\mathrm{e}^{\ln x\,\mathcal{H}_{n}}$). In order to obtain
the highest possible power of $x$, the derivative should only hit on the
${x\mathcal{G}_{n}}$ part of the exponent. In this way
$\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}$ produces a
contribution of order $x^{k_{1}+\dots+k_{r}}$. Hence, the largest possible
power of $x$ comes from derivatives such that $\Sigma k_{\ell}=N$, and then
$\partial_{z_{1}}^{k_{1}}\dots\partial_{z_{r}}^{k_{r}}\mathcal{F}_{\left|p\right|}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right)=x^{N}\prod\limits_{s=1}^{r}\left(\partial_{z_{k}}\mathcal{G}_{\left|p\right|}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right)\right)^{k_{s}}\mathcal{F}_{\left|p\right|}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right)+\mathrm{O}\left({\mathchoice{\dfrac{x^{N}\ln
x}{x}}{\dfrac{x^{N}\ln x}{x}}{\frac{x^{N}\ln x}{x}}{\frac{x^{N}\ln
x}{x}}}\right)\;.$ (4.20)
Note that the positive power $x^{N}$ simplifies with the prefactor in front of
$I_{n}^{\left(N,m\right)}$. Once the derivatives are computed, one should send
all of the $z$’s to $\sigma q$, $\sigma=\pm$. Hence, due to the conditions
$\Sigma\epsilon_{\ell}=0$ the upper and the lower variables in hypergeometric
notation will be equal. This allows one to apply the reduction properties to
$\mathcal{F}_{\left|p\right|}$, as well as to most of the variables appearing
in $\partial_{z_{k}}\mathcal{G}_{\left|p\right|}$. In particular, the
reduction properties imply that
$\partial_{z_{k}}\mathcal{G}_{\left|p\right|}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}^{p_{s\,0}}\\\
\left\\{z_{s}\right\\}^{p_{s\,0}+\epsilon_{s}}\end{array}\right)=\epsilon_{k}\partial_{\epsilon}\mathcal{G}_{\left|p\right|}\left(\begin{array}[]{c}\left\\{z_{s}\right\\}_{s\not=k}^{p_{s\,0}}\;;\;z_{k}\\\
\left\\{z_{s}\right\\}_{s\not=k}^{p_{s\,0}+\epsilon_{s}}\;;\;z_{k}+\epsilon\end{array}\right)_{\mid_{\epsilon=0}}\;.$
(4.21)
Hence, after implementing most of the operations, it follows that, up to
$\mathrm{o}\left(1\right)$ corrections,
$\mathcal{W}\,\mathrm{e}^{x\mathcal{G}+\ln x\,\mathcal{H}}$ can be pulled out
from the sum as it is a constant, and
$x^{-N}I_{n}^{\left(N,0\right)}\left[\mathcal{F}_{n}\right]=\mathcal{W}\mathrm{e}^{x\mathcal{G}+\ln
x\mathcal{H}}\mathfrak{S}^{\left(m\right)}_{N}*\sum\limits_{\Sigma
k_{\ell}=N}G^{\mathfrak{S}^{\left(m\right)}_{N}}_{\left\\{k_{\ell}\right\\}}\\\
\times\left\\{\prod\limits_{s=1}^{r}\left[\epsilon_{s}\mathcal{G}^{\prime}\left(z_{s}\right)\right]^{k_{s}}\left[\mathcal{V}\left(z_{s}\right)\right]^{p_{s\,0}}\prod\limits_{t=1}^{n}\left\\{\int\limits_{-q}^{q}\mathrm{d}\mu{\mathchoice{\dfrac{\left[\mathcal{V}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\dfrac{\left[\mathcal{V}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[\mathcal{V}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[\mathcal{V}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}}\right\\}^{p_{\ell\,t}}\right\\}+\mathrm{o}\left(1\right)$
(4.22)
Thus by applying the sum-rule (3.14) we get the claim. □
### 4.3 The action of $I_{n}^{\left(N,\pm 1\right)}$
We now focus on the action of the functional $x^{-N}I^{\left(N,m\right)}_{n}$,
$m=\pm$ and $N\geq 2$, on the class of $x$-dependent functions as in (4.1).
###### Proposition 4.2
Let $\mathcal{F}_{n}$ be as in (4.1), then
${\mathchoice{\dfrac{I_{n}^{\left(N;m\right)}\left[\mathcal{F}_{n}\right]}{x^{N}}}{\dfrac{I_{n}^{\left(N;m\right)}\left[\mathcal{F}_{n}\right]}{x^{N}}}{\frac{I_{n}^{\left(N;m\right)}\left[\mathcal{F}_{n}\right]}{x^{N}}}{\frac{I_{n}^{\left(N;m\right)}\left[\mathcal{F}_{n}\right]}{x^{N}}}}=x^{-2}\mathcal{W}^{\left(m\right)}\mathrm{e}^{x\mathcal{G}^{\left(m\right)}+\ln
x\mathcal{H}^{\left(m\right)}}\sum\limits_{k=0}^{N}\left({\mathchoice{\dfrac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{+}}{p^{\prime}_{+}}}{\dfrac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{+}}{p^{\prime}_{+}}}{\frac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{+}}{p^{\prime}_{+}}}{\frac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{+}}{p^{\prime}_{+}}}}\right)^{k}\left({\mathchoice{\dfrac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{-}}{p^{\prime}_{-}}}{\dfrac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{-}}{p^{\prime}_{-}}}{\frac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{-}}{p^{\prime}_{-}}}{\frac{i\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}_{-}}{p^{\prime}_{-}}}}\right)^{N-k}\partial_{\gamma}^{n}\widetilde{O}_{N,\,k}\left[\sigma\nu^{\left(m\right)}\right]_{\mid_{\gamma=0}}\;+\;\mathrm{o}\left(x^{-2}\right)\\\
\;\mathrm{with}\quad\nu^{\left(m\right)}\left(z\right)={\mathchoice{\dfrac{i}{2\pi}}{\dfrac{i}{2\pi}}{\frac{i}{2\pi}}{\frac{i}{2\pi}}}\ln\left(1+\gamma\mathcal{V}^{\left(m\right)}\left(z\right)\right)\;.$
(4.23)
There we have introduced a notation for the almost reduced functions
$\mathcal{W}^{\left(m\right)}=\mathcal{W}_{1}\left(\begin{array}[]{c}-mq\\\
mq\end{array}\right)\;,\quad\mathcal{H}^{\left(m\right)}\left(z\right)=\mathcal{H}_{1}\left(\begin{array}[]{c}-mq\\\
mq\end{array}\right)\;,\quad\mathcal{V}^{\left(m\right)}\left(z\right)=\mathcal{V}_{1}\left(z\left|\begin{array}[]{c}-mq\\\
mq\end{array}\right.\right)\;,\\\
\mathrm{as}\,\mathrm{well}\,\mathrm{as}\qquad\left(\mathcal{G}^{\left(m\right)}\right)^{\prime}\left(z\right)=\partial_{\epsilon}\mathcal{G}_{2}\left(\begin{array}[]{c}-mq,z\\\
mq,z+\epsilon\end{array}\right)\;.\hskip 142.26378pt$ (4.24)
Proof — If one restricts oneself to the leading $\mathrm{O}\left(1\right)$
part of this action, then, as before, one should only consider the part of the
sum corresponding to integers $k_{\ell}$ such that $\sum\limits k_{\ell}=N-2$.
All the others will produce subdominant contributions. Next, one should once
again act with the derivatives on the exponent
$\exp\left\\{x\mathcal{G}_{n}\right\\}$ exactly as it was done in (4.20).
Note that, for the oscillating functional we consider, there are at least $2$
variables $z_{\ell}$. Moreover, after the computation of derivatives one
should set $z_{1}=\dots=z_{p}=\sigma q$ and $z_{p+1}=\dots=z_{r}=-\sigma q$,
$p\in\left[\,1\,;r-1\,\right]$. Indeed, for the slowest oscillating terms when
a maximal number of derivatives $\Sigma k_{\ell}=N-2$ is considered, then
there is only one jump in the sequence $\sigma_{\ell}$.
The first step of the analysis, that is to say the determination of the action
of the $\eth^{\left(t\right)}_{z_{\ell}}\left(\mu_{\ell,\,t,\,j}\right)$
operators is the same as in the non-oscillating case. Then, in order to
recover an $\mathrm{O}\left(x^{-2}\right)$ contribution, one should act with
all of the derivatives only on the exponent $x\mathcal{G}_{n}$. The only
difference in the proof consists in to the specifications of the $z$’s. It is
not difficult to see that, for any function satisfying the reduction
properties (4.2):
$\mathcal{W}_{r}\left(\begin{array}[]{c}\left\\{\sigma
q\right\\}^{p}\cup\left\\{-\sigma q\right\\}^{r-p}\\\ \left\\{\sigma
q\right\\}^{p+\sigma m}\cup\left\\{-\sigma q\right\\}^{r-p-\sigma
m}\end{array}\right)=\mathcal{W}_{1}\left(\begin{array}[]{c}-mq\\\
mq\end{array}\right)\;.$ (4.25)
Thence,
$x^{-N}I_{n}^{\left(N;m\right)}\left[\mathcal{F}_{n}\right]=x^{-2}\mathcal{W}^{\left(m\right)}\mathrm{e}^{x\mathcal{G}^{\left(m\right)}+\ln
x\mathcal{H}^{\left(m\right)}}\;\;\mathfrak{S}^{\left(m\right)}_{N}\;*\\!\\!\sum\limits_{\Sigma
k_{\ell}=N-2}\\\ \hskip
56.9055ptG^{\mathfrak{S}^{\left(m\right)}_{N}}_{\left\\{k_{\ell}\right\\}}\left\\{\prod\limits_{s=1}^{r}\left[\epsilon_{s}\mathcal{G}^{\prime}\left(z_{s}\right)\right]^{k_{s}}\left[\mathcal{V}^{\left(m\right)}\left(z_{s}\right)\right]^{p_{s\,0}}\prod\limits_{t=1}^{n}\left\\{\int\limits_{-q}^{q}\mathrm{d}\mu{\mathchoice{\dfrac{\left[\mathcal{V}^{\left(m\right)}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}^{\left(m\right)}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\dfrac{\left[\mathcal{V}^{\left(m\right)}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}^{\left(m\right)}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[\mathcal{V}^{\left(m\right)}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}^{\left(m\right)}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}{\frac{\left[\mathcal{V}^{\left(m\right)}\left(z_{s}\right)\right]^{t}-\left[\mathcal{V}^{\left(m\right)}\left(\mu\right)\right]^{t}}{z_{s}-\mu}}}\right\\}^{p_{\ell\,t}}\right\\}+\mathrm{o}\left(1\right)\;.$
(4.26)
It is now clear that the above series is exactly of the type considered in the
sum-rule (3.15). □
## 5 Conclusion
In this article, we have given the explicit proof of the action of the
functionals involved in the AE of cyclic integrals. The form of this action
was used in [9] for the computation of the asymptotic behaviour of the
generating function of the spin-spin correlation functions in integrable
models. Our result was proven thanks to a precise determination of the
structure of a part of the AE of $\ln\operatorname{det}\left[I+V\right]$ to
any order in $x^{-N}$. These were obtained thanks to several equivalent ways
of representing the kernel of the generalized sine kernel. We have only
derived this action up to the first non-vanishing terms. However, the method
that we have developed can also be applied, with the price of increasing
complexity, to compute the sub-leading corrections to these terms. These, in
turn would allow one to access to corrections in the long-distance asymptotic
behavior of $\langle\mathrm{e}^{\beta\mathcal{Q}_{x}}\rangle$ that deviate
from the conformal field theory predictions.
## Acknowledgements
Karol K. Kozlowski is supported by the ANR program GIMP ANR-05-BLAN-0029-01
and by the French ministry of research. The author would like to thank the
Centre de recherche math matiques of the Universit de Montr al for its
hospitality during the conference "Integrable quantum systems and solvable
statistical models". The author is grateful to N.-Kitanine, J.-M. Maillet, N.
A. Slavnov and V. Terras for many stimulating discussions.
## References
* [1] I. Affleck, _Field theory methods and quantum critical phenomena_ , ed. E. Br zin and J. Zinn-Justin, North-Holland, Amsterdam.
* [2] H. E. Boos, M. Jimbo, T. Miwa, F. Smirnov, and Y. Takeyama, _"Hidden Grassmann structure in the XXZ model."_ , Comm. Math. Phys. 272 (2007), 263–281.
* [3] J. L. Cardy, _"Conformal invariance and universality in finite-size scaling."_ , J. Phys. A: Math. Gen. 17 (1984), L385–387.
* [4] J.-S. Caux, P. Calabrese, and N.A. Slavnov, _"One-particle dynamical correlations in the one-dimensional Bose gas ."_ , J. Stat. Mech. (2007), P01008.
* [5] J.-S. Caux, R. Hagemans, and J.-M. Maillet, _"Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime."_ , J. Stat. Mech. 95 (2005), P09003.
* [6] F. D. M. Haldane, _"General relation of correlation exponents and spectral properties of one-dimensional Fermi systems: Application to the anisotropic s = 1/2 Heisenberg chain."_ , Phys. Rev. Lett. 45 (1980), 1358–1362.
* [7] M. Jimbo, T. Miwa, Y. Mori, and M. Sato, _"Density matrix of an impenetrable Bose gas and the fifth Painlev transcendent."_ , Physica D 1 (1980), 80–158.
* [8] N. Kitanine, K. K. Kozlowski, J.-M. Maillet, N. A. Slavnov, and V. Terras, _"On correlation functions of integrable models associated with the six-vertex R-matrix."_ , J. Stat. Mech. (2007), P01022.
* [9] N. Kitanine, K. K. Kozlowski, J.-M. Maillet, N. A. Slavnov, and V. Terras, _"Algebraic Bethe Ansatz approach to the asymptotics behavior of correlation functions."_ , math-ph/ 08080227 (2008).
* [10] N. Kitanine, K. K. Kozlowski, J.-M. Maillet, N. A. Slavnov, and V. Terras, _"The Riemann-Hilbert approach to a generalized sine kernel and applications."_ , math-ph/ 08054586 (2008).
* [11] N. Kitanine, J.-M. Maillet, N.A. Slavnov, and V. Terras, _"Master equation for spin-spin correlation functions of the XXZ chain."_ , Nucl.Phys. B 712 (2005), 600–622.
* [12] N. Kitanine, J.-M. Maillet, and V. Terras, _"Form factors of the XXZ Heisenberg spin-1/2 finite chain."_ , J. Phys. A: Math. Gen. 35 (2002), L753–10502.
* [13] T. Kojima, V. E. Korepin, and N. A. Slavonv, _"Determinant representation for dynamical correlation functions of the quantum nonlinear Schr dinger equation."_ , Comm. Math. Phys. 188 (1997), 657–689.
|
arxiv-papers
| 2010-01-20T17:23:19 |
2024-09-04T02:49:07.874712
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. K. Kozlowski",
"submitter": "Karol Kozlowski Kajetan",
"url": "https://arxiv.org/abs/1001.3646"
}
|
1001.3912
|
# Titchmarsh-Sims-Weyl theory for complex Hamiltonian systems on Sturmian time
scales
Douglas R. Anderson Department of Mathematics and Computer Science, Concordia
College, Moorhead, MN 56562 USA
visiting the School of Mathematics, The University of New South Wales Sydney
2052, Australia andersod@cord.edu
###### Abstract.
We study non-self-adjoint Hamiltonian systems on Sturmian time scales,
defining Weyl-Sims sets, which replace the classical Weyl circles, and a
matrix-valued $M-$function on suitable cone-shaped domains in the complex
plane. Furthermore, we characterize realizations of the corresponding dynamic
operator and its adjoint, and construct their resolvents. Even-order scalar
equations and the Orr-Sommerfeld equation on time scales are given as examples
illustrating the theory, which are new even for difference equations. These
results unify previous discrete and continuous theories to dynamic equations
on Sturmian time scales.
###### Key words and phrases:
linear equations; non-self-adjoint operator; Orr-Sommerfeld equation; Sturm-
Liouville theory; even-order equations, Weyl-Sims theory.
###### 2000 Mathematics Subject Classification:
34N05; 34B20; 34B27; 47A10; 47B39
## 1\. Introduction
Brown, Evans, and Plum [13] study Titchmarsh-Sims-Weyl theory for the complex,
generally non-self-adjoint continuous Hamiltonian system
$Jy^{\prime}(t,\lambda)=\Big{(}\lambda A(t)+B(t)\Big{)}y(t,\lambda),\quad
t\in[0,\infty),$ (1.1)
where $A$ and $B$ are $2n\times 2n$ matrix-valued functions with matrix weight
function $A(t)\geq 0$, and $J=\left(\begin{smallmatrix}0_{n}&-I_{n}\\\
I_{n}&0_{n}\end{smallmatrix}\right)$. In a related work, Monaquel and Schmidt
[23] introduce the uniformly discrete (the step size of the domain is a
constant unit) counterpart to the continuous Hamiltonian system via
$J{\bf\Delta}y(t,\lambda)=(\lambda A(t)+B(t))y(t,\lambda)\quad\text{for}\quad
t\in[0,\infty)\cap\mathbb{Z},$ (1.2)
where ${\bf\Delta}$ is a mixed right-and-left difference operator described
via ${\bf\Delta}=\left(\begin{smallmatrix}\Delta&0_{n}\\\
0_{n}&\nabla\end{smallmatrix}\right)$ for the forward difference operator
$\Delta u(t)=u(t+1)-u(t)$ and the backward difference operator $\nabla
u(t)=u(t)-u(t-1)$, and where again $A$ and $B$ are $2n\times 2n$ matrix-valued
functions but with the assumption $A(t)>0$.
We seek to extend (1.1), (1.2) to Sturmian time scales (introduced in
Ahlbrandt, Bohner, and Voepel [3]), thus unifying the continuous (1.1) and
discrete (1.2) non-self-adjoint theories in a single setting. As we do so, the
robust nature of time-scale theory will offer more flexibility when
discretizing (1.1), for example allowing time-varying step sizes in the
domain, than that represented by (1.2). Just as in the translation of the
continuous theory to the uniformly discrete case, the unification and
extention of the two theories to Sturmian time scales require some care and
provide unexpected difficulties. The first such issue is defining an
appropriate time-scale dynamic operator that generalizes ${\bf\Delta}$ in
(1.2) and accounts for the shifts at scattered domain points. To construct
such an operator, we follow [23] by utilizing a partial left-shift operator
applied to $y$ defined on $[t_{0},\infty)_{\mathbb{X}}$ for some
$t_{0}\in\mathbb{X}$ by
$\widehat{y}(t):=\begin{pmatrix}y_{1}(t)\\\ y_{2}^{\rho}(t)\end{pmatrix},\quad
y_{1}:[t_{0},\infty)_{\mathbb{X}}\rightarrow\mathbb{C}^{n},\quad
y_{2}:[\rho(t_{0}),\infty)_{\mathbb{X}}\rightarrow\mathbb{C}^{n}.$ (1.3)
Then we introduce the complex Hamiltonian dynamic system on Sturmian time
scales given by
$J\widehat{y}^{\Delta}(t)=\Big{(}\lambda A(t)+B(t)\Big{)}y(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}}:=[t_{0},\infty)\cap\mathbb{X},\quad
J=\left(\begin{smallmatrix}0_{n}&-I_{n}\\\
I_{n}&0_{n}\end{smallmatrix}\right),$ (1.4)
where we assume that the Hermitian weight function
$A:[t_{0},\infty)_{\mathbb{X}}\rightarrow\mathbb{C}^{2n,2n}$ and the
(generally) non-Hermitian coefficient function
$B:[t_{0},\infty)_{\mathbb{X}}\rightarrow\mathbb{C}^{2n,2n}$ satisfy the block
forms
$A(t)=\begin{pmatrix}A_{1}(t)&0_{n}\\\ 0_{n}&A_{2}(t)\end{pmatrix}\geq
0\quad\text{and}\quad B(t)=\begin{pmatrix}B_{1}(t)&B_{2}(t)\\\
B_{3}(t)&B_{4}(t)\end{pmatrix}$ (1.5)
for $n\times n$ rd-continuous complex matrices $A_{1}$, $A_{2}$, $B_{1}$,
$B_{2}$, $B_{3}$, and $B_{4}$ such that
$E_{2}(t):=\Big{(}I_{n}+\mu(t)B_{2}(t)\Big{)}^{-1}\quad\text{and}\quad\Big{(}I_{n}+\mu(t)B_{3}(t)\Big{)}^{-1}\quad\text{exist.}$
(1.6)
Our positive semi-definite assumption in (1.5) on $A$ weakens the positive
definite assumption made on $A$ in the discrete case [23]; in compensation, we
make a certain definiteness assumption below in (3.9) that mirrors the
continuous case [13, (3.14)]. Using standard notation [18], $\mathbb{X}$ is a
nonempty unbounded closed subset of the set of real numbers $\mathbb{R}$ such
that the left jump operator $\rho$ and right jump operator $\sigma$ given by
$\rho(t)=\sup\\{s\in\mathbb{X}:s<t\\}\quad\text{and}\quad\sigma(t)=\inf\\{s\in\mathbb{X}:s>t\\}$
satisfy the Sturmian [3] time-scale condition
$\sigma(\rho(t))=\rho(\sigma(t))=t,\quad t\in[t_{0},\infty)_{\mathbb{X}},$
(1.7)
with the compositions $y\circ\rho$ and $z\circ\sigma$ denoted by $y^{\rho}$
and $z^{\sigma}$, respectively; the graininess functions are defined by
$\mu(t)=\sigma(t)-t$ and $\nu(t)=t-\rho(t)$; the delta derivative of $y$ at
$t\in\mathbb{X}$, denoted $y^{\Delta}(t)$, and the nabla derivative of $y$ at
$t\in\mathbb{X}$, denoted $y^{\nabla}(t)$, are the vectors (provided they
exist) given by, respectively,
$y^{\Delta}(t):=\lim_{s\rightarrow
t}\frac{y^{\sigma}(t)-y(s)}{\sigma(t)-s}\quad\text{and}\quad
y^{\nabla}(t):=\lim_{s\rightarrow t}\frac{y^{\rho}(t)-y(s)}{\rho(t)-s}.$
Note that if $\mathbb{X}=\mathbb{R}$ we have
$J\widehat{y}^{\Delta}(t)=Jy^{\prime}(t)$
as in (1.1), while if $\mathbb{X}=\mathbb{Z}$ we have
$J\widehat{y}^{\Delta}(t)=J\Delta\widehat{y}(t)=J\Delta\left(\begin{smallmatrix}y_{1}(t)\\\
y_{2}(t-1)\end{smallmatrix}\right)=J\left(\begin{smallmatrix}\Delta
y_{1}(t)\\\ \nabla y_{2}(t)\end{smallmatrix}\right)=J{\bf\Delta}y(t)$
as in (1.2), giving credibility to system (1.4) as a unifying vehicle for this
theory. To recover the case $\mathbb{X}=\mathbb{R}$, just set $\mu(t)=0$ and
$\sigma(t)=\rho(t)=t$, and to recover the case $\mathbb{X}=\mathbb{Z}$, set
$\mu(t)=1$, $\sigma(t)=t+1$, and $\rho(t)=t-1$.
For $A$ given in (1.5), let $L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$ denote the
Hilbert space of measurable $\mathbb{C}^{2n}-$valued functions $y$ for which
the delta integral exists and satisfies
$\|y\|_{A}^{2}:=\int_{t_{0}}^{\infty}y^{*}(t)A(t)y(t)\Delta t<\infty.$ (1.8)
Vector functions $x,y\in L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$ are said to be
$A-$square integrable, with the scalar product defined via
$(x,y)_{A}:=\int_{t_{0}}^{\infty}y^{*}(t)A(t)x(t)\Delta t,\quad x,y\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}.$ (1.9)
As $A$ in (1.5) may be singular, the inner product for
$L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$ in (1.9) may not be positive. To
account for this, we introduce the following quotient space. For $x,y\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$, $x$ and $y$ are said to be equal iff
$\|x-y\|_{A}=0$. In this context $L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$ is an
inner product space with inner product (1.9). In addition, a $2n\times n$
matrix is $A-$square integrable if and only if each of its columns is
$A-$square integrable. The same terminology will be used for other $2n\times
2n$ matrix weight functions.
Next we define the linear vector function space
$\mathbb{D}:=\left\\{y=(y_{1},y_{2})^{\operatorname{T}}\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}\Big{|}\;y_{1},y_{2}^{\rho}:[t_{0},\infty)_{\mathbb{X}}\rightarrow\mathbb{C}^{n}\;\text{are
delta differentiable}\right\\}.$ (1.10)
Then $y$ is a solution of (1.4) if and only if $y\in\mathbb{D}$ and $y$
satisfies (1.4). With (1.7) in mind, we point out that in (1.10) the
assumption $y_{2}^{\rho}$ is delta differentiable is equivalent to the
assumption that $y_{2}$ is delta differentiable. For more on time scales
generally, see Bohner and Peterson [10, 11].
In [12], Brown, Evans, McCormack, and Plum study the equation
$-(py^{\prime})^{\prime}+qy=\lambda wy,$
allowing the potentials to be complex-valued functions, and in so doing relax
some of the conditions used by Sims [28]; see (4.2) below. Atkinson [8],
Hinton and Shaw [20, 21], and Krall [22] all worked on (1.1) in the case of
symmetric coefficients for $\mathbb{X}=\mathbb{R}$. In the discrete symmetric
case, Clark and Gesztesy [15, 16] presented a Weyl-Titchmarsh theory for a
version of (1.2), while Shi [26] did the same for a version of (2.12) given
below. For more on related discrete theory for Hamiltonian systems see
Ahlbrandt [1], Ahlbrandt and Peterson [4], Bohner, Došlý, and Kratz [9], Shi
[25], Shi and Wu [27], and Sun, Shi, and Chen [29]. Preliminary work on
Hamiltonian systems on time scales includes Ahlbrandt, Bohner, and Ridenhour
[2], Anderson [5], Anderson and Buchholz [6], and Hilscher [19]. For some
specific examples where non-self-adjoint problems may arise see Chandrasekhar
[14].
Our notation and organization of the discussion are obviously largely based on
the continuous [13] and discrete [23] cases. The key to bringing these two
theories together is finding an operator with domain inside a weighted Hilbert
space such that existence and uniqueness of solutions can be shown for initial
value problems, and an integration by parts formula holds. In the next section
we will give a detailed analysis of these issues on Sturmian time scales. The
Sturmian assumption in (1.7) may from one point of view seem unduly
restrictive; from another it can be seen as surprising that the discrete and
continuous theories can be combined at all, as there is no a priori physical
reason why this theory should exist on all possible time scales, especially
pathological ones. Assumption (1.7) merely requires that all points be dense
from both sides or scattered from both sides. With this supposition in place,
we will see how unexpectedly harmonious the continuous and discrete cases can
be made when treated concurrently in this context.
## 2\. The homogeneous system
We begin our analysis of the linear Hamiltonian system (1.4) in this section
with an existence and uniqueness result.
###### Theorem 2.1 (Existence and Uniqueness).
Assume (1.6). Then the linear Hamiltonian system (1.4) with initial condition
$\widehat{y}(t_{0})=\left(\begin{smallmatrix}y_{1}(t_{0})\\\
y_{2}^{\rho}(t_{0})\end{smallmatrix}\right)$ (2.1)
has a unique solution $\widehat{y}=(y_{1},y_{2}^{\rho})^{\operatorname{T}}$
with $y\in\mathbb{D}$.
###### Proof.
Let $\widehat{y}$ be given as in (1.3). For any given $\lambda\in\mathbb{C}$,
using (1.7) and the assumptions on the block forms of $J$, $A$, and $B$, we
can rewrite (1.4) as the pair of $n$-vector equations
$\begin{cases}&y_{1}^{\Delta}(t)=B_{3}(t)y_{1}(t)+\big{(}\lambda
A_{2}(t)+B_{4}(t)\big{)}(y_{2}^{\rho})^{\sigma}(t),\\\
&(y_{2}^{\rho})^{\Delta}(t)=-\big{(}\lambda
A_{1}(t)+B_{1}(t)\big{)}y_{1}(t)-B_{2}(t)(y_{2}^{\rho})^{\sigma}(t).\end{cases}$
Using the simple useful formula
$y_{2}=(y_{2}^{\rho})^{\sigma}=(y_{2}^{\rho})+\mu(y_{2}^{\rho})^{\Delta}$, we
also have (suppressing the $t$)
$\begin{cases}&y_{1}^{\Delta}=\Big{(}B_{3}-\mu(\lambda
A_{2}+B_{4})E_{2}(\lambda A_{1}+B_{1})\Big{)}y_{1}+(\lambda
A_{2}+B_{4})E_{2}(y_{2}^{\rho}),\\\
&(y_{2}^{\rho})^{\Delta}=-E_{2}\big{(}\lambda
A_{1}+B_{1}\big{)}y_{1}-E_{2}B_{2}(y_{2}^{\rho}),\end{cases}$ (2.2)
where we have taken $E_{2}$ as in (1.6). Thus we may also view solutions
$y=(y_{1},y_{2})^{\operatorname{T}}$ of (1.4) and (2.2) via (1.3) as solutions
of
$\widehat{y}^{\Delta}(t)=\mathcal{K}(t,\lambda)\widehat{y}(t),\quad\mathcal{K}(\cdot,\lambda):=-J(\lambda
A+B)H,$ (2.3)
where on $[t_{0},\infty)_{\mathbb{X}}$ we use $E_{2}=(I_{n}+\mu B_{2})^{-1}$
and
$H:=\begin{pmatrix}I_{n}&0_{n}\\\ -\mu E_{2}(\lambda
A_{1}+B_{1})&E_{2}\end{pmatrix},\quad\text{with}\quad-J(\lambda
A+B)=\begin{pmatrix}B_{3}&\lambda A_{2}+B_{4}\\\ -(\lambda
A_{1}+B_{1})&-B_{2}\end{pmatrix},$ (2.4)
since $E_{2}B_{2}=B_{2}E_{2}$. Directly from the definition of
$\mathcal{K}(\cdot,\lambda)$ in (2.3) and (2.4) we have that
$I_{2n}+\mu\mathcal{K}(\cdot,\lambda)=\begin{pmatrix}I_{n}+\mu
B_{3}&\mu(\lambda A_{2}+B_{4})\\\
0_{n}&I_{n}\end{pmatrix}\begin{pmatrix}I_{n}&0_{n}\\\ -\mu E_{2}(\lambda
A_{1}+B_{1})&E_{2}\end{pmatrix},$
so that $I_{2n}+\mu(t)\mathcal{K}(t,\lambda)$ is invertible by (1.6) and thus
$\mathcal{K}(\cdot,\lambda)$ is regressive. By [10, Theorem 5.8], the matrix
equation $\widehat{y}^{\Delta}=\mathcal{K}(\cdot,\lambda)\widehat{y}$ with
initial condition (2.1) has a unique solution $\widehat{y}$. It follows that
(1.4), (1.3) with initial condition (2.1) has a unique solution
$y\in\mathbb{D}$. ∎
###### Theorem 2.2 (Green’s Formula).
For $y,z\in\mathbb{D}$ and $a,b\in[t_{0},\infty)_{\mathbb{X}}$ with $b>a$ we
have
$\int_{a}^{b}\left[z^{*}J\widehat{y}^{\Delta}-\left(J\widehat{z}^{\Delta}\right)^{*}y\right](t)\Delta
t=\widehat{z}^{*}(b)J\widehat{y}(b)-\widehat{z}^{*}(a)J\widehat{y}(a).$
###### Proof.
Using (1.7) as we expand out the integrand, we have (suppressing the variable
$t$)
$\displaystyle z^{*}J\widehat{y}^{\Delta}$ $\displaystyle=$ $\displaystyle-
z_{1}^{*}(y_{2}^{\rho})^{\Delta}+(z_{2}^{\rho})^{\sigma*}y_{1}^{\Delta},$
$\displaystyle\left(J\widehat{z}^{\Delta}\right)^{*}y$ $\displaystyle=$
$\displaystyle-(z_{2}^{\rho})^{\Delta*}y_{1}+z_{1}^{\Delta*}(y_{2}^{\rho})^{\sigma},$
so that when we subtract the second from the first, we obtain
$\displaystyle
z^{*}J\widehat{y}^{\Delta}-\left(J\widehat{z}^{\Delta}\right)^{*}y$
$\displaystyle=$ $\displaystyle-
z_{1}^{*}(y_{2}^{\rho})^{\Delta}+(z_{2}^{\rho})^{\sigma*}y_{1}^{\Delta}+(z_{2}^{\rho})^{\Delta*}y_{1}-z_{1}^{\Delta*}(y_{2}^{\rho})^{\sigma}$
$\displaystyle=$
$\displaystyle-(z_{1}^{*}y_{2}^{\rho})^{\Delta}+(z_{2}^{\rho*}y_{1})^{\Delta}$
$\displaystyle=$
$\displaystyle\left[\widehat{z}^{*}J\widehat{y}\right]^{\Delta}.$
The result follows from the fundamental theorem of calculus. ∎
If $y\in\mathbb{D}$ is a solution of (1.4), then it is related to
$\widehat{y}$ via
$y(t)=H(t)\widehat{y}(t),\quad t\in[t_{0},\infty)_{\mathbb{X}},$ (2.5)
for $H$ given in (2.4). As a result we may write (1.4) as the equivalent
equation (2.3), in other words as
$J\widehat{y}^{\Delta}(t)=\big{(}\lambda A(t)+B(t)\big{)}H(t)\widehat{y}(t).$
(2.6)
Moreover, the formal adjoint of (1.4) takes the form
$J\widehat{z}^{\Delta}(t)=\Big{(}\overline{\lambda}A(t)+B^{*}(t)\Big{)}z(t),\quad\lambda\in\mathbb{C},\quad
t\in[t_{0},\infty)_{\mathbb{X}};$ (2.7)
under the hypotheses of Theorem 2.1, an existence and uniqueness result holds
for (2.7). A solution $z$ of (2.7) is related to $\widehat{z}$ by the
transformation
$z(t)=\widetilde{H}(t)\widehat{z}^{\;\sigma}(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}},$ (2.8)
where on $[t_{0},\infty)_{\mathbb{X}}$ we have taken $E_{2}$ as in (1.6) and
$\widetilde{H}:=\begin{pmatrix}E_{2}^{*}&-\mu
E_{2}^{*}\Big{(}\overline{\lambda}A_{2}+B_{4}^{*}\Big{)}\\\
0_{n}&I_{n}\end{pmatrix},\quad\quad\widetilde{H}^{-1}=\begin{pmatrix}I+\mu
B_{2}^{*}&\mu\Big{(}\overline{\lambda}A_{2}+B_{4}^{*}\Big{)}\\\
0_{n}&I_{n}\end{pmatrix}.$ (2.9)
It follows for $\lambda\in\mathbb{C}$ and $t\in[t_{0},\infty)_{\mathbb{X}}$
that (2.7) is equivalent to the equation
$J\widehat{z}^{\Delta}(t)=\Big{(}\overline{\lambda}A(t)+B^{*}(t)\Big{)}\widetilde{H}(t)\widehat{z}^{\;\sigma}(t)=H^{*}(t)\Big{(}\overline{\lambda}A(t)+B^{*}(t)\Big{)}\widehat{z}^{\;\sigma}(t).$
(2.10)
###### Definition 2.3.
A matrix
$\widehat{Y}:[t_{0},\infty)_{\mathbb{X}}\rightarrow\mathbb{C}^{2n,2n}$ is
called a fundamental system of (2.6) if and only if its columns are linearly
independent solutions of (2.6), if and only if $\widehat{Y}(t)$ has rank $2n$
for some $t\in[t_{0},\infty)_{\mathbb{X}}$. In this case the columns of
$Y(t)=H(t)\widehat{Y}(t)$ form a fundamental system (system of linearly
independent solutions) of (1.4), where $H$ is given in (2.4).
###### Remark 2.4.
In the development below, let the matrix
$\widehat{Y}(t)=\left(\widehat{\theta}(t)|\widehat{\phi}(t)\right)$ for
$t\in[t_{0},\infty)_{\mathbb{X}}$ be the fundamental system of (2.6)
satisfying the initial condition
$\widehat{Y}(t_{0})=J.$ (2.11)
It follows from (2.5) that
$\widehat{Y}(t)=\left(\widehat{\theta}(t)|\widehat{\phi}(t)\right)$ for
$t\in[t_{0},\infty)_{\mathbb{X}}$ is the fundamental system of (1.4)
satisfying
$Y(t_{0})=H(t_{0})J=\begin{pmatrix}0_{n}&-I_{n}\\\
\Big{(}I+\mu(t_{0})B_{2}(t_{0})\Big{)}^{-1}&\mu(t_{0})\Big{(}I+\mu(t_{0})B_{2}(t_{0})\Big{)}^{-1}\Big{(}\lambda
A_{1}(t_{0})+B_{1}(t_{0})\Big{)}\end{pmatrix}.$
In a similar fashion take
$\widehat{Z}(t)=\left(\widehat{\eta}(t)|\widehat{\chi}(t)\right)$ for
$t\in[t_{0},\infty)_{\mathbb{X}}$ to be the fundamental system of (2.10)
satisfying $\widehat{Z}(t_{0})=J$, so that by (2.8) and (2.10) we have
$Z(t_{0})=[\mu J(\overline{\lambda}A+B^{*})+\widetilde{H}^{-1}]^{-1}J$, i.e.
$Z(t_{0})=\begin{pmatrix}0_{n}&-I_{n}\\\
\Big{(}I_{n}+\mu(t_{0})B_{3}^{*}(t_{0})\Big{)}^{-1}&\mu(t_{0})\Big{(}I_{n}+\mu(t_{0})B_{3}^{*}(t_{0})\Big{)}^{-1}\Big{(}\overline{\lambda}A_{1}(t_{0})+B_{1}^{*}(t_{0})\Big{)}\end{pmatrix}.$
###### Lemma 2.5.
For the fundamental systems $\widehat{Y}$ and $\widehat{Z}$ of (2.6) and
(2.10), respectively, the equality
$\widehat{Z}(t)=-J(\widehat{Y}^{-1})^{*}(t)J$
holds for all $t\in[t_{0},\infty)_{\mathbb{X}}$.
###### Proof.
Let $\widehat{U}(t):=-J(\widehat{Y}^{-1})^{*}(t)J$ for all
$t\in[t_{0},\infty)_{\mathbb{X}}$. As $\widehat{Y}(t_{0})=J$, we see that
$\widehat{U}(t_{0})=-J(\widehat{Y}^{-1})^{*}(t_{0})J=-JJJ=J=\widehat{Z}(t_{0}).$
If we can show that $\widehat{U}$ solves (2.10), then by uniqueness we will
have $\widehat{U}=\widehat{Z}$. By the product rule,
$0_{2n}=\left(\widehat{Y}^{-1}\widehat{Y}\right)^{\Delta}(t)=(\widehat{Y}^{-1})^{\sigma}(t)\widehat{Y}^{\Delta}(t)+(\widehat{Y}^{-1})^{\Delta}(t)\widehat{Y}(t);$
from this and (2.6) we have that
$(\widehat{Y}^{-1})^{\Delta}(t)=-(\widehat{Y}^{\sigma})^{-1}(t)\widehat{Y}^{\Delta}(t)\widehat{Y}^{-1}(t)=(\widehat{Y}^{\sigma})^{-1}(t)J\big{(}\lambda
A(t)+B(t)\big{)}H(t).$
Putting it all together we get
$\displaystyle J\widehat{U}^{\Delta}(t)$ $\displaystyle=$ $\displaystyle
J\left(-J(\widehat{Y}^{-1})^{*}J\right)^{\Delta}(t)=(\widehat{Y}^{-1})^{*\Delta}(t)J=(\widehat{Y}^{-1})^{\Delta*}(t)J$
$\displaystyle=$ $\displaystyle H^{*}(t)\big{(}\lambda
A(t)+B(t)\big{)}^{*}J^{*}(\widehat{Y}^{\sigma})^{-1*}(t)J$ $\displaystyle=$
$\displaystyle H^{*}(t)\big{(}\lambda
A(t)+B(t)\big{)}^{*}\widehat{U}^{\sigma}(t),$
so that $\widehat{U}$ solves (2.10). Thus
$\widehat{Z}(t)=\widehat{U}(t)=-J(\widehat{Y}^{-1})^{*}(t)J$ for all
$t\in[t_{0},\infty)_{\mathbb{X}}$. ∎
Throughout the paper, we will exhibit explicit dependence on $\lambda$ only
when necessary. We end this section with the following remark.
###### Remark 2.6.
As seen above in Theorems 2.1 and 2.2, the assumption that $\mathbb{X}$ is
Sturmian plays a decisive role in securing existence and uniqueness of
solutions, and an integration by parts formula. In this remark, we show that
there is no getting around this assumption. For example, instead of system
(1.4), consider the alternative system
$Jy^{\nabla}(t)=\Big{(}\lambda A(t)+B(t)\Big{)}\widehat{y}(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}},\quad J=\begin{pmatrix}0_{n}&-I_{n}\\\
I_{n}&0_{n}\end{pmatrix},$ (2.12)
where we have $\widehat{y}$ on the right-hand side, the nabla derivative is
used on the left, and we assume
$E_{2}^{\prime}(t):=\Big{(}I_{n}-\nu(t)B_{2}(t)\Big{)}^{-1}\quad\text{and}\quad\Big{(}I_{n}-\nu(t)B_{3}(t)\Big{)}^{-1}\quad\text{exist}$
(2.13)
in place of (1.6). System (2.12) may also be viewed as a generalization of
(1.1) and (1.2). As in (2.3), we can rewrite (2.12) as the system
$y^{\nabla}(t)=\mathcal{K}_{1}(t,\lambda)y(t),\quad\mathcal{K}_{1}(\cdot,\lambda):=-J(\lambda
A+B)H_{1},\quad H_{1}:=\begin{pmatrix}I_{n}&0_{n}\\\ \nu
E_{2}^{\prime}(\lambda A_{1}+B_{1})&E_{2}^{\prime}\end{pmatrix}.$ (2.14)
Directly from the definition of $\mathcal{K}_{1}(\cdot,\lambda)$ in (2.14) we
have that
$I_{2n}-\nu\mathcal{K}_{1}(\cdot,\lambda)=\begin{pmatrix}I_{n}-\nu
B_{3}&-\nu(\lambda A_{2}+B_{4})\\\ 0_{n}&I_{n}\end{pmatrix}H_{1},$
so that $I_{2n}-\nu(t)\mathcal{K}_{1}(t,\lambda)$ is invertible by (2.13),
$\mathcal{K}_{1}(\cdot,\lambda)$ is $\nu$-regressive, and the matrix equation
$y^{\nabla}=\mathcal{K}_{1}(\cdot,\lambda)y$ with proper initial condition has
a unique solution $y$. It follows that (2.12) has a unique solution.
Additionally, we have the integration by parts formula (Green’s formula)
$\int_{a}^{b}\left[\widehat{z}^{*}Jy^{\nabla}-\left(Jz^{\nabla}\right)^{*}\widehat{y}\right](t)\nabla
t=z^{*}(b)Jy(b)-z^{*}(a)Jy(a).$
To this point in the remark the analysis is valid on general time scales.
However, the scalar product in (1.9) is now replaced by
$(x,y)_{A}:=\int_{t_{0}}^{\infty}\widehat{y}^{*}(t)A(t)\widehat{x}(t)\nabla
t,\quad x,y\in L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}.$ (2.15)
To show that this defines an inner product on
$L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$, we would need the hat operator
$\widehat{\cdot}$ in (1.3) to be invertible, which is only possible if (1.7)
holds, i.e. on Sturmian time scales. In summary, to unify (1.1) and (1.2) on
time scales, systems equivalent to (1.4) or (2.12) must be used to account for
the shifts in the discrete case [23]. For those systems to admit the existence
and uniqueness of solutions to initial value problems, an integration by parts
formula, and a matrix weighted scalar product in a Hilbert space, the Sturmian
assumption (1.7) must used somewhere.
## 3\. Weyl-Sims nesting sets
Weyl-Sims sets $D(t,\lambda)$, $t\in[t_{0},\infty)_{\mathbb{X}}$, are the
analogue of the classical Weyl circles. Here the spectral parameter $\lambda$
varies in a set $\Lambda(\lambda_{0},\mathscr{U}_{2n})\subset\mathbb{C}$, a
cone-shaped set defined to parallel the construction in [13], which takes the
role of Sims’ rotated half-planes. The class of matrices $\mathscr{U}_{2n}$ is
one of many possible describing this rotation. The key result is the nesting
property of the Weyl-Sims sets (see Theorem 3.3 below). It will follow that
there is a limit set $D(\infty,\lambda)$ with the property that for any $l\in
D(\infty,\lambda)$, the Weyl solution $\psi=\theta+\phi l$ is square
integrable with respect to a certain matrix weight function $W$, while an
analogous statement holds for the adjoint equation. We will conclude this
section by noting conditions which imply $A-$square integrability of the Weyl
solution.
To begin, choose $\mathscr{U}\in\mathbb{C}^{n,n}$ invertible and define
$\mathscr{U}_{2n}=\begin{pmatrix}\mathscr{U}&0_{n}\\\
0_{n}&-\mathscr{U}^{*}\end{pmatrix}.$ (3.1)
Note that $\mathscr{U}_{2n}J$ is Hermitian and has precisely $n$ positive and
$n$ negative eigenvalues. Indeed, if $(\lambda,v)$ is an eigenpair associated
with $\mathscr{U}_{2n}J$, then
$(-\lambda,w),\quad\text{where}\quad w=\begin{pmatrix}0_{n}&-\mathscr{U}\\\
\mathscr{U}^{*}&0_{n}\end{pmatrix}v,$
is also an eigenpair associated with $\mathscr{U}_{2n}J$. Moreover, since
$J\mathscr{U}_{2n}^{*}=-\mathscr{U}_{2n}J$, a use of Theorem 2.2, (2.5) and
(2.6) yields that the fundamental system
$\widehat{Y}=\left(\widehat{\theta}|\widehat{\phi}\right)$ satisfies
$\displaystyle\widehat{Y}^{*}(t)\mathscr{U}_{2n}J\widehat{Y}(t)-\widehat{Y}^{*}(t_{0})\mathscr{U}_{2n}J\widehat{Y}(t_{0})$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\left\\{Y^{*}\left(\mathscr{U}_{2n}J\widehat{Y}^{\Delta}\right)+\left(\mathscr{U}_{2n}J\widehat{Y}^{\Delta}\right)^{*}Y\right\\}(s)\Delta
s$ (3.2) $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}Y^{*}\left\\{\mathscr{U}_{2n}(\lambda
A+B)+\left[\mathscr{U}_{2n}(\lambda A+B)\right]^{*}\right\\}Y(s)\Delta s$
$\displaystyle=$ $\displaystyle
2\int_{t_{0}}^{t}Y^{*}(s)W(s,\lambda)Y(s)\Delta s,$
where
$W(t,\lambda):=\operatorname{Re}\left[\mathscr{U}_{2n}(\lambda
A(t)+B(t))\right].$ (3.3)
Note that by using the notation $W(t,\lambda)$ in (3.3) we mimic the discrete
case [23, (3.1)], which uses $W_{k}(\lambda)$ for $k\in\mathbb{N}$; in the
continuous case [13, (3.5)] they use the notation $C_{\lambda}(x)$ for
$x\in\mathbb{R}$. For $M\in\mathbb{C}^{n,n}$ we take as its norm $\|M\|$ the
largest eigenvalue of $(M^{*}M)^{1/2}$, and define its real and imaginary
parts as
$\operatorname{Re}[M]=\frac{1}{2}(M+M^{*})\quad\text{and}\quad\operatorname{Im}[M]=\frac{1}{2i}(M-M^{*}).$
(3.4)
###### Definition 3.1.
Let $\lambda_{0}\in\mathbb{C}$, and let $\mathscr{U}_{2n}$ be given as above
in (3.1). Then $(\lambda_{0},\mathscr{U}_{2n})$ is called an admissible pair
for (1.4), and denoted $(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}$, if and
only if
$W(t,\lambda_{0})=\operatorname{Re}\left[\mathscr{U}_{2n}\big{(}\lambda_{0}A(t)+B(t)\big{)}\right]\geq
0,\quad t\in[t_{0},\infty)_{\mathbb{X}}.$ (3.5)
In this case, we define the set
$\Lambda(\lambda_{0},\mathscr{U}_{2n}):=\left\\{\lambda\in\mathbb{C}:\text{for
some}\;\delta>0,\operatorname{Re}[(\lambda-\lambda_{0})\mathscr{U}_{2n}A(t)]\geq\delta\mathscr{U}_{2n}A(t)\mathscr{U}_{2n}^{*}\;\forall\;t\in[t_{0},\infty)_{\mathbb{X}}\right\\}.$
(3.6)
Then the Weyl-Sims sets for (1.4) are defined for
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ and
$\widehat{Y}=\left(\widehat{\theta}|\widehat{\phi}\right)$ as
$D(t,\lambda):=\left\\{l\in\mathbb{C}^{n,n}:\left(\widehat{\theta}(t)+\widehat{\phi}(t)l\right)^{*}\mathscr{U}_{2n}J\left(\widehat{\theta}(t)+\widehat{\phi}(t)l\right)\leq
0\right\\}.$ (3.7)
Prior to proving the nesting property of the Weyl-Sims sets (see Theorem 3.3
below), we need the following. Since
$W(t,\lambda)\geq\delta\mathscr{U}_{2n}A(t)\mathscr{U}_{2n}^{*}\geq
0\quad\text{for}\quad
t\in[t_{0},\infty)_{\mathbb{X}}\quad\text{and}\quad\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n}),$
(3.8)
with $A\geq 0$, in addition to (3.5) we require the following definiteness
condition: for any $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ and
$\zeta\in\mathbb{C}^{n}$,
$W(t,\lambda)\phi(t)\zeta=0\quad\text{for all}\quad
t\in[t_{0},\infty)_{\mathbb{X}}\quad\implies\quad\zeta=0.$ (3.9)
Setting
$\widehat{\theta}(t)=\begin{pmatrix}\theta_{1}(t)\\\
\theta_{2}^{\rho}(t)\end{pmatrix}\quad\text{and}\quad\widehat{\phi}(t)=\begin{pmatrix}\phi_{1}(t)\\\
\phi_{2}^{\rho}(t)\end{pmatrix},$
where the blocks $\theta_{1}(t)$, $\theta_{2}(t)$, $\phi_{1}(t)$,
$\phi_{2}(t)$ are $\mathbb{C}^{n,n}-$valued matrices, we write
$\widehat{Y}^{*}(t)\mathscr{U}_{2n}J\widehat{Y}(t)=2\begin{pmatrix}S(t)&T(t)\\\
T^{*}(t)&P(t)\end{pmatrix},$ (3.10)
where
$\displaystyle S(t,\lambda)$ $\displaystyle=$
$\displaystyle-\operatorname{Re}\left[\theta_{1}^{*}(t)\mathscr{U}\theta_{2}^{\rho}(t)\right],$
$\displaystyle T(t,\lambda)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\theta_{1}^{*}(t)\mathscr{U}\phi_{2}^{\rho}(t)+\theta_{2}^{\rho*}(t)\mathscr{U}^{*}\phi_{1}(t)\right],$
$\displaystyle P(t,\lambda)$ $\displaystyle=$
$\displaystyle-\operatorname{Re}\left[\phi_{1}^{*}(t)\mathscr{U}\phi_{2}^{\rho}(t)\right].$
From the initial condition in (2.11) we see that $P(t_{0},\lambda)=0$. We will
show in Lemma 3.2 below that (3.9) implies $P(t,\lambda)>0$ for
$t\in\mathbb{X}$ sufficiently large. For $t\geq t_{1}=t_{1}(\lambda)$ given
below in Lemma 3.2, we employ the notation
$\mathscr{C}(t,\lambda):=-\left(P^{-1}T^{*}\right)(t,\lambda),\quad\mathscr{R}(t,\lambda):=\left(TP^{-1}T^{*}-S\right)(t,\lambda).$
(3.11)
The proof of the following lemma is the same, with slight modifications, as in
the discrete [23, Lemma 3.1] and continuous [13, Lemma 3.5] cases.
###### Lemma 3.2.
For $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ for some
$(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}$, there exists some
$t_{1}=t_{1}(\lambda)\in[t_{0},\infty)_{\mathbb{X}}$ such that
1. (i)
$P(t,\lambda)$ is non-decreasing in $t$, $P(t,\lambda)\geq 0$, and, for $t\geq
t_{1}$, $P(t,\lambda)>0$,
2. (ii)
$D(t,\lambda)\neq\emptyset$ for $t\geq t_{1}$,
3. (iii)
for $t\in[t_{1},\infty)_{\mathbb{X}}$, $\mathscr{R}(t,\lambda)$ is non-
increasing in $t$, and $\mathscr{R}(t,\lambda)>0$.
If we then multiply (3.10) on the left by $(I_{n},l^{*})$, and on the right by
$\left(\begin{smallmatrix}I_{n}\\\ l\end{smallmatrix}\right)$, we get the
expression
$\left(\widehat{\theta}+\widehat{\phi}\;l\right)^{*}(t)\mathscr{U}_{2n}J\left(\widehat{\theta}+\widehat{\phi}\;l\right)(t)=2\left[l^{*}Pl+Tl+l^{*}T^{*}+S\right](t),\quad
t\in[t_{0},\infty)_{\mathbb{X}}.$
It follows that
$D(t,\lambda)=\left\\{l\in\mathbb{C}^{n,n}:(l-\mathscr{C}(t,\lambda))^{*}P(t,\lambda)(l-\mathscr{C}(t,\lambda))\leq\mathscr{R}(t,\lambda)\right\\}.$
(3.12)
Again as in both the discrete and continuous cases (see [23, (3.7)] and [13,
Lemma 3.5(iii)]), we have that $\mathscr{R}(t,\lambda)\geq 0$ with
$\mathscr{R}(t,\lambda)>0$ for $t\geq t_{1}(\lambda)$, so that
$D(t,\lambda)=\left\\{l\in\mathbb{C}^{n,n}:l=\mathscr{C}(t,\lambda)+P^{-1/2}(t,\lambda)V\mathscr{R}^{1/2}(t,\lambda)\;\text{for
some}\;V\in\mathbb{C}^{n,n}\;\text{with}\;V^{*}V\leq I_{n}\right\\},$ (3.13)
where $V$ can be taken as
$V=P^{1/2}(t,\lambda)(l-\mathscr{C}(t,\lambda))\mathscr{R}^{-1/2}(t,\lambda)$.
###### Theorem 3.3.
For $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ for some
$(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}$, we have
1. (i)
$D(t,\lambda)\subseteq D(\tau,\lambda)$ for
$t,\tau\in[t_{0},\infty)_{\mathbb{X}}$ with $t>\tau$,
2. (ii)
$D(t,\lambda)-\mathscr{C}(t,\lambda)\subseteq
D(\tau,\lambda)-\mathscr{C}(\tau,\lambda)$ for
$t,\tau\in[t_{0},\infty)_{\mathbb{X}}$ with $t>\tau$,
3. (iii)
$D(t,\lambda)$ is compact and convex for
$t\in[t_{1}(\lambda),\infty)_{\mathbb{X}}$,
4. (iv)
$\mathscr{C}(\infty,\lambda):=\displaystyle\lim_{t\rightarrow\infty}\mathscr{C}(t,\lambda)$
exists,
5. (v)
the equality
$\bigcap_{t\in[t_{1},\infty)_{\mathbb{X}}}\left[D(t,\lambda)-\mathscr{C}(t,\lambda)\right]=D(\infty,\lambda)-\mathscr{C}(\infty,\lambda)$
(3.14)
holds, where
$D(\infty,\lambda):=\cap_{t\in[t_{1},\infty)_{\mathbb{X}}}D(t,\lambda)$,
6. (vi)
$\mathscr{C}(\infty,\lambda)\in D(\infty,\lambda)$.
###### Proof.
In (3.2), multiply on the left by $(I_{n}|l^{*})$ and on the right by
$\left(\begin{smallmatrix}I_{n}\\\ l\end{smallmatrix}\right)$ to obtain
$\displaystyle\left(\widehat{\theta}(t)+\widehat{\phi}(t)l\right)^{*}\mathscr{U}_{2n}J\left(\widehat{\theta}(t)+\widehat{\phi}(t)l\right)=\left(\widehat{\theta}(t_{0})+\widehat{\phi}(t_{0})l\right)^{*}\mathscr{U}_{2n}J\left(\widehat{\theta}(t_{0})+\widehat{\phi}(t_{0})l\right)$
$\displaystyle+2\displaystyle\int_{t_{0}}^{t}(\theta(s)+\phi(s)l)^{*}W(s,\lambda)(\theta(s)+\phi(s)l)\Delta
s.$
If $l\in D(t,\lambda)$, then
$\left(\widehat{\theta}(t_{0})+\widehat{\phi}(t_{0})l\right)^{*}\mathscr{U}_{2n}J\left(\widehat{\theta}(t_{0})+\widehat{\phi}(t_{0})l\right)+2\displaystyle\int_{t_{0}}^{t}(\theta(s)+\phi(s)l)^{*}W(s,\lambda)(\theta(s)+\phi(s)l)\Delta
s\leq 0;$ (3.15)
from (3.8) we have for $\tau\in[t_{0},t)_{\mathbb{X}}$ that
$\int_{t_{0}}^{\tau}(\theta(s)+\phi(s)l)^{*}W(s,\lambda)(\theta(s)+\phi(s)l)\Delta
s\leq\int_{t_{0}}^{t}(\theta(s)+\phi(s)l)^{*}W(s,\lambda)(\theta(s)+\phi(s)l)\Delta
s\leq 0,$
putting $l\in D(\tau,\lambda)$. Thus (i) holds. For the rest of the proof, see
[13, Theorem 3.6] and [23, Theorem 3.2] using the representation (3.13) of
$D(t,\lambda)$. ∎
By the nesting property above (Theorem 3.3(v)), there exists a limiting set
$D(\infty,\lambda)$ that may consist of a single point. If $l\in
D(\infty,\lambda)$, then from (3.15) we see for
$t\in[t_{0},\infty)_{\mathbb{X}}$ that
$\int_{t_{0}}^{t}(\theta(s)+\phi(s)l)^{*}W(s,\lambda)(\theta(s)+\phi(s)l)\Delta
s\leq-\frac{1}{2}\left(\widehat{\theta}(t_{0})+\widehat{\phi}(t_{0})l\right)^{*}\mathscr{U}_{2n}J\left(\widehat{\theta}(t_{0})+\widehat{\phi}(t_{0})l\right).$
Consequently the infinite integral satisfies
$\int_{t_{0}}^{\infty}(\theta(s)+\phi(s)l)^{*}W(s,\lambda)(\theta(s)+\phi(s)l)\Delta
s<\infty,$
ergo the function $\psi(\lambda):=\theta(\lambda)+\phi(\lambda)l$ is
$W(\lambda)-$square integrable. It then follows from (3.8) that
$\psi(\lambda)$ is $\widetilde{A}-$square integrable, where
$\widetilde{A}(t):=\mathscr{U}_{2n}A(t)\mathscr{U}_{2n}^{*}$.
###### Remark 3.4.
In the (3.12) representation of the set $D(t,\lambda)$ the matrix
$\mathscr{C}(t,\lambda)$ can be thought of playing the part of the center, and
$\mathscr{R}(t,\lambda)$ that of the radius of the Weyl circles.
Now consider the adjoint equation (2.7). Analogous to (3.2), the fundamental
system $\widehat{Z}$ satisfies
$\left(\widehat{Z}^{*}\mathscr{U}_{2n}^{-1}J\widehat{Z}\right)(t)-\left(\widehat{Z}^{*}\mathscr{U}_{2n}^{-1}J\widehat{Z}\right)(t_{0})=2\int_{t_{0}}^{t}Z^{*}(s)\widetilde{W}(s,\lambda)Z(s)\Delta
s,$ (3.16)
where
$\widetilde{W}(t,\lambda):=\operatorname{Re}\left[(\lambda
A(t)+B(t))\mathscr{U}_{2n}^{-1*}\right]=\operatorname{Re}\left[\mathscr{U}_{2n}^{-1}\left(\overline{\lambda}A(t)+B^{*}(t)\right)\right],$
and
$W(t,\lambda)=\mathscr{U}_{2n}\widetilde{W}(t,\lambda)\mathscr{U}_{2n}^{*}$
for $W$ given in (3.3).
###### Definition 3.5.
Let $\lambda_{0}\in\mathbb{C}$, and let $\mathscr{U}_{2n}$ be given as above
in (3.1). Then $(\lambda_{0},\mathscr{U}_{2n}^{-1})$ is called an admissible
pair for the adjoint equation (2.7) if and only if
$\widetilde{W}(t,\lambda)=\operatorname{Re}\left[(\lambda
A(t)+B(t))\mathscr{U}_{2n}^{-1*}\right]\geq 0,\quad
t\in[t_{0},\infty)_{\mathbb{X}},$
and define the set
$\displaystyle\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$
$\displaystyle:=$
$\displaystyle\left\\{\lambda\in\mathbb{C}:\;\exists\;\delta>0\;\ni\;\operatorname{Re}[(\lambda-\lambda_{0})A(t)\mathscr{U}_{2n}^{-1*}]\geq\delta\mathscr{U}_{2n}^{-1}A(t)\mathscr{U}_{2n}^{-1*}\;\forall\;t\in[t_{0},\infty)_{\mathbb{X}}\right\\}$
(3.17) $\displaystyle=$
$\displaystyle\left\\{\lambda\in\mathbb{C}:\;\exists\;\delta>0\;\ni\;\operatorname{Re}[(\lambda-\lambda_{0})\mathscr{U}_{2n}A(t)]\geq\delta
A(t)\;\forall\;t\in[t_{0},\infty)_{\mathbb{X}}\right\\}.$
Similar to the authors in [23, p. 89], we make the following remarks for the
sake of completeness.
###### Remark 3.6.
(i) Note that $(\lambda_{0},\mathscr{U}_{2n})$ is an admissible pair for (1.4)
if and only if $(\lambda_{0},\mathscr{U}_{2n}^{-1})$ is an admissible pair for
the adjoint equation (2.7). For
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})\cap\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$
and sufficiently large $t\in\mathbb{X}$, the Weyl-Sims sets for (2.7) are
$\widetilde{D}(t,\lambda)=\left\\{l^{*}\in\mathbb{C}^{n,n}:l\in
D(t,\lambda)\right\\}=D^{*}(t,\lambda);$
see the discussion in [13].
(ii) For $\lambda\in\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$ we
have from (3.16) that $l^{*}\in\widetilde{D}(t,\lambda)$ if and only if
$\int_{t_{0}}^{t}(\eta(s)+\chi(s)l^{*})^{*}\widetilde{W}(s,\lambda)(\eta(s)+\chi(s)l^{*})\Delta
s\leq-\frac{1}{2}\left(\widehat{\eta}(t_{0})+\widehat{\chi}(t_{0})l^{*}\right)^{*}\mathscr{U}_{2n}^{-1}J\left(\widehat{\eta}(t_{0})+\widehat{\chi}(t_{0})l^{*}\right).$
If $l^{*}\in\widetilde{D}(\infty,\lambda)$, then
$\zeta(\lambda)=\eta(\lambda)+\chi(\lambda)l^{*}$ satisfies
$\int_{t_{0}}^{\infty}\zeta^{*}(s)\widetilde{W}(s,\lambda)\zeta(s)\Delta
s<\infty,$
making $\zeta$ a $\widetilde{W}(\lambda)-$square integrable function.
(iii) If $W(t,\lambda)\geq\delta A(t)$ for all
$t\in[t_{0},\infty)_{\mathbb{X}}$, some $\delta>0$ and $\lambda\in\mathbb{C}$,
then $L_{W(\lambda)}^{2}(t_{0},\infty)_{\mathbb{X}}\subseteq
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$. This condition holds in the following
two cases:
* •
if $\lambda\in\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$, as seen
by (3.5) and (3.17), or
* •
if
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})\quad\text{and}\quad\widetilde{A}(t)\geq\gamma
A(t)\quad\text{for some}\quad\gamma>0$ (3.18)
by using (3.5) and (3.6).
(iv) If $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$, then
$\widetilde{W}(t,\lambda)\geq\delta A(t)$ for some $\delta>0$, ergo
$L_{\widetilde{W}(\lambda)}^{2}(t_{0},\infty)_{\mathbb{X}}\subseteq
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$. Consequently, if
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})\cap\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$,
then we have
$L_{W(\lambda)}^{2}(t_{0},\infty)_{\mathbb{X}}\cup
L_{\widetilde{W}(\lambda)}^{2}(t_{0},\infty)_{\mathbb{X}}\subseteq
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}},$
and $\psi$ and $\zeta$ are $A-$square integrable.
(v) Using (3.6) and (3.17), condition (3.18) above yields
$\Lambda(\lambda_{0},\mathscr{U}_{2n})\subseteq\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$
(3.19)
for admissible $(\lambda_{0},\mathscr{U}_{2n})$. Besides (3.18), if the
reverse inequality $A(t)\geq\tilde{\gamma}\widetilde{A}(t)$ holds for some
$\tilde{\gamma}>0$, that is to say if $\widetilde{A}(t)\asymp A(t)$, then we
have equality in (3.19).
(vi) As in [13] and [23], the structure of the shifted limit set
$D(\infty,\lambda)-\mathscr{C}(\infty,\lambda)$ gives information about the
number of $W(\lambda)-$square integrable solutions to system (1.4). To be more
explicit, let $\mathscr{N}\mathcal{N}(\lambda):=\bigcup_{N\in
D(\infty,\lambda)}\operatorname{range}(N-\mathscr{C}(\infty,\lambda))$ and $r$
be the dimension of the linear hull of $\mathcal{N}(\lambda)$. Then there are
at least $n+r$ linearly independent $W(\lambda)-$square integrable solutions
of (1.4), and if $\mathscr{R}(t,\lambda)\not\longrightarrow 0$ as
$t\rightarrow\infty$ in the time scale, the number is exactly $n+r$.
In addition, if
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})\cap\widetilde{\Lambda}(\lambda_{0},\mathscr{U}_{2n}^{-1})$,
then (2.7) has precisely $n$ linearly independent
$\widetilde{W}(\lambda)-$square integrable solutions if
$\mathscr{R}(t,\lambda)\rightarrow 0$ as $t\rightarrow\infty$ in the time
scale. If $r=0$, then we also have $\tilde{r}=0$, where $\tilde{r}$ is the
corresponding number for the adjoint equation, and at least one of (1.4),
(2.7) has exactly $n$ linearly independent solutions that are $W(\lambda)$,
$\widetilde{W}(\lambda)-$square integrable, respectively.
## 4\. Examples
The examples given in the continuous case [13, Section 3] require $A$ defined
in (1.5) to be positive semi-definite. In the discrete case [23], however, the
blanket positive definite assumption $A>0$ rules out giving the corresponding
examples for difference equations. Thus we are able in this section to extend
the examples given in [13, Section 3] to difference equations and other cases
on Sturmian time scales, and mention a new result for general even-order
dynamic equations.
###### Example 4.1.
For $n=1$, set
$\mathscr{U}_{2}=\begin{pmatrix}-u&0\\\ 0&\bar{u}\end{pmatrix}$ (4.1)
for some nonzero $u\in\mathbb{C}$. Then (3.10) yields
$2\begin{pmatrix}S(t)&T(t)\\\
T^{*}(t)&P(t)\end{pmatrix}=\begin{pmatrix}\theta_{1}(t)&\phi_{1}(t)\\\
\theta_{2}^{\rho}(t)&\phi_{2}^{\rho}(t)\end{pmatrix}^{*}\begin{pmatrix}0&u(t)\\\
\bar{u}(t)&0\end{pmatrix}\begin{pmatrix}\theta_{1}(t)&\phi_{1}(t)\\\
\theta_{2}^{\rho}(t)&\phi_{2}^{\rho}(t)\end{pmatrix},$
so that when $P=P(t,\lambda)>0$ we have, also using (3.11),
$P=\operatorname{Re}\left(u\overline{\phi}_{1}\phi_{2}^{\rho}\right),\quad\mathscr{R}=\frac{1}{4P}|u|^{2}|\Gamma|^{2},\quad\mathscr{C}=\frac{-1}{2P}\left(u\overline{\phi}_{1}\theta_{2}^{\rho}+\overline{u}\theta_{1}\overline{\phi}_{2}^{\rho}\right),$
where $\Gamma=\theta_{1}\phi_{2}^{\rho}-\theta_{2}^{\rho}\phi_{1}$. Thus
(3.12) is
$D(t,\lambda)=\left\\{l\in\mathbb{C}:|l-\mathscr{C}(t,\lambda)|\leq\frac{|u||\Gamma(t,\lambda)|}{2P(t,\lambda)}\right\\}.$
As a special case of this, consider on Sturmian time scales the second-order
scalar Sturm-Liouville problem
$-(pv^{\nabla})^{\Delta}(t)+q(t)v(t)=\lambda w(t)v(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}},$ (4.2)
where $p$ and $q$ are complex-valued functions, with $p^{-1},q,w\in
L^{1}_{\operatorname{loc}}[t_{0},\infty)_{\mathbb{X}}$ such that $p\neq 0$ and
$w>0$ on $[t_{0},\infty)_{\mathbb{X}}$. Then (4.2) can be written in the form
(1.4)
$y=\begin{pmatrix}v\\\
p^{\sigma}v^{\Delta}\end{pmatrix},\quad\widehat{y}=\begin{pmatrix}v\\\
pv^{\nabla}\end{pmatrix},\quad A=\begin{pmatrix}w&0\\\ 0&0\end{pmatrix},\quad
B=\begin{pmatrix}-q&0\\\ 0&1/p^{\sigma}\end{pmatrix};$
note that $A$ and $B$ satisfy (1.5) and (1.6), respectively. If we choose
$u=e^{i\eta}$ for some $\eta\in\mathbb{R}$ in (4.2), we see from (3.3) that
$W(t,\lambda)=\operatorname{Re}\left[\mathscr{U}_{2}(\lambda
A(t)+B(t))\right]=\begin{pmatrix}\operatorname{Re}[e^{i\eta}(q-\lambda
w)(t)]&0\\\
0&\displaystyle\frac{1}{|p^{\sigma}(t)|^{2}}\operatorname{Re}[e^{i\eta}p^{\sigma}(t)]\end{pmatrix}.$
(4.3)
Thus for this choice of $\mathscr{U}_{2}$ we have
$(\lambda_{0},\mathscr{U}_{2})\in\mathscr{S}\iff\operatorname{Re}[e^{i\eta}(q-\lambda
w)(t)],\;\displaystyle\frac{1}{|p^{\sigma}(t)|^{2}}\operatorname{Re}[e^{i\eta}p^{\sigma}(t)]\geq
0\quad\forall t\in[t_{0},\infty)_{\mathbb{X}}.$
In addition we see that for $(\lambda_{0},\eta)$ such that
$(\lambda_{0},\mathscr{U}_{2})\in\mathscr{S}$,
$\Lambda(\lambda_{0},\mathscr{U}_{2})=\left\\{\lambda\in\mathbb{C}:\operatorname{Re}[(\lambda-\lambda_{0})e^{i\eta}]<0\right\\},$
just as in [13, (3.25)]. It is easy to show that the definiteness condition
(3.9) holds, see [13, p. 425].
###### Example 4.2.
Consider the fourth-order scalar problem [7, (7.1)]
$(p_{2}v^{\Delta\nabla})^{\nabla\Delta}(t)-(p_{1}v^{\nabla})^{\Delta}(t)+p_{0}(t)v(t)=\lambda
w(t)v(t),\quad t\in[t_{0},\infty)_{\mathbb{X}},$ (4.4)
where $p_{0}$, $p_{1}$, and $p_{2}$ are complex-valued functions with
$p_{2}\neq 0$ on $[t_{0},\infty)_{\mathbb{X}}$, and $w\in
L^{2}_{\operatorname{loc}}[t_{0},\infty)_{\mathbb{X}}$ satisfies $w>0$ on
$[t_{0},\infty)_{\mathbb{X}}$. For the quasi-derivatives given by
$v^{[1]}=v^{\Delta},\quad v^{[2]}=p_{2}^{\sigma}v^{\Delta\Delta},\quad
v^{[3]}=p_{1}^{\sigma}v^{\Delta}-(v^{[2]\rho})^{\Delta},$
we introduce the vector
$y=\left(v,v^{[1]},v^{[3]},v^{[2]}\right)^{\operatorname{T}},\quad\text{with}\quad\widehat{y}=\left(v,v^{[1]},v^{[3]\rho},v^{[2]\rho}\right)^{\operatorname{T}}.$
Then (4.4) can be written in the form of (1.4) if we take
$A=\operatorname{diag}\\{w,0,0,0\\}\quad\text{and}\quad
B=\begin{pmatrix}-p_{0}&&&\\\ &-p_{1}^{\sigma}&1&\\\ &1&0&\\\
&&&1/p_{2}^{\sigma}\end{pmatrix},$
with the unstated entries being zero. Note that $A$ and $B$ satisfy (1.5) and
(1.6), respectively. For some $\eta\in\mathbb{R}$ choose
$\mathscr{U}_{4}:=\begin{pmatrix}-e^{i\eta}I_{2}&0_{2}\\\
0_{2}&e^{-i\eta}I_{2}\end{pmatrix};$ (4.5)
it follows from (3.3) and (3.4) that
$W(t,\lambda)=\operatorname{diag}\left\\{\operatorname{Re}\left[e^{i\eta}(p_{0}(t)-\lambda
w(t))\right],\operatorname{Re}\left[e^{i\eta}p_{1}^{\sigma}(t)\right],0,\operatorname{Re}\left[e^{i\eta}p_{2}^{\sigma}(t)\right]/|p_{2}^{\sigma}(t)|^{2}\right\\},$
so that for this choice of $\mathscr{U}_{4}$ we have
$(\lambda_{0},\mathscr{U}_{4})\in\mathscr{S}\iff\operatorname{Re}\left[e^{i\eta}(p_{0}(t)-\lambda_{0}w(t))\right],\operatorname{Re}\left[e^{i\eta}p_{1}^{\sigma}(t)\right],\operatorname{Re}\left[e^{i\eta}p_{2}^{\sigma}(t)\right]\geq
0\quad\forall t\in[t_{0},\infty)_{\mathbb{X}}.$
In addition we see that for $\lambda_{0}$ such that
$(\lambda_{0},\mathscr{U}_{4})\in\mathscr{S}$,
$\Lambda(\lambda_{0},\mathscr{U}_{4})=\left\\{\lambda\in\mathbb{C}:\operatorname{Re}\left[(\lambda-\lambda_{0})e^{i\eta}\right]<0\right\\}.$
(4.6)
As in the previous example, the definiteness condition (3.9) holds.
###### Example 4.3.
Using Example 4.2 as a guide, consider on $[t_{0},\infty)_{\mathbb{X}}$ the
formally self-adjoint $2n$th-order dynamic equation [7] of the form
(suppressing the independent variable)
$(-1)^{n}(p_{n}v^{\Delta^{n-1}\nabla})^{\nabla^{n-1}\Delta}+\dots-(p_{3}v^{\Delta^{2}\nabla})^{\nabla^{2}\Delta}+(p_{2}v^{\Delta\nabla})^{\nabla\Delta}-(p_{1}v^{\nabla})^{\Delta}+p_{0}v=\lambda
wv,$ (4.7)
where $p_{j}$ is a complex-valued function for $j=0,1,\ldots,n-1$ with
$p_{n}\neq 0$ on $[t_{0},\infty)_{\mathbb{X}}$, and $w\in
L^{2}_{\operatorname{loc}}[t_{0},\infty)_{\mathbb{X}}$ satisfies $w>0$ on
$[t_{0},\infty)_{\mathbb{X}}$. Let $A=\operatorname{diag}\\{w,0,\ldots,0\\}$,
and let $B$ be as in (1.5), where we take
$B_{1}=\operatorname{diag}\\{-p_{0},-p_{1}^{\sigma},-p_{2}^{\sigma},\ldots,-p_{n-1}^{\sigma}\\},\quad
B_{2}=\operatorname{subdiag}\\{1,1,\ldots,1\\},$
$B_{3}=\operatorname{superdiag}\\{1,1,\ldots,1\\}=B_{2}^{\operatorname{T}},\quad
B_{4}=\operatorname{diag}\\{0,\ldots,0,1/p_{n}^{\sigma}\\}.$
Here $\operatorname{subdiag}$ means the matrix with all zero entries except on
the subdiagonal; similarly $\operatorname{superdiag}$ has nonzero entries only
on the superdiagonal. Clearly the conditions in (1.5) and (1.6) are satisfied.
Set
$y=\left(v,v^{[1]},v^{[2]},\ldots,v^{[n-1]},v^{[2n-1]},v^{[2n-2]},\ldots,v^{[n]}\right)^{\operatorname{T}},$
where the quasi-derivatives are given by
$\displaystyle v^{[k]}$ $\displaystyle=$ $\displaystyle v^{\Delta^{k}},\quad
1\leq k\leq n-1,$ $\displaystyle v^{[n]}$ $\displaystyle=$ $\displaystyle
p_{n}^{\sigma}v^{\Delta^{n}},$ $\displaystyle v^{[n+k]}$ $\displaystyle=$
$\displaystyle
p_{n-k}^{\sigma}v^{[n-k]}-\left(v^{[n+k-1]\rho}\right)^{\Delta},\quad 1\leq
k\leq n-1.$
Then (4.7) can be written in the form of (1.4). For some $\eta\in\mathbb{R}$
choose
$\mathscr{U}_{2n}:=\begin{pmatrix}-e^{i\eta}I_{n}&0_{n}\\\
0_{n}&e^{-i\eta}I_{n}\end{pmatrix}.$
It follows from (3.3) and (3.4) that
$\displaystyle
W(t,\lambda)=\operatorname{diag}\left\\{\operatorname{Re}\left[e^{i\eta}(p_{0}(t)-\lambda
w(t))\right],\operatorname{Re}\left[e^{i\eta}p_{1}^{\sigma}(t)\right],\operatorname{Re}\left[e^{i\eta}p_{2}^{\sigma}(t)\right],\ldots,\operatorname{Re}\left[e^{i\eta}p_{n-1}^{\sigma}(t)\right],\right.$
$\displaystyle\left.0,\ldots,0,\operatorname{Re}\left[e^{i\eta}p_{n}^{\sigma}(t)\right]/|p_{n}^{\sigma}(t)|^{2}\right\\},$
so that for this choice of $\mathscr{U}_{2n}$ we have
$(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}\iff\operatorname{Re}\left[e^{i\eta}(p_{0}(t)-\lambda_{0}w(t))\right],\operatorname{Re}\left[e^{i\eta}p_{1}^{\sigma}(t)\right],\ldots,\operatorname{Re}\left[e^{i\eta}p_{n}^{\sigma}(t)\right]\geq
0$
for all $t\in[t_{0},\infty)_{\mathbb{X}}$. In addition we see that for
$\lambda_{0}$ such that $(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}$,
$\Lambda(\lambda_{0},\mathscr{U}_{2n})=\left\\{\lambda\in\mathbb{C}:\operatorname{Re}\left[(\lambda-\lambda_{0})e^{i\eta}\right]<0\right\\}.$
As in the previous examples, the definiteness condition (3.9) holds.
This example is completely new, as general even-order equations are not
mentioned in either [13] or [23]; clearly the special cases
$\mathbb{X}=\mathbb{R}$ and $\mathbb{X}=\mathbb{Z}$ are included here.
###### Example 4.4.
Consider the Orr-Sommerfeld equation on time scales given by
$(-D^{2}+a^{2})^{2}u+iaR\left[V(-D^{2}+a^{2})u+uD^{2}V\right]=\lambda(-D^{2}+a^{2})u,\quad
D^{2}u\equiv u^{\nabla\Delta}$ (4.8)
on some interval $I\subseteq[t_{0},\infty)_{\mathbb{X}}$, where $a>0$ is the
wave number, $R>0$ is the Reynolds number, and $V$ is a real-valued flow
velocity profile perpendicular to $I$; see [13, Example 3.4] and Orszag [24].
If we introduce the variables
$y_{1}=-u^{\nabla\Delta}+a^{2}u,\quad y_{2}=u,\quad
y_{3}=(-u^{\nabla\Delta}+a^{2}u)^{\Delta},\quad y_{4}=u^{\Delta},$
then
$\widehat{y}(t)=\left(-u^{\nabla\Delta}+a^{2}u,u,(-u^{\nabla\Delta}+a^{2}u)^{\nabla},u^{\nabla}\right)^{\operatorname{T}}(t),$
and we see that (4.8) is equivalent to the Hamiltonian system (1.4) with
$A=\operatorname{diag}\\{1,0,0,0\\}\quad\text{and}\quad
B=\begin{pmatrix}-a^{2}-iaRV&-iaRV^{\nabla\Delta}&0&0\\\ 1&-a^{2}&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{pmatrix}$
such that (1.5) and (1.6) are easily satisfied. We will choose the same matrix
$\mathscr{U}_{4}$ as in (4.5) for some $\eta\in\mathbb{R}$; then by (3.3) we
have
$W(t,\lambda)=\begin{pmatrix}a^{2}\cos\eta-
aRV\sin\eta-\operatorname{Re}[\lambda
e^{i\eta}]&\frac{1}{2}(aRV^{\nabla\Delta}ie^{i\eta}-e^{-i\eta})&0&0\\\
-\frac{1}{2}(aRV^{\nabla\Delta}ie^{-i\eta}+e^{i\eta})&a^{2}\cos\eta&0&0\\\
0&0&\cos\eta&0\\\ 0&0&0&\cos\eta\end{pmatrix}.$
As in the case $\mathbb{X}=\mathbb{R}$ [13], we thus have
$W(t,\lambda)\geq 0\iff\begin{cases}\cos\eta>0&\text{and}\\\
\operatorname{Re}(\lambda e^{i\eta})\leq&a^{2}\cos\eta-aRV(t)\sin\eta\\\
&-\displaystyle\frac{1+\left(aRV^{\nabla\Delta}(t)\right)^{2}+2aRV^{\nabla\Delta}(t)\sin(2\eta)}{4a^{2}\cos\eta},\end{cases}$
(4.9)
so that $W(t,\lambda)>0$ if the last inequality in (4.9) is strict.
Consequently, $(\lambda_{0},\mathscr{U}_{4})\in\mathscr{S}$ if and only if the
right-hand side of (4.9) holds for $\lambda_{0}$ in place of $\lambda$, for
all $t\in I$. We can also show that for such an $\eta$ and $\lambda_{0}$ we
get that $\Lambda(\lambda_{0},\mathscr{U}_{4})$ is again as in (4.6).
Similarly, the definiteness condition (3.9) holds.
## 5\. Definition of the operators $L_{\xi}$ and $\widetilde{L}_{\xi}$
We will show below that for fixed
$\xi\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ and $M_{0}\in D(\infty,\xi)$
there exists for $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ a matrix-
valued function $M(\lambda)$ such that $M(\xi)=M_{0}$. In addition, the Weyl
solutions satisfy condition (5.3) at infinity. This in turn will allow us to
introduce the operator $L_{\xi}$ associated with (1.4) and the operator
$\widetilde{L}_{\xi}$ associated with the adjoint system (2.7). Throughout the
discussion we will assume that
$\widetilde{A}(t)=\mathscr{U}_{2n}A(t)\mathscr{U}_{2n}^{*}\asymp A(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}}.$ (5.1)
Indeed this holds if, for example,
$A_{k}(t)=a_{k}(t)I_{n}+\widetilde{A}_{k}(t)$ for arbitrary $a_{k}(t)$ and
$1/c\leq\widetilde{A}_{k}(t)\leq c$ for some real constant $c>1$, for all
$t\in[t_{0},\infty)_{\mathbb{X}}$ and $k=1,2$.
###### Theorem 5.1.
Let $\xi\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ for
$(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}$, and $M_{0}\in D(\infty,\xi)$
be fixed. Then there exists a function
$M:\Lambda(\lambda_{0},\mathscr{U}_{2n})\rightarrow\mathbb{C}^{n,n}$ such that
$M(\xi)=M_{0}$, $M(\lambda)\in D(\infty,\lambda)$ and
$M(\lambda)-M_{0}=(\lambda-\xi)\int_{t_{0}}^{\infty}\zeta^{*}(t,\xi)A(t)\psi(t,\lambda)\Delta
t=(\lambda-\xi)\int_{t_{0}}^{\infty}\zeta^{*}(t,\lambda)A(t)\psi(t,\xi)\Delta
t$ (5.2)
for $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$, where
$\psi(t,\lambda):=\theta(t,\lambda)+\phi(t,\lambda)M(\lambda)$ and
$\zeta(t,\lambda):=\eta(t,\lambda)+\chi(t,\lambda)M(\lambda)^{*}$. In
addition,
$\lim_{t\rightarrow\infty}\widehat{\zeta}^{*}(t,\xi)J\widehat{\psi}(t,\lambda)=\lim_{t\rightarrow\infty}\widehat{\zeta}^{*}(t,\lambda)J\widehat{\psi}(t,\xi)=0.$
(5.3)
###### Proof.
See [13, Theorem 5.4] and [23, Theorem 4.1], as the proof is unchanged from
the continuous and discrete cases except for minor notational modifications. ∎
###### Remark 5.2.
Clearly, $M(\lambda)$ has only one possible value if $D(\infty,\lambda)$ has
only one element. Consequently, if there is at least one point
$\xi\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ such that $D(\infty,\lambda)$
has only one element, the function $M$ is uniquely determined.
From this point on we assume that
$(\lambda_{0},\mathscr{U}_{2n})\in\mathscr{S}$ is fixed, and that the function
$M$ and the selected value $\xi$ are as in Theorem 5.1 above. Corresponding to
system (1.4) is the inhomogeneous system
$J\widehat{y}^{\Delta}(t)=\Big{(}\lambda A(t)+B(t)\Big{)}y(t)+A(t)f(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}},$ (5.4)
or in its expanded form (suppressing the $t$),
$\begin{cases}-&(y_{2}^{\rho})^{\Delta}=\big{(}\lambda
A_{1}+B_{1}\big{)}y_{1}+B_{2}y_{2}+A_{1}f_{1},\\\
&y_{1}^{\Delta}=B_{3}y_{1}+(\lambda A_{2}+B_{4})y_{2}+A_{2}f_{2}.\end{cases}$
Note that if $\mu(t)\neq 0$, then
$y_{2}(t)=\left(I+\mu(t)B_{2}(t)\right)^{-1}\left[y_{2}^{\rho}(t)-\mu(t)(\lambda
A_{1}(t)+B_{1}(t))y_{1}(t)-\mu(t)A_{1}(t)f_{1}(t)\right].$
A solution $y$ of system (5.4) is thus related to $\widehat{y}$ (see (1.3))
using
$y(t)=H(t)\widehat{y}(t)+N(t)A(t)f(t),\quad t\in[t_{0},\infty)_{\mathbb{X}},$
(5.5)
where $H$ is given in (2.4) and
$N(t):=\begin{pmatrix}0_{n}&0_{n}\\\
-\mu(t)E_{2}(t)&0_{n}\end{pmatrix}=\begin{pmatrix}0_{n}&0_{n}\\\
-\mu(t)\Big{(}I_{n}+\mu(t)B_{2}(t)\Big{)}^{-1}&0_{n}\end{pmatrix}.$ (5.6)
As $(I_{n}-\mu B_{2}E_{2})=E_{2}$ we have $(\lambda
A+B)N+I_{2n}=\widetilde{H}^{*}$ from (2.9), so that (5.4) is equivalent to
$J\widehat{y}^{\Delta}(t)=\Big{(}\lambda
A(t)+B(t)\Big{)}H(t)\widehat{y}(t)+\widetilde{H}^{*}(t)A(t)f(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}}.$ (5.7)
In a similar way, for the adjoint problem to (5.4), namely
$J\widehat{z}^{\Delta}(t)=\Big{(}\overline{\lambda}A(t)+B^{*}(t)\Big{)}z(t)+A(t)f(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}},$ (5.8)
we have $z_{1}=E_{2}^{*}z_{1}^{\sigma}-\mu
E_{2}^{*}(\overline{\lambda}A_{2}+B^{*}_{4})z_{2}-\mu E_{2}^{*}A_{2}f_{2}$ and
thus the relation
$z(t)=\widetilde{H}(t)\widehat{z}^{\;\sigma}(t)+N^{*}(t)A(t)f(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}}.$ (5.9)
Moreover, since $(\overline{\lambda}A+B^{*})N^{*}+I_{2n}=H^{*}$ for $H$ in
(2.4), we see that (5.8) is equivalent to
$J\widehat{z}^{\;\Delta}(t)=H^{*}(t)\Big{(}\overline{\lambda}A(t)+B^{*}(t)\Big{)}\widehat{z}^{\;\sigma}(t)+H^{*}(t)A(t)f(t)$
as in (2.10), for $t\in[t_{0},\infty)_{\mathbb{X}}$. Define
$\displaystyle G(t,s,\lambda)$
$\displaystyle:=\begin{cases}\psi(t,\lambda)\chi^{*}(s,\lambda)&:t_{0}\leq
s<t<\infty,\\\ \phi(t,\lambda)\zeta^{*}(s,\lambda)+N(t)\delta_{ts}&:t_{0}\leq
t\leq s<\infty,\end{cases}$ $\displaystyle\widetilde{G}(t,s,\lambda)$
$\displaystyle:=\begin{cases}\chi(t,\lambda)\psi^{*}(s,\lambda)&:t_{0}\leq
t<s<\infty,\\\
\zeta(t,\lambda)\phi^{*}(s,\lambda)+N^{*}(t)\delta_{ts}&:t_{0}\leq s\leq
t<\infty,\end{cases}$ $\displaystyle=G^{*}(s,t,\lambda),$
where $\delta_{ts}$ is the Dirac delta function, i.e., the function that
satisfies the sifting property
$\int_{t_{0}}^{\infty}f(s)\delta_{ts}\Delta s=f(t).$
In a subsequent lemma we prove that $G$ and $\widetilde{G}$ are Green’s
matrices for (5.4) and (5.8), respectively. For $f\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$, define
$\displaystyle(R_{\lambda}f)(t):=\int_{t_{0}}^{\infty}G(t,s,\lambda)A(s)f(s)\Delta
s,$ (5.10)
$\displaystyle\left(\widetilde{R}_{\lambda}f\right)(t):=\int_{t_{0}}^{\infty}\widetilde{G}(t,s,\lambda)A(s)f(s)\Delta
s.$ (5.11)
We assume that $R_{\lambda}$ and $\widetilde{R}_{\lambda}$ defined in (5.10)
and (5.11) are one-to-one as operators from
$L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$ into itself, in other words,
$\displaystyle f\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}},\;A(R_{\xi}f)(t)=0\;\forall
t\in(t_{0},\infty)_{\mathbb{X}}\implies(Af)(t)=0\;\forall
t\in(t_{0},\infty)_{\mathbb{X}},$ (5.12) $\displaystyle g\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}},\;A(\widetilde{R}_{\xi}g)(t)=0\;\forall
t\in(t_{0},\infty)_{\mathbb{X}}\implies(Ag)(t)=0\;\forall
t\in(t_{0},\infty)_{\mathbb{X}}.$ (5.13)
As pointed out in [13, p. 444], it will become evident in the sequel that
these same conditions hold when $\xi$ is replaced by any
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$. We now define the following
operators $L_{\xi}$ and $\widetilde{L}_{\xi}$ in a natural way. Set
$\displaystyle\mathscr{D}(L_{\xi})$ $\displaystyle:=$
$\displaystyle\left\\{y\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}:y\in\operatorname{AC_{loc}}[t_{0},\infty)_{\mathbb{X}},\right.$
(5.15)
$\displaystyle(ly)(t):=\begin{cases}J\widehat{y}^{\Delta}(t)-B(t)y(t)=A(t)f(t)&:t\in[\sigma(t_{0}),\infty)_{\mathbb{X}},\\\
\left(\begin{smallmatrix}-y_{2}(t_{0})/\mu(t_{0})\\\
y_{1}^{\Delta}(t_{0})\end{smallmatrix}\right)-B(t_{0})y(t_{0})=A(t_{0})f(t_{0})&:t=t_{0}>\rho(t_{0}),\end{cases}$
$\displaystyle\left.\text{for some}\;f\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}},\;\text{ and
}\;\lim_{t\rightarrow\infty}\widehat{\zeta}^{*}(t,\xi)J\widehat{y}(t)=0\right\\},$
$\displaystyle L_{\xi}y$ $\displaystyle:=$ $\displaystyle
f\;\;\text{for}\;y\in\mathscr{D}(L_{\xi})\;\text{satisfying}\;ly=Af\;\text{on}\;[t_{0},\infty)_{\mathbb{X}},$
(5.16)
where the second line in the definition of $(ly)$ holds only if $t_{0}$ is a
left-scattered (and thus right-scattered) point, and if we take
$y_{2}^{\rho}(t_{0})=0$. Similarly, set
$\displaystyle\mathscr{D}\left(\widetilde{L}_{\xi}\right)$ $\displaystyle:=$
$\displaystyle\left\\{z\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}:z\in\operatorname{AC_{loc}}[t_{0},\infty)_{\mathbb{X}},\right.$
(5.17)
$\displaystyle(\widetilde{l}z)(t):=\begin{cases}J\widehat{z}^{\Delta}(t)-B^{*}(t)z(t)=A(t)g(t)&:t\in[\sigma(t_{0}),\infty)_{\mathbb{X}},\\\
\left(\begin{smallmatrix}-z_{2}(t_{0})/\mu(t_{0})\\\
z_{1}^{\Delta}(t_{0})\end{smallmatrix}\right)-B^{*}(t_{0})z(t_{0})=A(t_{0})g(t_{0})&:t=t_{0}>\rho(t_{0}),\end{cases}$
$\displaystyle\left.\text{for some}\;g\in
L_{A}^{2}(t_{0},\infty)_{\mathbb{X}},\;\text{ and
}\;\lim_{t\rightarrow\infty}\widehat{\psi}^{*}(t,\xi)J\widehat{z}(t)=0\right\\},$
$\displaystyle\widetilde{L}_{\xi}z$ $\displaystyle:=$ $\displaystyle
g\;\;\text{for}\;z\in\mathscr{D}\left(\widetilde{L}_{\xi}\right)\;\text{satisfying}\;\widetilde{l}z=Ag\;\text{on}\;[t_{0},\infty)_{\mathbb{X}},$
(5.18)
where the second line in the definition of $(\widetilde{l}y)$ holds only if
$t_{0}$ is a left-scattered point, and if we take $z_{2}^{\rho}(t_{0})=0$. We
remark that $\mathscr{D}(L_{\xi})$ and
$\mathscr{D}\left(\widetilde{L}_{\xi}\right)$ consist of all equivalence
classes in $L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$ such that at least one
representative of the class satisfies the conditions in the definitions of
$\mathscr{D}(L_{\xi})$ and $\mathscr{D}\left(\widetilde{L}_{\xi}\right)$,
respectively. In particular, using (5.12) and (5.13) we see that this
representative is always unique, and thus $L_{\xi}$ and $\widetilde{L}_{\xi}$
are well defined. For more details, see the discussion in [13, p. 445].
## 6\. The Resolvent Sets
In this section we analyze the operators $L_{\xi}$ and $\widetilde{L}_{\xi}$
defined in the previous section in (5.16) and (5.18), respectively, and
establish their resolvents, which turn out to be $R_{\lambda}$ and
$\widetilde{R}_{\lambda}$ from (5.10) and (5.11), respectively. This operator
$R_{\lambda}$ will have inverse operator properties relative to $L_{\xi}-\xi$,
in particular that $L_{\xi}$ has resolvent set
$\Lambda(\lambda_{0},\mathscr{U}_{2n})$ with resolvent operator $R_{\lambda}$.
In addition, we prove that $L_{\xi}$ and $\widetilde{L}_{\xi}$ are adjoints.
###### Lemma 6.1.
Let
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})\cap\widetilde{\Lambda}(\lambda_{0},\mathscr{U}^{-1}_{2n})$
and $f\in L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$. Then $R_{\lambda}f$ is a
solution of (5.4) and satisfies $(R_{\lambda}f)_{2}^{\rho}(t_{0})=0_{n}$. In
particular, it satisfies the boundary conditions
$\widehat{\chi}^{\;*}(t_{0})J\left(\widehat{R}_{\lambda}f\right)(t_{0})=0$ and
$\lim_{t\rightarrow\infty}\widehat{\zeta}^{\;*}(t,\lambda)J\left(\widehat{R}_{\lambda}f\right)(t)=0$,
where
$\left(\widehat{R}_{\lambda}f\right)(t)=\widehat{\psi}(t,\lambda)\int_{t_{0}}^{t}\chi^{*}(s,\lambda)A(s)f(s)\Delta
s+\widehat{\phi}(t,\lambda)\int_{t}^{\infty}\zeta^{*}(s,\lambda)A(s)f(s)\Delta
s,\quad t\in[t_{0},\infty)_{\mathbb{X}}.$ (6.1)
###### Proof.
We use the method of variation of parameters. Let
$\widehat{U}\in\mathbb{C}^{2n,2n}$ have the form
$\widehat{U}=\widehat{Y}\mathscr{M}$, where $\mathscr{M}\in\mathbb{C}^{2n,2n}$
is to be determined and $Y=(\theta|\phi)$ is a fundamental matrix solution for
the corresponding homogeneous system (2.6) satisfying initial condition
(2.11). By the delta product rule,
$\widehat{U}^{\Delta}(t)=\widehat{Y}^{\sigma}(t)\mathscr{M}^{\Delta}(t)+\widehat{Y}^{\Delta}(t)\mathscr{M}(t).$
Multiply by $J$, use (2.6), and assume $\widehat{U}$ satisfies (5.7) to obtain
$J\widehat{Y}^{\sigma}(t)\mathscr{M}^{\Delta}(t)=\widetilde{H}^{*}(t)A(t)f(t).$
Then by (2.8) and Lemma 2.5 we have
$\displaystyle\mathscr{M}^{\Delta}(t)$ $\displaystyle=$
$\displaystyle-(\widehat{Y}^{\sigma})^{-1}(t)J\widetilde{H}^{*}(t)A(t)f(t)$
$\displaystyle=$
$\displaystyle-J\widehat{Z}^{\sigma*}(t)\widetilde{H}^{*}(t)A(t)f(t)$
$\displaystyle=$ $\displaystyle-JZ^{*}(t)A(t)f(t)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0_{n}&I_{n}\\\
0_{n}&M(\lambda)\end{pmatrix}Z^{*}(t)A(t)f(t)-\begin{pmatrix}0_{n}&0_{n}\\\
I_{n}&M(\lambda)\end{pmatrix}Z^{*}(t)A(t)f(t).$
Consequently we have, up to addition of a constant matrix,
$\displaystyle\mathscr{M}(t)$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\left(\begin{smallmatrix}0_{n}&I_{n}\\\
0_{n}&M(\lambda)\end{smallmatrix}\right)Z^{*}(s)A(s)f(s)\Delta
s+\int_{t}^{\infty}\left(\begin{smallmatrix}0_{n}&0_{n}\\\
I_{n}&M(\lambda)\end{smallmatrix}\right)Z^{*}(s)A(s)f(s)\Delta s$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\left(\begin{smallmatrix}I_{n}\\\
M(\lambda)\end{smallmatrix}\right)\chi^{*}(s)A(s)f(s)\Delta
s+\int_{t}^{\infty}\left(\begin{smallmatrix}0_{n}\\\
\eta^{*}(s)+M(\lambda)\chi^{*}(s)\end{smallmatrix}\right)A(s)f(s)\Delta s,$
since $Z=(\eta|\chi)$, so that
$\displaystyle\widehat{U}(t)$ $\displaystyle=$
$\displaystyle\widehat{Y}(t)\mathscr{M}(t)=\left(\widehat{\theta}(t)|\widehat{\phi}(t)\right)\mathscr{M}(t)$
(6.2) $\displaystyle=$
$\displaystyle\widehat{\psi}(t)\int_{t_{0}}^{t}\chi^{*}(s)A(s)f(s)\Delta
s+\widehat{\phi}(t)\int_{t}^{\infty}\zeta^{*}(s)A(s)f(s)\Delta s,$
where $\psi$ and $\zeta$ are as defined in Theorem 5.1. It follows from (2.5)
and (5.5) that
$\displaystyle U(t)$ $\displaystyle=$ $\displaystyle
H(t)\widehat{U}(t)+N(t)A(t)f(t)$ (6.3) $\displaystyle=$
$\displaystyle\psi(t)\int_{t_{0}}^{t}\chi^{*}(s)A(s)f(s)\Delta
s+\phi(t)\int_{t}^{\infty}\zeta^{*}(s)A(s)f(s)\Delta s+N(t)A(t)f(t)$
$\displaystyle=$ $\displaystyle(R_{\lambda}f)(t).$
We show that $U$ solves (5.4). Using (6.2), we have (supressing the $t$)
$\displaystyle J\widehat{U}^{\Delta}$ $\displaystyle=$ $\displaystyle
J\left[\widehat{\psi}^{\sigma}\chi^{*}-\widehat{\phi}^{\sigma}\zeta^{*}\right]Af+J\left[\widehat{\psi}^{\Delta}\int_{t_{0}}^{t}\chi^{*}(s)A(s)f(s)\Delta
s+\widehat{\phi}^{\Delta}\int_{t}^{\infty}\zeta^{*}(s)A(s)f(s)\Delta s\right]$
$\displaystyle=$ $\displaystyle
J\left[\widehat{\theta}^{\sigma}\chi^{*}-\widehat{\phi}^{\sigma}\eta^{*}\right]Af+(\lambda
A+B)(U-NAf)$ $\displaystyle=$
$\displaystyle-J\widehat{Y}^{\sigma}JZ^{*}Af+(\lambda A+B)(U-NAf)$
$\displaystyle=$ $\displaystyle(\lambda A+B)U+Af,$
using (2.8), Lemma 2.5 and the fact that $(\lambda
A+B)N+I_{2n}=\widetilde{H}^{*}$ by (5.6).
Continuing from (2.11), (6.2) and (6.3), we have
$\widehat{(R_{\lambda}f)}(t_{0})=\left(\widehat{R}_{\lambda}f\right)(t_{0})=\widehat{\phi}(t_{0},\lambda)\int_{t_{0}}^{\infty}\zeta^{*}(s)A(s)f(s)\Delta
s=\begin{pmatrix}-\displaystyle\int_{t_{0}}^{\infty}\zeta^{*}(s)A(s)f(s)\Delta
s\\\ 0_{n}\end{pmatrix},$
so that we have $(R_{\lambda}f)_{2}^{\rho}(t_{0})=0_{n}$. The identity
$\widehat{\chi}^{*}(t_{0})J\left(\widehat{R}_{\lambda}f\right)(t_{0})=\widehat{\chi}^{*}(t_{0})J\widehat{\phi}(t_{0})\int_{t_{0}}^{\infty}\zeta^{*}(t)A(t)f(t)\Delta
t=0$
follows immediately from the initial values of $\chi$ and $\phi$ in (2.11).
Note that the integral is convergent as each column of $\zeta$ is $A-$square
integrable and $f\in L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$. In addition, since
$\widehat{\psi}(t)=\widehat{Y}(t)\left(\begin{smallmatrix}I_{n}\\\
M\end{smallmatrix}\right)$ and
$\widehat{\zeta}(t)=\widehat{Z}(t)\left(\begin{smallmatrix}I_{n}\\\
M^{*}\end{smallmatrix}\right)$ we obtain from Lemma 2.5 the equalities
$\widehat{\zeta}^{*}(t,\lambda)J\widehat{\psi}(t,\lambda)=(I_{n}|M)\widehat{Z}^{*}(t,\lambda)J\widehat{Y}(t,\lambda)\left(\begin{smallmatrix}I_{n}\\\
M\end{smallmatrix}\right)=0_{n}$
and
$\widehat{\zeta}^{*}(t,\lambda)J\widehat{\phi}(t,\lambda)=(I_{n}|M)\widehat{Z}^{*}(t,\lambda)J\widehat{Y}(t,\lambda)\left(\begin{smallmatrix}0_{n}\\\
I_{n}\end{smallmatrix}\right)=-I_{n}.$ (6.4)
It thus follows that
$\lim_{t\rightarrow\infty}\widehat{\zeta}^{*}(t,\lambda)J\left(\widehat{R}_{\lambda}f\right)(t)=-\lim_{t\rightarrow\infty}\int_{t}^{\infty}\zeta^{*}(s)A(s)f(s)\Delta
s=0,$
completing the proof. ∎
###### Remark 6.2.
Note that
$\widehat{(R_{\lambda}f)}(t)=\left(\widehat{R}_{\lambda}f\right)(t)=H^{-1}(t)\left[(R_{\lambda}f)-NAf\right](t)$
for $t\in[t_{0},\infty)_{\mathbb{X}}$ by (5.5). For the adjoint system (5.8),
we may show in a similar manner that
$J\left(\widetilde{R}_{\lambda}f\right)^{\Delta}(t)=\left(\overline{\lambda}A(t)+B^{*}(t)\right)\left(\widetilde{R}_{\lambda}f\right)(t)+A(t)f(t),$
with boundary conditions
$\widehat{\phi}^{*}(t_{0})J\left(\widehat{\widetilde{R}}_{\lambda}f\right)(t_{0})=0$
and
$\lim_{t\rightarrow\infty}\widehat{\psi}^{*}(t,\lambda)J\left(\widehat{\widetilde{R}}_{\lambda}f\right)(t)=0$.
###### Theorem 6.3.
Let
$\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})\cap\widetilde{\Lambda}(\lambda_{0},\mathscr{U}^{-1}_{2n})$
and $f\in L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$. If we set $\Phi=R_{\lambda}f$
and $\widetilde{A}(t)=\mathscr{U}_{2n}A(t)\mathscr{U}^{*}_{2n}$, then
$\|\Phi\|^{2}_{W(\lambda_{0})}+(\delta-\varepsilon)\|\Phi\|^{2}_{\widetilde{A}}\;\leq\;\frac{1}{4\varepsilon}\|f\|_{A}^{2}$
for any $0<\varepsilon<\delta$, with $\delta=\delta(\lambda)$ as in (3.6), and
$\|\Phi\|^{2}_{\widetilde{A}}\;\leq\;\frac{1}{\delta}\|f\|_{A}.$
In particular, since $\widetilde{A}(t)\asymp A(t)$ for
$t\in[t_{0},\infty)_{\mathbb{X}}$, $R_{\lambda}$ is bounded on
$L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$.
###### Proof.
The proof is similar to that given in the continuous case [13, Theorem 5.1]
and the discrete case [23, Theorem 5.3], and thus is omitted. ∎
###### Lemma 6.4.
If $\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$, then $R_{\lambda}$ is an
injective linear operator defined on $L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$.
In addition, for any $\xi\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$ and
$L_{\xi}$ as in (5.16), we have
$\operatorname{range}R_{\xi}\subset\mathscr{D}(L_{\xi})$ and
$(L_{\xi}-\xi)R_{\xi}f=f$ for all $f\in L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$.
A similar statement holds for $\widetilde{R}_{\xi}$.
###### Proof.
See the proof in [23, Lemma 5.4]. ∎
###### Remark 6.5.
In the following lemma, it is traditional to use the Greek letter $\rho$ to
represent a resolvent set, but since we already employ $\rho$ as a backward
jump operator on time scales, we will use $r$ instead.
###### Lemma 6.6.
Denoting the resolvent sets of $L_{\xi}$ and $\widetilde{L}_{\xi}$ by
$r(L_{\xi})$ and $r\left(\widetilde{L}_{\xi}\right)$, respectively, we have
$\xi\in r(L_{\xi})$, $\overline{\xi}\in r\left(\widetilde{L}_{\xi}\right)$,
$\operatorname{range}R_{\xi}=\mathscr{D}(L_{\xi})$,
$\operatorname{range}\widetilde{R}_{\xi}=\mathscr{D}\left(\widetilde{L}_{\xi}\right)$,
$(L_{\xi}-\xi)^{-1}=R_{\xi}$ and
$\left(\widetilde{L}_{\xi}-\overline{\xi}\right)^{-1}=\widetilde{R}_{\xi}$. In
addition, $L_{\xi}$ and $\widetilde{L}_{\xi}$ are closed.
###### Proof.
See [13, Lemma 5.7] and [23, Lemma 5.5]. ∎
###### Lemma 6.7.
The space $\mathscr{D}(L_{\xi})$ is dense in
$L_{A}^{2}(t_{0},\infty)_{\mathbb{X}}$. Also,
$\widetilde{L}_{\xi}=L_{\xi}^{*}$, the adjoint of $L_{\xi}$.
###### Proof.
See [13, Lemma 5.8] and [23, Lemma 5.6]. ∎
###### Theorem 6.8.
If $\xi\in\Lambda(\lambda_{0},\mathscr{U}_{2n})$, then
$\Lambda(\lambda_{0},\mathscr{U}_{2n})\subset r(L_{\xi})$, and
$(L_{\xi}-\lambda)^{-1}=R_{\lambda},\quad\lambda\in\Lambda(\lambda_{0},\mathscr{U}_{2n}).$
A corresponding statement holds for $\widetilde{R}_{\lambda}$ and
$\widetilde{L}_{\xi}$.
###### Proof.
See [13, Theorem 5.9] and [23, Theorem 5.7]. ∎
###### Theorem 6.9.
If all solutions of (1.4) and (2.7) are $A-$square integrable for some
$\lambda^{\prime}\in\mathbb{C}$, and if
$A^{1/2}(t)N^{*}(t)A(t)N(t)A^{1/2}(t)\rightarrow 0\quad\text{as}\quad
t\rightarrow\infty$
for $A$ in (1.5) and $N$ in (5.6), then all solutions of (1.4) are $A-$square
integrable for all $\lambda\in\mathbb{C}$.
###### Proof.
Note that by a modified variation of constants approach (see [13, Lemma 6.5],
[17, Chapter 9 Theorem 2.1], [23, Theorem A.1]), any solution of
$J\widehat{y}^{\Delta}(t)=\Big{(}\lambda
A(t)+B(t)\Big{)}y(t)=\Big{(}\lambda^{\prime}A(t)+B(t)\Big{)}y(t)+(\lambda-\lambda^{\prime})A(t)y(t),\quad
t\in[t_{0},\infty)_{\mathbb{X}},$
can be written as
$\displaystyle y(t,\lambda)$ $\displaystyle=$
$\displaystyle\theta(t,\lambda^{\prime})\gamma+\phi(t,\lambda^{\prime})\widetilde{\gamma}$
$\displaystyle+(\lambda-\lambda^{\prime})\left(\int_{c}^{t}\left(\psi(t,\lambda^{\prime})\chi^{*}(s,\lambda^{\prime})-\phi(t,\lambda^{\prime})\zeta^{*}(s,\lambda^{\prime})\right)A(s)y(s,\lambda)\Delta
s+N(t)A(t)y(t,\lambda)\right)$
for some $\gamma,\widetilde{\gamma}\in\mathbb{C}^{n}$, where $N$ is given in
(5.6). The remainder of the proof is similar to that given in [23, Theorem
A.1] and is omitted. ∎
## References
* [1] C.D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, _J. Math. Anal. Appl._ 180 (1993) 498–517.
* [2] C.D. Ahlbrandt, M. Bohner, and J. Ridenhour, Hamiltonian systems on time scales, _J. Math. Anal. Appl._ 250 (2000) 561–578.
* [3] C.D. Ahlbrandt, M. Bohner and T. Voepel, Variable change for Sturm-Liouville differential expressions on time scales, _J. Difference Eq. Appl._ 9(1) (2003) 93- 107.
* [4] C.D. Ahlbrandt and A.C. Peterson, _Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Ricatti Equations_ , Kluwer Academic, Boston, 1996.
* [5] D.R. Anderson, Dominant and recessive solutions of self-adjoint matrix systems on time scales, _Nonlinear Dyn. Sys. Theory_ , 9(3) (2009) 219- 238.
* [6] D.R. Anderson and B. Buchholz, Self-adjoint matrix equations on time scales, _PanAmerican Mathematical Journal_ , 17(2) (2007) 81–104.
* [7] D.R. Anderson, G.Sh. Guseinov and J. Hoffacker, Higher-order self adjoint boundary value problems on time scales, _J. Computational Appl. Math._ 194(2) (2006) 309–342.
* [8] F.V. Atkinson, _Discrete and Continuous Boundary Problems_ , Academic Press, New York, 1964.
* [9] M. Bohner, O. Došlý, and W. Kratz, Sturmian and spectral theory for discrete symplectic systems, _Trans. Amer. Math. Soc._ 361(6) (2009) 3109–3123.
* [10] M. Bohner and A. Peterson, _Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2001.
* [11] M. Bohner and A. Peterson, editors, _Advances in Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2003.
* [12] B.M. Brown, W.D. Evans, D.K.R. McCormack, and M. Plum, On the spectrum of second order differential operators with complex coefficients, _Proc. Roy. Soc. London Ser. A_ 455 (1999) 1235–1257.
* [13] B.M. Brown, W.D. Evans, and M. Plum, Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems, _Proc. London Math. Soc._ 87 (3) (2003) 419–450.
* [14] S. Chandrasekhar, On characteristic value problems in high order differential equations which arise in studies on hydrodynamic and hydromagnetic stability, _Amer. Math. Monthly_ 61 (1954) 32–45.
* [15] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M-$function asymptotics, local uniqueness results, trace formulas, and Borg–type theorems for Dirac operators, _Trans. Amer. Math. Soc._ 354 (2002), no. 9, 3475–3534.
* [16] S. Clark and F. Gesztesy, On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems, _J. Comput. Appl. Math._ 171 (2004), no. 1-2, 151–184.
* [17] E.A. Coddington and N. Levinson, _Theory of Ordinary Differential Equations_ , McGraw Hill, New York, 1955.
* [18] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, _Results Math._ 18 (1990) 18–56.
* [19] R. Hilscher, Linear Hamiltonian systems on time scales: positivity of quadratic functionals, _Math. Comput. Modelling_ 32 (2000) 507–527.
* [20] D. B. Hinton and J. K. Shaw, On Titchmarsh-Weyl $m(\lambda)-$functions for Hamiltonian systems, _J. Differential Equations_ 40 (1981) 316–342.
* [21] D. B. Hinton and J. K. Shaw, On the spectrum of a singular Hamiltonian system, _Quaestiones Math._ 5 (1982) 29–81.
* [22] A.M. Krall, $M(\lambda)$ theory for singular Hamiltonian systems with one singular point, _SIAM J. Math. Anal._ 20 (1989) 664-700.
* [23] S. J. Monaquel and K. M. Schmidt, On $M-$functions and operator theory for non-self-adjoint discrete Hamiltonian systems, _J. Computational Appl. Math._ 208 (2007) 82–101.
* [24] S. A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, _J. Fluid Mech._ 50(4) (1971) 689–703.
* [25] Y.M. Shi, Spectral theory of discrete linear Hamiltonian systems, _J. Math. Anal. Appl._ 289(2) (2004) 554–570.
* [26] Y.M. Shi, Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems, _Linear Algebra Appl._ 416(2-3) (2006) 452–519.
* [27] G.L. Shi and H.Y. Wu, Spectral theory of Sturm-Liouville difference operators, _Linear Algebra Appl._ 430(2-3) (2009) 830–846.
* [28] A.R. Sims, Secondary conditions for linear differential operators of the second order, _J. Math. Mech. Ann. Mat. Pure_ 6 (1957) 247–285.
* [29] S.R. Sun, Y.M. Shi, and S.Z. Chen, The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems, _J. Math. Anal. Appl._ 327(2) (2007) 1360–1380.
|
arxiv-papers
| 2010-01-22T16:41:07 |
2024-09-04T02:49:07.891917
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1001.3912"
}
|
1001.3939
|
# Spectral energy distributions and age estimates of 104 M31 globular clusters
Song Wang,11affiliation: National Astronomical Observatories, Chinese Academy
of Sciences, Beijing, 100012, P. R. China;
majun@bac.pku.edu.cn 22affiliation: Graduate University, Chinese Academy of
Sciences, Beijing, 100039, P. R. China 33affiliation: Key Laboratory of
Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, China Zhou Fan,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@bac.pku.edu.cn Jun Ma,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@bac.pku.edu.cn 33affiliation: Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100012, China Richard de Grijs,44affiliation: Kavli Institute for Astronomy
and Astrophysics, Peking University, Beijing, 100871, P. R. China
55affiliation: Department of Physics & Astronomy, University of Sheffield,
Sheffield S3 7RH, UK Xu Zhou11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@bac.pku.edu.cn
###### Abstract
We present photometry of 104 M31 globular clusters (GCs) and GC candidates in
15 intermediate-band filters of the Beijing-Arizona-Taiwan-Connecticut (BATC)
photometric system. The GCs and GC candidates were selected from the Revised
Bologna Catalog (v.3.5). We obtain the cluster ages by comparing the
photometric data with up-to-date theoretical synthesis models. The photometric
data used are GALEX far- and near-ultraviolet and 2MASS near-infrared
$JHK_{\rm s}$ magnitudes, combined with optical photometry. The ages of our
sample clusters cover a large range, although most clusters are younger than
10 Gyr. Combined with the ages obtained in our series of previous papers
focusing on the M31 GC system, we present the full M31 GC age distribution.
The M31 GC system contains populations of young and intermediate-age GCs, as
well as the ‘usual’ complement of well-known old GCs, i.e., GCs of similar age
as the majority of the Galactic GCs. In addition, young GCs (and GC
candidates) are distributed nearly uniformly in radial distance from the
center of M31, while most old GCs (and GC candidates) are more strongly
concentrated.
###### Subject headings:
galaxies: individual (M31) – galaxies: star clusters – galaxies: stellar
content
††slugcomment: AJ, in press
## 1\. Introduction
Globular clusters (GCs) are among the oldest known stellar systems in the
Universe. They typically have ages similar to those of their host galaxies,
thus making them fossils that may provide important information about the
formation and evolution of their parent galaxies. In addition, nearly all
types of galaxies contain GCs, from dwarfs to giants and from the earliest to
the latest types (Fusi Pecci et al., 2005). However, our most in-depth
understanding of GC systems has predominantly come from studies of the Milky
Way.
M31, located at a distance of $\sim 780$ kpc (Stanek & Garnavich, 1998; Macri,
2001), is the largest galaxy in the Local Group. By virtue of the natural
advantage of being located at a reasonable distance, the galaxy offers us an
ideal environment for detailed, resolved investigations of a large GC system,
using both Hubble Space Telescope (HST) (e.g., Grillmair et al., 1996; Holland
et al., 1997; Rich et al., 2005; Perina et al., 2009b) and ground-based
observations with large telescopes (e.g., Christian & Heasley, 1991).
A large number of studies focusing on the M31 GC system have been performed
since Hubble (1932)’s original identification of 140 GC candidates in M31. The
latest Revised Bologna Catalogue of M31 GCs and candidates (hereafter RBC
v.3.5) (Galleti et al., 2004, 2006, 2007) was updated on March 27, 2008, and
contains 1983 objects (509 confirmed and 1049 candidate GCs, 9 controversial
objects, 147 galaxies, 6 Hii regions, 245 stars, 5 asterisms, and 13 extended
clusters). These objects were observed and discovered by a large number of
authors using a variety of observational systems (see, e.g., Vetes̆nik, 1962;
Sargent et al., 1977; Battistini et al., 1980; Crampton et al., 1985; Barmby
et al., 2000). To obtain a homogeneous photometric data set, Galleti et al.
(2004) took the observed data of Barmby et al. (2000) as reference and
transformed other observations to this standard setup.
An accurate and reliable analysis of star clusters is important for our
understanding of the formation, buildup, and evolutionary processes in
galaxies. By comparing integrated photometry with models of simple stellar
populations (SSPs), recent studies have achieved some success in determining
ages and masses of extragalactic star clusters (e.g., de Grijs et al., 2003a,
b, c; de Grijs & Anders, 2006; Bik et al., 2003; Ma et al., 2006a; Fan et al.,
2006; Ma et al., 2007a, 2009a). Ma et al. (2006a) and Fan et al. (2006)
derived age estimates for M31 GCs by fitting SSP models (Bruzual & Charlot,
2003, henceforth BC03) to their photometric measurements in a large number of
intermediate- and broad-band filters spanning the spectral range from the
optical to the near-infrared (NIR). In particular, Ma et al. (2007a)
determined an age for the M31 GC S312 (B379), using multicolor photometry from
the near-ultraviolet (near-UV) to the NIR, of $9.5^{+1.2}_{-1.0}$ Gyr. S312
(B379) is, in fact, among the first extragalactic GCs for which the age was
estimated accurately and independently, using main-sequence photometry, at
$10^{+2.5}_{-1}$ Gyr (Brown et al., 2004). This provides a robust check on our
methodology to derive age constraints based on the spectral energy
distributions (SEDs) of (simple) stellar systems.
This paper is organized as follows. In §2 we present Beijing-Arizona-Taiwan-
Connecticut (BATC) observations of our sample GCs and GC candidates, the
relevant data-processing steps, and the GALEX (far- and near-UV), optical
broad-band, and Two-Micron All Sky Survey (2MASS) NIR data that are
subsequently used in our analysis. In §3 we derive the ages of our sample
clusters by comparing their SEDs with the galev SSP models. We then discuss
and summarize our results in §4.
## 2\. GC sample and BATC intermediate-band photometry
### 2.1. GC sample selection
To obtain photometry in 15 intermediate-band filters of the BATC photometric
system for 61 GCs and GC candidates in the RBC v.3.5, for which few
measurements are presently available in any photometric system, Fan et al.
(2009) mined the BATC survey archive for observations obtained between
February 1995 and March 2008. The resulting set of observations covers
approximately six square degrees. For the purpose of estimating accurate
cluster ages, we selected clusters for which the metallicities and reddening
values had been estimated accurately, independently, and homogeneously in
previous studies (Huchra et al., 1991; Barmby et al., 2000; Perrett et al.,
2002; Fan et al., 2008): see §2.6. We selected classes 1, 2, 3, and 8 (1580
objects) from column ‘f’ in the RBC v.3.5, which include GCs, candidate GCs,
controversial objects, and extended clusters. This resulted in an initial
selection of 366 objects. Jiang et al. (2003), Ma et al. (2006a), and Ma et
al. (2009a) obtained multicolor photometry for 180 of these GCs and GC
candidates. In this paper, we consider the remaining 186 objects. Caldwell et
al. (2009) published an updated catalog of 1300 objects in M31, including 670
likely star clusters, with the remaining objects being stars or background
galaxies once thought to be clusters (see Tables 3 and 5 of Caldwell et al.,
2009). From a comparison with Caldwell et al. (2009), we find that 66 objects
are either stars or background galaxies. Therefore, the final sample of M31
GCs and GC candidates analyzed in this paper includes 120 objects. However, we
cannot obtain accurate photometric measurements for 16 of these objects
because of either a nearby very bright object (B065 and B344D), very faint
fluxes superimposed onto a bright background (B119, B396, NB16, and V031), or
a location very close to (or blend with) another object (B150D, B176, B256D,
B302, B345, B366, B381, B391, and B397), leading to compromised photometric
measurements. Object B330 is both faint and located very close to a brighter
object. Thus, here we analyze the multicolor photometric properties of 104 GCs
and GC candidates. Figure 1 shows their spatial distribution across the M31
fields observed with the BATC multicolor system.
Figure 1.— BATC observations of our M31 fields. Each field is $58\arcmin\times
58\arcmin$ (size of the old CCD). The large ellipse is the boundary between
the M31 disk and halo (Racine, 1991), while the two small ellipses represent
the $D_{25}$ isophotes of NGC 205 (northwest) and M32 (southeast). Solid
circles indicate the sample GCs and GC candidates discussed in this paper.
### 2.2. BATC intermediate-band photometry
The observations of our sample GCs and GC candidates were carried out in the
BATC photometric system, using the 60/90 cm $f$/3 Schmidt telescope at
Xinglong Station of the National Astronomical Observatories of the Chinese
Academy Sciences (NAOC). The BATC system includes 15 intermediate-band
filters, covering a wavelength range from 3300 Å to 1 $\mu$m. The parameters
of the filters are given in Table 1, where column (1) gives the filter name,
column (2) is the central wavelength for each filter, and column (3) lists the
bandwidth for each filter. The 2k$\times$2k CCD used before February 2006 had
a pixel size of 15 $\mu$m and a resolution of $1.7^{\prime\prime}$ pixel-1.
After February 2006, a new 4k$\times$4k CCD with a pixel size of 12 $\mu$m was
used, with a resolution of $1.3^{\prime\prime}$ pixel-1 (Fan et al., 2009).
The new CCD camera is much more sensitive at short wavelengths.
We obtained 143.9 hours of imaging (447 images) of the M31 field, covering
about six square degrees, through the set of 15 filters in five observing runs
from 1995 to 2008, spanning 13 years (see for details Fan et al., 2009). The
data were reduced using standard procedures, including bias subtraction and
flat fielding of the CCD images, with an automatic data-reduction software
package (PIPELINE I) specifically developed for the BATC sky survey. BATC
magnitudes are defined and obtained in a similar way as for the
spectrophotometric AB magnitude system (see for details Ma et al., 2009a). For
the $a$ to $p$ filters of the central field of M31 (M31-1 in Figure 1), the
absolute flux of the combined images was obtained using calibrated standard
stars, while for the M31-2 to M31-7 fields we used the M31-1 field to derive
secondary transformations (see for details Fan et al., 2009).
We determined the magnitudes of our sample objects on the combined images
using standard aperture photometry, i.e., using the PHOT routine in DAOPHOT
(Stetson, 1987). To avoid contamination from nearby objects, we adopted
apertures with radii of 3 and 4 pixels on the 2k$\times$2k and 4k$\times$4k
CCDs, respectively. For the old CCD, we took 8 and 13 pixels from the object’s
center as the inner and outer radii of the sky annulus for background
determination, while for the new CCD, the corresponding radii were set at 10
and 17 pixels, respectively (Fan et al., 2009). We used isolated stars to
obtain point-source aperture corrections by measuring the magnitude
differences between the fluxes contained within radii of 3 (4) pixels on the
old (new) CCD images and the total stellar magnitudes in each of the 15 BATC
filters. The resulting aperture-corrected SEDs for the sample GCs and GC
candidates in M31 are provided in Table 2. Columns (2) to (16) represent the
magnitudes in the 15 BATC passbands used for our photometry. The $1\sigma$
magnitude uncertainties, from DAOPHOT, are listed for each object on the
second line for the corresponding passband. For some GCs and GC candidates,
the magnitudes in some filters could not obtained because of low signal-to-
noise ratios.
### 2.3. GALEX UV, optical broad-band, and 2MASS NIR photometry
To estimate the ages of the M31 GCs and GC candidates, we should ideally use
as many photometric data points covering as wide a wavelength range as
possible (cf. de Grijs et al., 2003b; Anders et al., 2004; Ma et al., 2009a).
The RBC v.3.5 includes GALEX (far- and near-UV) fluxes from Rey et al. (2007),
optical broad-band, and 2MASS NIR magnitudes for 1983 objects, which we use as
the basis for our analysis. Although the $UBVRI$ magnitudes of the objects
published by Barmby et al. (2000) are included in the RBC v.3.5 and as such
provide the most homogeneous set of photometric measurements available, the
relevant photometric uncertainties are not listed. Therefore, we adopt the
original $UBVRI$ measurements of Barmby et al. (2000), including their
published photometric errors. For the remaining objects we adopt the $UBVRI$
measurements from the RBC v.3.5. We assign photometric uncertainties following
Galleti et al. (2004), i.e., $\pm 0.05$ mag in $BVRI$ and $\pm 0.08$ mag in
$U$ (see for details Ma et al., 2009a).
In the RBC v.3.5, the 2MASS $JHK_{\rm s}$ magnitudes were transformed to the
CIT photometric system (Galleti et al., 2004). However, we need the original
2MASS $JHK_{\rm s}$ data to compare the observed SEDs with the SSP models, so
we reversed the transformation using the equations given by Carpenter (2001).
We obtained the magnitude errors in the $JHK_{\rm s}$ bands by comparing our
photometric data with fig. 2 of Carpenter et al. (2001), which shows the
generic photometric uncertainties as a function of magnitude for stars
brighter than their observational completeness limits (see for details Ma et
al., 2009a). We include the GALEX, optical broad-band, and 2MASS NIR
photometry of the sample clusters in Table 3 (columns 3 to 12), where the
photometric errors are listed for each object on the second line for the
corresponding passband. The GALEX photometric system is calibrated to match
the spectrophotometric AB system, while the optical broad-band and 2MASS
photometric data are given in Vega magnitudes. Finally, column 2 includes the
classification flags from the RBC v.3.5.
### 2.4. Comparison with previously published photometry
To check our photometry, we transformed the BATC intermediate-band system to
the $UBVRI$ broad-band system using the relationships between these two
systems derived by Zhou et al. (2003):
$B=m_{d}+0.2201(m_{c}-m_{e})+0.1278\pm 0.076\quad\mbox{and}$ (1)
$V=m_{g}+0.3292(m_{f}-m_{h})+0.0476\pm 0.027.$ (2)
$B$-band photometry can be derived from the BATC $c,d$, and $e$ bands, while
$V$-band magnitudes can be obtained from the BATC $f,g$, and $h$ bands. Figure
2 shows a comparison of the $B$ and $V$ photometry of our M31 sample objects
with previous measurements from Barmby et al. (2000) and Galleti et al.
(2004). The mean $B$ and $V$ magnitude differences—in the sense of this paper
minus Barmby et al. (2000) or Galleti et al. (2004)—are $\langle\Delta
B\rangle=-0.077\pm 0.022$ mag and $\langle\Delta V\rangle=-0.047\pm 0.033$.
Our magnitudes are in good agreement with previous $V$-band determinations.
However, a significant disagreement becomes apparent in the $B$ band for
objects with $B>17.5$ mag. This disagreement has its origin in the difference
between our photometry and that of Galleti et al. (2004). In fact, our
$B$-band photometry agrees well with that of Barmby et al. (2000), even for
$B>17.5$ mag objects (except for one sample cluster). Referring to Jiang et
al. (2003) and Ma et al. (2009a), we also see that our photometric values are
fully consistent with Barmby et al. (2000). In Ma et al. (2009a), the analysis
of the majority of the GCs was based on the photometric data of Barmby et al.
(2000), so even in the $B$ band the agreement is good: see fig. 3 of Ma et al.
(2009a). We excluded B257 from the comparison, because its $V$-band magnitude
is too faint compared with the magnitudes obtained in the other bands (see
Table 3). In fact, from the observed SEDs, this photometric measurement is
unusually far away from the best-fitting integrated SEDs (see §3 for more
details). This data point was taken from Table 4 of Barmby & Huchra (2001),
and the offset may be a typical error. Based on the BATC $f,g$, and $h$
magnitudes, we obtain $V=17.76$ mag for B257. Note that the $B$-band magnitude
of this cluster as listed in Table 4 of Barmby & Huchra (2001) is 11.907,
which is too bright for any reasonable SED and may also be a typical
uncertainty.
Figure 2.— Comparison of the photometry of our GCs and GC candidates with
previous measurements from Barmby et al. (2000) (triangles) and Galleti et al.
(2004) (crosses). The dashed lines enclose $\Delta V$ and $\Delta B=0.3$ mag.
### 2.5. Metallicities and reddening values
We require independently (but homogeneously) determined metallicities and
reddening values to robustly and accurately estimate the ages of our sample
objects. We used three homogeneous sources of spectroscopic metallicities
(Huchra et al., 1991; Barmby et al., 2000; Perrett et al., 2002) and one
homogenized reference (see for details Fan et al., 2008; Ma et al., 2009a).
Following Ma et al. (2009a), for reasons of consistency we ranked the sources
used to assign metallicities to our M31 GCs in order of preference.
Metallicities from Perrett et al. (2002) were chosen whenever available
because of the large number of their metallicity determinations, followed (in
order) by metallicity determinations from Barmby et al. (2000) and Huchra et
al. (1991). If none of these three sources included spectroscopic
metallicities for a given sample cluster, we used the corresponding value from
Fan et al. (2008).
For reddening values, we used Barmby et al. (2000) and Fan et al. (2008) as
our reference (see for details Ma et al., 2009a). Because the reddening values
from Fan et al. (2008) comprise a homogeneous data set and the number of GCs
included is greater than that of Barmby et al. (2000), we preferentially adopt
Fan et al. (2008) reddening values, followed by those of Barmby et al. (2000),
in a similar approach as adopted by Ma et al. (2009a). The metallicities and
reddening values adopted for our sample clusters are listed in Table 4.
## 3\. Age determination
An SSP is defined as a single generation of coeval stars characterized by the
same parameters, including metallicity, age, and stellar initial mass function
(IMF). SSP models are calculated on the basis of a set of evolutionary tracks
of stars of different initial masses, combined with stellar spectra at
different evolutionary stages. In this paper, and following Ma et al. (2009a),
we compare the SEDs of our sample objects with the galev SSP models (e.g.,
Kurth et al., 1999; Schulz et al., 2002; Anders & Fritze-v. Alvensleben, 2003)
to estimate their ages. The galev SSPs are based on the Padova isochrones
(covering wavelengths from 91 Å to 160 $\mu$m) and a Salpeter (1955) stellar
IMF with lower and upper mass limits of $0.10~{}{\rm M}_{\odot}$ and 50–70
${\rm M}_{\odot}$, respectively (the latter depending on metallicity). These
models cover ages from 4 Myr to 16 Gyr, with an age resolution of 4 Myr for
ages younger than 2.35 Gyr, and 20 Myr for older ages. We convolved the
theoretical SSP SEDs with the GALEX, broad-band $UBVRI$, BATC intermediate-
band, and 2MASS $JHK_{\rm s}$ filter response curves to obtain synthetic UV,
optical, and NIR photometry (Ma et al., 2009a). The synthetic magnitude in the
AB magnitude system for the $i^{\rm th}$ filter is
$m_{i}=-2.5\log\frac{\int_{\nu}F_{\nu}\varphi_{i}(\nu){\rm
d}\nu}{\int_{\nu}\varphi_{i}(\nu){\rm d}\nu}-48.60,$ (3)
where $F_{\nu}$ is the theoretical SSP SED (which is a function of age and
metallicity) and $\varphi_{i}$ is the response curve of the $i^{\rm th}$
filter. The galev SSP models include five initial metallicities,
$Z=0.0004,0.004,0.008,0.02$ (solar), and $0.05$. For other metallicities, the
relevant spectra can be obtained by linear interpolation of the appropriate
model spectra for any of these five metallicities. For metallicities below
$Z=0.0004$ we use the $Z=0.0004$ model (Ma et al., 2009a).
To determine the most compatible galev SSP model for a given observed SED, we
adopted a $\chi^{2}$ minimization test,
$\chi^{2}=\sum_{i=1}^{25}{\frac{[m_{\nu_{i}}^{\rm intr}-m_{\nu_{i}}^{\rm
mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (4)
where $m_{\nu_{i}}^{\rm mod}(t)$ is the magnitude in the $i^{\rm th}$ filter
of a theoretical SSP at age $t$, while $m_{\nu_{i}}^{\rm intr}$ is the
intrinsic (observed and corrected) magnitude in the same filter. The
interstellar extinction curve $A_{\nu}$ is taken from Cardelli et al. (1989),
$R_{\nu}=A_{V}/E_{B-V}=3.1$. $\sigma_{i}$ is the magnitude uncertainty in the
$i^{\rm th}$ filter, defined as
$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}.$ (5)
Here, $\sigma_{{\rm obs},i}$ and $\sigma_{{\rm mod},i}$ are the observational
uncertainty and that associated with the model itself, respectively. Charlot
et al. (1996) estimated $\sigma_{{\rm mod},i}$ by comparing the colors
obtained from different stellar evolutionary tracks and spectral libraries. We
adopt $\sigma_{{\rm mod},i}=0.05$ mag, following Wu et al. (2005), Ma et al.
(2006a, 2009a), and Fan et al. (2006). The SED fits of our sample GCs and GC
candidates are shown in Fig. 3.
Figure 3.— SED fits of the galev SSP models to our sample objects.
Figure 3.— Continued.
Figure 3.— Continued.
## 4\. Results and summary
In §3 we determined the ages of 104 GCs and GC candidates in M31. The results
are tabulated in Table 5. Figure 4 shows the age distribution of the sample
clusters, from which we conclude that, except for 20 clusters, the ages of
most sample GCs are between 1 and 5 Gyr, with a peak at $\sim 2$ Gyr. The
‘usual’ complement of well-known old GCs (i.e., GCs of similar age as the
majority of the Galactic GCs) is also present. In addition, while fitting SSP
models to the observed data, we found that some photometric data of a small
number of clusters cannot be fitted with any SSP models. We therefore did not
use these deviating photometric data points to obtain the best fits. This
applies to the GALEX far-UV data of B138D, the 2MASS $K_{\rm s},H$, and $J$
magnitudes of B142D, B181D, and B289D, respectively, the $B$-band and 2MASS
$H$ fluxes of B245D, and the $V$-band magnitude of B257.
Other authors have also considered the age distribution of the GCs in M31. For
example, Barmby et al. (2000) discovered that M31 contains GCs exhibiting
strong Balmer lines and A-type spectra, from which one infers that these
objects must be very young. Beasley et al. (2004) and Puzia et al. (2005)
confirmed this conclusion. Burstein et al. (2004) and Fusi Pecci et al. (2005)
carefully studied the sample of young M31 GCs. Very recently, Caldwell et al.
(2009) determined the ages and reddening values of 140 young clusters in M31
by comparing the observed spectra with models, and found that these clusters
are less than 2 Gyr old, while most clusters have ages between $10^{8}$ and
$10^{9}$ yr. Perina et al. (2009a) estimated an age for VDB0-B195D of $\sim
25$ Myr based on HST/WFPC2 color-magnitude diagrams (CMDs). The ages of the
M31 clusters determined in this paper are in general agreement with previous
determinations, which we will show in more detail below on the basis of
comparisons between our determinations and previous age estimates for
individual objects.
The most direct and most accurate method to determine a cluster’s age is by
means of main-sequence photometry, since the absolute magnitude of the main-
sequence turnoff is a strong function of age. Williams & Hodge (2001a, b)
estimated ages of many young disk clusters in M31 based on HST/WFPC2 CMDs and
isochrone fitting to either the main sequence or luminous evolved stars. Only
one of their clusters (B315) is in common with our sample. They obtained an
age of $\sim 0.1$ Gyr for this cluster, while we determined it to be
approximately 0.5 Gyr old. Both age determinations are mutually consistent
within the uncertainties. Caldwell et al. (2009) compared their ages with
those of Williams & Hodge (2001a, b) and concluded that both sets of age
determinations were in good agreement. We therefore compare our ages with
Caldwell et al. (2009) for the seven clusters we have in common with their
sample (B018, B307, B316, B448, B475, B483, and V031: see Table 6). It is
evident that they are largely internally consistent. Puzia et al. (2005) also
presented spectroscopic ages, metallicities, and ${\rm[\alpha/Fe]}$ ratios for
70 M31 GCs based on Lick line-index measurements. A cross correlation with
Puzia et al. (2005)’s sample shows that we have 21 clusters in common. A
direct comparison shows that the ages of Puzia et al. (2005) are
systematically older than ours. This surprising result prompted us to compare
the ages of clusters in common between Puzia et al. (2005) and other authors
(Williams & Hodge, 2001a, b; Beasley et al., 2004; Caldwell et al., 2009). We
found similar systematic offsets (see for details also Ma et al., 2009a).
Figure 4.— Age distribution of our sample GCs and GC candidates in M31.
We have determined the ages of M31 GCs and GC candidates in a series of
previous papers (Jiang et al., 2003; Ma et al., 2006a, b; Fan et al., 2006; Ma
et al., 2007a, 2009a, 2009b) based on the same method as used in the present
paper, i.e., by constructing SEDs of known M31 GC candidates and using the SED
shapes to estimate cluster ages. In the first paper of this series, Jiang et
al. (2003) estimated the ages of 172 M31 GC candidates based on photometric
measurements in 13 BATC intermediate-band filters and the SSP models of
Bruzual & Charlot (1996; unpublished, hereafter BC96). Subsequently, Fan et
al. (2006) obtained new age estimates for 91 GCs from the Jiang et al. (2003)
sample, based on improved photometric data including intermediate- and broad-
band magnitudes from the optical to the NIR, and the SSP models of BC03. Ma et
al. (2006b) then estimated the ages of 33 M31 GC candidates using photometry
in 13 BATC intermediate-band filters and the BC03 SSP models, while Ma et al.
(2009a) determined the ages of 35 M31 GCs and GC candidates based on
photometry including far- and near-UV GALEX observations, $UBVRI$, 13 BATC
intermediate-band filters, and 2MASS $JHK_{\rm s}$, combined with the galev
SSP models. Ma et al. (2006a, 2007a, 2009b) determined the ages of three
specific M31 GCs (037-B327, S312, and G1) based on the BC03 SSP models and a
large number of photometric measurements. We determined the ages of these
three M31 GCs for special reasons: S312 is among the first extragalactic GCs
whose age was estimated accurately using main-sequence photometry, while
037-B327 and G1 are among the most massive GCs in the Local Group. They have
been speculated to be nucleated dwarf galaxies instead of genuine GCs (see for
detailed discusions Ma et al., 2006c, 2007b). In this series of seven
articles, we published ages for 331 different M31 GCs and GC candidates.
Figure 5 show the age distribution of these 331 objects. We see that $\sim 40$
clusters are younger than 1 Gyr. The ages range from $<1$ to 20 Gyr (the upper
age limit in the BC96 and BC03 SSP models). A population of young clusters,
peaking at $\sim 3$ Gyr, is also apparent.
Figure 5.— Homogenized age distribution of the 331 M31 GCs and GC candidates
discussed in our series of papers.
Figure 6 shows the absolute magnitudes of our sample of M31 GCs and GC
candidates as a function of their age. The crosses indicate that the ages are
from Jiang et al. (2003), Ma et al. (2006a, b, 2007a, 2009a), and Fan et al.
(2006), which were obtained based on the SSP models of BC96 or BC03, while the
circles mean that the ages are from Ma et al. (2009a) and the present paper,
obtained on the basis of the galev SSP models. The dashed and solid lines
represent SSP models with $Z=0.004$ taken from BC03 and galev, respectively,
for masses of $10^{2}$, $10^{3}$, $10^{4}$, $10^{5}$, and $10^{6}M_{\odot}$
and assuming a Salpeter stellar IMF. The $V$-band photometry is from the RBC
v.3.5. The absolute magnitudes have been corrected for extinction (Barmby et
al., 2000; Fan et al., 2008), except for 037-B327, S312, and G1, the reddening
values of which are from Ma et al. (2006a), Ma et al. (2007a), and Ma et al.
(2009b), respectively. We adopt a distance modulus of $(m-M)_{0}=24.47$ mag
(McConnachie et al., 2005). Figure 6 shows that the majority of the clusters
have masses between $10^{3}$ and $10^{6}M_{\odot}$.
Figure 6.— Absolute $V$-band magnitudes for the M31 GCs and GC candidates as a
function of age. Overplotted are theoretical lines corresponding to (from
bottom to top) masses of $10^{2}$, $10^{3}$, $10^{4}$, $10^{5}$ and
$10^{6}M_{\odot}$ from BC03 (dashed line) and galev (solid line),
respectively.
The distribution of absolute $V$ magnitude of GCs in M31 is shown in Figure 7.
Overall, the distribution has a cutoff at the faint end with a magnitude limit
of about $-5.5$ mag (with a few fainter clusters still visible, probably
because of advantageous positions, e.g., observable through a hole in the
extinction distribution). The various cluster ages are separated in Figure 8,
which are: (i) very young ($t<1$ Gyr), (ii) young ($1\leq t<4$ Gyr), (iii)
intermediate-age ($4\leq t<10$ Gyr), and (iv) old GCs and GC candidates
($t\geq 10$ Gyr). We do not see a clear trend between age and brightness.
However, the youngest clusters are not the most massive objects, implying that
the conditions in the M31 have not been conducive to massive cluster formation
in the recent past.
Figure 7.— Histogram of the absolute $V$ magnitude for the 331 sample GCs and
GC candidates in M31.
Figure 8.— Histogram of the absolute $V$ magnitude for the M31 GCs and GC
candidates: (a) very young ($t<1$ Gyr), (b) young ($1\leq t<4$ Gyr), (c)
intermediate-age ($4\leq t<10$ Gyr), and (d) old GCs and GC candidates ($t\geq
10$ Gyr).
We converted the absolute magnitudes of our M31 GC sample to photometric
masses using the appropriate age-dependent mass-to-light ratios provided by
the BC03 and galev SSP models. The GC mass versus age diagram is shown in
Figure 9. The crosses indicate that the ages are from Jiang et al. (2003), Ma
et al. (2006a, b, 2007a, 2009a), and Fan et al. (2006), and the masses were
obtained based on the SSP models of BC03, while the circles mean that the ages
are from Ma et al. (2009a) and the present paper, and the masses were obtained
on the basis of the galev SSP models. Overplotted is the fading limit,
assuming $M_{V,{\rm limit}}=-5.5$ mag and evolutionary fading based on the
$Z=0.004$ BC03 (dashed line) and galev (solid line) models, assuming a
Salpeter stellar IMF. Figure 9 shows that our observational ($\sim 50$%)
completeness limit describes the lower mass limit of the entire GC sample up
to the oldest ages very well. Similarly, the upper envelope of the points in
Figure 9 is likely a result of the ‘size-of sample effect’ (e.g., Gieles &
Bastian, 2008, and references therein). It is clear, however, that massive
star cluster formation halted abruptly in the disk of M31 approximately 1 Gyr
ago. Given that massive ($>10^{4}M_{\odot}$) young ($<1$ Gyr-old) clusters
will be significantly brighter than the much older GC-type counterparts in
M31, we would have expected any such young massive clusters to have been
detected in M31, yet they have not.
Figure 9.— Distribution of the M31 GCs and GC candidates in the age-versus-
mass plane. Overplotted is the fading limit, based on the observed
$M_{V}=-5.5$ mag sample cutoff and the fading function from the cluster
evolutionary models with $Z=0.004$ taken from BC03 (dashed line) and galev
(solid line), assuming a Salpeter stellar IMF.
Using these 331 GCs and GC candidates with homogeneously determined ages, we
can now investigate their spatial distribution. We use an $X,Y$ projection to
refer to the relative positions of the objects. Our adopted $X$ coordinate
projects along M31’s major axis, where positive $X$ increases towards the
northeast, while the $Y$ coordinate extends along the minor axis of the M31
disk, increasing towards the northwest. To obtain the relative coordinates of
the M31 clusters, we adopted $\alpha_{0}=00^{\rm h}42^{\rm m}44^{\rm s}.30$
and $\delta_{0}=+41^{\circ}16^{\prime}09^{\prime\prime}.0$ (J2000.0) for M31’s
center, following Huchra et al. (1991) and Perrett et al. (2002). Formally,
$X=A\sin\theta+B\cos\theta\quad{\rm and}$ (6)
$Y=-A\cos\theta+B\sin\theta,$ (7)
where $A=\sin(\alpha-\alpha_{0})\cos\delta$ and
$B=\sin\delta\cos\delta_{0}-\cos(\alpha-\alpha_{0})\cos\delta\sin\delta_{0}$.
We adopt a position angle of $\theta=38^{\circ}$ for the major axis of M31
(Kent, 1989). We divided the GCs and GC candidates into four age groups: (i)
very young ($t<1$ Gyr), (ii) young ($1\leq t<4$ Gyr), (iii) intermediate-age
($4\leq t<10$ Gyr), and (iv) old GCs and GC candidates ($t\geq 10$ Gyr).
Figure 10 shows their spatial distributions. Although our sample of M31 GCs
and GC candidates is not complete (in spatial, radial terms, given that we are
limited by the six observed fields), we note that there is a tendency for
young GCs and GC candidates to be nearly uniformly distributed around M31. The
majority of old GCs appear to occupy the central regions of the galaxy,
although this restricted distribution may be caused by selection biases.
Figure 11 shows the number of GCs and GC candidates as a function of projected
radial distance from the M31 center, confirming our conclusions derived from
Figure 10. Figure 12 displays the cluster ages as a function of projected
radial distance. The crosses indicate that the ages are from Jiang et al.
(2003), Ma et al. (2006a, b, 2007a, 2009b), and Fan et al. (2006), which were
obtained using the BC96 or BC03 SSP models, while the circles indicate that
the ages are from Ma et al. (2009a) and the present paper, obtained on the
basis of the galev SSP models. Figure 12 shows that young GCs and GC
candidates are distributed nearly uniformly, and that most of the old GCs (and
candidates) are more concentrated.
Figure 10.— Spatial distribution of the M31 GCs and GC candidates: very young
($t<1$ Gyr), young ($1\leq t<4$ Gyr), intermediate-age ($4\leq t<10$ Gyr) and
old GCs and GC candidates ($t\geq 10$ Gyr).
Figure 11.— Radial distribution of the M31 GCs and GC candidates: (a) very
young ($t<1$ Gyr), (b) young ($1\leq t<4$ Gyr), (c) intermediate-age ($4\leq
t<10$ Gyr), and (d) old GCs and GC candidates ($t\geq 10$ Gyr).
Figure 12.— Age versus projected galactocentric radius for 331 M31 GCs and GC
candidates.
This paper presents photometry of 104 M31 globular clusters (GCs) and GC
candidates in 15 intermediate-band filters of the BATC photometric system. The
age of the clusters were obtained by comparing the photometric data with the
theoretical synthesis models. The ages of our sample clusters cover a large
range, although most clusters are younger than 10 Gyr. Combined with the ages
obtained in our series of previous papers focusing on the M31 GC system, we
present the full M31 GC age distribution. The results show that the M31 GC
system contains populations of young and intermediate-age GCs, as well as the
‘usual’ complement of well-known old GCs, i.e., GCs of similar age as the
majority of the Galactic GCs. In addition, young GCs (and GC candidates) are
distributed nearly uniformly in radial distance from the center of M31, while
most old GCs (and GC candidates) are more strongly concentrated.
We are indebted to the referee for thoughtful comments and insightful
suggestions that improved this paper significantly. This work was supported by
the Chinese National Natural Science Foundation (grants 10873016, 10633020,
10603006, and 10803007) and by National Basic Research Program of China (973
Program; grant 2007CB815403).
## References
* Anders & Fritze-v. Alvensleben (2003) Anders, P., & Fritze-v. Alvensleben, U. 2003, A&A, 401, 1063
* Anders et al. (2004) Anders, P., Bissantz, N., Fritze-v. Alvensleben, U., & de Grijs, R. 2004, MNRAS, 347, 196
* Barmby et al. (2000) Barmby, P., Huchra, J., Brodie, J., Forbes, D., Schroder, L., & Grillmair, C. 2000, AJ, 119, 727
* Barmby & Huchra (2001) Barmby, P., & Huchra, J. P. 2001, AJ, 122, 2458
* Battistini et al. (1980) Battistini, P., Bònoli, F., Braccesi, A., Fusi Pecci, F., Malagnini, M. L., & Marano, B. 1980, A&AS, 42, 357
* Beasley et al. (2004) Beasley, M. A., et al. 2004, AJ, 128, 1623
* Bik et al. (2003) Bik, A., Lamers, H. J. G. L. M., Bastian, N., Panagia, N., & Romaniello, M. 2003, A&A, 397, 473
* Brown et al. (2004) Brown, T. M., et al. 2004, ApJ, 613, L125
* Bruzual & Charlot (2003) Bruzual, A. G., & Charlot, S. 2003, MNRAS, 344, 1000 (BC03)
* Burstein et al. (2004) Burstein, D., et al. 2004, ApJ, 614, 158
* Caldwell et al. (2009) Caldwell, N., Harding, P., Morrison, H., Rose, J. A., Schiavon, R., & Kriessler, J. 2009, AJ, 137, 94
* Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* Carpenter (2001) Carpenter, J. M. 2001, AJ, 121, 2851
* Carpenter et al. (2001) Carpenter, J. M., Hillenbrand, L. A., & Skrutskie, M. F. 2001, AJ, 121, 3160
* Charlot et al. (1996) Charlot, S., Worthey, G., & Bressan, A. 1996, ApJ, 457, 625
* Christian & Heasley (1991) Christian, C. A., & Heasley, J. N. 1991, AJ, 101, 848
* Crampton et al. (1985) Crampton, D., Cowley, A. P., Schade, D., & Chayer, P. 1985, ApJ, 288, 494
* de Grijs et al. (2003a) de Grijs, R., Bastian, N., & Lamers, H. J. G. L. M. 2003a, MNRAS, 340, 197
* de Grijs et al. (2003b) de Grijs, R., Fritze-v. Alvensleben, U., Anders, P., Gallagher, J. S., Bastian, N., Taylor, V. A., & Windhorst, R. A. 2003b, MNRAS, 342, 259
* de Grijs et al. (2003c) de Grijs, R., Anders, P., Lynds, R., Bastian, N., Lamers, H. J. G. L. M., & O’Neill, E. J. Jr. 2003c, MNRAS, 343, 1285
* de Grijs & Anders (2006) de Grijs, R., & Anders, P. 2006, MNRAS, 366, 295
* Fan et al. (2006) Fan, Z., Ma, J., de Grijs, R., Yang, Y., & Zhou, X. 2006, MNRAS, 371, 1648
* Fan et al. (2008) Fan, Z., Ma, J., de Grijs, R., & Zhou, X. 2008, MNRAS, 385, 1973
* Fan et al. (2009) Fan, Z., Ma, J., & Zhou, X. 2009, RAA, 9, 993
* Fukugita et al. (1996) Fukugita, M., et al. 1996, AJ, 111, 1748
* Fusi Pecci et al. (2005) Fusi Pecci, F., Bellazzini, M., Buzzoni, A., De Simone, E., Federici, L., & Galleti, S. 2005, AJ, 130, 554
* Galleti et al. (2004) Galleti, S., Federici, L., Bellazzini, M., Fusi Pecci, F., & Macrina, S. 2004, A&A, 426, 917
* Galleti et al. (2006) Galleti, S., Federici, L., Bellazzini, M., Buzzoni, A., & Fusi Pecci, F. 2006, A&A, 456, 985
* Galleti et al. (2007) Galleti, S., Bellazzini, M., Federici, L., Buzzoni, A., & Fusi Pecci, F. 2007, A&A, 471, 127
* Gieles & Bastian (2008) Gieles, M., & Bastian, N. 2008, A&A, 482, 165
* Grillmair et al. (1996) Grillmair, C. J., Ajhar, E. A., Faber, S. M., Baum, W. A., Holtzman, J. A., Lauer, T. R., Lynds, C. R., & O’Neil, E. J., Jr. 1996, AJ, 111, 2293
* Harris (1991) Harris, W. E. 1991, ARA&A, 29, 543
* Holland et al. (1997) Holland, S., Fahlman, G. G., & Richer, H. B. 1997, AJ, 114, 1488
* Hubble (1932) Hubble, E. P. 1932, ApJ, 76, 44
* Huchra et al. (1991) Huchra, J. P., Brodie, J. P., & Kent, S. M. 1991, ApJ, 370, 495
* Jiang et al. (2003) Jiang, L., Ma, J., Zhou, X., Chen, J., Wu, H., & Jiang Z. 2003, AJ, 125, 727
* Kent (1989) Kent, S. 1989, AJ, 97, 1614
* Kurth et al. (1999) Kurth, O. M., Fritze-v. Alvensleben, U., & Fricke, K. J. 1999, A&AS, 138, 19
* Ma et al. (2006a) Ma, J., de Grijs, R., Yang, Y., Zhou, X., Chen, J., Jiang, Z., Wu, Z., & Wu, J. 2006a, MNRAS, 368, 1443
* Ma et al. (2006b) Ma, J., et al. 2006b, A&A, 449, 143
* Ma et al. (2006c) Ma, J., et al. 2006c, ApJ, 636, L93
* Ma et al. (2007a) Ma, J., et al. 2007a, ApJ, 659, 359
* Ma et al. (2007b) Ma, J., et al. 2007b, MNRAS, 376, 1621
* Ma et al. (2009a) Ma, J., et al. 2009a, AJ, 137, 4884
* Ma et al. (2009b) Ma, J., et al. 2009b, RAA, 9, 641
* McConnachie et al. (2005) McConnachie, A. W., Irwin, M. J., Ferguson, A. M. N., Ibata, R. A., Lewis, G. F., & Tanvir, N. 2005, MNRAS, 356, 979
* Macri (2001) Macri, L. M. 2001, ApJ, 549, 721
* Perrett et al. (2002) Perrett, K. M., Bridges, T. J., Hanes, D. A., Irwin, M. J., Brodie, J. P., Carter, D., Huchra, J. P., & Watson, F. G. 2002, AJ, 123, 2490
* Perina et al. (2009a) Perina, S. et al. 2009a, A&A, 494, 933
* Perina et al. (2009b) Perina, S. et al. 2009b, A&A, in press (arXiv:0910.0204)
* Puzia et al. (2005) Puzia, T. H., Perrett, K. M., & Bridges, T. J. 2005, A&A, 434, 909
* Racine (1991) Racine, R. 1991, AJ, 101, 865
* Rey et al. (2007) Rey, S. C., et al. 2007, ApJS, 173, 643
* Rich et al. (2005) Rich, R. M., Corsi, C. E., Cacciari, C., Federici, L., Fusi Pecci, F., Djorgovski, S. G., & Freedman, W. L. 2005, AJ, 129, 2670
* Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161
* Sargent et al. (1977) Sargent, W. L. W., Kowal, C. T., Hartwick, F. D. A., & van den Bergh, S. 1977, AJ, 82, 947
* Schulz et al. (2002) Schulz, J., Fritze-v. Alvensleben, U., Möller, C. S., & Fricke, K. J. 2002, A&A, 392, 1
* Stanek & Garnavich (1998) Stanek K. Z., & Garnavich, P. M. 1998, ApJ, 503, 131
* Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191
* Vetes̆nik (1962) Vetes̆nik, M. 1962, Bull. Astron. Inst. Czech., 13, 180
* Williams & Hodge (2001a) Williams, B. F., & Hodge, P. W. 2001a, ApJ, 548, 190
* Williams & Hodge (2001b) Williams, B. F., & Hodge, P. W. 2001b, ApJ, 559, 851
* Wu et al. (2005) Wu, H., Shao, Z. Y., Mo, H. J., Xia, X. Y., & Deng, Z. G. 2005, ApJ, 622, 244
* Zhou et al. (2003) Zhou, X., et al. 2003, A&A, 397, 361
Table 1BATC filter parameters Filter | Central wavelength | Bandwidth
---|---|---
| (Å) | (Å)
$a$ | 3360 | 360
$b$ | 3890 | 340
$c$ | 4210 | 320
$d$ | 4550 | 340
$e$ | 4925 | 390
$f$ | 5270 | 340
$g$ | 5795 | 310
$h$ | 6075 | 310
$i$ | 6660 | 480
$j$ | 7050 | 300
$k$ | 7490 | 330
$m$ | 8020 | 260
$n$ | 8480 | 180
$o$ | 9190 | 260
$p$ | 9745 | 270
Table 2BATC intermediate-band photometry (magnitudes) of 104 M31 GCs and GC candidates. Object | $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $m$ | $n$ | $o$ | $p$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
B001 | 19.08 | 18.94 | 18.32 | 17.68 | 17.49 | 17.23 | 16.78 | 16.65 | 16.37 | 16.28 | 16.10 | 15.89 | 15.89 | 15.72 | 15.67
| 0.190 | 0.045 | 0.022 | 0.038 | 0.032 | 0.016 | 0.021 | 0.011 | 0.008 | 0.013 | 0.012 | 0.011 | 0.016 | 0.021 | 0.019
B003 | … | 18.67 | 18.35 | 18.01 | 17.93 | 17.74 | 17.38 | 17.28 | 17.15 | 17.10 | 16.98 | 16.85 | 16.80 | 16.77 | 16.69
| … | 0.039 | 0.022 | 0.045 | 0.046 | 0.023 | 0.031 | 0.019 | 0.015 | 0.024 | 0.024 | 0.021 | 0.043 | 0.044 | 0.051
B005 | 18.07 | 17.23 | 16.66 | 16.01 | 15.94 | 15.78 | 15.43 | 15.32 | 15.08 | 15.05 | 14.90 | 14.86 | 14.86 | 14.68 | 14.62
| 0.037 | 0.019 | 0.017 | 0.016 | 0.012 | 0.010 | 0.010 | 0.008 | 0.006 | 0.007 | 0.009 | 0.007 | 0.008 | 0.011 | 0.013
B018 | 19.68 | 18.58 | 18.18 | 17.84 | 17.77 | 17.69 | 17.36 | 17.31 | 17.12 | 17.07 | 16.89 | 16.90 | 16.94 | 16.82 | 16.87
| 0.144 | 0.041 | 0.054 | 0.050 | 0.030 | 0.031 | 0.038 | 0.027 | 0.025 | 0.032 | 0.034 | 0.030 | 0.040 | 0.058 | 0.077
B020D | … | … | 18.43 | 18.09 | 17.80 | 17.52 | 17.16 | 17.09 | 16.85 | 16.68 | 16.55 | 16.40 | 16.37 | 16.13 | 16.12
| … | … | 0.056 | 0.044 | 0.037 | 0.033 | 0.027 | 0.024 | 0.023 | 0.021 | 0.022 | 0.018 | 0.033 | 0.024 | 0.037
B024 | … | 18.55 | 17.87 | 17.66 | 17.16 | 16.84 | 16.48 | 16.43 | 16.30 | 16.20 | 16.02 | 15.83 | 15.94 | 15.80 | 15.77
| … | 0.043 | 0.020 | 0.044 | 0.025 | 0.016 | 0.025 | 0.011 | 0.015 | 0.022 | 0.014 | 0.012 | 0.031 | 0.033 | 0.027
B046 | … | 19.23 | 18.66 | 18.36 | 18.21 | 17.85 | 17.39 | 17.45 | 17.29 | 17.15 | 17.02 | 16.81 | 17.03 | 17.20 | 16.70
| … | 0.076 | 0.034 | 0.097 | 0.085 | 0.036 | 0.053 | 0.028 | 0.076 | 0.086 | 0.039 | 0.035 | 0.081 | 0.104 | 0.070
B058 | 17.11 | 16.14 | 15.75 | 15.49 | 15.23 | 15.12 | 14.81 | 14.79 | 14.55 | 14.50 | 14.41 | 14.38 | 14.44 | 14.27 | 14.25
| 0.016 | 0.005 | 0.007 | 0.010 | 0.006 | 0.005 | 0.005 | 0.004 | 0.003 | 0.004 | 0.006 | 0.004 | 0.005 | 0.008 | 0.011
B083 | 19.10 | 18.73 | 17.97 | 17.90 | 17.32 | 17.03 | 16.70 | 16.63 | 16.55 | 16.43 | 16.20 | 15.98 | 16.20 | 16.09 | 15.91
| 0.138 | 0.045 | 0.020 | 0.084 | 0.039 | 0.021 | 0.031 | 0.015 | 0.026 | 0.037 | 0.017 | 0.014 | 0.044 | 0.046 | 0.041
B085 | 18.89 | 17.73 | 17.48 | 17.27 | 17.08 | 16.97 | 16.67 | 16.60 | 16.41 | 16.38 | 16.31 | 16.23 | 16.33 | 16.10 | 16.14
| 0.071 | 0.014 | 0.027 | 0.035 | 0.017 | 0.015 | 0.018 | 0.011 | 0.008 | 0.013 | 0.021 | 0.012 | 0.020 | 0.034 | 0.032
B138D | … | 18.51 | 18.39 | 17.61 | 17.04 | 16.84 | 16.48 | 16.38 | 16.16 | 16.07 | 15.94 | 15.70 | 15.69 | 15.66 | 15.47
| … | 0.050 | 0.021 | 0.035 | 0.023 | 0.012 | 0.015 | 0.009 | 0.007 | 0.012 | 0.010 | 0.010 | 0.013 | 0.019 | 0.018
B140D | … | 19.27 | 18.99 | 18.45 | 18.04 | 17.64 | 17.39 | 17.25 | 16.99 | 16.93 | 16.80 | 16.64 | 16.55 | 16.40 | 16.27
| … | 0.098 | 0.036 | 0.075 | 0.053 | 0.019 | 0.028 | 0.015 | 0.010 | 0.021 | 0.017 | 0.014 | 0.022 | 0.029 | 0.035
B141D | … | 19.22 | 18.76 | 18.22 | 17.87 | 17.33 | 17.12 | 16.91 | 16.67 | 16.65 | 16.52 | 16.32 | 16.18 | 16.09 | 16.03
| … | 0.096 | 0.030 | 0.063 | 0.046 | 0.015 | 0.022 | 0.012 | 0.008 | 0.017 | 0.015 | 0.012 | 0.018 | 0.024 | 0.028
B142D | 18.84 | 19.83 | 19.61 | 19.33 | 18.78 | 18.71 | 18.27 | 18.32 | 18.09 | 18.07 | 18.05 | 17.63 | 18.18 | 17.70 | 17.51
| 0.164 | 0.151 | 0.072 | 0.186 | 0.131 | 0.049 | 0.066 | 0.044 | 0.024 | 0.051 | 0.055 | 0.032 | 0.125 | 0.104 | 0.093
B144D | … | 19.93 | 19.05 | 18.25 | 18.04 | 18.06 | 17.55 | 17.42 | 17.20 | 17.10 | 16.95 | 16.79 | 16.84 | 16.70 | 16.67
| … | 0.176 | 0.041 | 0.060 | 0.050 | 0.025 | 0.032 | 0.017 | 0.012 | 0.024 | 0.019 | 0.017 | 0.034 | 0.038 | 0.048
B156D | … | 18.91 | 18.80 | 18.23 | 18.04 | 17.89 | 17.74 | 17.65 | 17.53 | 17.48 | 17.23 | 17.24 | 17.13 | 17.29 | 17.54
| … | 0.079 | 0.032 | 0.063 | 0.053 | 0.023 | 0.041 | 0.022 | 0.014 | 0.030 | 0.023 | 0.024 | 0.043 | 0.072 | 0.115
B157D | … | 19.62 | 19.23 | 18.46 | 18.06 | 17.78 | 17.40 | 17.32 | 17.13 | 17.06 | 16.91 | 16.66 | 16.70 | 16.58 | 16.47
| … | 0.107 | 0.044 | 0.067 | 0.053 | 0.020 | 0.030 | 0.017 | 0.011 | 0.021 | 0.019 | 0.015 | 0.029 | 0.044 | 0.035
B165D | … | 19.14 | 18.79 | 18.17 | 17.97 | 17.78 | 17.60 | 17.48 | 17.28 | 17.07 | 17.10 | 16.87 | 16.79 | 16.90 | 16.64
| … | 0.061 | 0.030 | 0.061 | 0.045 | 0.021 | 0.033 | 0.016 | 0.012 | 0.020 | 0.022 | 0.018 | 0.029 | 0.058 | 0.037
B166D | … | 19.53 | 19.43 | 18.33 | 18.33 | 18.18 | 17.73 | 17.62 | 17.41 | 17.30 | 17.18 | 17.03 | 17.00 | 16.97 | 17.07
| … | 0.055 | 0.147 | 0.065 | 0.037 | 0.039 | 0.038 | 0.022 | 0.012 | 0.020 | 0.035 | 0.020 | 0.029 | 0.052 | 0.089
B167D | … | 19.07 | 18.55 | 18.33 | 18.26 | 17.89 | 17.56 | 17.52 | 17.43 | 17.38 | 17.31 | 17.09 | 17.15 | 17.05 | 17.11
| … | 0.051 | 0.031 | 0.084 | 0.086 | 0.030 | 0.054 | 0.023 | 0.036 | 0.050 | 0.034 | 0.027 | 0.058 | 0.103 | 0.092
B172D | 19.55 | 18.76 | 18.52 | 17.82 | 18.11 | 18.10 | 17.92 | 17.89 | 17.82 | 17.89 | 17.97 | 17.79 | 17.99 | 18.05 | 17.89
| 0.094 | 0.043 | 0.063 | 0.048 | 0.033 | 0.035 | 0.045 | 0.025 | 0.025 | 0.044 | 0.078 | 0.043 | 0.112 | 0.160 | 0.134
B177D | 20.11 | 19.37 | 18.97 | 18.50 | 18.40 | 18.34 | 18.09 | 17.98 | 17.78 | 17.87 | 17.74 | 17.81 | 18.04 | 17.76 | 17.75
| 0.158 | 0.062 | 0.095 | 0.074 | 0.045 | 0.041 | 0.051 | 0.028 | 0.021 | 0.034 | 0.055 | 0.040 | 0.074 | 0.099 | 0.143
B181D | … | 19.12 | 18.64 | 18.19 | 17.94 | 17.85 | 17.65 | 17.51 | 17.38 | 17.30 | 17.26 | 17.16 | 17.25 | 17.34 | 17.03
| … | 0.074 | 0.028 | 0.055 | 0.038 | 0.023 | 0.036 | 0.017 | 0.017 | 0.026 | 0.025 | 0.030 | 0.054 | 0.057 | 0.062
B196 | … | 18.43 | 18.15 | 17.77 | 17.58 | 17.40 | 17.16 | 17.05 | 16.79 | 16.74 | 16.64 | 16.65 | 16.62 | 16.58 | 16.60
| … | 0.071 | 0.021 | 0.027 | 0.025 | 0.022 | 0.039 | 0.021 | 0.011 | 0.034 | 0.020 | 0.015 | 0.037 | 0.048 | 0.035
B223D | … | … | 19.31 | 18.64 | 18.10 | 17.94 | 17.39 | 17.29 | 17.03 | 16.94 | 16.70 | 16.60 | 16.48 | 16.44 | 16.20
| … | … | 0.132 | 0.069 | 0.044 | 0.083 | 0.040 | 0.019 | 0.009 | 0.020 | 0.027 | 0.016 | 0.024 | 0.035 | 0.045
B240 | 17.13 | 16.39 | 15.93 | 15.69 | 15.50 | 15.35 | 15.11 | 15.00 | 14.85 | 14.78 | 14.68 | 14.59 | 14.61 | 14.56 | 14.48
| 0.035 | 0.009 | 0.004 | 0.009 | 0.006 | 0.005 | 0.006 | 0.003 | 0.003 | 0.005 | 0.005 | 0.004 | 0.007 | 0.006 | 0.017
B244 | … | 19.00 | 18.95 | 18.36 | 18.43 | 18.38 | 17.97 | 17.98 | 17.82 | 17.72 | 17.63 | 17.55 | 17.46 | 17.52 | 17.40
| … | 0.058 | 0.043 | 0.067 | 0.066 | 0.037 | 0.054 | 0.033 | 0.028 | 0.037 | 0.039 | 0.032 | 0.071 | 0.077 | 0.092
B245D | … | 19.86 | 19.27 | 18.69 | 18.60 | 18.36 | 18.00 | 18.14 | 18.00 | 17.88 | 17.88 | 17.74 | 17.88 | 18.01 | 18.11
| … | 0.177 | 0.135 | 0.083 | 0.083 | 0.131 | 0.081 | 0.054 | 0.040 | 0.058 | 0.076 | 0.061 | 0.089 | 0.169 | 0.251
B257 | … | … | 19.32 | 18.62 | 18.34 | 17.99 | 17.50 | 17.35 | 16.99 | 16.83 | 16.62 | 16.52 | 16.40 | 16.16 | 16.11
| … | … | 0.085 | 0.056 | 0.056 | 0.049 | 0.042 | 0.037 | 0.032 | 0.032 | 0.033 | 0.029 | 0.043 | 0.032 | 0.039
B260 | … | … | 20.51 | 19.76 | 19.53 | 18.95 | 18.25 | 18.06 | 17.57 | 17.35 | 17.02 | 16.82 | 16.62 | 16.27 | 16.12
| … | … | 0.399 | 0.235 | 0.226 | 0.147 | 0.105 | 0.087 | 0.066 | 0.058 | 0.050 | 0.043 | 0.058 | 0.035 | 0.036
B261D | 19.10 | 18.32 | 18.10 | 17.82 | 17.65 | 17.45 | 17.28 | 17.34 | 17.27 | 17.17 | 17.09 | 16.93 | 17.09 | 16.98 | 17.32
| 0.120 | 0.034 | 0.047 | 0.026 | 0.035 | 0.062 | 0.041 | 0.022 | 0.013 | 0.026 | 0.039 | 0.023 | 0.042 | 0.048 | 0.115
Table 2Continued. Object | $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $m$ | $n$ | $o$ | $p$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
B266 | … | … | 20.00 | 19.47 | 18.86 | 18.67 | 18.05 | 17.93 | 17.63 | 17.49 | 17.30 | 17.14 | 17.02 | 16.91 | 16.83
| … | … | 0.150 | 0.103 | 0.065 | 0.056 | 0.042 | 0.037 | 0.029 | 0.031 | 0.030 | 0.031 | 0.052 | 0.038 | 0.059
B270D | … | 18.62 | 18.25 | 18.05 | 17.84 | 17.62 | 17.42 | 17.27 | 17.15 | 17.05 | 16.98 | 16.92 | 16.87 | 16.93 | 16.83
| … | 0.052 | 0.020 | 0.043 | 0.026 | 0.019 | 0.030 | 0.012 | 0.012 | 0.018 | 0.019 | 0.017 | 0.031 | 0.033 | 0.129
B272 | … | … | 19.75 | 19.01 | 18.61 | 18.39 | 17.91 | 17.82 | 17.58 | 17.40 | 17.16 | 17.02 | 16.95 | 16.74 | 16.56
| … | … | 0.084 | 0.054 | 0.039 | 0.031 | 0.033 | 0.028 | 0.026 | 0.026 | 0.035 | 0.027 | 0.056 | 0.034 | 0.046
B281 | … | … | 18.62 | 18.23 | 17.99 | 17.80 | 17.53 | 17.42 | 17.23 | 17.16 | 17.00 | 16.90 | 16.90 | 16.67 | 16.67
| … | … | 0.034 | 0.025 | 0.022 | 0.025 | 0.024 | 0.024 | 0.025 | 0.026 | 0.030 | 0.027 | 0.042 | 0.034 | 0.057
B283D | 18.97 | 18.39 | 18.07 | 17.82 | 17.58 | 17.59 | 17.30 | 17.29 | 17.15 | 17.14 | 17.00 | 16.86 | 16.92 | 16.73 | 16.79
| 0.105 | 0.037 | 0.042 | 0.030 | 0.033 | 0.058 | 0.036 | 0.024 | 0.014 | 0.026 | 0.031 | 0.022 | 0.041 | 0.046 | 0.071
B283 | … | 19.45 | 18.66 | 18.42 | 18.03 | 17.89 | 17.55 | 17.47 | 17.27 | 17.18 | 17.04 | 16.93 | 17.07 | 16.78 | 16.80
| … | 0.132 | 0.034 | 0.062 | 0.036 | 0.030 | 0.040 | 0.023 | 0.021 | 0.027 | 0.031 | 0.025 | 0.071 | 0.028 | 0.134
B289D | 19.63 | 19.33 | 19.07 | 18.62 | 18.51 | 18.75 | 18.02 | 18.04 | 17.96 | 17.95 | 17.71 | 17.62 | 17.72 | 17.62 | 17.47
| 0.180 | 0.068 | 0.100 | 0.053 | 0.076 | 0.169 | 0.077 | 0.039 | 0.025 | 0.053 | 0.071 | 0.042 | 0.083 | 0.078 | 0.142
B292D | … | 19.06 | 19.01 | 18.34 | 18.00 | 17.91 | 17.62 | 17.51 | 17.38 | 17.37 | 17.33 | 17.05 | 17.04 | 17.12 | 16.85
| … | 0.055 | 0.102 | 0.041 | 0.047 | 0.092 | 0.050 | 0.028 | 0.016 | 0.032 | 0.047 | 0.025 | 0.044 | 0.052 | 0.077
B292 | 19.01 | 18.18 | 17.81 | 17.43 | 17.29 | 17.22 | 16.96 | 16.90 | 16.76 | 16.70 | 16.70 | 16.54 | 16.58 | 16.43 | 16.46
| 0.183 | 0.026 | 0.014 | 0.034 | 0.030 | 0.015 | 0.019 | 0.012 | 0.008 | 0.016 | 0.016 | 0.013 | 0.022 | 0.031 | 0.030
B293D | 19.58 | 18.58 | 18.36 | 18.08 | 18.14 | 17.79 | 17.76 | 17.60 | 17.59 | 17.50 | 17.36 | 17.32 | 17.38 | 17.08 | 17.32
| 0.186 | 0.065 | 0.022 | 0.034 | 0.047 | 0.040 | 0.065 | 0.033 | 0.024 | 0.036 | 0.041 | 0.033 | 0.135 | 0.076 | 0.099
B297D | … | … | 19.46 | 18.64 | 18.11 | 17.77 | 17.26 | 17.14 | 16.85 | 16.75 | 16.48 | 16.16 | 16.15 | 16.12 | 15.77
| … | … | 0.149 | 0.049 | 0.051 | 0.080 | 0.040 | 0.019 | 0.010 | 0.017 | 0.024 | 0.013 | 0.022 | 0.025 | 0.030
B301 | 19.56 | 18.62 | 18.14 | 17.36 | 17.40 | 17.28 | 16.96 | 16.85 | 16.60 | 16.57 | 16.52 | 16.44 | 16.41 | 16.34 | 16.18
| 0.093 | 0.028 | 0.043 | 0.030 | 0.020 | 0.018 | 0.022 | 0.013 | 0.008 | 0.013 | 0.021 | 0.015 | 0.019 | 0.029 | 0.041
B303 | 19.52 | 18.62 | 18.26 | 17.96 | 18.06 | 18.00 | 18.04 | 17.88 | 17.84 | 18.24 | 18.56 | 18.18 | 18.41 | 18.28 | 18.10
| 0.104 | 0.033 | 0.053 | 0.055 | 0.032 | 0.035 | 0.057 | 0.034 | 0.029 | 0.074 | 0.166 | 0.072 | 0.117 | 0.211 | 0.244
B304 | 18.92 | 17.84 | 17.53 | 17.19 | 17.13 | 16.93 | 16.65 | 16.58 | 16.43 | 16.40 | 16.34 | 16.19 | 16.19 | 16.14 | 16.14
| 0.189 | 0.021 | 0.011 | 0.025 | 0.030 | 0.013 | 0.017 | 0.010 | 0.007 | 0.013 | 0.013 | 0.011 | 0.022 | 0.028 | 0.030
B305 | 19.89 | 19.06 | 18.67 | 17.98 | 18.11 | 18.03 | 17.74 | 17.61 | 17.51 | 17.53 | 17.45 | 17.44 | 17.46 | 17.29 | 16.94
| 0.120 | 0.054 | 0.076 | 0.059 | 0.031 | 0.034 | 0.041 | 0.026 | 0.017 | 0.027 | 0.054 | 0.034 | 0.049 | 0.066 | 0.080
B306 | 19.47 | 18.17 | 17.69 | 16.71 | 16.78 | 16.54 | 16.07 | 15.88 | 15.57 | 15.48 | 15.29 | 15.17 | 15.15 | 14.91 | 14.79
| 0.113 | 0.023 | 0.033 | 0.020 | 0.013 | 0.011 | 0.011 | 0.008 | 0.004 | 0.006 | 0.008 | 0.006 | 0.007 | 0.010 | 0.015
B307 | 19.79 | 18.52 | 18.12 | 17.58 | 17.59 | 17.51 | 17.26 | 17.17 | 16.97 | 16.97 | 16.74 | 16.74 | 16.85 | 16.58 | 16.46
| 0.152 | 0.033 | 0.051 | 0.038 | 0.023 | 0.021 | 0.028 | 0.018 | 0.012 | 0.019 | 0.029 | 0.019 | 0.030 | 0.039 | 0.054
B309 | 19.98 | 18.78 | 18.55 | 17.80 | 17.88 | 17.75 | 17.44 | 17.31 | 17.12 | 17.10 | 17.15 | 17.03 | 17.04 | 17.01 | 17.05
| 0.148 | 0.054 | 0.077 | 0.054 | 0.033 | 0.034 | 0.037 | 0.021 | 0.020 | 0.028 | 0.056 | 0.035 | 0.044 | 0.088 | 0.118
B310 | 18.99 | 18.05 | 17.77 | 17.45 | 17.32 | 17.16 | 16.85 | 16.76 | 16.63 | 16.57 | 16.50 | 16.35 | 16.33 | 16.27 | 16.24
| 0.153 | 0.027 | 0.014 | 0.031 | 0.025 | 0.014 | 0.019 | 0.010 | 0.009 | 0.015 | 0.016 | 0.014 | 0.024 | 0.027 | 0.033
B311 | 17.64 | 16.62 | 16.33 | 15.81 | 15.79 | 15.61 | 15.32 | 15.20 | 14.98 | 14.95 | 14.83 | 14.79 | 14.79 | 14.63 | 14.61
| 0.024 | 0.007 | 0.011 | 0.012 | 0.007 | 0.006 | 0.007 | 0.005 | 0.003 | 0.005 | 0.008 | 0.005 | 0.007 | 0.010 | 0.011
B312 | 18.04 | 16.91 | 16.47 | 16.01 | 15.86 | 15.68 | 15.28 | 15.22 | 15.01 | 14.94 | 14.82 | 14.65 | 14.67 | 14.59 | 14.53
| 0.082 | 0.009 | 0.006 | 0.012 | 0.010 | 0.006 | 0.007 | 0.005 | 0.003 | 0.006 | 0.005 | 0.005 | 0.007 | 0.010 | 0.009
B313 | 18.85 | 17.99 | 17.47 | 16.82 | 16.74 | 16.57 | 16.17 | 16.08 | 15.82 | 15.76 | 15.58 | 15.49 | 15.47 | 15.24 | 15.22
| 0.063 | 0.026 | 0.034 | 0.024 | 0.016 | 0.012 | 0.013 | 0.009 | 0.006 | 0.009 | 0.011 | 0.008 | 0.012 | 0.014 | 0.018
B315 | 17.56 | 16.63 | 16.51 | 16.20 | 16.42 | 16.38 | 16.23 | 16.23 | 16.10 | 16.13 | 16.08 | 16.09 | 16.18 | 16.03 | 15.93
| 0.024 | 0.008 | 0.013 | 0.016 | 0.012 | 0.013 | 0.014 | 0.012 | 0.010 | 0.013 | 0.018 | 0.015 | 0.025 | 0.028 | 0.034
B316 | … | 18.28 | 17.75 | 17.25 | 17.24 | 17.21 | 16.95 | 16.82 | 16.67 | 16.65 | 16.55 | 16.57 | 16.60 | 16.48 | 16.42
| … | 0.028 | 0.038 | 0.030 | 0.020 | 0.021 | 0.025 | 0.018 | 0.014 | 0.019 | 0.025 | 0.019 | 0.027 | 0.038 | 0.049
B317 | 18.66 | 17.67 | 17.28 | 17.37 | 16.93 | 16.65 | 16.37 | 16.33 | 16.29 | 16.21 | 16.12 | 15.94 | 16.02 | 15.93 | 15.98
| 0.097 | 0.019 | 0.014 | 0.044 | 0.022 | 0.013 | 0.023 | 0.012 | 0.017 | 0.021 | 0.015 | 0.012 | 0.021 | 0.025 | 0.038
B325 | 18.50 | 17.81 | 17.55 | 17.12 | 17.27 | 17.19 | 16.92 | 16.86 | 16.70 | 16.71 | 16.64 | 16.63 | 16.69 | 16.42 | 16.55
| 0.053 | 0.027 | 0.034 | 0.036 | 0.026 | 0.023 | 0.024 | 0.017 | 0.014 | 0.019 | 0.034 | 0.019 | 0.030 | 0.050 | 0.053
B332D | … | … | 18.87 | 18.46 | 17.85 | 17.34 | 17.10 | 16.88 | 16.66 | 16.56 | 16.32 | 16.27 | 16.16 | 16.03 | 16.02
| … | … | 0.036 | 0.064 | 0.027 | 0.015 | 0.023 | 0.010 | 0.008 | 0.014 | 0.016 | 0.013 | 0.021 | 0.014 | 0.067
B335 | … | 19.60 | 19.15 | 18.77 | 18.36 | 18.15 | 17.64 | 17.49 | 17.12 | 17.07 | 16.76 | 16.71 | 16.69 | 16.52 | 16.23
| … | 0.101 | 0.132 | 0.116 | 0.053 | 0.050 | 0.047 | 0.033 | 0.024 | 0.031 | 0.040 | 0.024 | 0.029 | 0.051 | 0.046
B337 | 18.77 | 17.85 | 17.51 | 17.43 | 17.04 | 16.76 | 16.43 | 16.35 | 16.32 | 16.20 | 16.09 | 15.89 | 16.01 | 15.93 | 15.89
| 0.100 | 0.027 | 0.015 | 0.046 | 0.021 | 0.014 | 0.021 | 0.010 | 0.012 | 0.028 | 0.013 | 0.011 | 0.018 | 0.031 | 0.031
B338 | 16.31 | 15.36 | 15.02 | 14.59 | 14.53 | 14.41 | 14.11 | 14.05 | 13.84 | 13.82 | 13.71 | 13.71 | 13.73 | 13.57 | 13.54
| 0.011 | 0.005 | 0.005 | 0.006 | 0.004 | 0.003 | 0.004 | 0.003 | 0.002 | 0.003 | 0.004 | 0.003 | 0.003 | 0.005 | 0.008
B340D | … | … | 19.39 | 19.59 | 18.15 | 18.64 | 17.79 | 17.58 | 17.23 | 17.31 | 16.93 | 16.83 | 16.74 | 16.66 | 16.76
| … | … | 0.135 | 0.644 | 0.144 | 0.304 | 0.062 | 0.025 | 0.020 | 0.056 | 0.026 | 0.024 | 0.055 | 0.030 | 0.107
Table 2Continued. Object | $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $m$ | $n$ | $o$ | $p$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
B343 | 18.22 | 17.19 | 16.94 | 16.53 | 16.55 | 16.48 | 16.20 | 16.11 | 15.91 | 15.92 | 15.79 | 15.80 | 15.84 | 15.74 | 15.72
| 0.039 | 0.012 | 0.016 | 0.019 | 0.012 | 0.010 | 0.012 | 0.008 | 0.005 | 0.009 | 0.012 | 0.008 | 0.013 | 0.017 | 0.028
B344 | 18.17 | 17.32 | 16.79 | 16.62 | 16.22 | 15.96 | 15.59 | 15.49 | 15.40 | 15.32 | 15.16 | 14.98 | 15.07 | 14.95 | 14.93
| 0.063 | 0.015 | 0.009 | 0.018 | 0.015 | 0.008 | 0.012 | 0.006 | 0.006 | 0.010 | 0.007 | 0.006 | 0.008 | 0.015 | 0.015
B347 | 18.35 | 17.54 | 17.13 | 17.15 | 16.71 | 16.51 | 16.17 | 16.09 | 16.03 | 15.94 | 15.84 | 15.63 | 15.75 | 15.60 | 15.71
| 0.064 | 0.017 | 0.011 | 0.026 | 0.023 | 0.012 | 0.018 | 0.009 | 0.011 | 0.016 | 0.011 | 0.010 | 0.020 | 0.030 | 0.028
B348 | 19.28 | 18.49 | 17.86 | 17.63 | 17.18 | 16.84 | 16.47 | 16.48 | 16.36 | 16.26 | 16.08 | 15.89 | 15.98 | 15.83 | 15.84
| 0.172 | 0.042 | 0.024 | 0.054 | 0.036 | 0.018 | 0.030 | 0.014 | 0.020 | 0.023 | 0.017 | 0.013 | 0.027 | 0.039 | 0.033
B350 | 18.61 | 17.61 | 17.28 | 16.84 | 16.91 | 16.89 | 16.51 | 16.48 | 16.25 | 16.23 | 16.10 | 16.10 | 16.15 | 15.95 | 16.00
| 0.049 | 0.013 | 0.022 | 0.024 | 0.015 | 0.013 | 0.015 | 0.010 | 0.006 | 0.010 | 0.016 | 0.009 | 0.017 | 0.021 | 0.034
B351 | 19.34 | 18.70 | 18.33 | 18.12 | 17.84 | 17.93 | 17.36 | 17.11 | 17.14 | 17.13 | 17.04 | 16.78 | 16.85 | 16.77 | 16.77
| 0.134 | 0.053 | 0.055 | 0.040 | 0.036 | 0.102 | 0.039 | 0.019 | 0.011 | 0.023 | 0.032 | 0.018 | 0.030 | 0.051 | 0.066
B352 | 18.59 | 17.61 | 17.18 | 17.01 | 16.81 | 16.59 | 16.32 | 16.31 | 16.14 | 16.08 | 16.02 | 15.83 | 15.89 | 15.86 | 15.79
| 0.070 | 0.022 | 0.020 | 0.018 | 0.016 | 0.032 | 0.017 | 0.010 | 0.005 | 0.011 | 0.014 | 0.010 | 0.015 | 0.024 | 0.029
B354 | 19.34 | 18.86 | 18.52 | 18.31 | 18.02 | 18.08 | 17.49 | 17.52 | 17.38 | 17.34 | 17.27 | 17.06 | 17.11 | 16.96 | 16.92
| 0.139 | 0.065 | 0.065 | 0.060 | 0.039 | 0.099 | 0.042 | 0.026 | 0.014 | 0.030 | 0.041 | 0.022 | 0.042 | 0.056 | 0.073
B356 | 19.34 | 18.31 | 17.85 | 17.59 | 17.32 | 17.08 | 16.77 | 16.71 | 16.47 | 16.41 | 16.30 | 16.07 | 16.15 | 16.00 | 16.00
| 0.146 | 0.046 | 0.036 | 0.034 | 0.025 | 0.044 | 0.026 | 0.015 | 0.008 | 0.015 | 0.018 | 0.014 | 0.018 | 0.031 | 0.034
B357 | 18.84 | 18.02 | 17.52 | 17.00 | 16.86 | 16.68 | 16.47 | 16.33 | 16.12 | 16.02 | 15.92 | 15.94 | 15.88 | 15.75 | 15.74
| 0.094 | 0.041 | 0.012 | 0.014 | 0.014 | 0.013 | 0.022 | 0.011 | 0.005 | 0.014 | 0.012 | 0.009 | 0.023 | 0.024 | 0.023
B361 | 18.91 | 17.88 | 17.67 | 17.32 | 17.23 | 17.12 | 16.95 | 16.82 | 16.63 | 16.63 | 16.53 | 16.58 | 16.48 | 16.49 | 16.42
| 0.089 | 0.034 | 0.014 | 0.017 | 0.019 | 0.017 | 0.029 | 0.017 | 0.007 | 0.023 | 0.017 | 0.012 | 0.029 | 0.047 | 0.033
B365 | 18.77 | 17.75 | 17.44 | 17.24 | 16.93 | 16.76 | 16.52 | 16.45 | 16.29 | 16.26 | 16.04 | 15.92 | 15.97 | 15.95 | 15.88
| 0.088 | 0.023 | 0.027 | 0.019 | 0.019 | 0.035 | 0.021 | 0.012 | 0.006 | 0.012 | 0.022 | 0.011 | 0.018 | 0.023 | 0.034
B370 | 18.23 | 17.43 | 17.11 | 16.84 | 16.52 | 16.31 | 16.00 | 15.95 | 15.73 | 15.67 | 15.52 | 15.33 | 15.39 | 15.29 | 15.20
| 0.052 | 0.020 | 0.021 | 0.016 | 0.018 | 0.027 | 0.016 | 0.011 | 0.007 | 0.010 | 0.012 | 0.009 | 0.012 | 0.016 | 0.021
B372 | 18.82 | 17.71 | 17.43 | 17.07 | 16.81 | 16.69 | 16.36 | 16.30 | 16.10 | 16.09 | 15.91 | 15.76 | 15.82 | 15.77 | 15.69
| 0.095 | 0.024 | 0.026 | 0.020 | 0.020 | 0.033 | 0.020 | 0.012 | 0.007 | 0.011 | 0.015 | 0.011 | 0.016 | 0.023 | 0.027
B373 | 18.06 | 17.37 | 16.73 | 16.27 | 15.94 | 15.75 | 15.40 | 15.37 | 15.13 | 15.06 | 14.84 | 14.67 | 14.72 | 14.58 | 14.48
| 0.049 | 0.018 | 0.015 | 0.010 | 0.011 | 0.018 | 0.009 | 0.007 | 0.004 | 0.006 | 0.007 | 0.006 | 0.008 | 0.010 | 0.013
B375 | … | 18.92 | 18.51 | 18.34 | 17.92 | 17.73 | 17.38 | 17.32 | 17.13 | 17.02 | 16.87 | 16.78 | 16.85 | 16.66 | 16.48
| … | 0.068 | 0.033 | 0.061 | 0.037 | 0.026 | 0.035 | 0.022 | 0.020 | 0.026 | 0.031 | 0.028 | 0.043 | 0.041 | 0.106
B377 | 19.28 | 18.08 | 17.70 | 17.45 | 17.34 | 17.16 | 17.03 | 16.82 | 16.67 | 16.67 | 16.56 | 16.63 | 16.54 | 16.49 | 16.47
| 0.127 | 0.049 | 0.014 | 0.019 | 0.023 | 0.017 | 0.033 | 0.019 | 0.009 | 0.023 | 0.021 | 0.013 | 0.036 | 0.042 | 0.041
B378 | … | 18.49 | 18.38 | 18.18 | 17.80 | 17.65 | 17.36 | 17.42 | 17.23 | 17.17 | 17.05 | 16.91 | 17.09 | 16.97 | 17.05
| … | 0.044 | 0.059 | 0.047 | 0.043 | 0.070 | 0.045 | 0.036 | 0.021 | 0.036 | 0.044 | 0.032 | 0.052 | 0.071 | 0.092
B382 | … | 18.51 | 18.01 | 17.87 | 17.59 | 17.42 | 17.19 | 17.10 | 16.96 | 16.89 | 16.77 | 16.66 | 16.71 | 16.70 | 16.80
| … | 0.050 | 0.021 | 0.037 | 0.026 | 0.019 | 0.027 | 0.013 | 0.015 | 0.021 | 0.029 | 0.022 | 0.040 | 0.039 | 0.121
B383 | 18.06 | 17.30 | 16.72 | 16.30 | 16.05 | 15.88 | 15.57 | 15.43 | 15.24 | 15.16 | 14.96 | 14.87 | 14.87 | 14.75 | 14.72
| 0.088 | 0.017 | 0.007 | 0.013 | 0.009 | 0.007 | 0.008 | 0.004 | 0.004 | 0.006 | 0.006 | 0.005 | 0.008 | 0.007 | 0.020
B384 | 18.14 | 17.18 | 16.78 | 16.34 | 16.17 | 16.02 | 15.69 | 15.57 | 15.34 | 15.26 | 15.10 | 15.12 | 15.16 | 14.97 | 14.97
| 0.046 | 0.018 | 0.007 | 0.009 | 0.012 | 0.008 | 0.012 | 0.007 | 0.003 | 0.008 | 0.007 | 0.005 | 0.010 | 0.011 | 0.011
B386 | 17.75 | 16.84 | 16.48 | 16.17 | 15.86 | 15.70 | 15.38 | 15.35 | 15.14 | 15.11 | 14.97 | 14.79 | 14.84 | 14.79 | 14.71
| 0.035 | 0.009 | 0.012 | 0.008 | 0.009 | 0.017 | 0.009 | 0.006 | 0.003 | 0.005 | 0.008 | 0.005 | 0.008 | 0.011 | 0.014
B387 | 19.14 | 17.73 | 17.63 | 17.47 | 17.30 | 17.14 | 16.97 | 16.85 | 16.66 | 16.65 | 16.54 | 16.55 | 16.65 | 16.44 | 16.32
| 0.106 | 0.037 | 0.013 | 0.021 | 0.023 | 0.017 | 0.032 | 0.015 | 0.010 | 0.020 | 0.021 | 0.013 | 0.035 | 0.046 | 0.029
B393 | … | 18.25 | 17.83 | 17.54 | 17.26 | 17.09 | 16.80 | 16.69 | 16.53 | 16.49 | 16.33 | 16.25 | 16.21 | 16.14 | 16.03
| … | 0.039 | 0.016 | 0.029 | 0.018 | 0.014 | 0.019 | 0.009 | 0.008 | 0.013 | 0.013 | 0.011 | 0.026 | 0.015 | 0.046
B399 | … | 18.38 | 18.05 | 18.01 | 17.58 | 17.45 | 17.23 | 17.14 | 17.03 | 16.92 | 16.82 | 16.79 | 16.91 | 16.78 | 16.90
| … | 0.049 | 0.019 | 0.044 | 0.025 | 0.018 | 0.028 | 0.013 | 0.011 | 0.019 | 0.024 | 0.018 | 0.047 | 0.029 | 0.134
B401 | 18.67 | 17.70 | 17.39 | 17.43 | 17.07 | 16.92 | 16.69 | 16.59 | 16.47 | 16.42 | 16.29 | 16.20 | 16.23 | 16.20 | 16.22
| 0.149 | 0.030 | 0.012 | 0.029 | 0.017 | 0.013 | 0.020 | 0.010 | 0.009 | 0.014 | 0.014 | 0.013 | 0.023 | 0.019 | 0.085
B403 | 18.77 | 17.75 | 17.20 | 17.07 | 16.54 | 16.40 | 16.06 | 15.93 | 15.75 | 15.66 | 15.47 | 15.35 | 15.39 | 15.27 | 15.29
| 0.133 | 0.032 | 0.009 | 0.023 | 0.012 | 0.009 | 0.011 | 0.006 | 0.005 | 0.008 | 0.009 | 0.007 | 0.013 | 0.009 | 0.030
B405 | 17.33 | 16.26 | 15.85 | 15.87 | 15.43 | 15.29 | 15.01 | 14.90 | 14.74 | 14.69 | 14.52 | 14.43 | 14.47 | 14.41 | 14.42
| 0.043 | 0.010 | 0.004 | 0.010 | 0.006 | 0.005 | 0.005 | 0.003 | 0.003 | 0.005 | 0.005 | 0.004 | 0.006 | 0.005 | 0.018
B420 | … | 19.36 | 19.18 | 18.45 | 17.91 | 17.67 | 17.36 | 17.20 | 16.98 | 16.90 | 16.79 | 16.56 | 16.50 | 16.48 | 16.20
| … | 0.110 | 0.046 | 0.073 | 0.039 | 0.020 | 0.030 | 0.015 | 0.011 | 0.021 | 0.018 | 0.015 | 0.026 | 0.035 | 0.032
B448 | 19.60 | 18.24 | 18.15 | 17.92 | 17.95 | 17.89 | 17.59 | 17.58 | 17.44 | 17.40 | 17.45 | 17.38 | 17.40 | 17.44 | 17.18
| 0.136 | 0.037 | 0.060 | 0.067 | 0.052 | 0.058 | 0.055 | 0.050 | 0.048 | 0.054 | 0.069 | 0.059 | 0.071 | 0.121 | 0.088
B457 | 19.20 | 18.25 | 17.91 | 17.90 | 17.48 | 17.21 | 16.91 | 16.84 | 16.76 | 16.66 | 16.55 | 16.35 | 16.42 | 16.44 | 16.43
| 0.139 | 0.041 | 0.020 | 0.072 | 0.034 | 0.021 | 0.035 | 0.015 | 0.023 | 0.039 | 0.019 | 0.017 | 0.045 | 0.061 | 0.051
Table 2Continued. Object | $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $m$ | $n$ | $o$ | $p$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
B461 | 19.41 | 18.91 | 18.38 | 18.15 | 17.67 | 17.49 | 17.09 | 16.99 | 16.85 | 16.78 | 16.62 | 16.41 | 16.54 | 16.39 | 16.43
| 0.157 | 0.056 | 0.027 | 0.067 | 0.047 | 0.020 | 0.036 | 0.016 | 0.018 | 0.034 | 0.017 | 0.014 | 0.026 | 0.050 | 0.054
B462 | … | 19.03 | 18.77 | 18.67 | 18.42 | 18.19 | 17.80 | 17.74 | 17.66 | 17.52 | 17.45 | 17.30 | 17.44 | 17.29 | 17.37
| … | 0.063 | 0.040 | 0.112 | 0.092 | 0.036 | 0.065 | 0.030 | 0.036 | 0.061 | 0.032 | 0.027 | 0.058 | 0.090 | 0.130
B467 | … | 18.57 | 18.25 | 18.00 | 17.83 | 17.51 | 17.35 | 17.25 | 17.11 | 17.02 | 17.08 | 16.82 | 16.86 | 16.75 | 16.84
| … | 0.052 | 0.052 | 0.044 | 0.035 | 0.060 | 0.039 | 0.021 | 0.012 | 0.024 | 0.038 | 0.019 | 0.032 | 0.046 | 0.063
B472 | 17.60 | 16.43 | 16.00 | 15.68 | 15.40 | 15.25 | 14.95 | 14.89 | 14.75 | 14.66 | 14.58 | 14.46 | 14.40 | 14.35 | 14.35
| 0.025 | 0.024 | 0.009 | 0.007 | 0.006 | 0.006 | 0.006 | 0.006 | 0.006 | 0.006 | 0.007 | 0.006 | 0.009 | 0.008 | 0.012
B475 | 19.38 | 18.08 | 17.74 | 17.66 | 17.50 | 17.36 | 17.28 | 17.37 | 17.21 | 17.17 | 17.00 | 16.97 | 16.95 | 16.71 | 16.57
| 0.156 | 0.047 | 0.050 | 0.054 | 0.062 | 0.074 | 0.066 | 0.069 | 0.053 | 0.061 | 0.076 | 0.061 | 0.067 | 0.068 | 0.071
B476 | … | … | 19.26 | 18.87 | 18.72 | 18.50 | 18.14 | 18.12 | 17.95 | 17.87 | 17.70 | 17.49 | 17.68 | 17.46 | 17.68
| … | … | 0.064 | 0.093 | 0.076 | 0.050 | 0.068 | 0.037 | 0.037 | 0.053 | 0.070 | 0.054 | 0.096 | 0.083 | 0.254
B483 | … | 18.86 | 18.62 | 18.49 | 18.32 | 18.17 | 18.33 | 18.11 | 18.13 | 18.20 | 18.11 | 17.86 | 18.03 | 18.07 | 17.58
| … | 0.059 | 0.071 | 0.054 | 0.062 | 0.103 | 0.096 | 0.044 | 0.039 | 0.069 | 0.095 | 0.055 | 0.099 | 0.130 | 0.146
B486 | … | 18.68 | 18.19 | 18.03 | 17.83 | 17.65 | 17.48 | 17.40 | 17.23 | 17.15 | 17.11 | 17.05 | 17.03 | 16.89 | 16.86
| … | 0.055 | 0.019 | 0.042 | 0.027 | 0.019 | 0.031 | 0.014 | 0.014 | 0.023 | 0.021 | 0.020 | 0.058 | 0.036 | 0.123
B489 | … | 19.32 | 18.88 | 18.01 | 17.65 | 17.33 | 17.12 | 16.91 | 16.70 | 16.57 | 16.40 | 16.32 | 16.26 | 16.15 | 15.98
| … | 0.112 | 0.033 | 0.035 | 0.036 | 0.030 | 0.041 | 0.020 | 0.010 | 0.021 | 0.022 | 0.013 | 0.052 | 0.034 | 0.035
B495 | … | … | 19.59 | 18.82 | 18.16 | 17.95 | 17.38 | 17.19 | 16.94 | 16.85 | 16.58 | 16.34 | 16.26 | 16.22 | 15.93
| … | … | 0.160 | 0.075 | 0.046 | 0.077 | 0.039 | 0.020 | 0.009 | 0.019 | 0.020 | 0.013 | 0.021 | 0.028 | 0.034
G260 | 19.07 | 18.05 | 17.75 | 17.57 | 17.22 | 17.11 | 16.79 | 16.74 | 16.62 | 16.58 | 16.45 | 16.25 | 16.29 | 16.28 | 16.26
| 0.112 | 0.027 | 0.034 | 0.025 | 0.023 | 0.041 | 0.025 | 0.013 | 0.007 | 0.015 | 0.023 | 0.013 | 0.023 | 0.031 | 0.047
Table 3GALEX, optical broad-band, and 2MASS NIR photometry of 104 M31 GCs and
GC candidates.
Object | $c$† | FUV | NUV | $U$ | $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---|---|---|---|---|---
B001 | 1 | … | … | 18.82 | 18.33 | 17.06 | 16.47 | 15.41 | 14.69 | 13.73 | 13.84
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B003 | 1 | 23.06 | 21.55 | 18.40 | 18.35 | 17.57 | 17.07 | 16.41 | 15.95 | 15.15 | 15.51
| | 0.20 | 0.04 | 0.04 | 0.02 | 0.01 | 0.03 | 0.02 | 0.08 | 0.10 | 0.12
B005 | 1 | … | 21.08 | 16.12 | 16.04 | 15.44 | 14.99 | 14.66 | 13.40 | 12.69 | 12.51
| | … | 0.12 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B018 | 1 | … | 22.09 | 18.47 | 18.25 | 17.53 | 17.00 | 16.38 | 15.47 | 14.77 | 14.59
| | … | 0.15 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B020D | 1 | … | … | 18.99 | 18.43 | 17.44 | … | 16.04 | 14.91 | 14.62 | 13.98
| | … | … | 0.08 | 0.05 | 0.05 | … | 0.05 | 0.04 | 0.07 | 0.05
B024 | 1 | … | 22.51 | 18.40 | 17.75 | 16.80 | 16.27 | 15.65 | 14.80 | 14.25 | 13.98
| | … | 0.07 | 0.08 | 0.02 | 0.01 | 0.01 | 0.01 | 0.04 | 0.07 | 0.05
B046 | 1 | 22.80 | 21.71 | 18.70 | 18.65 | 17.81 | 17.37 | 16.87 | 15.91 | 14.83 | 14.93
| | 0.16 | 0.05 | 0.09 | 0.03 | 0.01 | 0.04 | 0.03 | 0.08 | 0.07 | 0.10
B058 | 1 | 20.23 | 19.09 | 16.05 | 15.81 | 15.01 | 14.48 | 13.91 | 13.13 | 12.50 | 12.36
| | 0.04 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B083 | 1 | 22.60 | 21.45 | 17.91 | 17.85 | 17.09 | 16.55 | 15.93 | 15.32 | 14.61 | 14.59
| | 0.14 | 0.03 | 0.04 | 0.02 | 0.01 | 0.02 | 0.01 | 0.08 | 0.07 | 0.10
B085 | 1 | 22.24 | 20.60 | 17.92 | 17.55 | 16.84 | 16.38 | 15.63 | 15.03 | 14.72 | 14.29
| | 0.11 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B138D | 2 | 23.11 | 22.38 | 18.70 | 17.92 | 16.87 | 16.14 | … | 14.24 | 13.48 | 13.18
| | 0.18 | 0.06 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.04 | 0.04 | 0.05
B140D | 2 | 22.31 | 21.73 | 18.74 | 18.87 | 17.67 | 17.04 | … | 15.17 | 14.45 | 13.99
| | 0.08 | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | 0.05
B141D | 2 | … | … | 18.89 | 18.73 | 17.43 | 16.69 | … | 14.75 | 14.04 | 13.55
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.04 | 0.07 | 0.05
B142D | 2 | 21.21 | 21.21 | 19.06 | 19.80 | 18.63 | 17.93 | … | 16.99 | … | 14.98
| | 0.10 | 0.07 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.10 | … | 0.10
B144D | 2 | … | … | 19.37 | 18.85 | 17.72 | 17.06 | … | 15.70 | 15.02 | 14.96
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.10 | 0.10
B156D | 2 | 20.97 | 20.29 | 18.57 | 18.88 | 18.16 | 17.49 | … | 16.15 | … | 15.18
| | 0.04 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.10 | … | 0.12
B157D | 2 | … | … | 19.37 | 19.06 | 17.83 | 17.24 | … | 15.19 | 14.62 | 14.03
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | 0.10
B165D | 2 | 21.06 | 20.82 | 18.92 | 18.68 | 17.62 | 17.34 | … | 15.63 | 14.72 | 14.19
| | 0.06 | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | 0.10
B166D | 2 | … | … | 19.11 | 18.90 | 17.93 | 17.34 | … | 15.47 | 14.83 | 14.41
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | 0.10
B167D | 1 | 22.09 | 21.34 | 18.41 | 18.54 | 17.79 | 17.31 | … | 16.36 | 15.65 | 15.28
| | 0.07 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.10 | 0.10 | 0.12
B172D | 2 | … | 21.71 | 18.44 | 18.64 | 17.97 | 17.61 | … | … | … | …
| | … | 0.08 | 0.08 | 0.05 | 0.05 | 0.05 | … | … | … | …
B177D | 2 | … | … | 19.11 | 18.89 | 18.09 | 17.63 | … | … | … | …
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | … | … | …
B181D | 2 | … | … | 18.73 | 18.71 | 17.72 | 17.14 | … | 15.24 | 16.41 | …
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.10 | …
B196 | 1 | … | 21.60 | 18.35 | 18.05 | 17.40 | 16.75 | 16.10 | 15.54 | 14.57 | 14.82
| | … | 0.10 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B223D | 2 | … | 22.86 | 19.30 | 18.74 | 17.75 | … | … | 15.25 | 14.19 | 13.81
| | … | 0.13 | 0.08 | 0.05 | 0.05 | … | … | 0.08 | 0.07 | 0.05
B240 | 1 | 19.90 | 18.94 | 16.03 | 15.93 | 15.23 | 14.73 | 14.20 | 13.47 | 12.90 | 12.76
| | 0.04 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B244 | 2 | … | 22.23 | 19.13 | 19.05 | 18.27 | 17.66 | 17.22 | 15.46 | 15.36 | …
| | … | 0.07 | 0.08 | 0.04 | 0.01 | 0.05 | 0.04 | 0.08 | 0.10 | …
B245D | 2 | … | … | 18.91 | 19.08 | 18.22 | 17.56 | … | 16.20 | 16.65 | …
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.10 | 0.10 | …
B257 | 2 | … | … | 19.75 | 19.43 | 20.96 | … | 16.31 | 14.79 | 13.98 | 13.69
| | … | … | 0.08 | 0.05 | 0.05 | … | 0.05 | 0.04 | 0.04 | 0.05
B260 | 2 | … | … | … | … | 18.50 | 17.52 | 16.42 | 14.69 | 13.90 | 13.74
| | … | … | … | … | 0.01 | 0.01 | 0.07 | 0.04 | 0.04 | 0.05
††footnotetext: New classification flag (RBC v.3.5): 1 = GC, 2 = candidate GC.
Table 3Continued.
Object | $c$† | FUV | NUV | $U$ | $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---|---|---|---|---|---
B261D | 2 | … | 21.31 | 18.02 | 18.06 | 17.60 | 16.97 | … | 15.35 | 15.38 | …
| | … | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.10 | …
B266 | 1 | … | … | 18.76 | 19.29 | 18.32 | 17.40 | 16.50 | 15.64 | 15.06 | 14.45
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.10
B270D | 2 | … | 21.81 | 18.24 | 17.90 | 17.50 | 16.94 | … | 15.43 | 15.09 | …
| | … | 0.07 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.10 | …
B272 | 1 | … | … | 19.94 | 19.52 | 18.20 | 17.50 | 16.61 | 14.70 | 14.40 | …
| | … | … | 0.08 | 0.06 | 0.01 | 0.05 | 0.05 | 0.04 | 0.07 | …
B281 | 1 | … | … | 19.01 | 18.51 | 17.67 | 17.11 | 16.51 | 15.87 | 14.88 | 14.26
| | … | … | 0.06 | 0.03 | 0.01 | 0.03 | 0.03 | 0.08 | 0.07 | 0.10
B283D | 2 | … | 21.02 | 18.11 | 18.12 | 17.51 | 16.97 | … | 15.16 | 14.87 | …
| | … | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | …
B283 | 1 | … | … | 19.63 | 18.70 | 17.64 | 17.20 | 16.53 | 15.73 | 14.90 | 15.05
| | … | … | 0.08 | 0.04 | 0.01 | 0.03 | 0.02 | 0.08 | 0.07 | 0.12
B289D | 1 | … | … | 18.94 | 18.94 | 18.09 | 17.74 | … | 15.49 | … | …
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | … | …
B292D | 1 | … | … | 18.95 | 18.89 | 17.81 | … | … | 15.39 | 15.25 | …
| | … | … | 0.08 | 0.05 | 0.05 | … | … | 0.08 | 0.10 | …
B292 | 1 | 21.80 | 21.02 | 17.87 | 17.89 | 17.00 | 16.62 | 16.06 | 15.41 | 15.03 | 14.71
| | 0.09 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.10
B293D | 2 | … | 21.32 | 18.33 | 18.45 | 17.91 | 17.29 | … | 16.52 | 15.99 | 15.41
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.10 | 0.10 | 0.12
B297D | 2 | … | … | 20.06 | 19.05 | 17.63 | … | … | 14.66 | 13.90 | 13.70
| | … | … | 0.08 | 0.05 | 0.05 | … | … | 0.04 | 0.04 | 0.05
B301 | 1 | … | … | 18.48 | 18.13 | 17.12 | 16.53 | 15.71 | 15.08 | 14.31 | 14.06
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B303 | 1 | … | 21.15 | 18.92 | 18.46 | 18.22 | 17.50 | 17.03 | … | … | …
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | … | … | …
B304 | 1 | 22.03 | 20.69 | 17.78 | 17.54 | 16.83 | 16.32 | 15.74 | 15.12 | 14.46 | 14.27
| | 0.08 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B305 | 1 | … | 22.69 | 18.68 | 18.82 | 17.93 | 17.21 | 17.00 | 15.80 | 15.48 | 15.34
| | … | 0.18 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.12
B306 | 1 | … | 22.31 | 18.19 | 17.55 | 16.30 | 15.59 | 14.63 | 13.55 | 12.74 | 12.58
| | … | 0.15 | 0.08 | 0.05 | 0.01 | 0.05 | 0.01 | 0.03 | 0.03 | 0.03
B307 | 1 | … | 21.84 | 19.20 | 18.19 | 17.32 | 17.15 | 16.10 | 15.22 | 14.56 | 14.18
| | … | 0.08 | 0.08 | 0.05 | 0.01 | 0.05 | 0.01 | 0.08 | 0.07 | 0.10
B309 | 1 | … | 22.00 | 18.63 | 18.47 | 17.50 | 17.00 | 15.80 | 16.01 | … | 15.92
| | … | 0.10 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.10 | … | 0.12
B310 | 1 | 22.03 | 20.87 | 18.03 | 17.80 | 17.04 | 16.54 | 15.98 | 15.24 | 14.65 | 14.63
| | 0.08 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B311 | 1 | 21.18 | 19.72 | 16.55 | 16.41 | 15.44 | 14.94 | 14.22 | 13.50 | 12.86 | 12.70
| | 0.06 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.03 | 0.03 | 0.03
B312 | 1 | 21.06 | 20.08 | 16.68 | 16.34 | 15.52 | 14.96 | 14.16 | 13.56 | 12.85 | 12.78
| | 0.10 | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.03 | 0.03 | 0.03
B313 | 1 | … | 20.77 | 18.11 | 17.41 | 16.37 | 15.79 | 14.96 | 14.06 | 13.30 | 13.11
| | … | 0.12 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B315 | 1 | 18.70 | 18.36 | 16.57 | 16.55 | 16.47 | 16.03 | 15.58 | 14.82 | 14.67 | 13.98
| | 0.02 | 0.01 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | 0.05
B316 | 1 | … | … | 18.10 | 17.77 | 17.06 | 16.39 | 16.14 | 14.71 | 14.43 | …
| | … | … | 0.08 | 0.02 | 0.01 | 0.02 | 0.02 | 0.04 | 0.07 | …
B317 | 1 | 21.67 | 20.27 | 17.58 | 17.30 | 16.57 | 16.13 | 15.71 | 14.98 | 14.53 | 14.50
| | 0.05 | 0.01 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | 0.10
B325 | 1 | 19.93 | … | … | 17.54 | 16.94 | 16.42 | 15.91 | 15.22 | 14.57 | 14.38
| | 0.06 | … | … | 0.02 | 0.01 | 0.02 | 0.02 | 0.08 | 0.07 | 0.10
B332D | 2 | … | … | 19.02 | 18.70 | 17.21 | 16.60 | … | 14.81 | 13.86 | 13.55
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.04 | 0.04 | 0.05
B335 | 1 | … | … | 19.99 | 19.04 | 17.89 | 16.91 | 16.14 | 14.94 | 14.24 | 13.94
| | … | … | 0.08 | 0.06 | 0.01 | 0.04 | 0.03 | 0.04 | 0.07 | 0.05
B337 | 1 | 21.89 | 20.70 | 17.69 | 17.52 | 16.73 | 16.24 | 15.63 | 15.06 | 14.47 | 14.06
| | 0.06 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
††footnotetext: New classification flag (RBC v.3.5): 1 = GC, 2 = candidate GC.
Table 3Continued.
Object | $c$† | FUV | NUV | $U$ | $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---|---|---|---|---|---
B338 | 1 | 19.05 | 18.28 | 15.21 | 15.06 | 14.30 | 13.67 | 13.28 | 12.45 | 11.79 | 11.67
| | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.05 | 0.01 | 0.02 | 0.02 | 0.02
B340D | 2 | … | … | 20.22 | 19.93 | 17.92 | 17.22 | … | 15.53 | 14.55 | 14.45
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | 0.10
B343 | 1 | 20.80 | 19.90 | 17.23 | 17.08 | 16.31 | 15.91 | 15.27 | 14.63 | 13.94 | 13.90
| | 0.06 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B344 | 1 | … | 20.75 | 17.01 | 16.77 | 15.95 | 15.33 | 14.87 | 14.02 | 13.40 | 13.35
| | … | 0.02 | 0.08 | 0.05 | 0.01 | 0.05 | 0.01 | 0.04 | 0.04 | 0.05
B347 | 1 | 22.10 | 20.26 | 17.38 | 17.23 | 16.50 | 15.97 | 15.49 | 14.75 | 14.04 | 14.11
| | 0.08 | 0.02 | 0.08 | 0.05 | 0.01 | 0.05 | 0.01 | 0.04 | 0.07 | 0.10
B348 | 1 | … | 22.06 | 18.33 | 17.98 | 16.79 | 16.33 | 15.64 | 14.78 | 14.34 | 13.93
| | … | 0.07 | 0.08 | 0.05 | 0.01 | 0.05 | 0.01 | 0.04 | 0.07 | 0.05
B350 | 1 | 21.77 | 20.47 | 17.47 | 17.36 | 16.74 | 16.20 | 15.64 | 14.84 | 14.46 | 14.19
| | 0.08 | 0.03 | 0.08 | 0.05 | 0.01 | 0.05 | 0.01 | 0.04 | 0.07 | 0.10
B351 | 1 | … | 21.67 | 18.61 | 18.40 | 17.55 | 17.07 | 16.43 | 15.98 | 15.22 | 14.88
| | … | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.10
B352 | 1 | 21.11 | 20.08 | 17.42 | 17.25 | 16.54 | 15.96 | 15.57 | 14.93 | 14.28 | 14.11
| | 0.05 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | 0.10
B354 | 1 | … | 21.69 | 18.63 | 17.94 | 17.81 | 17.19 | 16.74 | 16.01 | 15.21 | 15.25
| | … | 0.05 | 0.08 | 0.05 | 0.01 | 0.05 | 0.02 | 0.10 | 0.10 | 0.12
B356 | 1 | 22.33 | 21.43 | 18.42 | 18.12 | 17.34 | 16.43 | 15.71 | 14.93 | 14.29 | 14.24
| | 0.17 | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | 0.10
B357 | 1 | … | 21.81 | 17.98 | 17.49 | 16.61 | 15.97 | 15.39 | 14.48 | 13.80 | 13.68
| | … | 0.07 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B361 | 1 | 21.40 | 20.54 | 17.92 | 17.78 | 17.05 | 16.57 | 16.01 | 15.32 | 14.71 | 14.56
| | 0.07 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B365 | 1 | 21.64 | 20.58 | 17.72 | 17.51 | 16.72 | 15.90 | 15.57 | 14.97 | 14.24 | 14.10
| | 0.05 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | 0.10
B370 | 1 | 21.48 | 20.53 | 17.28 | 17.15 | 16.30 | 15.63 | 15.14 | 14.17 | 13.44 | 13.34
| | 0.05 | 0.02 | 0.03 | 0.01 | 0.01 | 0.01 | 0.01 | 0.04 | 0.04 | 0.05
B372 | 1 | … | 20.97 | 17.71 | 17.48 | 16.60 | 15.84 | 15.50 | 14.68 | 13.95 | 13.80
| | … | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B373 | 1 | … | 21.23 | 17.07 | 16.58 | 15.64 | 15.13 | 14.39 | 13.36 | 12.57 | 12.46
| | … | 0.07 | 0.03 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B375 | 1 | … | 22.44 | 19.87 | 18.51 | 17.61 | 17.07 | 16.59 | 15.49 | 14.68 | 14.32
| | … | 0.19 | 0.08 | 0.03 | 0.01 | 0.03 | 0.04 | 0.08 | 0.07 | 0.10
B377 | 1 | … | 20.70 | 17.83 | 17.77 | 17.14 | 16.64 | 16.20 | 15.43 | 15.09 | 14.44
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.10
B378 | 1 | … | 21.49 | 18.85 | 18.52 | 17.83 | 16.94 | 16.50 | 15.99 | 15.40 | 15.19
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.12
B382 | 1 | 21.91 | 20.97 | 18.28 | 18.15 | 17.36 | 16.74 | 16.08 | 15.69 | 15.09 | 15.52
| | 0.10 | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.12
B383 | 1 | … | 21.21 | 17.23 | 16.17 | 15.33 | 14.87 | 14.43 | 13.58 | 12.85 | 12.66
| | … | 0.06 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.03 | 0.03 | 0.03
B384 | 1 | … | 21.32 | 17.37 | 16.74 | 15.75 | 15.26 | 14.56 | 13.76 | 13.01 | 12.91
| | … | 0.07 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.03 | 0.04 | 0.03
B386 | 1 | 21.64 | 20.00 | 16.68 | 16.45 | 15.55 | 14.75 | 14.39 | 13.70 | 12.95 | 12.83
| | 0.08 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.03 | 0.03 | 0.03
B387 | 1 | … | 20.69 | 17.84 | 17.66 | 16.98 | 16.48 | 16.05 | 15.32 | 14.94 | 14.43
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B393 | 1 | … | 21.75 | 17.97 | 17.72 | 16.93 | 16.40 | 16.00 | 15.18 | 14.23 | 14.21
| | … | 0.09 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B399 | 1 | … | 21.00 | 18.25 | 18.05 | 17.28 | 16.85 | 16.23 | 15.80 | 15.50 | 15.35
| | … | 0.08 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.12
B401 | 1 | … | 20.43 | 17.83 | 17.51 | 16.83 | 16.50 | 15.68 | 15.09 | 14.60 | 14.56
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B403 | 1 | … | 21.59 | 17.62 | 17.19 | 16.22 | 15.68 | 15.00 | 14.15 | 13.48 | 13.24
| | … | 0.11 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B405 | 1 | 19.72 | 18.89 | 16.22 | 15.93 | 15.19 | 14.66 | … | 13.41 | 12.77 | 12.66
| | 0.06 | 0.02 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.03 | 0.03 | 0.03
††footnotetext: New classification flag (RBC v.3.5): 1 = GC, 2 = candidate GC.
Table 3Continued.
Object | $c$† | FUV | NUV | $U$ | $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---|---|---|---|---|---
B420 | 2 | … | … | 19.68 | 18.96 | 17.85 | 17.12 | 16.13 | 15.16 | 14.68 | 14.15
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B448 | 1 | 21.90 | 21.14 | 18.11 | 18.10 | 17.49 | 17.06 | 16.77 | … | … | …
| | 0.12 | 0.05 | 0.09 | 0.04 | 0.01 | 0.04 | 0.03 | … | … | …
B457 | 1 | … | 21.21 | 18.05 | 17.73 | 16.91 | 16.38 | 16.87 | 14.68 | 14.78 | …
| | … | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | …
B461 | 1 | … | 22.82 | 18.73 | 18.77 | 17.52 | 16.89 | 16.04 | 15.55 | 14.67 | 14.61
| | … | 0.14 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.07 | 0.10
B462 | 1 | … | 21.96 | 18.82 | 18.83 | 18.06 | 17.38 | 17.01 | 15.45 | … | …
| | … | 0.06 | 0.08 | 0.05 | 0.01 | 0.05 | 0.02 | 0.08 | … | …
B467 | 1 | 22.49 | 21.33 | 18.36 | 18.31 | 17.43 | 16.85 | 16.15 | 15.96 | 15.73 | 15.18
| | 0.10 | 0.03 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.12
B472 | 1 | 19.93 | 19.03 | 16.16 | 15.97 | 15.19 | 14.67 | 14.12 | 13.34 | 12.69 | 12.60
| | 0.04 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B475 | 1 | … | 21.07 | 17.97 | 17.87 | 17.56 | 17.03 | 16.91 | 16.04 | 15.01 | 14.63
| | … | 0.06 | 0.10 | 0.03 | 0.01 | 0.04 | 0.04 | 0.10 | 0.10 | 0.10
B476 | 2 | … | … | 19.52 | 19.12 | 18.12 | 17.69 | 17.00 | 15.34 | 14.68 | …
| | … | … | 0.09 | 0.05 | 0.01 | 0.06 | 0.05 | 0.08 | 0.07 | …
B483 | 1 | … | 21.73 | 18.81 | 18.73 | 18.46 | 17.75 | 17.82 | … | … | …
| | … | 0.05 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | … | … | …
B486 | 1 | … | 21.36 | 18.80 | 17.87 | 17.52 | 17.03 | 16.49 | 15.90 | 15.54 | 15.46
| | … | 0.09 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.08 | 0.10 | 0.12
B489 | 2 | … | … | 19.32 | 18.47 | 17.35 | 16.66 | 15.64 | 14.76 | 13.99 | 13.80
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05
B495 | 2 | … | … | 19.60 | 19.02 | 17.61 | 17.04 | 15.82 | 14.83 | 14.02 | 13.74
| | … | … | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.07 | 0.05
G260 | 1 | 22.69 | 21.11 | 17.64 | 17.81 | 17.01 | 16.53 | … | 15.25 | 14.80 | 14.59
| | 0.13 | 0.04 | 0.08 | 0.05 | 0.05 | 0.05 | … | 0.08 | 0.07 | 0.10
††footnotetext: New classification flag (RBC v.3.5): 1 = GC, 2 = candidate GC.
Table 4Reddening values (magnitudes) and metallicities (dex) for 104 M31 GCs
and GC candidates.
Object | $E(B-V)$ | ref.a | $\rm[Fe/H]$ | ref.b
---|---|---|---|---
B001 | 0.25$\pm$ 0.02 | 1 | $-0.58\pm$ 0.18 | 1
B003 | 0.19$\pm$ 0.02 | 1 | $-2.08\pm$ 0.07 | 4
B005 | 0.28$\pm$ 0.02 | 1 | $-1.18\pm$ 0.17 | 1
B018 | 0.20$\pm$ 0.01 | 1 | $-1.63\pm$ 0.77 | 1
B020D | 0.22$\pm$ 0.06 | 1 | $-0.76\pm$ 0.08 | 4
B024 | 0.03$\pm$ 0.02 | 1 | $-0.48\pm$ 0.30 | 3
B046 | 0.19$\pm$ 0.03 | 1 | $-1.84\pm$ 0.61 | 3
B058 | 0.13$\pm$ 0.01 | 1 | $-1.45\pm$ 0.24 | 3
B083 | 0.12$\pm$ 0.02 | 1 | $-1.18\pm$ 0.44 | 1
B085 | 0.14$\pm$ 0.02 | 1 | $-1.83\pm$ 0.40 | 3
B138D | 0.23$\pm$ 0.04 | 1 | $-0.36\pm$ 0.04 | 4
B140D | 0.45$\pm$ 0.11 | 1 | $-1.57\pm$ 0.12 | 4
B141D | 0.43$\pm$ 0.09 | 1 | $-1.07\pm$ 0.08 | 4
B142D | 0.58$\pm$ 0.03 | 1 | $-2.59\pm$ 0.24 | 4
B144D | 0.33$\pm$ 0.02 | 1 | $-1.62\pm$ 0.09 | 4
B156D | 0.45$\pm$ 0.09 | 1 | $-2.58\pm$ 0.14 | 4
B157D | 0.15$\pm$ 0.01 | 1 | $-0.09\pm$ 0.08 | 4
B165D | 0.21$\pm$ 0.07 | 1 | $-1.05\pm$ 0.13 | 4
B166D | 0.26$\pm$ 0.04 | 1 | $-1.02\pm$ 0.08 | 4
B167D | 0.23$\pm$ 0.03 | 1 | $-2.34\pm$ 0.08 | 4
B172D | 0.11$\pm$ 0.05 | 1 | $-2.51\pm$ 0.12 | 4
B177D | 0.08$\pm$ 0.02 | 1 | $-1.32\pm$ 0.01 | 4
B181D | 0.36$\pm$ 0.09 | 1 | $-2.21\pm$ 0.20 | 4
B196 | 0.26$\pm$ 0.04 | 1 | $-1.94\pm$ 0.08 | 4
B223D | 0.20$\pm$ 0.05 | 1 | $-0.23\pm$ 0.08 | 4
B240 | 0.13$\pm$ 0.00 | 1 | $-1.76\pm$ 0.18 | 3
B244 | 0.27$\pm$ 0.03 | 1 | $-1.50\pm$ 0.21 | 4
B245D | 0.52$\pm$ 0.03 | 1 | $-2.88\pm$ 0.09 | 4
B257 | 1.17$\pm$ 0.03 | 1 | $-2.05\pm$ 0.82 | 4
B260 | 0.67$\pm$ 0.02 | 1 | $-0.36\pm$ 0.10 | 4
B261D | 0.27$\pm$ 0.06 | 1 | $-2.45\pm$ 0.19 | 4
B266 | 0.98$\pm$ 0.09 | 1 | $-2.80\pm$ 0.15 | 4
B270D | 0.25$\pm$ 0.02 | 1 | $-2.28\pm$ 0.19 | 4
B272 | 0.57$\pm$ 0.04 | 1 | $-1.25\pm$ 0.16 | 1
B281 | 0.12$\pm$ 0.02 | 1 | $-0.87\pm$ 0.52 | 1
B283D | 0.16$\pm$ 0.04 | 1 | $-1.55\pm$ 0.17 | 4
B283 | 0.08$\pm$ 0.06 | 2 | $-0.06\pm$ 0.20 | 1
B289D | 0.23$\pm$ 0.05 | 1 | $-1.71\pm$ 0.63 | 1
B292D | 0.23$\pm$ 0.14 | 2 | $-0.47\pm$ 0.54 | 1
B292 | 0.23$\pm$ 0.14 | 2 | $-1.42\pm$ 0.16 | 2
B293D | 0.27$\pm$ 0.06 | 1 | $-2.57\pm$ 0.11 | 4
B297D | 0.30$\pm$ 0.07 | 1 | $0.10\pm$ 0.08 | 4
B301 | 0.17$\pm$ 0.02 | 1 | $-0.76\pm$ 0.25 | 1
B303 | 0.14$\pm$ 0.06 | 1 | $-2.09\pm$ 0.41 | 1
B304 | 0.07$\pm$ 0.01 | 1 | $-1.32\pm$ 0.22 | 2
B305 | 0.38$\pm$ 0.29 | 2 | $-0.90\pm$ 0.61 | 1
B306 | 0.42$\pm$ 0.02 | 1 | $-0.85\pm$ 0.71 | 1
B307 | 0.08$\pm$ 0.02 | 1 | $-0.41\pm$ 0.36 | 1
B309 | 0.17$\pm$ 0.04 | 1 | $-2.03\pm$ 0.26 | 4
B310 | 0.09$\pm$ 0.01 | 1 | $-1.43\pm$ 0.28 | 2
B311 | 0.29$\pm$ 0.02 | 1 | $-1.96\pm$ 0.07 | 1
B312 | 0.16$\pm$ 0.01 | 1 | $-1.41\pm$ 0.08 | 1
aafootnotetext: The reddening values are taken from Fan et al. (2008) (ref=1)
and Barmby et al. (2000) (ref=2).
bbfootnotetext: The metallicities are taken from Perrett et al. (2002)
(ref=1), Barmby et al. (2000) (ref=2), Huchra et al. (1991) (ref=3), and Fan
et al. (2008) (ref=4).
Table 4Continuted.
Object | $E(B-V)$ | ref.a | $\rm[Fe/H]$ | ref.b
---|---|---|---|---
B313 | 0.21$\pm$ 0.02 | 1 | $-1.09\pm$ 0.10 | 1
B315 | 0.07$\pm$ 0.02 | 1 | $-2.35\pm$ 0.54 | 1
B316 | 0.21$\pm$ 0.03 | 1 | $-1.47\pm$ 0.23 | 1
B317 | 0.11$\pm$ 0.02 | 1 | $-2.12\pm$ 0.36 | 3
B325 | 0.14$\pm$ 0.02 | 1 | $-1.77\pm$ 0.08 | 4
B332D | 0.33$\pm$ 0.13 | 1 | $-0.65\pm$ 0.09 | 4
B335 | 0.65$\pm$ 0.02 | 1 | $-1.05\pm$ 0.26 | 1
B337 | 0.06$\pm$ 0.02 | 1 | $-1.09\pm$ 0.32 | 2
B338 | 0.14$\pm$ 0.02 | 1 | $-1.46\pm$ 0.12 | 1
B340D | 0.23$\pm$ 0.06 | 1 | $0.19\pm$ 0.29 | 4
B343 | 0.06$\pm$ 0.01 | 1 | $-1.49\pm$ 0.17 | 3
B344 | 0.11$\pm$ 0.02 | 1 | $-1.13\pm$ 0.21 | 3
B347 | 0.14$\pm$ 0.02 | 1 | $-1.71\pm$ 0.03 | 4
B348 | 0.25$\pm$ 0.04 | 1 | $-1.38\pm$ 0.07 | 4
B350 | 0.10$\pm$ 0.02 | 1 | $-1.47\pm$ 0.17 | 2
B351 | 0.15$\pm$ 0.02 | 1 | $-1.60\pm$ 0.05 | 4
B352 | 0.14$\pm$ 0.02 | 1 | $-1.88\pm$ 0.83 | 3
B354 | 0.05$\pm$ 0.02 | 1 | $-1.46\pm$ 0.38 | 2
B356 | 0.31$\pm$ 0.01 | 1 | $-1.46\pm$ 0.28 | 1
B357 | 0.12$\pm$ 0.02 | 1 | $-0.80\pm$ 0.42 | 3
B361 | 0.11$\pm$ 0.01 | 1 | $-1.61\pm$ 0.02 | 4
B365 | 0.19$\pm$ 0.02 | 1 | $-1.78\pm$ 0.19 | 1
B370 | 0.34$\pm$ 0.01 | 1 | $-1.80\pm$ 0.02 | 1
B372 | 0.20$\pm$ 0.02 | 1 | $-1.42\pm$ 0.17 | 1
B373 | 0.10$\pm$ 0.01 | 1 | $-0.50\pm$ 0.22 | 3
B375 | 0.29$\pm$ 0.03 | 1 | $-1.23\pm$ 0.22 | 3
B377 | 0.16$\pm$ 0.02 | 1 | $-2.19\pm$ 0.65 | 3
B378 | 0.14$\pm$ 0.02 | 1 | $-1.64\pm$ 0.26 | 1
B382 | 0.10$\pm$ 0.02 | 1 | $-1.52\pm$ 0.27 | 1
B383 | 0.00$\pm$ 0.02 | 2 | $-0.48\pm$ 0.20 | 2
B384 | 0.04$\pm$ 0.02 | 1 | $-0.66\pm$ 0.22 | 3
B386 | 0.21$\pm$ 0.01 | 1 | $-1.62\pm$ 0.14 | 1
B387 | 0.12$\pm$ 0.02 | 1 | $-1.96\pm$ 0.29 | 3
B393 | 0.14$\pm$ 0.02 | 1 | $-1.41\pm$ 0.05 | 4
B399 | 0.03$\pm$ 0.02 | 1 | $-1.69\pm$ 0.09 | 4
B401 | 0.06$\pm$ 0.06 | 2 | $-1.75\pm$ 0.29 | 2
B403 | 0.07$\pm$ 0.02 | 1 | $-0.45\pm$ 0.78 | 3
B405 | 0.14$\pm$ 0.02 | 1 | $-1.80\pm$ 0.31 | 3
B420 | 0.30$\pm$ 0.03 | 1 | $-0.63\pm$ 0.07 | 4
B448 | 0.05$\pm$ 0.01 | 1 | $-2.16\pm$ 0.19 | 1
B457 | 0.14$\pm$ 0.02 | 1 | $-1.60\pm$ 0.21 | 4
B461 | 0.58$\pm$ 0.07 | 1 | $-2.56\pm$ 0.07 | 4
B462 | 0.39$\pm$ 0.04 | 1 | $-2.28\pm$ 0.34 | 4
B467 | 0.27$\pm$ 0.02 | 1 | $-2.49\pm$ 0.47 | 1
B472 | 0.13$\pm$ 0.00 | 1 | $-1.45\pm$ 0.02 | 1
B475 | 0.16$\pm$ 0.03 | 1 | $-2.00\pm$ 0.14 | 1
B476 | 0.08$\pm$ 0.05 | 1 | $-0.03\pm$ 0.13 | 4
B483 | 0.08$\pm$ 0.06 | 1 | $-2.96\pm$ 0.35 | 1
B486 | 0.17$\pm$ 0.02 | 1 | $-2.28\pm$ 0.98 | 3
B489 | 0.17$\pm$ 0.04 | 1 | $-0.04\pm$ 0.10 | 4
B495 | 0.34$\pm$ 0.08 | 1 | $-0.35\pm$ 0.05 | 4
G260 | 0.30$\pm$ 0.05 | 1 | $-2.45\pm$ 0.06 | 4
aafootnotetext: The reddening values are taken from Fan et al. (2008) (ref=1)
and Barmby et al. (2000) (ref=2).
bbfootnotetext: The metallicities are taken from Perrett et al. (2002)
(ref=1), Barmby et al. (2000) (ref=2), Huchra et al. (1991) (ref=3), and Fan
et al. (2008) (ref=4).
Table 5Ages estimates for 104 GCs and GC candidates in M31. Object | Age | $\chi_{\rm min}^{2}$ | Object | Age | $\chi_{\rm min}^{2}$
---|---|---|---|---|---
| (Gyr) | (per degree of freedom) | | (Gyr) | (per degree of freedom)
B001 | $10.51\pm 0.80$ | 2.79 | B313 | $7.28\pm 0.70$ | 5.48
B003 | $2.01\pm 0.20$ | 2.10 | B315 | $0.50\pm 0.10$ | 7.88
B005 | $1.60\pm 0.10$ | 18.92 | B316 | $1.06\pm 0.10$ | 7.81
B018 | $1.79\pm 0.20$ | 2.74 | B317 | $2.06\pm 0.15$ | 2.59
B020D | $8.41\pm 1.70$ | 1.29 | B325 | $0.40\pm 0.10$ | 5.28
B024 | $15.25\pm 0.75$ | 2.96 | B332D | $11.30\pm 0.85$ | 4.33
B046 | $1.30\pm 0.10$ | 5.71 | B335 | $1.17\pm 0.15$ | 3.55
B058 | $2.02\pm 0.10$ | 2.41 | B337 | $2.03\pm 0.10$ | 5.66
B083 | $2.89\pm 0.20$ | 10.25 | B338 | $1.70\pm 0.10$ | 2.43
B085 | $2.18\pm 0.20$ | 2.39 | B340D | $13.57\pm 1.45$ | 9.68
B138D | $2.95\pm 0.35$ | 8.09 | B343 | $1.82\pm 0.10$ | 2.82
B140D | $0.39\pm 0.10$ | 12.18 | B344 | $12.68\pm 2.35$ | 2.60
B141D | $4.76\pm 1.00$ | 3.78 | B347 | $2.53\pm 0.15$ | 3.23
B142D | $0.03\pm 0.01$ | 18.34 | B348 | $6.58\pm 0.70$ | 3.28
B144D | $14.36\pm 0.95$ | 3.36 | B350 | $1.99\pm 0.10$ | 2.25
B156D | $0.10\pm 0.01$ | 22.86 | B351 | $3.20\pm 0.40$ | 2.26
B157D | $8.00\pm 1.05$ | 3.81 | B352 | $1.51\pm 0.10$ | 1.16
B165D | $0.50\pm 0.10$ | 20.62 | B354 | $5.24\pm 0.65$ | 3.85
B166D | $3.73\pm 0.90$ | 3.26 | B356 | $1.21\pm 0.10$ | 3.16
B167D | $0.90\pm 0.10$ | 3.22 | B357 | $4.98\pm 0.60$ | 2.94
B172D | $1.00\pm 0.10$ | 10.73 | B361 | $1.57\pm 0.10$ | 1.40
B177D | $1.70\pm 0.30$ | 3.32 | B365 | $1.73\pm 0.10$ | 1.89
B181D | $0.63\pm 0.15$ | 7.72 | B370 | $1.10\pm 0.10$ | 3.17
B196 | $1.62\pm 0.10$ | 2.39 | B372 | $2.34\pm 0.35$ | 1.54
B223D | $4.03\pm 0.35$ | 6.13 | B373 | $7.79\pm 0.40$ | 1.02
B240 | $1.79\pm 0.10$ | 1.05 | B375 | $2.19\pm 0.30$ | 7.62
B244 | $0.90\pm 0.10$ | 6.33 | B377 | $1.09\pm 0.10$ | 1.81
B245D | $0.10\pm 0.01$ | 7.58 | B378 | $2.12\pm 0.30$ | 3.70
B257 | $0.10\pm 0.01$ | 6.16 | B382 | $1.83\pm 0.15$ | 3.33
B260 | $14.30\pm 0.50$ | 1.26 | B383 | $13.99\pm 1.05$ | 4.73
B261D | $0.57\pm 0.10$ | 6.91 | B384 | $10.27\pm 0.95$ | 4.29
B266 | $0.03\pm 0.01$ | 10.36 | B386 | $2.54\pm 0.15$ | 1.93
B270D | $1.00\pm 0.10$ | 4.01 | B387 | $1.62\pm 0.10$ | 2.66
B272 | $3.73\pm 0.90$ | 5.10 | B393 | $6.76\pm 1.10$ | 1.17
B281 | $5.97\pm 1.30$ | 1.97 | B399 | $5.59\pm 0.50$ | 2.03
B283D | $1.09\pm 0.10$ | 3.84 | B401 | $3.49\pm 0.40$ | 2.55
B283 | $2.83\pm 0.35$ | 4.99 | B403 | $6.39\pm 0.40$ | 2.56
B289D | $0.81\pm 0.25$ | 2.20 | B405 | $1.30\pm 0.10$ | 2.97
B292D | $1.15\pm 0.15$ | 4.49 | B420 | $10.90\pm 0.40$ | 4.86
B292 | $1.00\pm 0.10$ | 2.53 | B448 | $1.70\pm 0.10$ | 6.19
B293D | $0.50\pm 0.10$ | 3.90 | B457 | $3.16\pm 0.35$ | 12.00
B297D | $15.18\pm 0.85$ | 3.37 | B461 | $0.56\pm 0.10$ | 9.51
B301 | $2.20\pm 0.30$ | 4.06 | B462 | $0.50\pm 0.10$ | 4.38
B303 | $0.50\pm 0.10$ | 17.86 | B467 | $1.00\pm 0.10$ | 5.95
B304 | $2.20\pm 0.10$ | 1.36 | B472 | $1.30\pm 0.10$ | 5.15
B305 | $0.40\pm 0.10$ | 9.52 | B475 | $0.97\pm 0.10$ | 6.69
B306 | $3.39\pm 0.50$ | 4.61 | B476 | $3.14\pm 0.35$ | 7.65
B307 | $1.61\pm 0.10$ | 7.46 | B483 | $1.00\pm 0.10$ | 6.49
B309 | $4.66\pm 0.55$ | 10.39 | B486 | $1.61\pm 0.10$ | 3.31
B310 | $2.07\pm 0.15$ | 1.35 | B489 | $9.07\pm 1.30$ | 3.56
B311 | $1.62\pm 0.10$ | 4.21 | B495 | $14.54\pm 0.55$ | 2.81
B312 | $2.56\pm 0.25$ | 7.27 | G260 | $1.00\pm 0.10$ | 5.84
Table 6Age comparison Object | Age (Gyr) | Age (Gyr)
---|---|---
| (Caldwell et al., 2009) | (this paper)
B018 | 1.00 | $1.79\pm 0.20$
B303 | 0.40 | $0.50\pm 0.10$
B307 | 1.00 | $1.61\pm 0.10$
B315 | 0.16 | $0.50\pm 0.10$
B316 | 1.00 | $1.06\pm 0.10$
B325 | 0.63 | $0.40\pm 0.10$
B448 | 0.25 | $1.70\pm 0.10$
B475 | 0.32 | $0.97\pm 0.10$
B483 | 0.50 | $1.00\pm 0.10$
|
arxiv-papers
| 2010-01-22T08:42:05 |
2024-09-04T02:49:07.904317
|
{
"license": "Public Domain",
"authors": "Song Wang (1,2,3), Zhou Fan (1), Jun Ma (1,3), Richard de Grijs (4,5),\n Xu Zhou (1) ((1) National Astronomical Observatories, Chinese Academy of\n Sciences; (2) Graduate University, Chinese Academy of Sciences; (3) Key\n Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese\n Academy of Sciences; (4) Kavli Institute for Astronomy and Astrophysics,\n Peking University; (5) Department of Physics and Astronomy, University of\n Sheffield)",
"submitter": "Jun Ma",
"url": "https://arxiv.org/abs/1001.3939"
}
|
1001.4010
|
# Spectral Theory for Second-Order Vector Equations on Finite Time-Varying
Domains
Douglas R. Anderson Department of Mathematics and Computer Science, Concordia
College, Moorhead, MN 56562 United States
visiting the School of Mathematics, The University of New South Wales Sydney
2052, Australia andersod@cord.edu
###### Abstract.
In this study, we are concerned with spectral problems of second-order vector
dynamic equations with two-point boundary value conditions and mixed
derivatives, where the matrix-valued coefficient of the leading term may be
singular, and the domain is non-uniform but finite. A concept of self-
adjointness of the boundary conditions is introduced. The self-adjointness of
the corresponding dynamic operator is discussed on a suitable admissible
function space, and fundamental spectral results are obtained. The dual
orthogonality of eigenfunctions is shown in a special case. Extensions to
even-order Sturm-Liouville dynamic equations, linear Hamiltonian and
symplectic nabla systems on general time scales are also discussed.
###### Key words and phrases:
boundary value problem; linear equations; self-adjoint operator; time scales;
mixed derivatives; Sturm-Liouville theory; linear Hamiltonian systems;
symplectic systems; nabla derivative.
###### 2000 Mathematics Subject Classification:
34B24, 39A10, 15A24
## 1\. Introduction
Shi and Chen [27] provide an analysis for second-order vector difference
equations of the form
$-\nabla\left(C_{n}\Delta x_{n}\right)+B_{n}x_{n}=\lambda\omega_{n}x_{n},\quad
n\in[1,N]\cap\mathbb{Z},\quad N\geq 2,$
where the forward difference operator $\Delta x_{n}:=x_{n+1}-x_{n}$ and the
backward difference operator $\nabla x_{n}:=x_{n}-x_{n-1}$ are utilized. This
is one vector form related to the well-known self-adjoint scalar form, see
Kelley and Peterson [20, Chapters $6-7$],
$-\Delta\left(c_{n-1}\Delta x_{n-1}\right)+b_{n}x_{n}=\lambda\omega_{n}x_{n}.$
Recently, however, there has been a growing interest in time-varying systems
such as by Davis, Gravagne, Jackson, and Marks [12] or Gravagne, Davis, and
DaCunha [14], that is discrete systems with varying step-size between domain
points; these systems are sometimes also called isolated time scales. An
important example in this class of problems would be quantum ($q$-difference)
equations; see Simmons [31, Chapter B.5] and Kac and Cheung [19] for quantum
calculus, and Erbe and Hilger [13] and Bohner and Peterson [10] for more on
time scales, including isolated time scales.
Motivated by the appeal of time-varying domains and the ability to
simultaneously unify and generalize recent and classic results, we introduce
here a finite-dimensional analysis of second-order vector dynamic equations of
the form
(1.1) $-(Px^{\Delta})^{\nabla}(t)+Q(t)x(t)=\lambda\omega(t)x(t),\quad
t\in[a,b]_{\mathbb{T}},\quad b\geq\sigma(a),$
with boundary conditions
(1.2) $R\left(\begin{smallmatrix}-x^{\rho}(a)\\\
x(b)\end{smallmatrix}\right)+S\left(\begin{smallmatrix}P^{\rho}(a)x^{\nabla}(a)\\\
P(b)x^{\Delta}(b)\end{smallmatrix}\right)=0.$
A solution $x$ of (1.1) is defined on $[\rho(a),\sigma(b)]_{\mathbb{T}}$. Here
$\mathbb{T}$ is a (finite) isolated time scale, the discrete interval is given
by
$[a,b]_{\mathbb{T}}:=\\{a,\sigma(a),\sigma^{2}(a),\cdots,\rho(b),b\\},\quad
b=\sigma^{N-1}(a),$
the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$, $Q(t)$ and
$\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with $P^{\rho}(a)$
and $P(b)$ invertible, $\omega(t)>0$ (positive definite) for
$t\in[a,b]_{\mathbb{T}}$, and $R$ and $S$ in (1.2) are $2d\times 2d$ matrices
with $\operatorname{rank}(R,S)=2d$. The differential operators in (1.1) are
given, respectively, by
$x^{\Delta}(t)=\frac{x^{\sigma}(t)-x(t)}{\mu_{\sigma}(t)}\quad\text{and}\quad
x^{\nabla}(t)=\frac{x(t)-x^{\rho}(t)}{\mu_{\rho}(t)}=x^{\Delta\rho}(t),$
where time-varying step-sizes are given by $\mu_{\sigma}(t):=\sigma(t)-t>0$,
$\mu_{\rho}(t):=t-\rho(t)>0$, with $\sigma(t)$ the immediate domain point to
the right of $t$, $\rho(t)$ the immediate domain point to the left of $t$, and
where compositions are often denoted $f(\alpha(t))=f^{\alpha}(t)$. Note that
the coefficient matrix $P(t)$ in (1.1) may be singular for
$t\in[a,\rho(b)]_{\mathbb{T}}$. Moreover, the second-order dynamic operator
given in (1.1) may not be formally self-adjoint in the space of vector series
of the form
$\left\\{x(\sigma^{n}(a))\right\\}_{n=-1}^{N},\quad\sigma^{-1}(a):=\rho(a),\quad\sigma^{0}(a):=a,\quad\sigma^{n}(a):=\sigma(\sigma^{n-1}(a)),$
subject to the boundary conditions in (1.2) unless those conditions themselves
are self-adjoint in some sense to be defined. On the other hand, even if the
boundary conditions (1.2) are self-adjoint, the second-order dynamic operator
may still not be self-adjoint. As in the difference equations case [27], self-
adjointness depends on whether the vectors at the boundaries, namely
$x^{\rho}(a)$ and $x^{\sigma}(b)$, can be solved from (1.2) in terms of $x(a)$
and $x(b)$. Indeed if this is the case, we will call (1.2) proper self-adjoint
boundary conditions. If (1.2) happen to be improper, we can find a smaller
space where the second-order dynamic operator is self-adjoint. Once this is
sorted out, we will give the appropriate analysis of the eigenvalue problem
for (1.1) and (1.2).
The particular appeal of (1.1) is that it is still a discrete problem, but
with non-constant step-size between domain points. As
$\lim_{\mu_{\sigma}\rightarrow 0}x^{\Delta}(t)=\lim_{\mu_{\rho}\rightarrow
0}x^{\nabla}(t)=x^{\prime}(t),\quad^{\prime}=\frac{d}{dt},$
for differentiable functions $x$, these dynamic results serve as an
alternative discrete analog to the differential equations case. There are many
papers on the spectral theory for difference equations, including recent
papers such as Ji and Yang [17, 18], Shi [25, 26], Shi and Chen [27, 28], Shi
and Wu [29], Shi and Yan [30], Sun and Shi [32], Sun, Shi, and Chen [33], Wang
and Shi [34], and Bohner, Došlý, and Kratz [9], but none on dynamic vector
equations. Thus this work will continue and extend the discussion of self-
adjoint equations on time scales found in [4, 5, 11, 15, 23]. As in the
uniformly-discrete case [27], we use mixed derivatives, as $-\nabla$ is a
natural adjoint of $\Delta$. To emphasis the dimensional analysis we focus on
isolated time scales, but a similar treatment can be made on general time
scales.
The analysis of (1.1) and its solutions will unfold as follows, largely
motivated by Shi and Chen [27]. In Section 2 we will give a definition of the
self-adjointness of the boundary conditions in (1.2). Section 3 contains the
introduction of a suitable admissible function space in which the
corresponding dynamic operator is self-adjoint, followed in Section 4 by some
fundamental spectral results. In the event of proper self-adjoint boundary
conditions, the dual orthogonality of eigenfunctions will be given. In Section
5 we discuss the possibility of extending these results to even-order Sturm-
Liouville equations, linear Hamiltonian and symplectic nabla systems on time
scales.
## 2\. Self-Adjoint Boundary Conditions
In this section we are concerned with the second-order vector dynamic operator
$\ell$ given by
(2.1) $\ell
x(t)=\omega^{-1}(t)\left[-(Px^{\Delta})^{\nabla}(t)+Q(t)x(t)\right],\quad
t\in[a,b]_{\mathbb{T}},$
where we will write $x\in\mathscr{R}$ if
$\left\\{x(\sigma^{n}(a))\right\\}_{n=-1}^{N}$ satisfies the boundary
conditions given in (1.2). If we set
$\ell[\rho(a),\sigma(b)]:=\left\\{x=\left\\{x(\sigma^{n}(a))\right\\}_{n=-1}^{N}:x(t)\in\mathbb{C}^{d},t\in[\rho(a),\sigma(b)])_{\mathbb{T}}\right\\},$
then $\dim\ell[\rho(a),\sigma(b)]=d(N+2)$. Define the (weighted) inner product
to be given by
(2.2)
$\big{\langle}x,y\big{\rangle}=\sum_{t\in[a,b]_{\mathbb{T}}}y^{*}(t)\omega(t)x(t)\mu_{\rho}(t)\quad\text{for}\quad
x,y\in\ell[\rho(a),\sigma(b)],$
where $\omega(t)$ is as in (1.1), $\mu_{\rho}(t)=t-\rho$ is the left-
graininess at $t$, and $y^{*}(t)$ denotes the complex conjugate transpose of
$y(t)$. Using standard notation, $x\perp y$ will mean $\langle x,y\rangle=0$.
###### Theorem 2.1 (Lagrange Identity).
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, and $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$. Then for $x,y\in\ell[\rho(a),\sigma(b)]$ we have
that
(2.3) $\big{\langle}\ell x,y\big{\rangle}-\big{\langle}x,\ell
y\big{\rangle}=\left[(Py^{\Delta})^{*}x-y^{*}Px^{\Delta}\right](b)-\left[(Py^{\Delta})^{*}x-y^{*}Px^{\Delta}\right]^{\rho}(a),$
where the inner product is given in (2.2).
###### Proof.
Let $x,y\in\ell[\rho(a),\sigma(b)]$. Using summation by parts [8, Theorem 2.7]
we have
$\displaystyle\langle\ell x,y\rangle-\langle x,\ell y\rangle$ $\displaystyle=$
$\displaystyle\sum_{t\in[a,b]_{\mathbb{T}}}y^{*}(t)\omega(t)\ell
x(t)\mu_{\rho}(t)-\sum_{t\in[a,b]_{\mathbb{T}}}(\ell
y)^{*}(t)\omega(t)x(t)\mu_{\rho}(t)$ $\displaystyle=$
$\displaystyle-\sum_{t\in[a,b]_{\mathbb{T}}}y^{*}(t)\left((Px^{\Delta})^{\nabla}-Qx\right)(t)\mu_{\rho}(t)$
$\displaystyle+\sum_{t\in[a,b]_{\mathbb{T}}}\left((Py^{\Delta})^{\nabla}-Qy\right)^{*}(t)x(t)\mu_{\rho}(t)$
$\displaystyle=$
$\displaystyle-\sum_{t\in[a,b]_{\mathbb{T}}}y^{*}(t)(Px^{\Delta})^{\nabla}(t)\mu_{\rho}(t)+\sum_{t\in[a,b]_{\mathbb{T}}}(Py^{\Delta})^{\nabla*}(t)x(t)\mu_{\rho}(t)$
$\displaystyle=$
$\displaystyle-\left[y^{*}Px^{\Delta}\right](t)\Big{|}^{b}_{\rho(a)}+\sum_{t\in[a,b]_{\mathbb{T}}}(Py^{\Delta})^{*\rho}(t)x^{\nabla}(t)\mu_{\rho}(t)$
$\displaystyle+\sum_{t\in[a,b]_{\mathbb{T}}}(Py^{\Delta})^{\nabla*}(t)x(t)\mu_{\rho}(t)$
$\displaystyle=$
$\displaystyle-\left[y^{*}Px^{\Delta}\right](t)\Big{|}^{b}_{\rho(a)}+\left[(Py^{\Delta})^{*}x\right](t)\Big{|}_{\rho(a)}^{b}$
$\displaystyle=$
$\displaystyle\left[(Py^{\Delta})^{*}x-y^{*}Px^{\Delta}\right](t)\Big{|}^{b}_{\rho(a)}$
and the result follows. ∎
The above theorem, Theorem 2.1, is also valid on general time scales; see
Anderson and Buchholz [6, Theorem 2.13].
###### Definition 2.2.
The boundary conditions given in (1.2) are called self-adjoint iff
(2.4)
$\left[(Py^{\Delta})^{*}x-y^{*}Px^{\Delta}\right](t)\Big{|}^{b}_{\rho(a)}=0$
for $x,y\in\mathscr{R}$.
The next theorem then follows easily from the definition and Theorem 2.1.
###### Theorem 2.3.
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$, and $R$ and $S$ in (1.2) are $2d\times 2d$ matrices
with $\operatorname{rank}(R,S)=2d$. If the boundary conditions in (1.2) are
self-adjoint, then
$\langle\ell x,y\rangle=\langle x,\ell y\rangle$
for $x,y\in\ell[\rho(a),\sigma(b)]$ with $x,y\in\mathscr{R}$.
###### Lemma 2.4.
Assume that $R$ and $S$ are $2d\times 2d$ matrices with
$\operatorname{rank}(R,S)=2d$. Then
(2.5) $SR^{*}=RS^{*}$
if and only if the boundary conditions (1.2) are self-adjoint.
###### Proof.
We follow Kratz [22, Proposition 2.1.1]. Let $x\in\ell[\rho(a),\sigma(b)]$
with $x\in\mathscr{R}$. By the definition of $\mathscr{R}$, $x$ satisfies the
boundary conditions (1.2). Let $\mathcal{K}=\left(\begin{smallmatrix}B\\\
C\end{smallmatrix}\right)$ be any $4d\times 2d$ matrix with
$\operatorname{Im}\mathcal{K}=\ker(B,C)$, so that $SB+RC=0$,
$\operatorname{rank}\mathcal{K}=2d$, and so that $x\in\mathscr{R}$ if and only
if $\left(\begin{smallmatrix}x^{\rho}(a)\\\
-x(b)\end{smallmatrix}\right)=B\eta$,
$\left(\begin{smallmatrix}P^{\rho}(a)x^{\nabla}(a)\\\
P(b)x^{\Delta}(b)\end{smallmatrix}\right)=C\eta$ for some
$\eta\in\mathbb{C}^{2d}$. This results in (2.4) holding for all
$x,y\in\mathscr{R}$ if and only if $\eta_{1}^{*}[B^{*}C-C^{*}B]\eta_{2}=0$ for
all $\eta_{1},\eta_{2}\in\mathbb{C}^{2d}$, that is to say $B^{*}C=C^{*}B$.
First, assume that $RS^{*}=SR^{*}$. Since $\operatorname{rank}(R,S)=2d$, we
see that $\operatorname{Im}\left(\begin{smallmatrix}S^{*}\\\
-R^{*}\end{smallmatrix}\right)=\ker(R,S)$. Consequently, the matrix
$\mathcal{K}$ given above can be taken to be
$\left(\begin{smallmatrix}S^{*}\\\ -R^{*}\end{smallmatrix}\right)$; by the
Hermitian assumption on $RS^{*}$ we see that $B^{*}C$ is Hermitian, and thus
the boundary conditions (1.2) are self-adjoint by Definition 2.2. The rest of
the proof is identical to [22, Proposition 2.1.1] and is omitted. ∎
###### Lemma 2.5.
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$, and $R$ and $S$ in (1.2) are $2d\times 2d$ matrices
with $\operatorname{rank}(R,S)=2d$. If the boundary conditions in (1.2) are
self-adjoint, then $x\in\mathscr{R}$ if and only if there exists a unique
vector $\eta\in\mathbb{C}^{2d}$ such that
(2.6) $\left(\begin{smallmatrix}-x^{\rho}(a)\\\
x(b)\end{smallmatrix}\right)=-S^{*}\eta,\qquad\left(\begin{smallmatrix}P^{\rho}(a)x^{\nabla}(a)\\\
P(b)x^{\Delta}(b)\end{smallmatrix}\right)=R^{*}\eta.$
###### Proof.
See Kratz [22, Proposition 2.1.2] and Shi and Chen [27, Lemma 2.2]. ∎
## 3\. Admissible Function Space and Self-Adjointness of the Dynamic Operator
Throughout this section we assume the $d\times d$ matrices $P(t)$ for
$t\in[\rho(a),b]_{\mathbb{T}}$, $Q(t)$ and $\omega(t)$ for
$t\in[a,b]_{\mathbb{T}}$, are all Hermitian with $P^{\rho}(a)$ and $P(b)$
invertible, $\omega(t)>0$ for $t\in[a,b]_{\mathbb{T}}$, the matrices $R$ and
$S$ in (1.2) are $2d\times 2d$ matrices with $\operatorname{rank}(R,S)=2d$,
and the boundary conditions in (1.2) are self-adjoint. We begin the
construction of an admissible function space to establish the self-adjointness
of a dynamic operator related to (1.1), following the development in the
uniformly discrete case presented in [27].
Let $R=(R_{1},R_{2})$ and $S=(S_{1},S_{2})$, where $R_{k}$ and $S_{k}$ are
$2d\times d$ matrices, $k=1,2$. By Lemma 2.5, $x\in\mathscr{R}$ if and only if
there exists a unique vector $\eta\in\mathbb{C}^{2d}$ such that
$\displaystyle x^{\rho}(a)=S_{1}^{*}\eta,\quad
x(a)=(S_{1}^{*}+\mu_{\rho}(a)P^{\rho-1}(a)R_{1}^{*})\eta,$ (3.1)
$\displaystyle x(b)=-S_{2}^{*}\eta,\quad
x^{\sigma}(b)=(-S_{2}^{*}+\mu_{\sigma}(b)P^{-1}(b)R_{2}^{*})\eta,$
where $P^{\rho-1}=(P^{\rho})^{-1}$. Rewrite the boundary conditions (1.2) as
(3.2) $\Gamma\left(\begin{smallmatrix}P^{\rho}(a)&0\\\
0&-P(b)\end{smallmatrix}\right)\left(\begin{smallmatrix}x^{\rho}(a)\\\
x^{\sigma}(b)\end{smallmatrix}\right)=\left(\frac{1}{\mu_{\rho}(a)}S_{1}P^{\rho}(a),R_{2}-\frac{1}{\mu_{\sigma}(b)}S_{2}P(b)\right)\left(\begin{smallmatrix}x(a)\\\
x(b)\end{smallmatrix}\right),$
where the coefficient matrix $\Gamma$ is given by
$\Gamma=\left(R_{1}P^{\rho-1}(a)+\frac{1}{\mu_{\rho}(a)}S_{1},\frac{1}{\mu_{\sigma}(b)}S_{2}\right).$
Set
$r=\operatorname{rank}\left(R_{1}+\frac{1}{\mu_{\rho}(a)}S_{1}P^{\rho}(a),\frac{1}{\mu_{\sigma}(b)}S_{2}\right)\in\\{0,1,\cdots,2d\\},$
where this value $r$ will play a key role in the development to follow. If the
matrix $\Gamma$ is invertible, in other words if $r=2d$, then the boundary
conditions (1.2) are proper, since (3.2) is solvable for $x^{\rho}(a)$ and
$x^{\sigma}(b)$, respectively. We then see that, as $x^{\rho}(a)$ and
$x^{\sigma}(b)$ are not weighted with respect to the weight function $\omega$,
the $dN$-dimensional space
$L[\rho(a),\sigma(b)]:=\left\\{x\in\ell[\rho(a),\sigma(b)]:x\in\mathscr{R}\right\\}$
might serve as a suitable admissible space. If $r<2d$, however, then from
(3.2) we have that the $2d$ components of $x(a)$ and $x(b)$ are themselves
linked by $2d-r$ relations. From (1.1) we see that $x(a)$ and $x(b)$ are
weighted by $\omega(a)$ and $\omega(b)$ in the first and last vector
equations. In the two vector equations, via some transformation, only $r$
scalar equations are really weighted, while the remaining $2d-r$ ones involve
$x^{\rho}(a)$, $x(a)$, $x^{\sigma}(a)$, $x^{\rho}(b)$, $x(b)$, and
$x^{\sigma}(b)$ but not $\lambda$; consequently they can be viewed as extra
conditions for the admissible functions. We will now show this as follows.
Using standard matrix theory [16], there exist $2d\times 2d$-unitary matrices
$U$ and $V$ such that
(3.3) $U^{*}\Gamma V=\operatorname{diag}\\{0,M\\},$
where $M$ is an $r\times r$ matrix with $\operatorname{rank}M=r$. Let
$U=(U_{1},U_{2})$ and $V=(V_{1},V_{2})$, where $U_{1}$ and $V_{1}$ are
$2d\times(2d-r)$ matrices, $U_{2}$ and $V_{2}$ are $2d\times r$ matrices.
Since $V$ is unitary, from (3.3) we have that
(3.4) $V_{1}^{*}V_{1}=I_{2d-r},\quad V_{1}^{*}V_{2}=0_{(2d-r)\times r},\quad
V_{2}^{*}V_{2}=I_{r},$ (3.5) $U_{1}^{*}\Gamma=0_{(2d-r)\times 2d},\quad
V_{2}=\Gamma^{*}U_{2}M^{*-1}.$
Using (3.1) and (3.3), we see that
(3.6) $\left(\begin{smallmatrix}x(a)\\\
-x(b)\end{smallmatrix}\right)=V\operatorname{diag}\\{0,M^{*}\\}U^{*}\eta=(0_{2d\times(2d-r)},V_{2}M^{*})U^{*}\eta,$
so that by (3.4) we have
(3.7) $V_{1}^{*}\left(x^{T}(a),-x^{T}(b)\right)=0,\qquad x^{T}=\text{transpose
of $x$}.$
The first and last relations in (1.1) can be written as
$\lambda\begin{pmatrix}x(a)\\\
-x(b)\end{pmatrix}=\begin{pmatrix}\omega^{-1}(a)&0\\\
0&\omega^{-1}(b)\end{pmatrix}\begin{pmatrix}\frac{-1}{\left(\mu_{\rho}(a)\right)^{2}}P^{\rho}(a)x^{\rho}(a)+\tilde{P}(a)x(a)-\frac{1}{\mu_{\rho}(a)\mu_{\sigma}(a)}P(a)x^{\sigma}(a)\\\
\frac{1}{\left(\mu_{\rho}(b)\right)^{2}}P^{\rho}(b)x^{\rho}(b)-\tilde{P}(b)x(b)+\frac{1}{\mu_{\rho}(b)\mu_{\sigma}(b)}P(b)x^{\sigma}(b)\end{pmatrix},$
where we have taken
$\tilde{P}(t)=Q(t)+\frac{1}{\mu_{\rho}(t)\mu_{\sigma}(t)}P(t)+\frac{1}{\left(\mu_{\rho}(t)\right)^{2}}P^{\rho}(t).$
From (3.7) we obtain the $2d-r$ scalar equations
(3.8) $V_{1}^{*}\begin{pmatrix}\omega^{-1}(a)&0\\\
0&\omega^{-1}(b)\end{pmatrix}\begin{pmatrix}\frac{-1}{\left(\mu_{\rho}(a)\right)^{2}}P^{\rho}(a)x^{\rho}(a)+\tilde{P}(a)x(a)-\frac{1}{\mu_{\rho}(a)\mu_{\sigma}(a)}P(a)x^{\sigma}(a)\\\
\frac{1}{\left(\mu_{\rho}(b)\right)^{2}}P^{\rho}(b)x^{\rho}(b)-\tilde{P}(b)x(b)+\frac{1}{\mu_{\rho}(b)\mu_{\sigma}(b)}P(b)x^{\sigma}(b)\end{pmatrix}=0.$
If $x$ satisfies (3.8) we will write $x\in\mathscr{A}$. Noting the absence of
$\lambda$ from (3.8), we can think of (3.8) as additional conditions together
with the boundary conditions (1.2). Thus, we define the admissible function
space
$L^{2}_{\omega}[\rho(a),\sigma(b)]:=\\{x\in\ell[\rho(a),\sigma(b)]:x\in\mathscr{R}\cap\mathscr{A}\\}.$
From (3.4) we see that $x\in\mathscr{A}$ if and only if $x$ satisfies
(3.9)
$\begin{pmatrix}\frac{-1}{\left(\mu_{\rho}(a)\right)^{2}}P^{\rho}(a)x^{\rho}(a)+\tilde{P}(a)x(a)-\frac{1}{\mu_{\rho}(a)\mu_{\sigma}(a)}P(a)x^{\sigma}(a)\\\
\frac{1}{\left(\mu_{\rho}(b)\right)^{2}}P^{\rho}(b)x^{\rho}(b)-\tilde{P}(b)x(b)+\frac{1}{\mu_{\rho}(b)\mu_{\sigma}(b)}P(b)x^{\sigma}(b)\end{pmatrix}=\begin{pmatrix}\omega(a)&0\\\
0&\omega(b)\end{pmatrix}V_{2}\gamma$
for some $\gamma\in\mathbb{C}^{r}$. As $P^{\rho}(a)$ and $P(b)$ are
invertible, (3.9) implies that
(3.10) $\displaystyle\begin{pmatrix}x^{\rho}(a)\\\ x^{\sigma}(b)\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}P^{\rho-1}(a)\left(\left(\mu_{\rho}(a)\right)^{2}\tilde{P}(a)x(a)-\frac{\mu_{\rho}(a)}{\mu_{\sigma}(a)}P(a)x^{\sigma}(a)\right)\\\
P^{-1}(b)\left(-\frac{\mu_{\sigma}(b)}{\mu_{\rho}(b)}P^{\rho}(b)x^{\rho}(b)+\mu_{\rho}(b)\mu_{\sigma}(b)\tilde{P}(b)x(b)\right)\end{pmatrix}$
$\displaystyle+\begin{pmatrix}-\left(\mu_{\rho}(a)\right)^{2}P^{\rho-1}(a)\omega(a)&0\\\
0&\mu_{\rho}(b)\mu_{\sigma}(b)P^{-1}(b)\omega(b)\end{pmatrix}V_{2}\gamma.$
Let $x\in L^{2}_{\omega}[\rho(a),\sigma(b)]$. Inserting (3.10) into boundary
conditions (1.2) or (3.2) yields
(3.12)
$\displaystyle\Gamma\operatorname{diag}\\{\left(\mu_{\rho}(a)\right)^{2}\omega(a),\mu_{\rho}(b)\mu_{\sigma}(b)\omega(b)\\}V_{2}\gamma$
$\displaystyle=$
$\displaystyle\Gamma\begin{pmatrix}\left(\mu_{\rho}(a)\right)^{2}\tilde{P}(a)x(a)-\frac{\mu_{\rho}(a)}{\mu_{\sigma}(a)}P(a)x^{\sigma}(a)\\\
\frac{\mu_{\sigma}(b)}{\mu_{\rho}(b)}P^{\rho}(b)x^{\rho}(b)-\mu_{\rho}(b)\mu_{\sigma}(b)\tilde{P}(b)x(b)\end{pmatrix}$
$\displaystyle-\left(\frac{1}{\mu_{\rho}(a)}S_{1}P^{\rho}(a),R_{2}-\frac{1}{\mu_{\sigma}(b)}S_{2}P(b)\right)\left(\begin{smallmatrix}x(a)\\\
x(b)\end{smallmatrix}\right)$
$\displaystyle=:f\left(x(a),x^{\sigma}(a),x^{\rho}(b),x(b)\right).$
From (3.3) we have
(3.13)
$\Gamma\operatorname{diag}\\{\left(\mu_{\rho}(a)\right)^{2}\omega(a),\mu_{\rho}(b)\mu_{\sigma}(b)\omega(b)\\}V_{2}=U\left(0_{r\times(2d-r)},J^{T}\right)^{T},$
where
$J=MV_{2}^{*}\operatorname{diag}\\{\left(\mu_{\rho}(a)\right)^{2}\omega(a),\mu_{\rho}(b)\mu_{\sigma}(b)\omega(b)\\}V_{2}$
is an $r\times r$ invertible matrix, as
$\operatorname{rank}M=\operatorname{rank}V_{2}=r$, and $\omega(a)$ and
$\omega(b)$ are positive definite by assumption. Multiplying (3.12) on the
left by $U^{*}$, we have from (3.5) (in particular $U_{1}^{*}S_{2}=0$) and
(3.13) that
(3.14)
$U_{1}^{*}\left(\frac{1}{\mu_{\rho}(a)}S_{1}P^{\rho}(a)x(a)+R_{2}x(b)\right)=0$
(3.15)
$\gamma=J^{-1}U_{2}^{*}f\left(x(a),x^{\sigma}(a),x^{\rho}(b),x(b)\right).$
We are now in a position to get a useful characterization for a function $x$
to be a member of $L^{2}_{\omega}[\rho(a),\sigma(b)]$. As we have the
necessary structure in place and in parallel with the discrete case presented
by [27], we omit the proofs of the following key results; for more details,
please see [27].
###### Lemma 3.1.
We have $x\in L^{2}_{\omega}[\rho(a),\sigma(b)]$ if and only if $x$ satisfies
(3.10) and (3.14) in which $\gamma$ is determined by (3.15).
###### Lemma 3.2.
We have $m:=\dim L^{2}_{\omega}[\rho(a),\sigma(b)]=d(N-2)+r$.
###### Theorem 3.3.
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$, and $R$ and $S$ in (1.2) are $2d\times 2d$ matrices
with $\operatorname{rank}(R,S)=2d$. If the boundary conditions in (1.2) are
self-adjoint, then $L^{2}_{\omega}[\rho(a),\sigma(b)]$ is an $m$-dimensional
Hilbert space with the inner product defined in (2.2).
Let
$L:=\ell\big{|}_{L_{\omega}^{2}[\rho(a),\sigma(b)]},\quad\text{where $\ell$ is
given in \eqref{saeq}.}$
We will see the self-adjointness of this dynamic operator.
###### Lemma 3.4.
The dynamic operator $L$ maps $L_{\omega}^{2}[\rho(a),\sigma(b)]$ into itself.
###### Proof.
See [27, Proposition 3.3], which uses (3.10) with $x$ replaced by $Lx$ as in
(3.16) $\displaystyle\begin{pmatrix}Lx^{\rho}(a)\\\
Lx^{\sigma}(b)\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}P^{\rho-1}(a)\left(\left(\mu_{\rho}(a)\right)^{2}\tilde{P}(a)Lx(a)-\frac{\mu_{\rho}(a)}{\mu_{\sigma}(a)}P(a)Lx^{\sigma}(a)\right)\\\
P^{-1}(b)\left(-\frac{\mu_{\sigma}(b)}{\mu_{\rho}(b)}P^{\rho}(b)Lx^{\rho}(b)+\mu_{\rho}(b)\mu_{\sigma}(b)\tilde{P}(b)Lx(b)\right)\end{pmatrix}$
$\displaystyle+\begin{pmatrix}-\left(\mu_{\rho}(a)\right)^{2}P^{\rho-1}(a)\omega(a)&0\\\
0&\mu_{\rho}(b)\mu_{\sigma}(b)P^{-1}(b)\omega(b)\end{pmatrix}V_{2}\hat{\gamma},$
where $\hat{\gamma}$ is determined by (3.15) with $x(a)$, $x^{\sigma}(a)$,
$x^{\rho}(b)$, and $x(b)$ replaced by $Lx(a)$, $Lx^{\sigma}(a)$,
$Lx^{\rho}(b)$, and $Lx(b)$. ∎
###### Theorem 3.5.
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$, and $R$ and $S$ in (1.2) are $2d\times 2d$ matrices
with $\operatorname{rank}(R,S)=2d$. If the boundary conditions in (1.2) are
self-adjoint, then the dynamic operator $L$ is self-adjoint.
## 4\. Dynamic Spectral Theory
As in the previous section, the machinery for a dynamic spectral theory is in
place. All proofs of the following results are modifications of those found in
the discrete case in [27], and thus are omitted here. For completeness,
however, we list the main ideas and theorems that hold in this setting as
well.
###### Definition 4.1.
A complex number $\lambda\in\mathbb{C}$ is an eigenvalue of (1.1), (1.2) if
and only if there exists $x\in\ell[\rho(a),\sigma(b)]$ that is nonzero and
solves the boundary value problem (1.1), (1.2). The nonzero solution $x$ is a
corresponding eigenfunction for $\lambda$, denoted $x(\cdot,\lambda)$.
Following the discussion in the previous section, any eigenfunction of (1.1),
(1.2) is in the space $L_{\omega}^{2}[\rho(a),\sigma(b)]$. Therefore, the
following fundamental spectral results for (1.1), (1.2) can be directly
concluded by Theorem 3.5 and by employing the spectral theory of self-adjoint
linear operators in Hilbert spaces. spaces.
###### Theorem 4.2.
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$, $R$ and $S$ in (1.2) are $2d\times 2d$ matrices with
$\operatorname{rank}(R,S)=2d$, and the boundary conditions in (1.2) are self-
adjoint. Let
$r=\operatorname{rank}\left(R_{1}+\frac{1}{\mu_{\rho}(a)}S_{1}P^{\rho}(a),\frac{1}{\mu_{\sigma}(b)}S_{2}\right)$
and $m=d(N-2)+r$.
1. (i)
The eigenvalue problem (1.1), (1.2) has $m$ real eigenvalues
(4.1) $\lambda_{1},\lambda_{2},\cdots,\lambda_{m},$
including multiplicities, and $m$ linearly independent eigenfunctions
(4.2) $x(\cdot,\lambda_{1}),x(\cdot,\lambda_{2}),\cdots,x(\cdot,\lambda_{m}),$
that form an orthonormal set, to wit
(4.3) $\langle
x(\cdot,\lambda_{j}),x(\cdot,\lambda_{k})\rangle=\sum_{t\in[a,b]_{\mathbb{T}}}x^{*}(t,\lambda_{k})\omega(t)x(t,\lambda_{j})\mu_{\sigma}(t)=\delta_{jk},\quad
j,k\in\\{1,2,\cdots,m\\}.$
2. (ii)
The eigenfunction basis for (1.1), (1.2) consists of $m$ linearly independent
eigenfunctions (4.2) and is complete for admissible function space
$L_{\omega}^{2}[\rho(a),\sigma(b)]$, to wit for each $x\in
L_{\omega}^{2}[\rho(a),\sigma(b)]$, there exists a unique set of scalars
$\\{c_{j}\\}_{j=1}^{m}\subset\mathbb{C}$ such that, for
$t\in[\rho(a),\sigma(b)]_{\mathbb{T}}$, we have
(4.4) $x(t)=\sum_{j=1}^{m}c_{j}x(t,\lambda_{j}),$
where
(4.5) $c_{j}=\langle
x,x(\cdot,\lambda_{j}\rangle=\sum_{t\in[a,b]_{\mathbb{T}}}x^{*}(t,\lambda_{j})\omega(t)x(t)\mu_{\sigma}(t),\quad
j\in\\{1,2,\cdots,m\\},$
and the following Parseval equality holds, namely
(4.6) $\langle x,x\rangle=\sum_{j=1}^{m}|c_{j}|^{2}.$
3. (iii)
The dynamic operator $L$ has the spectral resolution
(4.7)
$Lx(t)=\sum_{j=1}^{m}\lambda_{j}\pi_{j}x(t)=\int_{-\infty}^{\infty}\lambda
dE_{\lambda}x(t),\quad x\in L_{\omega}^{2}[\rho(a),\sigma(b)],\quad
t\in[\rho(a),\sigma(b)]_{\mathbb{T}},$
with $Lx^{\rho}(a)$ and $Lx^{\sigma}(b)$ given as in (3.16), the projective
operators are given by
(4.8) $\pi_{j}x(t,\lambda_{j})=\delta_{jk}x(t,\lambda_{k}),\quad
j,k\in\\{1,2,\cdots,m\\},\quad t\in[\rho(a),\sigma(b)]_{\mathbb{T}},$
and the projective operator-valued function is given by
(4.9)
$E_{\lambda}=\begin{cases}\sum_{0<\lambda_{j}\leq\lambda}\pi_{j},&:\lambda\geq
0,\\\ -\sum_{\lambda<\lambda_{j}\leq 0}\pi_{j},&:\lambda<0.\end{cases}$
###### Theorem 4.3 (Dual Orthogonality).
Assume the $d\times d$ matrices $P(t)$ for $t\in[\rho(a),b]_{\mathbb{T}}$,
$Q(t)$ and $\omega(t)$ for $t\in[a,b]_{\mathbb{T}}$, are all Hermitian with
$P^{\rho}(a)$ and $P(b)$ invertible, $\omega(t)>0$ for
$t\in[a,b]_{\mathbb{T}}$, $R$ and $S$ in (1.2) are $2d\times 2d$ matrices with
$\operatorname{rank}(R,S)=2d$, and the boundary conditions in (1.2) are self-
adjoint. Let
$r=\operatorname{rank}\left(R_{1}+\frac{1}{\mu_{\rho}(a)}S_{1}P^{\rho}(a),\frac{1}{\mu_{\sigma}(b)}S_{2}\right)$
and $m=d(N-2)+r$. If the boundary conditions (1.2) are proper, that is $r=2d$,
then $m=dN$ and the $dN$ eigenfunctions (4.2) in Theorem 4.2 satisfy the dual
orthogonality condition
(4.10)
$\sum_{j=1}^{dN}x(t,\lambda_{j})x^{*}(\tau,\lambda_{j})=\delta(t,\tau)\omega^{-1}(\tau),\quad
t,\tau\in[a,b]_{\mathbb{T}},\quad\delta(t,\tau)=\begin{cases}0&:t\neq\tau,\\\
1&:t=\tau.\end{cases}$
###### Remark 4.4.
There are additional variational properties of eigenvalues and comparison
results, including Rayleigh’s principle and a minimax theorem, that carry over
verbatim from the discrete case. We refer the interested reader to [27,
Section 5]. One could also use the recent results by Shi and Wu [29] in the
discrete case to improve our results above.
## 5\. Further Extensions to Sturmian Theory
For the discussion that follows, assume the time scale $\mathbb{T}$ is
general, that is to say not necessarily isolated. In the classic continuous
case $(\mathbb{T}=\mathbb{R})$, scalar, vector, and matrix equations of the
form
$-(px^{\prime})^{\prime}(t)+q(t)x(t)=0$
are embedded in a robust theory that extends to even-order Sturm-Liouville
equations, linear Hamiltonian and symplectic systems; see, for example, Reid
[24] and Kratz [22]. It is common on general time scales to consider these
same connections using the basic second-order scalar equation
(5.1) $-(px^{\Delta})^{\Delta}(t)+q(t)x^{\sigma}(t)=0.$
Erbe and Hilger [13] began a Sturmian study of this equation, followed by
Agarwal, Bohner, and Wong [1]. Recently Kong [21] extended some of these
results. To the best of our knowledge, however, it is an open problem on how
to extend (5.1) to even-order self-adjoint Sturm-Liouville problems using only
delta derivatives.
As seen above there is a natural alternative theory that builds on the scalar
equation
$-(px^{\Delta})^{\nabla}(t)+q(t)x(t)=0.$
After Atici and Guseinov introduced the nabla derivative [8] as a left-hand
counterpart on time scales to the delta derivative, there has been an attempt
to formulate a Sturmian theory using both delta and nabla derivatives. Messer
[23] continued the work of [8] by studying second-order scalar self-adjoint
equations; this work was extended to higher order equations, first by Guseinov
[15], and then by Anderson, Guseinov, and Hoffacker [5]; see also Davidson and
Rynne [11]. A Reid roundabout theorem and an analysis of dominant and
recessive solutions is given in [6, 7] for the related matrix equation (5.2),
given below.
A next step might be to extend the results of this paper to even-order Sturm-
Liouville equations and linear Hamiltonian systems on arbitrary time scales,
or more generally to symplectic systems on time scales. We illustrate this in
one possible treatment not found in the literature. For example, we reconsider
(2.1) rewritten in the related homogeneous second-order self-adjoint matrix
dynamic equation in the form
(5.2) $-\left(PX^{\Delta}\right)^{\nabla}(t)+Q(t)X(t)=0,\quad
t\in[a,b]_{\mathbb{T}}.$
Let $P$ be invertible Hermitian, $Q$ ld-continuous Hermitian, and let
$\mathbb{D}$ denote the set of all $n\times n$ matrix-valued functions $X$
defined on $\mathbb{T}$ such that $X^{\Delta}$ is continuous on
$[\rho(a),b]_{\mathbb{T}}$ and $(PX^{\Delta})^{\nabla}$ is left-dense
continuous on $[a,b]_{\mathbb{T}}$. Then $X$ is a solution of (5.2) on
$[\rho(a),\sigma(b)]_{\mathbb{T}}$ provided $X\in\mathbb{D}$ and $X(t)$
satisfies the equation for all $t\in[a,b]_{\mathbb{T}}$.
###### Lemma 5.1.
Let $X\in\mathbb{D}$. If we set
(5.3) $Z:=\left(\begin{smallmatrix}X\\\
PX^{\Delta}\end{smallmatrix}\right)\quad\text{and}\quad
S:=\left(\begin{smallmatrix}-\mu_{\rho}P^{\rho-1}Q&P^{\rho-1}\\\
Q&0\end{smallmatrix}\right),$
then $X$ solves (5.2) on $[a,b]_{\mathbb{T}}$ if and only if $Z$ solves
(5.4) $Z^{\nabla}=S(t)Z,\quad t\in[a,\sigma(b)]_{\mathbb{T}}.$
###### Proof.
We give an idea of the proof. Let $X\in\mathbb{D}$ and $Z,S$ be given as in
(5.3); multiplying out, we get
$\displaystyle SZ$ $\displaystyle=$
$\displaystyle\left(\begin{smallmatrix}-\mu_{\rho}P^{\rho-1}Q&P^{\rho-1}\\\
Q&0\end{smallmatrix}\right)\left(\begin{smallmatrix}X\\\
PX^{\Delta}\end{smallmatrix}\right)=\left(\begin{smallmatrix}-\mu_{\rho}P^{\rho-1}QX+P^{\rho-1}PX^{\Delta}\\\
QX\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{smallmatrix}-\mu_{\rho}P^{\rho-1}(PX^{\Delta})^{\nabla}+P^{\rho-1}PX^{\Delta}\\\
QX\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{smallmatrix}-P^{\rho-1}((PX^{\Delta})-(PX^{\Delta})^{\rho})+P^{\rho-1}PX^{\Delta}\\\
QX\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{smallmatrix}P^{\rho-1}(PX^{\Delta})^{\rho}\\\
QX\end{smallmatrix}\right)=\left(\begin{smallmatrix}X^{\nabla}\\\
QX\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{smallmatrix}X^{\nabla}\\\
(PX^{\Delta})^{\nabla}\end{smallmatrix}\right)=Z^{\nabla},$
where $(PX^{\Delta})^{\rho}=P^{\rho}X^{\nabla}$ since $X\in\mathbb{D}$ implies
$X^{\Delta}$ is continuous. ∎
Next we introduce a new type of Hamiltonian for the nabla differential
operator, not previously discussed in the literature, although clearly in
parallel with that available for the delta differential operator. In the
discrete case see Ahlbrandt and Peterson [3], and Shi [25, 26], and see Bohner
and Peterson [10, Chapter 7] for the delta operator on time scales.
###### Definition 5.2.
A $2n\times 2n$-matrix-valued function $\mathcal{H}$ is Hamiltonian with
respect to $\mathbb{T}$ provided
(5.5) $\mathcal{H}(t)\mbox{ is Hamiltonian and
}I+\mu_{\rho}(t)\mathcal{H}(t)\mathcal{M}^{*}\mathcal{M}\mbox{ is invertible
for all }t\in\mathbb{T},$
where we have used the $2n\times 2n$-matrix
$\mathcal{M}=\left(\begin{smallmatrix}0&I\\\ 0&0\end{smallmatrix}\right)$. For
$\mathcal{H}$ satisfying (5.5), the system
(5.6)
$z^{\nabla}=\mathcal{H}(t)[\mathcal{M}^{*}\mathcal{M}z^{\rho}+\mathcal{M}\mathcal{M}^{*}z]$
is then called a linear Hamiltonian dynamic nabla system.
###### Remark 5.3.
Take $\mathcal{J}$ to be the $2n\times 2n$-matrix
$\mathcal{J}=\left(\begin{smallmatrix}0&I\\\ -I&0\end{smallmatrix}\right)$ and
$\mathcal{H}=\left(\begin{smallmatrix}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&-\mathcal{A}^{*}\end{smallmatrix}\right)$ for some $n\times
n$-matrix-valued functions $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ such
that $\mathcal{B}$ and $\mathcal{C}$ are Hermitian. Then
$\mathcal{H}^{*}\mathcal{J}+\mathcal{J}\mathcal{H}=0$, so that $\mathcal{H}$
is Hamiltonian, and
$I+\mu_{\rho}\mathcal{H}\mathcal{M}^{*}\mathcal{M}=\left(\begin{smallmatrix}I&\mu_{\rho}\mathcal{B}\\\
0&I-\mu_{\rho}\mathcal{A}^{*}\end{smallmatrix}\right).$
Therefore, condition (5.5) may be rewritten as
$\mathcal{H}=\left(\begin{smallmatrix}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&-\mathcal{A}^{*}\end{smallmatrix}\right)$, where
$I-\mu_{\rho}\mathcal{A}^{*}$ is invertible and $\mathcal{B},\mathcal{C}$ are
Hermitian. If we let $z=\left(\begin{smallmatrix}x\\\
u\end{smallmatrix}\right)$, we can rewrite the system (5.6) as
(5.7) $x^{\nabla}=\mathcal{A}(t)x+\mathcal{B}(t)u^{\rho},\quad
u^{\nabla}=\mathcal{C}(t)x-\mathcal{A}^{*}(t)u^{\rho}.$
Also note that equation (5.2) is equivalent to a Hamiltonian system by taking
$\mathcal{A}=0$, $\mathcal{B}=P^{\rho-1}$, $\mathcal{C}=Q$, $x=X$, and
$u=PX^{\Delta}$. Finally, note that (5.7) is the form of the discrete
Hamiltonian introduced by Ahlbrandt [2].
###### Theorem 5.4.
If we set
(5.8)
$\mathcal{S}=(I+\mu_{\rho}\mathcal{H}\mathcal{M}^{*}\mathcal{M})^{-1}\mathcal{H},$
then every Hamiltonian system (5.6) is also a symplectic nabla system
$z^{\nabla}=\mathcal{S}(t)z$.
###### Proof.
We assume first that $z$ solves (5.6). Using the simple formula
$z^{\rho}=z-\mu_{\rho}z^{\nabla}$ and observing that
$\mathcal{M}^{*}\mathcal{M}+\mathcal{M}\mathcal{M}^{*}=I$, we obtain
$(I+\mu_{\rho}\mathcal{H}\mathcal{M}^{*}\mathcal{M})z^{\nabla}=\mathcal{H}z$,
so that (5.8) follows from (5.5). To verify that $\mathcal{S}$ defined by
(5.8) is symplectic with respect to $\mathbb{T}$, videlicet satisfies
condition
$\mathcal{S}^{*}(t)\mathcal{J}+\mathcal{J}\mathcal{S}(t)=\mu_{\rho}(t)\mathcal{S}^{*}(t)\mathcal{J}\mathcal{S}(t)$
for all $t\in\mathbb{T}$, note that
$\mathcal{S}=\left(\begin{smallmatrix}\mathcal{A}-\mu_{\rho}\mathcal{B}(I-\mu_{\rho}\mathcal{A}^{*})^{-1}\mathcal{C}&\mathcal{B}(I-\mu_{\rho}\mathcal{A}^{*})^{-1}\\\
(I-\mu_{\rho}\mathcal{A}^{*})^{-1}\mathcal{C}&-(I-\mu_{\rho}\mathcal{A}^{*})^{-1}\mathcal{A}^{*}\end{smallmatrix}\right).$
It is straightforward to check that
$\mathcal{S}^{*}\mathcal{J}+\mathcal{J}\mathcal{S}=\mu_{\rho}\mathcal{S}^{*}\mathcal{J}\mathcal{S}$
holds on $\mathbb{T}$. ∎
As a final important illustration, consider even-order Sturm-Liouville dynamic
equations; see a related discussion by Shi and Chen [28] on higher-order
discrete Sturm-Liouville problems. This is a new formulation not previously
presented, but clearly in the spirit of the classical Sturm-Liouville analysis
in the continuous case; see Kratz [22, (6.2.1)] and Shi and Chen [28, p. 8].
###### Example 5.5.
Consider the even-order Sturm-Liouville dynamic equation
$\displaystyle My(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{n}(-1)^{n-k}\left(p_{n-k}y^{\nabla^{n-k-1}\Delta}\right)^{\Delta^{n-k-1}\nabla}(t)$
$\displaystyle=$
$\displaystyle(-1)^{n}\left(p_{n}y^{\nabla^{n-1}\Delta}\right)^{\Delta^{n-1}\nabla}(t)+\dots-\left(p_{3}y^{\Delta^{2}\nabla}\right)^{\nabla^{2}\Delta}(t)$
$\displaystyle+\left(p_{2}y^{\nabla\Delta}\right)^{\Delta\nabla}(t)-\left(p_{1}y^{\Delta}\right)^{\nabla}(t)+p_{0}(t)y(t),$
which is formally self-adjoint [5], $p_{n}\neq 0$. We will show (5.5) can be
written in the form of (5.7), where
$\displaystyle\mathcal{A}=(a_{ij})_{1\leq i,j\leq n}\quad\text{with}\quad
a_{ij}=\begin{cases}1:&\text{if }j=i+1,\;1\leq i\leq n-1,\\\
0:&\text{otherwise,}\end{cases}$
$\displaystyle\mathcal{B}=\operatorname{diag}\left\\{0,\dots,0,\frac{1}{p_{n}^{\rho}}\right\\},\quad\mathcal{C}=\operatorname{diag}\left\\{p_{0},p_{1}^{\rho},p_{2}^{\rho},\ldots,p_{n-1}^{\rho}\right\\}.$
To do this, we introduce the pseudo-derivatives of the function $y$ given by
(5.10) $\displaystyle y^{[k]}$ $\displaystyle=$ $\displaystyle
y^{\nabla^{k}},\quad 0\leq k\leq n-1,\quad y^{[0]}=y^{\nabla^{0}}=y,$
$\displaystyle y^{[n]}$ $\displaystyle=$ $\displaystyle
p_{n}y^{\nabla^{n-1}\Delta},$ $\displaystyle y^{[n+k]}$ $\displaystyle=$
$\displaystyle
p_{n-k}y^{\nabla^{n-k-1}\Delta}-\left(y^{[n+k-1]}\right)^{\Delta}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{k}(-1)^{k-i}\left(p_{k-i}y^{\nabla^{n-i-1}\Delta}\right)^{\Delta^{k-i}},\quad
1\leq k\leq n-1,$ (5.11) $\displaystyle y^{[2n]}$ $\displaystyle=$
$\displaystyle p_{0}y-\left(y^{[2n-1]}\right)^{\nabla}=My.$
From (5.10) and (5.11) we have
$\displaystyle\left(y^{[k]}\right)^{\nabla}=y^{[k+1]},\quad 0\leq k\leq n-2,$
$\displaystyle\left(y^{[n-1]}\right)^{\nabla}=y^{\nabla^{n}}=\frac{1}{p_{n}^{\rho}}\left(y^{[n]}\right)^{\rho},$
$\displaystyle\left(y^{[n+k-1]}\right)^{\nabla}=p_{n-k}^{\rho}y^{[n-k]}-\left(y^{[n+k]}\right)^{\rho},\quad
1\leq k\leq n-1,$ $\displaystyle\left(y^{[2n-1]}\right)^{\nabla}=p_{0}y-My.$
For more details on the establishment of related (but different) types of
formulas, please see [5, Section 3]. Then using the substitution
$x=\left(\begin{smallmatrix}y^{[0]}\\\ y^{[1]}\\\ \vdots\\\
y^{[n-1]}\end{smallmatrix}\right),\quad
u=\left(\begin{smallmatrix}y^{[2n-1]}\\\ y^{[2n-2]}\\\ \vdots\\\
y^{[n]}\end{smallmatrix}\right),$
and the matrices $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ above, we
have that
$x^{\nabla}=\mathcal{A}(t)x+\mathcal{B}(t)u^{\rho},\quad
u^{\nabla}=\mathcal{C}(t)x-\mathcal{A}^{*}(t)u^{\rho},$
and the example is complete.
Consider extending the discrete linear Hamiltonian analysis given by Shi [25]
to an eigenvalue problem for the Hamiltonian nabla system (5.7) on general
time scales (including non-isolated time scales), namely
$x^{\nabla}=\mathcal{A}(t)x+\mathcal{B}(t)u^{\rho},\quad
u^{\nabla}=\left[\mathcal{C}(t)-\lambda\omega(t)\right]x-\mathcal{A}^{*}(t)u^{\rho},\quad
t\in[a,b]_{\mathbb{T}}$
with boundary conditions
(5.12) $R\left(\begin{smallmatrix}-x^{\rho}(a)\\\
x(b)\end{smallmatrix}\right)+S\left(\begin{smallmatrix}u^{\rho}(a)\\\
u(b)\end{smallmatrix}\right)=0.$
Here $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ and $\omega$ are $d\times d$
matrices, $\mathcal{B}$ and $\mathcal{C}$ are Hermitian, $\omega>0$ is
positive definite, $A^{*}$ denotes the complex conjugate transpose of $A$, and
$R$ and $S$ are as in the first part of the paper, to wit $2d\times 2d$
matrices with $\operatorname{rank}(R,S)=2d$. Write
$H(t)=\left(\begin{smallmatrix}-\mathcal{C}(t)&\mathcal{A}^{*}(t)\\\
\mathcal{A}(t)&\mathcal{B}(t)\end{smallmatrix}\right),\quad
W(t)=\left(\begin{smallmatrix}\omega(t)&0\\\ 0&0\end{smallmatrix}\right),\quad
J=\left(\begin{smallmatrix}0&-I\\\ I&0\end{smallmatrix}\right).$
Then $H$ and $W$ are $2d\times 2d$ Hermitian matrices with $W(t)\geq 0$ for
$t\in[a,b]_{\mathbb{T}}$, and the system (5.7), (5.12) can be recast in the
form
(5.13) $J\left(\begin{smallmatrix}x^{\nabla}(t)\\\
u^{\nabla}(t)\end{smallmatrix}\right)=\left[H(t)+\lambda
W(t)\right]\left(\begin{smallmatrix}x(t)\\\
u^{\rho}(t)\end{smallmatrix}\right).$
Define the natural dynamic nabla differential operator for (5.7), (5.12) via
(5.14) $\ell(x,u)(t)=J\left(\begin{smallmatrix}x^{\nabla}(t)\\\
u^{\nabla}(t)\end{smallmatrix}\right)-H(t)\left(\begin{smallmatrix}x(t)\\\
u^{\rho}(t)\end{smallmatrix}\right).$
Clearly this includes (1.1), (1.2) as a special case, if we replace $b$ by
$\sigma(b)$ as the right-hand endpoint to accommodate $u=Px^{\Delta}$. As
before we introduce the linear space
$\ell[\rho(a),b]:=\left\\{\left\\{(x(t),u(t))\right\\}_{t\in[\rho(a),b]_{\mathbb{T}}}:x(t),u(t)\in\mathbb{C}^{d},t\in[\rho(a),b])_{\mathbb{T}}\right\\},$
and the weighted inner product
$\langle x,y\rangle=\int_{\rho(a)}^{b}y^{*}(t)\omega(t)x(t)\nabla
t,\quad(x,u),(y,v)\in\ell[\rho(a),b].$
Then we have the following key result.
###### Theorem 5.6 (Lagrange Identity).
For all $(x,u),(y,v)\in\ell[\rho(a),b]$, we have
$\int_{\rho(a)}^{b}\left\\{\left(y^{*},v^{\rho*}\right)\ell(x,u)-\ell(y,v)^{*}\left(\begin{smallmatrix}x\\\
u^{\rho}\end{smallmatrix}\right)\right\\}(t)\nabla
t=\left(y^{*}(t),v^{*}(t)\right)J\left(\begin{smallmatrix}x(t)\\\
u(t)\end{smallmatrix}\right)\Big{|}_{\rho(a)}^{b}.$
###### Proof.
Suppressing the variable $t$, we have
$(y^{*},v^{\rho*})\ell(x,u)=-y^{*}u^{\nabla}+y^{*}\mathcal{C}x-y^{*}\mathcal{A}^{*}u^{\rho}+v^{\rho*}x^{\nabla}-v^{\rho*}\mathcal{A}x-v^{\rho*}\mathcal{B}u^{\rho},$
$\ell(y,v)^{*}\left(\begin{smallmatrix}x\\\
u^{\rho}\end{smallmatrix}\right)=-v^{\nabla*}x+y^{*}\mathcal{C}x-v^{\rho*}\mathcal{A}x+y^{\nabla*}u^{\rho}-y^{*}\mathcal{A}^{*}u^{\rho}-v^{\rho*}\mathcal{B}u^{\rho},$
so that when we subtract the second from the first, we obtain
$\displaystyle(y^{*},v^{\rho*})\ell(x,u)-\ell(y,v)^{*}\left(\begin{smallmatrix}x\\\
u^{\rho}\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle-y^{*}u^{\nabla}+v^{\rho*}x^{\nabla}+v^{\nabla*}x-y^{\nabla*}u^{\rho}$
$\displaystyle=$ $\displaystyle(v^{*}x)^{\nabla}-(y^{*}u)^{\nabla}.$
The result follows from the fundamental theorem of calculus. ∎
A spectral theory for dynamic linear Hamiltonian nabla systems should now be
possible; the interested reader is encouraged to see [25] for the details in
the discrete case.
## References
* [1] R.P. Agarwal, M. Bohner, and P.J.Y. Wong, Sturm-Liouville eigenvalue problems on time scales, _Appl. Math Computation_ 99 (1999) 153–166.
* [2] C.D. Ahlbrandt Equivalence of discrete Euler equations and discrete Hamiltonian systems, _J. Math. Anal. Appl._ 180 (1993) 498–517.
* [3] C.D. Ahlbrandt and A.C. Peterson, _Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Ricatti Equations_ , Kluwer Academic, Boston, 1996.
* [4] D.R. Anderson and J. Hoffacker, Even-order self-adjoint time-scale problems, _Electronic J. Diff. Eq._ 2005(24) (2005) 1 -9.
* [5] D.R. Anderson, G.Sh. Guseinov and J. Hoffacker, Higher-order self adjoint boundary value problems on time scales, _J. Computational Appl. Math._ 194(2) (2006) 309–342.
* [6] D.R. Anderson and B. Buchholz, Self-adjoint matrix equations on time scales, _PanAmerican Mathematical Journal_ , 17(2) (2007) 81–104.
* [7] D.R. Anderson, Dominant and recessive solutions of self-adjoint matrix systems on time scales, _Nonlinear Dyn. Sys. Theory_ , 9(3) (2009) 219- 238.
* [8] F.M. Atici and G.Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, _J. Comput. Appl. Math._ 141 (2002) 75–99.
* [9] M. Bohner, O. Došlý, and W. Kratz, Sturmian and spectral theory for discrete symplectic systems, _Trans. Amer. Math. Soc._ 361(6) (2009) 3109–3123.
* [10] M. Bohner and A. Peterson, _Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2001.
* [11] F.A. Davidson and B.P. Rynne, Self-adjoint boundary value problems on time scales, _Electronic J. Diff. Eq._ 2007(175) (2007) 1–10.
* [12] J.M. Davis, I.A. Gravagne, B.J. Jackson, and R.J. Marks II, Controllability, observability, realizability, and stability of dynamic linear systems, _Electron. J. Diff. Eq._ 2009(37) (2009) 1–32.
* [13] L. Erbe and S. Hilger, Sturmian theory on measure chains, _Differential Equations Dynam. Systems_ 1(3) (1993) 223–244.
* [14] I.A. Gravagne, J.M. Davis, and J.J. DaCunha, A unified approach to high-gain adaptive controllers, submitted. arXiv:0901.3873v1
* [15] G.Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green s functions, _Turkish J. Math._ 29 (2005) 365- 380.
* [16] R.A. Horn and C.R. Johnson, _Topics in Matrix Analysis_ , Cambridge University Press, Cambridge, 1991.
* [17] J. Ji and B. Yang, Eigenvalue comparisons for a class of boundary value problems of second order difference equations, _Linear Algebra Appl._ 420(1) (2007) 218–227.
* [18] J. Ji and B. Yang, Eigenvalue comparisons for second order difference equations with Neumann boundary conditions, _Linear Algebra Appl._ 425(1) (2007) 171–183.
* [19] V. Kac and P. Cheung, _Quantum Calculus_ , Springer, New York, 2002.
* [20] W.G. Kelley and A.C. Peterson, _Difference Equations: An Introduction with Applications_ , 2nd edition, Academic Press, Boston, 2001.
* [21] Q.K. Kong, Sturm-Liouville problems on time scales with separated boundary conditions, _Results Math._ 52(1-2) (2008) 111–121.
* [22] W. Kratz, _Quadratic Functionals in Variational Analysis and Control Theory_ , Akademie Verlag, Berlin, 1995.
* [23] K. Messer, A second-order self-adjoint equation on a time scale, _Dyn. Sys. Appl._ 12 (2003) 201–215.
* [24] W. T. Reid, _Sturmian Theory for Ordinary Differential Equations_ , Springer -Verlag, New York, 1980.
* [25] Y.M. Shi, Spectral theory of discrete linear Hamiltonian systems, _J. Math. Anal. Appl._ 289(2) (2004) 554–570.
* [26] Y.M. Shi, Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems, _Linear Algebra Appl._ 416(2-3) (2006) 452–519.
* [27] Y.M. Shi and S.Z. Chen, Spectral theory of second-order vector difference equations, _J. Math. Anal. Appl._ 239 (1999) 195–212.
* [28] Y.M. Shi and S.Z. Chen, Spectral theory of higher-order discrete vector Sturm-Liouville problems, _Linear Algebra Appl._ 323(1-3) (2001) 7–36.
* [29] G.L. Shi and H.Y. Wu, Spectral theory of Sturm-Liouville difference operators, _Linear Algebra Appl._ 430(2-3) (2009) 830–846.
* [30] G.L. Shi and R. Yan, Spectral theory of left definite difference operators, _J. Math. Anal. Appl._ 337(1) (2008) 116–122.
* [31] G.F. Simmons, _Calculus Gems: Brief Lives and Memorable Mathematics_ , McGraw-Hill, New York, 1992.
* [32] H.Q. Sun and Y.M. Shi, Eigenvalues of second-order difference equations with coupled boundary conditions, _Linear Algebra Appl._ 414(1) (2006) 361–372.
* [33] S.R. Sun, Y.M. Shi, and S.Z. Chen, The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems, _J. Math. Anal. Appl._ 327(2) (2007) 1360–1380.
* [34] Y. Wang and Y.M. Shi, Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, _J. Math. Anal. Appl._ 309(1) (2005) 56–69.
|
arxiv-papers
| 2010-01-22T14:54:42 |
2024-09-04T02:49:07.940063
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1001.4010"
}
|
1001.4040
|
# Unifying discrete and continuous Weyl -Titchmarsh theory via a class of
linear Hamiltonian systems on Sturmian time scales
Douglas R. Anderson Department of Mathematics and Computer Science, Concordia
College, Moorhead, MN 56562 USA
visiting the School of Mathematics, the University of New South Wales, Sydney
NSW 2052, Australia andersod@cord.edu
###### Abstract.
In this study, we are concerned with introducing Weyl-Titchmarsh theory for a
class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian
time scales. After developing fundamental properties of solutions and regular
spectral problems, we introduce the corresponding maximal and minimal
operators for the system. Matrix disks are constructed and proved to be nested
and converge to a limiting set. Some precise relationships among the rank of
the matrix radius of the limiting set, the number of linearly independent
square summable solutions, and the defect indices of the minimal operator are
established. Using the above results, a classification of singular dynamic
linear Hamiltonian nabla systems is given in terms of the defect indices of
the minimal operator, and several equivalent conditions on the cases of limit
point and limit circle are obtained, respectively. These results unify and
extend certain classic and recent results on the subject in the continuous and
discrete cases, respectively, to Sturmian time scales.
###### Key words and phrases:
limit point; limit circle; spectral problem; Sturmian time scales; Sturm-
Liouville theory; linear Hamiltonian systems; nabla derivative.
###### 2000 Mathematics Subject Classification:
34B20; 34N05; 47A10; 47B25
## 1\. Introduction
Ahlbrandt [2] introduced the following class of linear discrete Hamiltonian
equations
$\nabla x(t)=H_{y}(t,x(t),y(t-1)),\quad\nabla y(t)=-H_{x}(t,x(t),y(t-1)),$
where $\nabla x(t)=x(t)-x(t-1)$, which yields a linear discrete Hamiltonian
system [5] of the form
$\nabla x(t)=A(t)x(t)+B(t)u(t-1),\quad\nabla u(t)=C(t)x(t)-A^{*}(t)u(t-1),$
where $A$, $B$, and $C$ are $d\times d$ matrices, $A^{*}$ is the complex
conjugate transpose of $A$, and $B$ and $C$ are Hermitian. Shi [41] shifted
the points one unit right and extended the analysis to develop a Weyl-
Titchmarsh theory for linear discrete Hamiltonian systems
(1.1) $J\Delta y(t)=(\lambda W(t)+P(t))R(y)(t),\quad
t\in[0,\infty)\cap\mathbb{Z},$
where $W$ and $P$ are $2d\times 2d$ complex Hermitian matrices with weight
function $W(t)\geq 0$, which is one possible discrete version (see also Clark
and Gesztesy [16]) of the classic form studied by Atkinson [11]
(1.2) $Jy^{\prime}(t)=(\lambda W(t)+P(t))y(t),\quad t\in[0,\infty).$
Shi used the partial right uniform shift operator
$R(y)(t)=(y_{1}^{\operatorname{T}}(t+1),y_{2}^{\operatorname{T}}(t))^{\operatorname{T}}$
with the vector
$y(t)=(y_{1}^{\operatorname{T}}(t),y_{2}^{\operatorname{T}}(t))^{\operatorname{T}}$
for $y_{1},y_{2}\in\mathbb{C}^{d}$ and the canonical symplectic matrix
$J=\left(\begin{smallmatrix}0&-I_{d}\\\ I_{d}&0\end{smallmatrix}\right)$. With
a view toward extending (1.1) to Sturmian time scales, we introduce the linear
Hamiltonian nabla system on Sturmian time scales
(1.3) $Jy^{\nabla}(t)=(\lambda W(t)+P(t))\Upsilon y(t),\quad
t\in[t_{0},\infty)_{\mathbb{T}}:=[t_{0},\infty)\cap\mathbb{T},\quad
J=\left(\begin{smallmatrix}0&-I_{d}\\\ I_{d}&0\end{smallmatrix}\right),$
where $W$ and $P$ are $2d\times 2d$ complex Hermitian left-dense continuous
matrix functions, $y:\mathbb{T}\rightarrow\mathbb{C}^{2d}$ is a nabla
differentiable vector function, and we use the partial left-shift operator
$\Upsilon$ defined by
(1.4) $\Upsilon y(t):=\begin{pmatrix}y_{1}(t)\\\
y_{2}^{\rho}(t)\end{pmatrix}\quad\text{for}\quad
y(t)=\begin{pmatrix}y_{1}(t)\\\ y_{2}(t)\end{pmatrix},\quad
y_{1},y_{2}:[\rho(t_{0}),\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{d}.$
Throughout this work we have the following key assumptions: following Atkinson
[11, Chapter 9], the weight function $W$ satisfies the definiteness conditions
(1.5) $\begin{cases}W(t)=\operatorname{diag}\\{W_{1}(t),W_{2}(t)\\}\geq
0,&W_{j}(t)\geq 0\;\text{is}\;d\times d\;\text{Hermitian},\quad j=1,2,\\\
\displaystyle\int_{t_{0}}^{t}(\Upsilon y)^{*}(s)W(s)\Upsilon y(s)\nabla
s>0,&\forall\;t\in[t_{1},\infty)_{\mathbb{T}}\;\text{for
some}\;t_{1}\in\mathbb{T},\end{cases}$
for every nontrivial solution $y$ of (1.3), and $P$ satisfies the block form
(1.6) $P(t)=\begin{pmatrix}-C(t)&A^{*}(t)\\\ A(t)&B(t)\end{pmatrix},\quad
I_{d}-\nu(t)A(t)\;\text{invertible},\quad t\in[t_{0},\infty)_{\mathbb{T}},$
where $A$, $B$, and $C$ are left-dense continuous $d\times d$ complex matrices
with $B$ and $C$ Hermitian. Using standard notation, $\mathbb{T}$ is an
unbounded Sturmian time scale; the left jump operator $\rho$ is given by
$\rho(t)=\sup\\{s\in\mathbb{T}:s<t\\}$ with the composition $u\circ\rho$
denoted $u^{\rho}$; the graininess function is defined by $\nu(t)=t-\rho(t)$;
and the nabla derivative of $x$ at $t\in\mathbb{T}$, denoted $x^{\nabla}(t)$,
is the vector (provided it exists) given by
$x^{\nabla}(t):=\lim_{s\rightarrow t}\frac{x^{\rho}(t)-x(s)}{\rho(t)-s}.$
Sturmian time scales, introduced in [4], are a specialized class of time
scales (closed, nonempty sets of real numbers) with the property that
(1.7) $\sigma(\rho(t))=\rho(\sigma(t))\quad\text{for all}\quad
t\in[t_{0},\infty)_{\mathbb{T}}.$
This crucial assumption allows us to identify a partial right-shift operator
$\Upsilon^{-1}$ in terms of (1.4), namely
(1.8) $\Upsilon^{-1}y(t):=\begin{pmatrix}y_{1}(t)\\\
y_{2}^{\sigma}(t)\end{pmatrix}\quad\text{for}\quad
y(t)=\begin{pmatrix}y_{1}(t)\\\ y_{2}(t)\end{pmatrix},\quad
y_{1},y_{2}:[\rho(t_{0}),\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{d};$
note that on our Sturmian time scale $\mathbb{T}$, we have
$\Upsilon(\Upsilon^{-1}y)(t)=\Upsilon^{-1}(\Upsilon y)(t)=y(t)$ for all
$t\in[t_{0},\infty)_{\mathbb{T}}$. For more on general time scales using the
nabla derivative, see [10] and [15, Chapter 3].
###### Remark 1.1.
We employ the nabla version in (1.3) for two reasons. One, it reflects the
original form introduced by Ahlbrandt [2], and two, it contains the following
two important dynamic models [7, 9]. The first is the linear Hamiltonian nabla
system on general time scales
(1.9) $x^{\nabla}(t)=A(t)x(t)+B(t)u^{\rho}(t),\quad
u^{\nabla}(t)=\left[C(t)-\lambda\omega(t)\right]x(t)-A^{*}(t)u^{\rho}(t)$
for $t\in[a,b]_{\mathbb{T}}$, where $A$, $B$, $C$ and $\omega$ are $d\times d$
matrices, $B$ and $C$ are Hermitian, $\omega>0$ is positive definite. It is
straightforward to write (1.9) in the form (1.3), by taking
$J=\left(\begin{smallmatrix}0&-I_{d}\\\ I_{d}&0\end{smallmatrix}\right),\quad
y(t)=\left(\begin{smallmatrix}x(t)\\\ u(t)\end{smallmatrix}\right),\quad
P(t)=\left(\begin{smallmatrix}-C(t)&A^{*}(t)\\\
A(t)&B(t)\end{smallmatrix}\right),\quad
W(t)=\left(\begin{smallmatrix}\omega(t)&0\\\ 0&0\end{smallmatrix}\right).$
This model includes the second-order self-adjoint matrix equation [6, 8]
$-(P_{0}X^{\Delta})^{\nabla}(t)+Q(t)X(t)=0$
for Hermitian $P_{0}$ and $Q$ with $P_{0}$ invertible, by taking $A=0$,
$B=(P_{0}^{\rho})^{-1}$, $C=Q$, and $\lambda=0$ in (1.9).
The second important dynamic model is the even-order self-adjoint Sturm-
Liouville dynamic equation
$\displaystyle My(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{n}(-1)^{n-k}\left(p_{n-k}y^{\nabla^{n-k-1}\Delta}\right)^{\Delta^{n-k-1}\nabla}(t)$
$\displaystyle=$
$\displaystyle(-1)^{n}\left(p_{n}y^{\nabla^{n-1}\Delta}\right)^{\Delta^{n-1}\nabla}(t)+\dots-\left(p_{3}y^{\Delta^{2}\nabla}\right)^{\nabla^{2}\Delta}(t)$
$\displaystyle+\left(p_{2}y^{\nabla\Delta}\right)^{\Delta\nabla}(t)-\left(p_{1}y^{\Delta}\right)^{\nabla}(t)+p_{0}(t)y(t),$
which is formally self-adjoint [9], where $p_{n}\neq 0$. We will show (1.1)
can be written in the form of (1.9), where
(1.11) $\displaystyle A=(a_{ij})_{1\leq i,j\leq n}\quad\text{with}\quad
a_{ij}=\begin{cases}1:&\text{if }j=i+1,\;1\leq i\leq n-1,\\\
0:&\text{otherwise,}\end{cases}$ $\displaystyle
B=\operatorname{diag}\left\\{0,\dots,0,\frac{1}{p_{n}^{\rho}}\right\\},\quad
C=\operatorname{diag}\left\\{p_{0},p_{1}^{\rho},p_{2}^{\rho},\ldots,p_{n-1}^{\rho}\right\\}.$
To do this, we introduce the pseudo-derivatives of the function $y$ given by
$\displaystyle y^{[k]}$ $\displaystyle=$ $\displaystyle y^{\nabla^{k}},\quad
0\leq k\leq n-1,\quad y^{[0]}=y^{\nabla^{0}}=y,$ $\displaystyle y^{[n]}$
$\displaystyle=$ $\displaystyle p_{n}y^{\nabla^{n-1}\Delta},$ $\displaystyle
y^{[n+k]}$ $\displaystyle=$ $\displaystyle
p_{n-k}y^{\nabla^{n-k-1}\Delta}-\left(y^{[n+k-1]}\right)^{\Delta}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{k}(-1)^{k-i}\left(p_{k-i}y^{\nabla^{n-i-1}\Delta}\right)^{\Delta^{k-i}},\quad
1\leq k\leq n-1,$ $\displaystyle y^{[2n]}$ $\displaystyle=$ $\displaystyle
p_{0}y-\left(y^{[2n-1]}\right)^{\nabla}=My.$
Then using the substitution
$x=\left(\begin{smallmatrix}y^{[0]}\\\ y^{[1]}\\\ \vdots\\\
y^{[n-1]}\end{smallmatrix}\right),\quad
u=\left(\begin{smallmatrix}y^{[2n-1]}\\\ y^{[2n-2]}\\\ \vdots\\\
y^{[n]}\end{smallmatrix}\right),$
and the matrices $A$, $B$, and $C$ above in (1.11), we have that
$x^{\nabla}=A(t)x+B(t)u^{\rho},\quad u^{\nabla}=C(t)x-A^{*}(t)u^{\rho},$
and the example is complete.
There is a vast literature on continuous linear Hamiltonian systems, and of
late a growing collection of results on corresponding discrete Hamiltonian
systems. For a few of the many relevant papers in the continuous case, see
[12],[17]$-$[21],[38],[53], and see [2, 5, 13, 16, 33, 34],[40]$-$[49] for
recent work on discrete second-order difference equations and linear
Hamiltonian systems. Some of the fundamental continuous and disrete results
alluded to above have been unified and extended via dynamic equations on time
scales, introduced by Hilger [26]. For scalar Sturm-Liouville results on time
scales, see [1, 4, 35, 37], and for systems see [3, 14, 27]. Turning to Weyl
[52] and Titchmarsh [48] specifically, much has been published on the
continuous Weyl-Titchmarsh theory, for example [23, 24, 25],[28, 29],[30, 31,
32], and three substantial works on the corresponding discrete theory,
Atkinson [11], Clark and Gesztesy [16] and Shi [41]. Recently [51] made a
first start on the scalar theory on time scales. There is yet to be any
published work on a unified continuous and discrete Weyl-Titchmarsh theory,
however, for linear Hamiltonian systems; via this paper we hope to initiate
such an investigation. With Sturmian time scales there is a much broader scope
of discretization options other than just the uniform step size offered by
difference equations. In the analysis that follows, we will largely follow a
development of the theory along the lines of Shi [41].
We will proceed as follows. In Section 2, we introduce fundamental properties
for system (1.3) and introduce a Lagrange identity. Regular spectral problems
are discussed in Section 3 via separated boundary conditions, and a result on
eigenpairs is given. In Section 4 we introduce a weighted Hilbert space to
facilitate a study of maximal and minimal operators. Weyl disks and their
limiting set are the focus of Section 5, while in Section 6 we introduce the
concept of square summable solutions. In Section 7, we give a classification
of singular linear Hamiltonian nabla systems. In the final section we discuss
an alternative form to (1.3) that may also serve as a generalization of (1.1)
and (1.2) on Sturmian time scales.
## 2\. Fundamental Properties
For any given $\lambda\in\mathbb{C}$, using the assumptions on the block forms
of $W$ and $P$ in (1.5) and (1.6), respectively, we can rewrite (1.3) as the
pair of $d$-vector equations
(2.1) $\begin{cases}y_{1}^{\nabla}(t)=A(t)y_{1}(t)+\big{(}B(t)+\lambda
W_{2}(t)\big{)}y_{2}^{\rho}(t),\\\ y_{2}^{\nabla}(t)=\big{(}C(t)-\lambda
W_{1}(t)\big{)}y_{1}(t)-A^{*}(t)y_{2}^{\rho}(t).\end{cases}$
By (1.6), we have that
(2.2) $E(t):=\big{(}I_{d}-\nu(t)A(t)\big{)}^{-1}$
exists. Then we may also view solutions $y=(y_{1},y_{2})^{\operatorname{T}}$
of (1.3) and (2.1) as solutions of
(2.3)
$y^{\nabla}(t)=\mathcal{S}(t,\lambda)y(t),\quad\mathcal{S}(\cdot,\lambda):=\begin{pmatrix}A-\nu\big{(}B+\lambda
W_{2}\big{)}E^{*}\big{(}C-\lambda W_{1}\big{)}&\big{(}B+\lambda
W_{2}\big{)}E^{*}\\\ E^{*}\big{(}C-\lambda
W_{1}\big{)}&-E^{*}A^{*}\end{pmatrix}$
for $E$ in (2.2), where it is straightforward to check that
$\mathcal{S}(\cdot,\lambda)$ satisfies
(2.4)
$\mathcal{S}^{*}\left(\cdot,\overline{\lambda}\right)J+J\mathcal{S}(\cdot,\lambda)=\nu\mathcal{S}^{*}\left(\cdot,\overline{\lambda}\right)J\mathcal{S}(\cdot,\lambda)$
for $t\in\mathbb{T}$. Directly from (2.4) we have that
(2.5)
$\left(I_{2d}-\nu(t)\mathcal{S}(t,\lambda)\right)^{*}J\left(I_{2d}-\nu(t)\mathcal{S}(t,\lambda)\right)=J,$
so that $I_{2d}-\nu(t)\mathcal{S}(t,\lambda)$ is invertible and thus
$\mathcal{S}(\cdot,\lambda)$ is $\nu-$regressive. Given the results above, we
now have the following lemma.
###### Lemma 2.1.
Assume $I_{d}-\nu A$ is invertible on $\mathbb{T}$, and $\lambda\in\mathbb{C}$
is arbitrary. Then for any vector solution $y(\cdot,\lambda)$ of (1.3)λ and
for any vector solution $z\left(\cdot,\overline{\lambda}\right)$ of
(1.3)${}_{\overline{\lambda}}$ we have
(2.6)
$z^{*}\left(t,\overline{\lambda}\right)Jy(t,\lambda)=\operatorname{const}.$
###### Proof.
Let $t\in\mathbb{T}$. By using the simple useful formula $y^{\rho}=y-\nu
y^{\nabla}$ and (2.3) we have that
$y^{\rho}(t,\lambda)=y(t,\lambda)-\nu(t)y^{\nabla}(t,\lambda)=\left(I_{d}-\nu(t)\mathcal{S}(t,\lambda)\right)y(t,\lambda).$
From the nabla product rule we subsequently obtain
$\displaystyle\left(z^{*}\left(t,\overline{\lambda}\right)Jy(t,\lambda)\right)^{\nabla}$
$\displaystyle=$ $\displaystyle
z^{*\nabla}\left(t,\overline{\lambda}\right)Jy^{\rho}(t,\lambda)+z^{*}\left(t,\overline{\lambda}\right)Jy^{\nabla}(t,\lambda)$
$\displaystyle=$ $\displaystyle
z^{*}\left(t,\overline{\lambda}\right)\mathcal{S}^{*}\left(t,\overline{\lambda}\right)J\left(I_{d}-\nu(t)\mathcal{S}(t,\lambda)\right)y(t,\lambda)+z^{*}\left(t,\overline{\lambda}\right)J\mathcal{S}(t,\lambda)y(t,\lambda)$
$\displaystyle=$ $\displaystyle
z^{*}\left(t,\overline{\lambda}\right)\left[\mathcal{S}^{*}\left(t,\overline{\lambda}\right)J+J\mathcal{S}(t,\lambda)-\nu(t)\mathcal{S}^{*}\left(t,\overline{\lambda}\right)J\mathcal{S}(t,\lambda)\right]y(t,\lambda)$
$\displaystyle=$ $\displaystyle 0,$
where the last line follows from (2.4). ∎
We now define a natural dynamic nabla differential operator for (1.3) via
(2.7) $\mathscr{L}y(t):=Jy^{\nabla}(t)-P(t)\Upsilon y(t),\quad y\in{\rm
C}_{\rm ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right),$
where ${\rm C}_{\rm ld}^{1}$ is the space of $2d$-vector functions with left-
dense continuous nabla derivatives on the given time scale interval. Then we
have the following key result.
###### Theorem 2.2 (Lagrange Identity).
For all $x,y\in{\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right)$, where
$x=\left(\begin{smallmatrix}x_{1}\\\ x_{2}\end{smallmatrix}\right)$ and
$y=\left(\begin{smallmatrix}y_{1}\\\ y_{2}\end{smallmatrix}\right)$, we have
$\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon
x)^{*}\mathscr{L}y-(\mathscr{L}x)^{*}\Upsilon y\big{\\}}(t)\nabla
t=x^{*}(t)Jy(t)\Big{|}_{\rho(t_{0})}^{b}.$
###### Proof.
Suppressing the variable $t$, we have
$(\Upsilon
x)^{*}\mathscr{L}y=-x_{1}^{*}y_{2}^{\nabla}+x_{1}^{*}Cy_{1}-x_{1}^{*}A^{*}y_{2}^{\rho}+x_{2}^{\rho*}y_{1}^{\nabla}-x_{2}^{\rho*}Ay_{1}-x_{2}^{\rho*}By_{2}^{\rho},$
$(\mathscr{L}x)^{*}\Upsilon
y=-x_{2}^{\nabla*}y_{1}+x_{1}^{*}Cy_{1}-x_{2}^{\rho*}Ay_{1}+x_{1}^{\nabla*}y_{2}^{\rho}-x_{1}^{*}A^{*}y_{2}^{\rho}-x_{2}^{\rho*}By_{2}^{\rho},$
so that when we subtract the second from the first, we obtain
$\displaystyle(\Upsilon x)^{*}\mathscr{L}y-(\mathscr{L}x)^{*}\Upsilon y$
$\displaystyle=$ $\displaystyle-
x_{1}^{*}y_{2}^{\nabla}+x_{2}^{\rho*}y_{1}^{\nabla}+x_{2}^{\nabla*}y_{1}-x_{1}^{\nabla*}y_{2}^{\rho}$
$\displaystyle=$
$\displaystyle-(x_{1}^{*}y_{2})^{\nabla}+(x_{2}^{*}y_{1})^{\nabla}=\left(x^{*}Jy\right)^{\nabla}(t).$
The result follows from the fundamental theorem of calculus. ∎
###### Lemma 2.3.
Assume $I_{d}-\nu A$ is invertible on $\mathbb{T}$. For all
$\lambda,\eta\in\mathbb{C}$, let $y(\cdot,\lambda)$ and $z(\cdot,\eta)$ be any
solutions of (1.3)λ and (1.3)η , respectively. Then for any
$t\in(t_{0},\infty)_{\mathbb{T}}$, we have
$\left(\eta-\overline{\lambda}\right)\int_{\rho(t_{0})}^{t}(\Upsilon
y)^{*}(s,\lambda)W(s)\Upsilon z(s,\eta)\nabla
s=y^{*}(t,\lambda)Jz(t,\eta)\Big{|}_{\rho(t_{0})}^{t}.$
###### Proof.
By (2.7), $\mathscr{L}y(\cdot,\lambda)=\lambda W\Upsilon y(\cdot,\lambda)$ and
$\mathscr{L}z(\cdot,\eta)=\eta W\Upsilon z(\cdot,\eta)$. Use of Theorem 2.2
yields the result. ∎
## 3\. Regular Spectral Problems
In this section we analyze (1.3) over the time scale interval
$[\rho(t_{0}),b]_{\mathbb{T}}$ of finite measure, obtaining some fundamental
spectral results. To this end, consider the regular spectral problem for
system (1.3) on $[\rho(t_{0}),b]_{\mathbb{T}}$ with the separated (homogeneous
Dirichlet) boundary conditions
(3.1) $\alpha y^{\rho}(t_{0})=0,\quad\beta y(b)=0,\qquad y\in{\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right),$
where $\alpha$ and $\beta$ are (normalized) $d\times 2d$ matrices that satisfy
the following self-adjoint boundary conditions
(3.2) $\displaystyle\operatorname{rank}\alpha=d,$
$\displaystyle\alpha\alpha^{*}=I_{d},$ $\displaystyle\alpha J\alpha^{*}=0,$
(3.3) $\displaystyle\operatorname{rank}\beta=d,$
$\displaystyle\beta\beta^{*}=I_{d},$ $\displaystyle\beta J\beta^{*}=0.$
We call these boundary conditions self-adjoint as they cause the Lagrange
identity in Theorem 2.2 to equal zero for $y\in{\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right)$.
###### Lemma 3.1.
Let $\alpha$ and $\beta$ satisfy (3.2) and (3.3), respectively. Then $y\in{\rm
C}_{\rm ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right)$
satisfies (3.1) if and only if there exists a unique vector
$\xi\in\mathbb{C}^{2d}$ such that
(3.4) $y^{\rho}(t_{0})=M\xi,\quad y(b)=N\xi,$
where $M=(-J\alpha^{*},0)$ and $N=(0,J\beta^{*})$. Additionally,
(3.5) $M^{*}JM=0=N^{*}JN,\quad\operatorname{rank}\left(\begin{smallmatrix}M\\\
N\end{smallmatrix}\right)=2d.$
###### Proof.
For (3.4), see the continuous case in Kratz [36, Proposition 2.1.1] or Zettl
[53, Theorem 10.4.3], or in the discrete case in Shi [40, Lemma 2.1]; the
time-scales case is unchanged [6, Lemma 2.4]. Equation (3.5) follows
immediately from (3.2) and (3.3), respectively. ∎
Assume that (1.6) holds on $[\rho(t_{0}),b]_{\mathbb{T}}$. For each
$\lambda\in\mathbb{C}$, let $\Phi(\cdot,\lambda)$ be a fundamental matrix
solution for (1.3)λ. It follows from (2.3) that we can view
$\Phi(\cdot,\lambda)$ as the solution of the initial value problem
$\Phi^{\nabla}=\mathcal{S}(\cdot,\lambda)\Phi,\qquad\Phi^{\rho}(t_{0})=I_{2d},$
and that a general solution of (1.3)λ can be written as
(3.6) $y(t,\lambda)=\Phi(t,\lambda)y^{\rho}(t_{0},\lambda).$
###### Theorem 3.2.
Assume (1.5) and (1.6), and let $\alpha$ and $\beta$ satisfy (3.2) and (3.3),
respectively. Then for each $b\in[t_{1},\infty)_{\mathbb{T}}$ $($where $t_{1}$
is specified in (1.5)$)$, $\lambda$ is an eigenvalue of the boundary value
problem (1.3), (3.1) if and only if
(3.7) $\det\big{(}\Phi(b,\lambda)M-N\big{)}=0$
for the fundamental matrix solution $\Phi$ in (3.6). Moreover, all the
eigenvalues of (1.3) and (3.1) are real and can be numbered serially as in
(3.8) $|\lambda_{0}(b)|\leq|\lambda_{1}(b)|\leq|\lambda_{2}(b)|\leq\cdots,$
such that the corresponding eigenfunctions $y(\cdot,\lambda_{j}(b))$ satisfy
the orthonormality relation
(3.9)
$\Big{\langle}y(\cdot,\lambda_{j}(b)),y(\cdot,\lambda_{k}(b))\Big{\rangle}_{b}:=\int_{\rho(t_{0})}^{b}(\Upsilon
y)^{*}(t,\lambda_{k}(b))W(t)\Upsilon y(t,\lambda_{j}(b))\nabla t=\delta_{jk}.$
###### Proof.
To show (3.7), recall that $\lambda$ is an eigenvalue for the boundary value
problem (1.3), (3.1) with nontrivial eigenfunction $y(\cdot,\lambda)$ if and
only if there exists a vector $\xi\in\mathbb{C}^{2d}$, $\xi\neq 0$, such that
$y^{\rho}(t_{0},\lambda)=M\xi$ and $y(b,\lambda)=N\xi$ by Lemma 3.1 and (3.4),
if and only if $\left(\Phi(b,\lambda)M-N\right)\xi=0$ by (3.6). That the
eigenvalues are real follows from Theorem 2.2 and Lemma 3.1. As in Atkinson
[11, Theorem 9.2.1], we note that $\Phi(b,\lambda)$ consists of entire
functions of $\lambda$, so that the left-hand side of (3.7) is also an entire
function. Thus it has no complex zeros, and its zeros have no finite limit
point, whence we have (3.8), which may be a finite or infinite list and
includes possible multiplicities.
If $\lambda_{j}(b)\neq\lambda_{k}(b)$, then the corresponding eigenfunctions
$y(\cdot,\lambda_{j}(b))$ and $y(\cdot,\lambda_{k}(b))$ are orthogonal by
Theorem 2.2 and Lemma 3.1. These functions can be normalized by taking
$y(t,\lambda_{j}(b))/\|y(\cdot,\lambda_{j}(b))\|_{b}$, where
$\|x\|_{b}:=\big{(}\langle x,x\rangle_{b}\big{)}^{1/2}$
in terms of (3.9). Following Atkinson [11, Section 9.3] and Shi [41, Theorem
2.3], suppose $\lambda_{j}(b)$ is an eigenvalue with multiplicity $d_{j}$. Set
(3.10)
$V_{j}=\left\\{\xi\in\mathbb{C}^{2d}:\left(\Phi(b,\lambda_{j}(b))M-N\right)\xi=0\right\\}.$
It follows that $V_{j}$ is a subspace of $\mathbb{C}^{2d}$ with $\dim
V_{j}=d_{j}$, and $\lambda_{j}(b)$ appears exactly $d_{j}$ times in (3.8), say
$\lambda_{j}(b),\quad j=j^{\prime}+1,\cdots,j^{\prime}+d_{j}.$
We will choose a basis
$\left\\{\xi_{j}\right\\}_{j=j^{\prime}+1}^{j^{\prime}+d_{j}}$ of the set
$V_{j}$ in (3.10) such that the corresponding eigenfunctions
$y(t,\lambda_{j}(b))=\Phi(t,\lambda_{j}(b))M\xi_{j},\quad
j=j^{\prime}+1,\cdots,j^{\prime}+d_{j},$
are mutually orthonormal. We apply a process of orthogonalization ala Atkinson
[11, 9.3.13]. If we write
$u_{j}(b)=y(\rho(t_{0}),\lambda_{j}(b))=M\xi_{j},\quad
y(t,\lambda_{j}(b))=\Phi(t,\lambda_{j}(b))u_{j}(b),$
then (3.9) is equivalent to
(3.11) $u^{*}_{r}(b)K(b,\lambda_{j}(b))u_{s}(b)=\delta_{rs},\quad
j^{\prime}+1\leq r,s\leq j^{\prime}+d_{j},$
where
$K(t,\lambda):=\int_{\rho(t_{0})}^{t}(\Upsilon\Phi)^{*}(\tau,\lambda)W(\tau)\Upsilon\Phi(\tau,\lambda)\nabla\tau,$
and $\Upsilon\Phi(t,\lambda)$ denotes the partial left-shift operator
$\Upsilon$ from (1.4) acting on the last $d$ rows of the fundamental matrix
$\Phi(t,\lambda)$ with respect to the variable $t$. Using the definiteness
condition in (1.5) and the invertibility of $\Phi(\cdot,\lambda_{j}(b))$, we
see that $K(b,\lambda_{j}(b))>0$. On the other hand, the space
$\widetilde{V}_{j}=\\{u:u=M\xi\\}$ has the same dimension $d_{j}$ as $V_{j}$,
as $M\xi=N\xi=0$ always implies $\xi=0$ by (3.5). By the invertibility of
$K(b,\lambda_{j}(b))$, the space
$\widehat{V}_{j}=\\{v:v=(K(b,\lambda_{j}(b)))^{1/2}u,\;u\in\widetilde{V}_{j}\\}$
has dimension $d_{j}$ as well, and an orthonormal basis
$\\{v_{r}\\}_{r=j^{\prime}+1}^{j^{\prime}+d_{j}}$, that is
$v_{r}^{*}v_{s}=\delta_{rs},\quad j^{\prime}+1\leq r,s\leq j^{\prime}+d_{j}.$
From this we recover a basis for $\widetilde{V}_{j}$, namely
$u_{r}=(K(b,\lambda_{j}(b)))^{-1/2}v_{r},\quad j^{\prime}+1\leq r,s\leq
j^{\prime}+d_{j},$
that satisfies (3.11), and (3.9) follows. ∎
## 4\. Maximal and Minimal Operators
In this section we introduce a weighted Hilbert space
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, define minimal and
maximal operators corresponding to system (1.3), and show that the minimal
operator is symmetric, the maximal operator is densely defined, and the
adjoint of the minimal operator is precisely the maximal operator. To clarify
the notation to follow, we will denote the domain, range, and kernel of an
operator $K$ by $\operatorname{Dom}(K)$, $\operatorname{Ran}(K)$, and
$\operatorname{Ker}(K)$, respectively. Some concepts for linear operators in
Hilbert spaces are first introduced; see [50].
###### Definition 4.1.
Let $X$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$, and
let $K:\operatorname{Dom}(K)\subset X\rightarrow X$ be a linear operator.
1. (i)
$K$ is said to be densely defined if $\operatorname{Dom}(K)$ is dense.
2. (ii)
$K$ is said to be Hermitian if it is formally self-adjoint, i.e., $\langle
Kf,g\rangle=\langle f,Kg\rangle$ for all $f,g\in\operatorname{Dom}(K)$.
3. (iii)
$K$ is said to be symmetric if it is Hermitian and densely defined.
4. (iv)
Let $K$ be a densely defined linear operator. The adjoint operator $K^{*}$ of
$K$ is defined as $\operatorname{Dom}(K^{*})=\\{g\in X:\text{the
functional}\;f\mapsto\langle g,Kf\rangle\;\text{is continuous
on}\;\operatorname{Dom}(K)\\}$ and $\langle K^{*}g,f\rangle=\langle
g,Kf\rangle$ for all $f\in\operatorname{Dom}(K)$ and
$g\in\operatorname{Dom}(K^{*})$.
5. (v)
For given $\lambda\in\mathbb{C}$, the subspace
$\operatorname{Ran}\left(\overline{\lambda}-K\right)^{\perp}$ is called the
defect space of $K$ and $\lambda$, and
$d(\lambda)=\dim\operatorname{Ran}\left(\overline{\lambda}-K\right)^{\perp}$
is called the defect index of $K$ and $\lambda$.
If $K$ is Hermitian, then $d(\lambda)$ is constant in the upper and lower half
planes, respectively. Denote $d_{+}=d(i)$ and $d_{-}=d(-i)$. Then $d_{+}$ and
$d_{-}$ are called the positive and negative defect indices of $K$,
respectively. Further, if $K$ is densely defined, then
$\overline{\operatorname{Ran}K}\oplus\operatorname{Ker}K^{*}=X$. Hence, if $K$
is symmetric, then
$\operatorname{Ran}\left(\overline{\lambda}-K\right)^{\perp}=\operatorname{Ker}(\lambda-K^{*})$.
It follows that, for the symmetric operator $K$,
$\operatorname{Ran}(-i-K)^{\perp}=\operatorname{Ker}(i-K^{*})$ and
$\operatorname{Ran}(i-K)^{\perp}=\operatorname{Ker}(i+K^{*})$, whereby
$d_{+}=\dim\operatorname{Ker}(i-K^{*})$ and
$d_{-}=\dim\operatorname{Ker}(i+K^{*})$.
We now introduce the following linear spaces. On the time scale half line, let
$L^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):=\left\\{y:[\rho(t_{0}),\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{2d}:\;y\;\text{is
integrable on}\;[\rho(t_{0}),\infty)_{\mathbb{T}}\right\\}$
and let
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):=\left\\{y\in
L^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):\int_{\rho(t_{0})}^{\infty}\big{(}(\Upsilon
y)^{*}W\Upsilon y\big{)}(t)\nabla t<\infty\right\\}$
for the partial left-shift operator $\Upsilon$ given in (1.4), with inner
product given by
(4.1) $\langle y,z\rangle:=\int_{\rho(t_{0})}^{\infty}\big{(}(\Upsilon
z)^{*}W\Upsilon y\big{)}(t)\nabla t,$
where the weight function $W$ is the $2d\times 2d$ nonnegative Hermitian left-
dense continuous matrix satisfying (1.5). In a similar manner, define on the
finite-length interval the linear space
(4.2)
$L^{1}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):=\left\\{y:[\rho(t_{0}),b]_{\mathbb{T}}\rightarrow\mathbb{C}^{2d}:\;y\;\text{is
integrable on}\;[\rho(t_{0}),b]_{\mathbb{T}}\right\\},$
and let
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):=\left\\{y\in
L^{1}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):\int_{\rho(t_{0})}^{b}\big{(}(\Upsilon
y)^{*}W\Upsilon y\big{)}(t)\nabla t<\infty\right\\}$
be the space with weighted inner product $\langle\cdot,\cdot\rangle_{b}$
defined in (3.9).
We will use the notation $\|y\|_{W}=(\langle y,y\rangle)^{1/2}$ for $y\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, and
$\|y\|_{b}=(\langle y,y\rangle_{b})^{1/2}$ for $y\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$. As $W$ may be singular,
the inner products for
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ and
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ may not be positive. To
account for this, we introduce the following quotient spaces. For $y,z\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, $y$ and $z$ are said
to be equal iff $\|y-z\|_{W}=0$. In this context
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is an inner product
space with inner product $\langle\cdot,\cdot\rangle$. Likewise functions
$y,z\in L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ are said to be
equal iff $\|y-z\|_{b}=0$, making
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ into an inner product
space with inner product $\langle\cdot,\cdot\rangle_{b}$.
###### Lemma 4.2.
The space $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is a
weighted Hilbert space with inner product $\langle\cdot,\cdot\rangle$ given in
(4.1), and $L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ is a weighted
Hilbert space with inner product $\langle\cdot,\cdot\rangle_{b}$ given in
(3.9). In addition,
$\dim
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)=\int_{\rho(t_{0})}^{b}\operatorname{rank}W(t)\nabla
t.$
###### Proof.
The style of proof is based on that given in the discrete case by Shi [41]. We
will show that $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is
complete, as the proof for
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ is similar and is
omitted. First, let us consider a simpler case, namely where
$W(t)=\widetilde{W}(t)=\operatorname{diag}\\{0,W_{2}(t)\\}$ for the
$r(t)\times r(t)$ matrix $W_{2}(t)>0$, $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$
and $0\leq r(t)\leq 2d$. For convenience, set
$\widetilde{W}^{\pm 1/2}(t):=\begin{pmatrix}0&0\\\ 0&W_{2}^{\pm
1/2}(t)\end{pmatrix},$
$y(t)=\left(y_{1}^{\operatorname{T}}(t),y_{2}^{\operatorname{T}}(y)\right)^{\operatorname{T}}$
for $y_{i}\in\mathbb{C}^{d}$, $i=1,2$, and
$y(t)=\left(y^{(1)\operatorname{T}}(t),y^{(2)\operatorname{T}}(t)\right)^{\operatorname{T}}$
with $y^{(1)}(t)\in\mathbb{C}^{r(t)}$ and $y^{(2)}(t)\in\mathbb{C}^{2d-r(t)}$.
To prove completeness, assume $\\{f_{n}\\}$ is a Cauchy sequence in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, i.e.,
given $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that
$\|f_{n}-f_{m}\|_{\widetilde{W}}<\varepsilon$ for all $n,m\geq N$. Define the
functions $g_{n}:[t_{0},\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{2d}$ via
(4.3) $g_{n}(t):=\widetilde{W}^{1/2}(t)\Upsilon f_{n}(t),\qquad n\geq 1,\qquad
t\in[t_{0},\infty)_{\mathbb{T}}.$
Since $f_{n}\in
L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, by (4.3)
we have $g_{n}\in L^{2}[t_{0},\infty)=\left\\{g\in
L^{1}[t_{0},\infty):(g^{*}g)\in L^{1}[t_{0},\infty)\right\\}$, where
$L^{2}[t_{0},\infty)$ is a known Hilbert space with $L^{2}$ norm
$\|g\|_{L^{2}}=\left(\int_{t_{0}}^{\infty}(g^{*}g)(t)\nabla t\right)^{1/2}$
for $g\in L^{2}[t_{0},\infty)$; see [39, Section 4]. It follows that
$\|g_{n}-g_{m}\|_{L^{2}}=\|f_{n}-f_{m}\|_{\widetilde{W}}$. Thus $\\{g_{n}\\}$
is a Cauchy sequence in $L^{2}[t_{0},\infty)$, so by the completeness of
$L^{2}$ there exists an integrable function $g\in L^{2}[t_{0},\infty)$ such
that $\|g_{n}-g\|_{L^{2}}\rightarrow 0$ as $n\rightarrow\infty$. Set
(4.4) $f(t)=\Upsilon^{-1}(\widetilde{W}^{-1/2}g)(t),$
where $\Upsilon^{-1}$ is given in (1.8). Then $\Upsilon
f=\Upsilon(\Upsilon^{-1}\widetilde{W}^{-1/2}g)=\widetilde{W}^{-1/2}g$, so that
$f\in L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. To
see that $f_{n}$ converges to $f$ in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, note
that by (4.3) we have $g_{n}^{(1)}(t)=0$ for all
$t\in[t_{0},\infty)_{\mathbb{T}}$, whence $g^{(1)}(t)=0$ for
$t\in[t_{0},\infty)_{\mathbb{T}}$. As a result, from (4.3) and (4.4) we see
that (suppressing the $t$)
$\displaystyle\left[\Upsilon(f_{n}-f)\right]^{*}\widetilde{W}\Upsilon(f_{n}-f)$
$\displaystyle=$ $\displaystyle\left(\begin{smallmatrix}0\\\
W_{2}^{-1/2}(g_{n}^{(2)}-g^{(2)})\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0\\\
0&W_{2}\end{smallmatrix}\right)\left(\begin{smallmatrix}0\\\
W_{2}^{-1/2}(g_{n}^{(2)}-g^{(2)})\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle(g_{n}^{(2)}-g^{(2)})^{*}(g_{n}^{(2)}-g^{(2)}),$
which implies that $\|f_{n}-f\|_{\widetilde{W}}=\|g_{n}-g\|_{L^{2}}$, ergo
$f_{n}\rightarrow f$ in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ as
$n\rightarrow\infty$. Consequently,
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is
complete.
For the general case, assume $\operatorname{rank}W(t)=r(t)$ for
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. As $W(t)\geq 0$ and Hermitian, there
exists a unitary matrix $U$ such that
$U^{*}(t)W(t)U(t)=\operatorname{diag}\\{0,W_{2}(t)\\}=:\widetilde{W}(t),\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}},$
where $W_{2}(t)$ is an $r(t)\times r(t)$ positive definite matrix for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. Suppose $\\{f_{n}\\}$ is a Cauchy
sequence in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, and set
$h_{n}(t):=\Upsilon^{-1}U^{*}(\Upsilon f_{n})(t)$. Then
$\displaystyle(\Upsilon h_{n})^{*}\widetilde{W}(\Upsilon h_{n})$
$\displaystyle=$ $\displaystyle(U^{*}(\Upsilon
f_{n}))^{*}\widetilde{W}U^{*}(\Upsilon f_{n})$ $\displaystyle=$
$\displaystyle(\Upsilon f_{n})^{*}U\widetilde{W}U^{*}(\Upsilon
f_{n})=(\Upsilon f_{n})^{*}W(\Upsilon f_{n}),$
so that $h_{n}\in
L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ and
$\|h_{n}-h_{m}\|_{\widetilde{W}}=\|f_{n}-f_{m}\|_{W}$, making $\\{h_{n}\\}$ a
Cauchy sequence in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. From
the discussion earlier for the simpler case, there exists a function $h\in
L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ such that
$h_{n}\rightarrow h$ in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ as
$n\rightarrow\infty$. If we set $f(t)=\Upsilon^{-1}U(\Upsilon h)(t)$, then
$f\in L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ with
$\|f_{n}-f\|_{W}=\|h_{n}-h\|_{\widetilde{W}}$. It follows that
$f_{n}\rightarrow f$ in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ as
$n\rightarrow\infty$, and
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is complete.
To calculate the dimension of
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$, note that for $y\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$, $y=0$ if and only if
$\|y\|^{2}_{b}=\int_{\rho(t_{0})}^{b}(\Upsilon y)^{*}(t)W(t)\Upsilon
y(t)\nabla t=0,$
which is equivalent to $W(t)\Upsilon y(t)=0$ for all
$t\in[t_{0},b]_{\mathbb{T}}$. Thus $\widetilde{W}U^{*}(\Upsilon y)(t)=0$ for
all $t\in[t_{0},b]_{\mathbb{T}}$, and $U(\Upsilon y)(t)$ has exactly $r(t)$
components taking effect on the inner product. As $U(t)$ is invertible, we
have that $\dim
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)=\int_{\rho(t_{0})}^{b}r(t)\nabla
t$, and the proof is complete. ∎
###### Remark 4.3.
The above result establishes that
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is a Hilbert space
with weighted norm
$\|y\|_{W}^{2}=\int_{\rho(t_{0})}^{\infty}\big{(}(\Upsilon y)^{*}W(\Upsilon
y)\big{)}(t)\nabla t.$
Following [39, Section 4.1], given ${\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right)$ with
the weighted norm
(4.5) $\|y\|_{1}^{2}:=\|y\|_{W}^{2}+\|y^{\nabla}\|_{W}^{2},\quad y\in{\rm
C}_{\rm
ld}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right),$
if we define $\mathcal{H}^{1}_{W}([\rho(t_{0}),\infty)_{\mathbb{T}})\subset
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ to be the completion
of ${\rm C}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right)$
with respect to the norm $\|\cdot\|_{1}$ in (4.5), then
$\mathcal{H}^{1}_{W}([\rho(t_{0}),\infty)_{\mathbb{T}})$ is a time-scale
analogue of the usual Sobolev space $H^{1}(I)$ on a real interval $I$, and
${\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right)\subset\mathcal{H}^{1}_{W}([\rho(t_{0}),\infty)_{\mathbb{T}})$.
We turn now to definitions of maximal and minimal operators corresponding to
system (1.3). We use $H$ and $H_{0}$ to denote the maximal and minimal
operators over $[\rho(t_{0}),\infty)_{\mathbb{T}}$, respectively, where
$\displaystyle D(H)$ $\displaystyle:=$ $\displaystyle\left\\{y\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):\;\text{there
exists}\;f\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)\;\text{such
that}\;\right.$ $\displaystyle\left.(\mathscr{L}y)(t)=W(t)(\Upsilon
f)(t),\;t\in[\rho(t_{0}),\infty)_{\mathbb{T}}\right\\},$ (4.6) $\displaystyle
Hy$ $\displaystyle:=$ $\displaystyle f,$ $\displaystyle D(H_{0})$
$\displaystyle:=$ $\displaystyle\left\\{y\in D(H):\;\text{there
exists}\;b\in[\rho(t_{0}),\infty)_{\mathbb{T}}\;\text{such that}\;\right.$
$\displaystyle\left.y^{\rho}(t_{0})=y(t)=0\;\text{for
all}\;t\in[b,\infty)_{\mathbb{T}}\right\\},$ (4.7) $\displaystyle H_{0}y$
$\displaystyle:=$ $\displaystyle Hy.$
In a similar manner, we use $H^{b}$ and $H^{b}_{0}$ to denote the maximal and
minimal operators over $[\rho(t_{0}),b]_{\mathbb{T}}$, respectively, where
$\displaystyle D(H^{b})$ $\displaystyle:=$ $\displaystyle\left\\{y\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):\;\text{there exists}\;f\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)\;\text{such that}\;\right.$
$\displaystyle\left.(\mathscr{L}y)(t)=W(t)(\Upsilon
f)(t),\;t\in[\rho(t_{0}),b]_{\mathbb{T}}\right\\},$ (4.8) $\displaystyle
H^{b}y$ $\displaystyle:=$ $\displaystyle f,$ $\displaystyle D(H^{b}_{0})$
$\displaystyle:=$ $\displaystyle\left\\{y\in
D(H^{b}):y^{\rho}(t_{0})=y(b)=0\right\\},$ (4.9) $\displaystyle H^{b}_{0}y$
$\displaystyle:=$ $\displaystyle H^{b}y.$
By these definitions it is clear that $H_{0}\subset H$ and $H_{0}^{b}\subset
H^{b}$.
###### Lemma 4.4.
The operators $H_{0}$ and $H^{b}_{0}$ are Hermitian.
###### Proof.
Since the proof is similar for $H^{b}_{0}$, we focus on just $H_{0}$. For any
$y,z\in D(H_{0})$, there exists $b\in[\rho(t_{0}),\infty)_{\mathbb{T}}$ such
that
(4.10) $y^{\rho}(t_{0})=z^{\rho}(t_{0})=0,\quad y(t)=z(t)=0\quad\text{for
all}\quad t\in[b,\infty)_{\mathbb{T}},$
and there exists $f,g\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ such that $H_{0}y=f$
and $H_{0}z=g$, that is to say
$(\mathscr{L}y)(t)=W(t)(\Upsilon f)(t),\quad(\mathscr{L}z)(t)=W(t)(\Upsilon
g)(t),\quad t\in[\rho(t_{0}),\infty)_{\mathbb{T}}.$
From Theorem 2.2 and (4.10) we have that
$\displaystyle\langle H_{0}y,z\rangle-\langle y,H_{0}z\rangle$
$\displaystyle=$ $\displaystyle\langle f,z\rangle-\langle y,g\rangle$
$\displaystyle=$ $\displaystyle\int_{\rho(t_{0})}^{\infty}\big{\\{}(\Upsilon
z)^{*}W(\Upsilon f)-(\Upsilon g)^{*}W(\Upsilon y)\big{\\}}(t)\nabla t$
$\displaystyle=$ $\displaystyle\int_{\rho(t_{0})}^{\infty}\big{\\{}(\Upsilon
z)^{*}\mathscr{L}y-(\mathscr{L}z)^{*}\Upsilon y\big{\\}}(t)\nabla t$
$\displaystyle=$
$\displaystyle\lim_{b\rightarrow\infty}z^{*}(t)Jy(t)\Big{|}_{\rho(t_{0})}^{b}=0.$
Therefore $\langle H_{0}y,z\rangle=\langle y,H_{0}z\rangle$, so that $H_{0}$
is Hermitian. ∎
The following lemma has a similar proof to that just completed.
###### Lemma 4.5.
The relation $\langle H^{b}_{0}y,z\rangle_{b}=\langle y,H^{b}z\rangle_{b}$
holds for all $y\in D(H^{b}_{0})$ and for all $z\in D(H^{b})$, while $\langle
H_{0}y,z\rangle=\langle y,Hz\rangle$ holds for all $y\in D(H_{0})$ and for all
$z\in D(H)$.
###### Lemma 4.6.
If (1.6) holds, then
$\operatorname{Ran}(H^{b}_{0})=\operatorname{Ker}(H^{b})^{\perp}$.
###### Proof.
Given any $f\in\operatorname{Ran}(H_{0}^{b})$, there exists $y\in
D(H_{0}^{b})$ such that $H_{0}^{b}y=f$. For each
$z\in\operatorname{Ker}(H^{b})$, it follows from the previous lemma that
$\langle f,z\rangle_{b}=\langle H_{0}^{b}y,z\rangle_{b}=\langle
y,H^{b}z\rangle_{b}=\langle y,0\rangle_{b}=0$, so that
$f\in\operatorname{Ker}(H^{b})^{\perp}$. Thus
$\operatorname{Ran}(H_{0}^{b})\subset\operatorname{Ker}(H^{b})^{\perp}$. If
$f\in\operatorname{Ker}(H^{b})^{\perp}$, then $\langle f,z\rangle_{b}=0$ for
all $z\in\operatorname{Ker}(H^{b})$. Consider the following initial value
problem:
$Jy^{\nabla}(t)=P(t)\Upsilon y(t)+W(t)\Upsilon f(t),\quad
y^{\rho}(t_{0})=0,\qquad t\in[\rho(t_{0}),b]_{\mathbb{T}}.$
By (1.6), this problem has a unique solution $y$ on
$[\rho(t_{0}),b]_{\mathbb{T}}$. Let
$\Phi(t)=(\varphi_{1},\varphi_{2},\cdots,\varphi_{2d})(t)$ be the fundamental
solution matrix of the homogeneous system
$Jx^{\nabla}(t)=P(t)\Upsilon x(t),\quad\Phi(b)=J,\qquad
t\in[\rho(t_{0}),b]_{\mathbb{T}}.$
Clearly $\varphi_{k}\in\operatorname{Ker}(H^{b})$ for $1\leq k\leq 2d$, so by
Theorem 2.2 and (3.9),
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle
f,\varphi_{k}\rangle_{b}=\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon\varphi_{k})^{*}W\Upsilon
f\big{\\}}(t)\nabla t$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon\varphi_{k})^{*}\mathscr{L}y\big{\\}}(t)\nabla
t$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon\varphi_{k})^{*}\mathscr{L}y-(\mathscr{L}\varphi_{k})^{*}\Upsilon
f\big{\\}}(t)\nabla t$ $\displaystyle=$
$\displaystyle\varphi^{*}_{k}(b)Jy(b)-\varphi^{*}_{k}(\rho(t_{0}))Jy(\rho(t_{0}))$
$\displaystyle=$ $\displaystyle\varphi^{*}_{k}(b)Jy(b).$
Thus we have that $\Phi^{*}(b)Jy(b)=y(b)=0$, and
$\operatorname{Ker}(H^{b})^{\perp}\subset\operatorname{Ran}(H_{0}^{b})$. ∎
###### Theorem 4.7.
If (1.6) holds, then $H_{0}$ is symmetric and $H$ is densely defined.
###### Proof.
By Lemma 4.4 and the fact that $H_{0}\subset H$, it suffices to show that
$D(H_{0})$ is dense in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, i.e., that
$D(H_{0})^{\perp}=\\{0\\}$. If $f\in D(H_{0})^{\perp}$, then
$D(H^{b}_{0})\subset D(H_{0})$ for all $t\in(\rho(t_{0}),\infty)_{\mathbb{T}}$
in the sense that all the functions of $D(H^{b}_{0})$ are considered to have
been extended by zero on to $[\rho(t_{0}),\infty)_{\mathbb{T}}$. Then, for all
$t\in(\rho(t_{0}),\infty)_{\mathbb{T}}$ and for all $z\in D(H^{b}_{0})$,
$\langle f,z\rangle_{b}=\langle f,z\rangle=0$. Set $H^{b}_{0}(z)(t)=g(t)$ for
$t\in[\rho(t_{0}),b]_{\mathbb{T}}$, and let $y$ be any solution of the system
$Jy^{\nabla}(t)=P(t)\Upsilon y(t)+W(t)\Upsilon f(t),\quad
t\in[\rho(t_{0}),b]_{\mathbb{T}}.$
By Theorem 2.2, we have
$\displaystyle\langle y,g\rangle_{b}-\langle f,z\rangle_{b}$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\left\\{(\Upsilon g)^{*}W\Upsilon
y-(\Upsilon z)^{*}W\Upsilon f\right\\}(t)\nabla t$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\left\\{(\mathscr{L}z)^{*}\Upsilon
y-(\Upsilon z)^{*}\mathscr{L}y\right\\}(t)\nabla t$ $\displaystyle=$
$\displaystyle-z^{*}(b)Jy(b)+z^{*}(\rho(t_{0}))Jy(\rho(t_{0}))=0.$
We then have that $\langle y,g\rangle_{b}=\langle f,z\rangle_{b}=0$. It
follows that
$y\in\operatorname{Ran}(H^{b}_{0})^{\perp}=\operatorname{Ker}(H^{b})$ by Lemma
4.6. Therefore, $H^{b}y=0$, and thus
$f\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}=0$ in
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$, that is $\|f\|_{b}=0$.
Since $b>\rho(t_{0})$ is arbitrary, it follows that $f=0$ in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. Consequently,
$D(H_{0})^{\perp}=\\{0\\}$. ∎
###### Theorem 4.8.
If (1.6) holds, then $H_{0}^{*}=H$.
###### Proof.
It suffices to show $D(H_{0}^{*})=D(H)$, and $H_{0}^{*}y=Hy$ for all $y\in
D(H_{0}^{*})$. If $y\in D(H)$, then $\langle y,H_{0}z\rangle=\langle
Hy,z\rangle$ for all $z\in D(H_{0})$ by Lemma 4.5. From this we see that the
functional $\langle y,H_{0}(\cdot)\rangle$ is continuous on $D(H_{0})$. Then
$y\in D(H_{0}^{*})$ by (iv) in Definition 4.1 and thus $D(H)\subset
D(H_{0}^{*})$. We now show $D(H_{0}^{*})\subset D(H)$. If $y\in D(H_{0}^{*})$,
then $y$ and $g:=H_{0}^{*}y$ are both in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. If $x$ is a
solution of the system
(4.11) $Jx^{\nabla}(t)=P(t)\Upsilon x(t)+W(t)\Upsilon g(t),$
then for each $z\in D(H_{0})$ there exists
$b\in[\rho(t_{0}),\infty)_{\mathbb{T}}$ such that $z^{\rho}(t_{0})=z(t)=0$ for
all $t\in[b,\infty)_{\mathbb{T}}$. Then
$z\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}\in D(H^{b}_{0})$. This implies from
Lemma 4.5 that
$\langle g,z\rangle=\langle g,z\rangle_{b}=\langle H^{b}x,z\rangle_{b}=\langle
x,H^{b}_{0}z\rangle_{b}=\langle x,H_{0}z\rangle.$
It follows that
$\langle y-x,H_{0}z\rangle=\langle y,H_{0}z\rangle-\langle
x,H_{0}z\rangle=\langle H_{0}^{*}y,z\rangle-\langle g,z\rangle=0.$
Additionally, $\langle y-x,H^{b}_{0}z\rangle_{b}=\langle y-x,H_{0}z\rangle=0$.
Thus
$(y-x)\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}\in\operatorname{Ran}(H^{b}_{0})^{\perp}$.
By Lemma 4.6 we have that
$(y-x)\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}\in\operatorname{Ker}(H^{b})$
whereby $H^{b}(y-x)=0$, in other words, $\mathscr{L}(y-x)(t)=0$ for
$t\in[\rho(t_{0}),b]_{\mathbb{T}}$. This, together with (4.11), implies that
$Jy^{\nabla}(t)-P(t)\Upsilon y(t)=Jx^{\nabla}(t)-P(t)\Upsilon
x(t)=W(t)\Upsilon g(t),\quad t\in[\rho(t_{0}),b]_{\mathbb{T}}.$
Since $b\geq\rho(t_{0})$ may be chosen arbitrarily large and $y,g\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, we have that $y\in
D(H)$ and $Hy=g=H_{0}^{*}y$. As a result, $D(H_{0}^{*})\subset D(H)$, whence
$D(H_{0}^{*})=D(H)$ and $H^{*}_{0}y=Hy$ for all $y\in D(H_{0}^{*})$. ∎
## 5\. Weyl disks and their limiting set
In this section, we first construct matrix disks for system (1.3) over time-
scale intervals of finite length. These matrix disks are called Weyl disks
[16], which turn out to be nested and converge to a limiting set. This
limiting set will play a major role in the discussions of square summable
solutions of (1.3). Again we will rely heavily on the organization of the
topic as done by Shi [41] in the discrete case.
Suppose that $\theta(t,\lambda)$ and $\phi(t,\lambda)$ are $2d\times d$
matrix-valued solutions of (1.3) satisfying
$\theta(\rho(t_{0}),\lambda)=\alpha^{*}$ and
$\phi(\rho(t_{0}),\lambda)=J\alpha^{*}$, respectively, where $\alpha$
satisfies (3.2). Then we have from (3.2) that
(5.1)
$\alpha\theta(\rho(t_{0}),\lambda)=I_{d},\quad\alpha\phi(\rho(t_{0}),\lambda)=0.$
Set
(5.2) $Y(t,\lambda):=(\theta,\phi)(t,\lambda),$
so that
(5.3) $Y(\rho(t_{0}),\lambda)=(\alpha^{*},J\alpha^{*})=:\Omega.$
Then we have from (3.2) that $\Omega$ is symplectic and unitary; in other
words,
(5.4) $\Omega^{*}J\Omega=J\quad\text{and}\quad\Omega^{*}\Omega=I_{2d}.$
Therefore, $Y(\cdot,\lambda)$ is a fundamental solution matrix of (1.3) and
satisfies, by (2.6) and from (5.4), that
(5.5) $Y^{*}\left(t,\overline{\lambda}\right)JY(t,\lambda)=J,\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}},$
and thus
(5.6) $Y(t,\lambda)JY^{*}\left(t,\overline{\lambda}\right)=J,\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}}.$
###### Lemma 5.1.
Let $\alpha$ and $\beta$ satisfy (3.2) and (3.3), respectively. Then $\lambda$
is an eigenvalue of the problem (1.3) with boundary conditions (3.1) if and
only if $\det(\beta\phi(b,\lambda))=0$. Additionally, $y(t,\lambda)$ is an
eigenfunction with respect to $\lambda$ if and only if there exists
$\xi\in\mathbb{C}^{d}$ such that $y(t,\lambda)=\phi(t,\lambda)\xi$, where
$\xi\neq 0$ is a solution of the homogeneous linear algebraic system
(5.7) $\beta\phi(b,\lambda)\xi=0.$
###### Proof.
The proof is unchanged from the discrete case, see [41, Lemma 3.1] and [16,
Lemma 2.8]. ∎
In the subsequent development we will be interested in the function $\chi$
given via
(5.8) $\chi(t,\lambda,b):=Y(t,\lambda)\begin{pmatrix}I_{d}\\\
M(\lambda,b)\end{pmatrix},$
where $M(\lambda,b)$ is a $d\times d$ matrix such that
$\beta\chi(b,\lambda,b)=0$, in other words
(5.9) $\beta\theta(b,\lambda)+\beta\phi(b,\lambda)M(\lambda,b)=0.$
Using Lemma 5.1, if $\lambda\in\mathbb{C}$ is not an eigenvalue of the problem
(1.3), (3.1), then $\beta\phi(b,\lambda)$ is invertible, and from (5.9) we
have that
(5.10)
$M(\lambda,b)=-\big{(}\beta\phi(b,\lambda)\big{)}^{-1}\beta\theta(b,\lambda).$
###### Lemma 5.2.
Let $\alpha$ satisfy (3.2). Then for each $b\in[t_{1},\infty)_{\mathbb{T}}$
$($where $t_{1}$ is specified in (1.5)$)$, we have the following.
1. (i)
$M(\lambda,b)$ is analytic on the upper and lower half planes and at all non-
eigenvalues of the problem (1.3), (3.1) on the real axis;
2. (ii)
for all $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
(5.11) $M^{*}\left(\overline{\lambda},b\right)=M(\lambda,b)$
and
(5.12)
$\operatorname{Im}M(\lambda,b):=\frac{M(\lambda,b)-M^{*}(\lambda,b)}{2i}\lessgtr
0\quad\text{for}\quad\operatorname{Im}\lambda\lessgtr 0.$
###### Proof.
The proof is unchanged from the discrete case, see [41, Lemma 3.2] and [16,
Lemma 2.14]. ∎
###### Lemma 5.3.
Let $\operatorname{Im}\lambda\neq 0$. If $\beta$ satisfies (3.3) and
$\chi(t,\lambda,b)$ satisfies
(5.13) $\beta\chi(b,\lambda,b)=0,$
then
(5.14) $\chi^{*}(b,\lambda,b)J\chi(b,\lambda,b)=0.$
Conversely, if $\chi(t,\lambda,b)$ satisfies (5.14) for some $d\times d$
matrix $M$, then there exists a $d\times 2d$ matrix $\beta$ satisfying (3.3)
such that (5.13) holds.
###### Proof.
See [16, Lemma 2.13]. ∎
Set
$C(M,b):=\mp
i(I_{d},M^{*})Y^{*}(b,\lambda)JY(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right),$
where “$+$” holds if $\operatorname{Im}\lambda<0$ and “$-$” holds when
$\operatorname{Im}\lambda>0$. Using the definition of $\chi$, it follows that
$C\big{(}M(\lambda,b),b\big{)}=\mp i\chi^{*}(b,\lambda,b)J\chi(b,\lambda,b).$
Consequently we have from Lemma 5.3 that $M$ satisfies the matrix equation
(5.15) $C(M,b)=0$
if and only if there exists a $d\times 2d$ matrix $\beta$ satisfying (3.3)
such that (5.13) holds, ergo (5.10) holds. Let
(5.16) $F(b,\lambda):=\mp iY^{*}(b,\lambda)JY(b,\lambda).$
Then $F(b,\lambda)$ is a $2d\times 2d$ Hermitian matrix such that
(5.17) $C(M,b)=(I_{d},M^{*})F(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right).$
For later use we block $F(b,\lambda)$ as in
(5.18) $F(b,\lambda):=\begin{pmatrix}F_{11}&F_{12}\\\
F^{*}_{12}&F_{22}\end{pmatrix}(b,\lambda),$
where $F_{mn}(b,\lambda)$ are $d\times d$ matrices for $m,n=1,2$. Then (5.15)
can be recast in the form
(5.19)
$M^{*}F_{22}(b,\lambda)M+F_{12}(b,\lambda)M+M^{*}F^{*}_{12}(b,\lambda)+F_{11}(b,\lambda)=0.$
Using Lemma 2.3, (5.3), and (5.4), we see that
$Y^{*}(b,\lambda)JY(b,\lambda)=J+2i\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla t,$
which in tandem with (5.16) yields
(5.20)
$F(b,\lambda)=\begin{cases}-iJ+2\operatorname{Im}\lambda\displaystyle\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla
t:&\operatorname{Im}\lambda>0,\\\
iJ-2\operatorname{Im}\lambda\displaystyle\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla
t:&\operatorname{Im}\lambda<0.\end{cases}$
###### Theorem 5.4.
The matrix sets $C(M,b)\leq 0$ are closed, convex, and nested in the sense
that for fixed $M$ and fixed $\lambda\in\mathbb{C}$, $C^{\nabla}(M,t)\geq 0$,
where the nabla derivative is with respect to $t$.
###### Proof.
The proof follows from (5.17) and (5.20). See also the discussion in [16,
Remark 2.16]. ∎
As noted in the discrete case [41, Section 3], which actually refers to [16,
Remark 2.16], the intersection of the matrix sets $C(M,b)\leq 0$ is a limiting
set that is nonempty, closed, and convex. In what follows here we present a
detailed analysis of the properties of $F(b,\lambda)$, which will play a vital
role in the next section as we obtain precise relationships among the rank of
the matrix radius of the limiting set, asymptotic behavior of eigenvalues of
the Weyl disks, and the number of linearly independent square summable
solutions of system (1.3). Proceeding with this in mind, from (5.16), (5.18),
and (5.20) we see that
$\displaystyle F_{11}(b,\lambda)$ $\displaystyle=$ $\displaystyle\mp
i\theta^{*}(b,\lambda)J\theta(b,\lambda)$ $\displaystyle=$ $\displaystyle\pm
2\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon\theta)^{*}(t,\lambda)W(t)\Upsilon\theta(t,\lambda)\nabla
t,$ $\displaystyle F_{22}(b,\lambda)$ $\displaystyle=$ $\displaystyle\mp
i\phi^{*}(b,\lambda)J\phi(b,\lambda)$ $\displaystyle=$ $\displaystyle\pm
2\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon\phi)^{*}(t,\lambda)W(t)\Upsilon\phi(t,\lambda)\nabla
t,$ $\displaystyle F_{12}(b,\lambda)$ $\displaystyle=$ $\displaystyle\mp
i\theta^{*}(b,\lambda)J\phi(b,\lambda)$ $\displaystyle=$ $\displaystyle\pm
iI_{d}\pm
2\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon\theta)^{*}(t,\lambda)W(t)\Upsilon\phi(t,\lambda)\nabla
t.$
Assuming (1.5), we can obtain the following from (5).
###### Theorem 5.5.
For any $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
$F_{11}(b,\lambda)>0$ and $F_{22}(b,\lambda)>0$ for all $b\geq t_{1}$.
Additionally, $F_{11}(b,\lambda)$ and $F_{22}(b,\lambda)$ are non-decreasing
with respect to $b$.
Using Theorem 5.5, equation (5.19) can be stated as
(5.22)
$\left(M+F_{22}^{-1}(b,\lambda)F_{12}^{*}(b,\lambda)\right)^{*}F_{22}(b,\lambda)\left(M+F_{22}^{-1}(b,\lambda)F_{12}^{*}(b,\lambda)\right)-\left(F_{12}F_{22}^{-1}F^{*}_{12}-F_{11}\right)(b,\lambda)=0.$
###### Theorem 5.6.
For any $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
$(F_{12}F_{22}^{-1}F^{*}_{12}-F_{11})(b,\lambda)=F_{22}^{-1}\left(b,\overline{\lambda}\right)>0$.
###### Proof.
From (5.5), (5.6), and (5.16) we see that
$F(b,\lambda)JF\left(b,\overline{\lambda}\right)=Y^{*}(b,\lambda)JY(b,\lambda)JY^{*}\left(b,\overline{\lambda}\right)JY\left(b,\overline{\lambda}\right)=-J.$
The rest of the proof is identical to [41, Proposition 3.2] and is omitted. ∎
Let
(5.23)
$\mathscr{C}(b,\lambda):=-F_{22}^{-1}(b,\lambda)F_{12}^{*}(b,\lambda),\quad\mathscr{R}(b,\lambda):=F_{22}^{-1/2}(b,\lambda).$
It follows from Theorem 5.6 that (5.15), that is to say (5.22), can be recast
as
(5.24)
$(M-\mathscr{C}(b,\lambda))^{*}\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))-\mathscr{R}^{2}\left(b,\overline{\lambda}\right)=0$
or as
(5.25)
$\left\\{\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))\mathscr{R}^{-1}\left(b,\overline{\lambda}\right)\right\\}^{*}\left\\{\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))\mathscr{R}^{-1}\left(b,\overline{\lambda}\right)\right\\}=I_{d}.$
###### Remark 5.7.
If the dimension $d=1$, then (5.24) is the equation of a circle. For this
reason we call (5.15) and/or (5.24) a Weyl circle equation, and call the
matrix set $C(M,b)\leq 0$ a Weyl disk; see [16, Definition 2.11].
Notice that if
$U:=\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))\mathscr{R}^{-1}\left(b,\overline{\lambda}\right)$,
then (5.25) can be written as $U^{*}U=I_{d}$, and $U$ is unitary. We then have
the following results.
###### Theorem 5.8.
The Weyl circle equation (5.24) and/or (5.15) can be expressed via
(5.26)
$E_{b}(\lambda):M=\mathscr{C}(b,\lambda)+\mathscr{R}(b,\lambda)U\mathscr{R}\left(b,\overline{\lambda}\right),$
and the Weyl disk $C(M,b)\leq 0$ can be expressed via
(5.27)
$\overline{E}_{b}(\lambda):M=\mathscr{C}(b,\lambda)+\mathscr{R}(b,\lambda)V\mathscr{R}\left(b,\overline{\lambda}\right),$
where $U$ is any matrix on the unit matrix circle $\partial
D=\\{U:U\in\mathbb{C}^{d\times d}\;\text{is a unitary matrix}\\}$ and $V$ is
any matrix on the unit matrix disk $D=\\{V:V\in\mathbb{C}^{d\times
d}\;\text{satisfies}\;V^{*}V\leq I_{d}\\}$.
###### Definition 5.9.
The matrix $\mathscr{C}(b,\lambda)$ is called the center, and the matrices
$\mathscr{R}(b,\lambda)$ and $\mathscr{R}\left(b,\overline{\lambda}\right)$
are called the matrix radii, respectively, of the Weyl circle (5.26) and the
Weyl disk (5.27).
###### Theorem 5.10.
For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
the matrix sequence $\\{\mathscr{R}(b,\lambda)\\}$ converges, and
$\mathscr{R}_{0}(\lambda):=\lim_{b\rightarrow\infty}\mathscr{R}(b,\lambda)\geq
0$.
###### Proof.
From Theorem 5.5, $F_{22}(b,\lambda)>0$ for all $b\geq t_{1}$, and
$F_{22}(b,\lambda)$ is non-decreasing with respect to $b$. Recall from (5.23)
that $\mathscr{R}(b,\lambda)=F_{22}^{-1/2}(b,\lambda)$, whence
$\\{\mathscr{R}(b,\lambda)\\}$ is a non-increasing sequence of positive
definite matrices. The conclusion follows from the fact that any non-
increasing sequence of Hermitian matrices that is bounded below converges to a
Hermitian matrix. ∎
###### Lemma 5.11.
For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
and $b>\tau\geq t_{1}$, there exists $V_{0}\in D$ such that
(5.28)
$\mathscr{C}(b,\lambda)-\mathscr{C}(\tau,\lambda)=\mathscr{R}(\tau,\lambda)V_{0}\mathscr{R}\left(\tau,\overline{\lambda}\right)-\mathscr{R}(b,\lambda)V_{0}\mathscr{R}\left(b,\overline{\lambda}\right).$
###### Proof.
See Shi [41, Lemma 3.4]. ∎
###### Theorem 5.12.
For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
the matrix sequence $\\{\mathscr{C}(b,\lambda)\\}$ converges, i.e. the matrix
$\mathscr{C}_{0}(\lambda):=\lim_{b\rightarrow\infty}\mathscr{C}(b,\lambda)$ is
well defined.
###### Proof.
The result follows from Theorem 5.10 and Lemma 5.11. ∎
###### Theorem 5.13.
For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
the matrix circle sequence $\\{E_{b}(\lambda)\\}$ and the matrix disk sequence
$\left\\{\overline{E}_{b}(\lambda)\right\\}$ converge as $b\rightarrow\infty$,
and their limiting sets can be represented, respectively, as
$\displaystyle E_{0}(\lambda):$ $\displaystyle
M=\mathscr{C}_{0}(\lambda)+\mathscr{R}_{0}(\lambda)U\mathscr{R}_{0}\left(\overline{\lambda}\right),\quad
U\in\partial D,$ $\displaystyle\overline{E}_{0}(\lambda):$ $\displaystyle
M=\mathscr{C}_{0}(\lambda)+\mathscr{R}_{0}(\lambda)V\mathscr{R}_{0}\left(\overline{\lambda}\right),\quad
V\in D.$
###### Proof.
The result follows from Theorems 5.8, 5.10, and 5.12. ∎
###### Remark 5.14.
Since $\mathscr{R}_{0}(\lambda)$ and
$\mathscr{R}_{0}\left(\overline{\lambda}\right)$ may be singular, the set
$\overline{E}_{0}(\lambda)$ may be a reduced matrix disk. We see that
$\overline{E}_{0}(\lambda)$ contains only one element if
$\mathscr{R}_{0}(\lambda)=0$ or
$\mathscr{R}_{0}\left(\overline{\lambda}\right)=0$, and it contains interior
points if and only if $\mathscr{R}_{0}(\lambda)$ and
$\mathscr{R}_{0}\left(\overline{\lambda}\right)$ are both invertible. Although
the limiting sets $E_{0}(\lambda)$ and $\overline{E}_{0}(\lambda)$ may be a
reduced matrix circle and a reduced matrix disk, respectively, we still give
the following definition for convenience.
###### Definition 5.15.
The matrix $\mathscr{C}_{0}(\lambda)$ is called the center, and the matrices
$\mathscr{R}_{0}(\lambda)$ and
$\mathscr{R}_{0}\left(\overline{\lambda}\right)$ are called the matrix radii
of the limiting sets $E_{0}(\lambda)$ and $\overline{E}_{0}(\lambda)$,
respectively.
###### Theorem 5.16.
Assume (1.5). For any given $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$ and for each $M\in\overline{E}_{0}(\lambda)$,
if $\operatorname{Im}\lambda\lessgtr 0$ then $\operatorname{Im}M\lessgtr 0$.
###### Proof.
Assume that $\operatorname{Im}\lambda\neq 0$ and
$M\in\overline{E}_{0}(\lambda)$. Set
(5.29) $\chi(t,\lambda):=Y(t,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right).$
Then from (5.20) we have that
(5.30)
$\displaystyle\int_{\rho(t_{0})}^{b}(\Upsilon\chi)^{*}(t,\lambda)W(t)\Upsilon\chi(t,\lambda)\nabla
t$ $\displaystyle=$ $\displaystyle(I_{d},M^{*})\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla
t\left(\begin{smallmatrix}I_{d}\\\ M\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2|\operatorname{Im}\lambda|}(I_{d},M^{*})\left(F(b,\lambda)\pm
iJ\right)\left(\begin{smallmatrix}I_{d}\\\ M\end{smallmatrix}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2|\operatorname{Im}\lambda|}(I_{d},M^{*})F(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right)\pm\frac{1}{|\operatorname{Im}\lambda|}\operatorname{Im}M,$
for $\operatorname{Im}\lambda\lessgtr 0$. Because the sets $C(M,b)\leq 0$ are
nested, $\overline{E}_{0}(\lambda)$ is a subset of $C(M,b)\leq 0$ for any
$b\geq t_{1}$. Consequently we have from (5.17) that
$(I_{d},M^{*})F(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right)\leq 0.$
This in tandem with (5.30) implies that for $b\geq t_{1}$ we have
(5.31)
$\int_{\rho(t_{0})}^{b}(\Upsilon\chi)^{*}(t,\lambda)W(t)\Upsilon\chi(t,\lambda)\nabla
t\leq\pm\frac{1}{|\operatorname{Im}\lambda|}\operatorname{Im}M,\quad\operatorname{Im}\lambda\lessgtr
0.$
The result then follows from the above relation and the assumed definiteness
condition (1.5). ∎
## 6\. Square summable solutions
We will call $y(\cdot,\lambda)$ a square summable solution of (1.3) if it is a
solution of (1.3) in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. In this section we
will make a connection between square summable solutions of (1.3) and the
elements of the limiting set $\overline{E}_{0}(\lambda)$ from Theorem 5.13,
obtaining precise relationships among the rank of the matrix radius
$\mathscr{R}_{0}(\lambda)=\lim_{b\rightarrow\infty}F_{22}^{-1/2}(b,\lambda)$
(see Theorem 5.10) of the limiting set $\overline{E}_{0}(\lambda)$, the number
of linearly independent square summable solutions of (1.3), and the asymptotic
behavior of the eigenvalues of the matrix radius $F_{22}(b,\lambda)$ of the
Weyl disk $\overline{E}_{b}(\lambda)$ from (5.27). Given the structure and
notation established in the previous sections that generalizes the discrete
results in Shi [41], the proofs of the following results are omitted, as there
is no change necessary from [41, Section 4] except for minor notational
adjustments.
###### Theorem 6.1.
For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$ and for
each $M\in\overline{E}_{0}(\lambda)$, all the columns of $\chi(\cdot,\lambda)$
are in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, where $\chi$
is given in (5.29).
###### Corollary 6.2.
For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$, system
(1.3) has at least $d$ linearly independent square summable solutions.
###### Theorem 6.3.
For
$\mathscr{R}_{0}(\lambda)=\lim_{b\rightarrow\infty}F_{22}^{-1/2}(b,\lambda)$,
set
$r(\lambda):=\operatorname{rank}\mathscr{R}_{0}(\lambda),\quad\operatorname{Im}\lambda\neq
0,$
and let
$k=d+\min\left\\{r(\lambda),r\left(\overline{\lambda}\right)\right\\}$. Then
system (1.3) has at least $k$ linearly independent square summable solutions.
By Theorem 5.5, for each $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$ and for any $b\geq t_{1}$,
$F_{22}(b,\lambda)>0$. If $\mu_{j}(b)$ are the eigenvalues of
$F_{22}(b,\lambda)$ for $1\leq j\leq d$, then $\mu_{j}(b)>0$ for $1\leq j\leq
d$, and they can be arranged as
$\mu_{1}(b)\leq\mu_{2}(b)\leq\cdots\leq\mu_{d}(b)$. We have the following
result.
###### Theorem 6.4.
For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
$\operatorname{rank}\mathscr{R}_{0}(\lambda)=r(\lambda)$ if and only if
$\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j},\quad 1\leq j\leq r(\lambda),$
are finite and positive, and
$\lim_{b\rightarrow\infty}\mu_{j}(b)=\infty,\quad r(\lambda)+1\leq j\leq d.$
In addition,
$\gamma_{1}^{-1/2},\gamma_{2}^{-1/2},\cdots,\gamma_{r(\lambda)}^{-1/2}$ are
the $r(\lambda)$ positive eigenvalues of $\mathscr{R}_{0}(\lambda)$.
###### Lemma 6.5.
For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$, system
(1.3) has exactly $d+l$ linearly independent square summable solutions if and
only if there exists a $d\times l$ matrix $\Lambda$ with
$\operatorname{rank}\Lambda=l$ such that $\phi(\cdot,\lambda)\Lambda\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, and
$\eta\in\operatorname{Ran}\Lambda=\left\\{\Lambda
v:v\in\mathbb{C}^{l}\right\\}$ if $\phi(\cdot,\lambda)\eta\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\eta\in\mathbb{C}^{d}$.
###### Theorem 6.6.
If $\operatorname{rank}\mathscr{R}_{0}(\lambda)=r(\lambda)$ for
$\operatorname{Im}\lambda\neq 0$, then system (1.3) has exactly $d+r(\lambda)$
linearly independent square summable solutions and thus, $r(\lambda)$ is
independent of the coefficient matrix $\alpha$ of the left boundary condition
in (3.1).
###### Theorem 6.7.
For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$, system
(1.3) has exactly $d+r(\lambda)$ linearly independent square summable
solutions if and only if $\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j}$ are
finite and positive for $1\leq j\leq r(\lambda)$, and
$\lim_{b\rightarrow\infty}\mu_{j}(b)=\infty$ for $r(\lambda)+1\leq j\leq d$.
In addition,
$\gamma_{1}^{-1/2},\gamma_{2}^{-1/2},\cdots,\gamma_{r(\lambda)}^{-1/2}$ are
the $r(\lambda)$ positive eigenvalues of $\mathscr{R}_{0}(\lambda)$.
## 7\. Classification of singular linear Hamiltonian nabla systems
In this section, we introduce the defect index $d(\lambda)$ of the minimal
operator $H_{0}$ (defined in Section $4$) and $\lambda$. We establish a
precise correspondence between $d(\lambda)$ and the number of linearly
independent square summable solutions of (1.3). Based on this correspondence,
we show that the defect indices $d_{\pm}$ of $H_{0}$ are not less than $d$. In
addition, we obtain a precise correspondence between $d(\lambda)$ and
$\operatorname{rank}\mathscr{R}_{0}(\lambda)$. Moreover, we discuss the defect
index problem for the special case where $P(t)$ and $W(t)$ are both real, and
the largest defect index problem for the general case. Building on the above
results, we present a suitable classification for singular Hamiltonian nabla
systems by using the positive and negative defect indices of $H_{0}$. Lastly,
we derive several equivalent conditions on the limit circle and the limit
point cases. The proofs of the first few results below carry over from the
discrete case unchanged [41, Section 5].
###### Theorem 7.1.
For all $\lambda\in\mathbb{C}$, the defect index $d(\lambda)$ of the minimal
operator $H_{0}$ and $\lambda$ is equal to the number of linearly independent
square summable solutions of system (1.3).
###### Remark 7.2.
As $H_{0}$ is Hermitian by Lemma 4.4, the defect index $d(\lambda)$ of $H_{0}$
is constant in the upper and lower half planes, respectively. Set
(7.1) $d_{+}=d(i),\quad d_{-}=d(-i);$
these are called the positive and negative defect indices of the minimal
operator, respectively. The following result then follows directly from
Corollary 6.2 and Theorem 7.1.
###### Theorem 7.3.
The number of linearly independent square summable solutions of system (1.3)
in the upper half plane is $d_{+}$, and in the lower half plane is $d_{-}$.
These numbers are independent of $\lambda$, with $d_{\pm}\geq d$.
###### Theorem 7.4.
The rank of $\mathscr{R}_{0}(\lambda)$ is equal to a constant $r_{+}$ for all
$\lambda$ with $\operatorname{Im}\lambda>0$, and equal to a constant $r_{-}$
for all $\lambda$ with $\operatorname{Im}\lambda<0$. Moreover, these ranks
satisfy the equations
(7.2) $d_{+}=d+r_{+},\qquad d_{-}=d+r_{-}.$
###### Lemma 7.5.
For any $\lambda_{0}\in\mathbb{C}$, the fundamental matrix solution
$\Phi(\cdot,\lambda_{0})$ of (1.3)${}_{\lambda_{0}}$ with initial condition
$\Phi(\rho(t_{0}),\lambda_{0})=I_{2d}$ satisfies
(7.3) $\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)=1$
for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$.
###### Proof.
By the initial condition, (7.3) holds at $t=\rho(t_{0})$. If
$t\in[t_{0},\infty)_{\mathbb{T}}$ is a left-scattered point, then from (2.3)
we have
(7.4)
$\left(I_{2d}-\nu(t)\mathcal{S}(t,\lambda_{0})\right)\Phi(t,\lambda_{0})=\Phi^{\rho}(t,\lambda_{0}).$
Consequently by (2.5) and (7.4) we have
$\det\big{(}\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\big{)}=\det\big{(}\Phi^{\rho*}(t,\lambda_{0})\Phi^{\rho}(t,\lambda_{0})\big{)}$,
so that
$\left[\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)\right]^{\nabla}(t)=0$
if $t$ is a left-scattered point.
By Liouville’s formula on time scales [22], we have
(7.5)
$\det\Phi(t,\lambda_{0})=\hat{e}_{q}(t,\rho(t_{0}))\det\Phi(\rho(t_{0}),\lambda_{0})=\hat{e}_{q}(t,\rho(t_{0})),\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}},$
where
$q(t)=\lambda_{1}\oplus_{\nu}\lambda_{2}\oplus_{\nu}+\cdots+\oplus_{\nu}\lambda_{2d}$
for eigenvalues $\lambda_{1},\cdots,\lambda_{2d}$ of
$\mathcal{S}(\cdot,\lambda)$ given in (2.3), and
$x=\hat{e}_{q}(\cdot,\rho(t_{0}))$ is the nabla exponential function [15,
Chapter 3] that uniquely solves the initial value problem
$x^{\nabla}(t)=q(t)x(t),\quad x^{\rho}(t_{0})=1.$
It follows that
$\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)=\hat{e}_{(q\oplus_{\nu}q^{*})}(t,\rho(t_{0}))$.
Suppose $t\in[t_{0},\infty)_{\mathbb{T}}$ is a left-dense point. Then
$\nu(t)=0$,
$(q\oplus_{\nu}q^{*})=(q+q^{*})=\operatorname{tr}(\mathcal{S}+\mathcal{S}^{*})=0$
from (2.3), and
$\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)=\hat{e}_{(q+q^{*})}(t,\rho(t_{0}))=\hat{e}_{0}(t,\rho(t_{0}))\equiv
1.$
Therefore (7.3) holds for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. ∎
###### Theorem 7.6 (The Largest Defect Index Theorem).
If there exists $\lambda_{0}\in\mathbb{C}$ such that all the solutions of
(1.3)${}_{\lambda_{0}}$ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, then all solutions
of (1.3)λ are in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$,
for any $\lambda\in\mathbb{C}$.
###### Proof.
Assume that all solutions of (1.3)${}_{\lambda_{0}}$ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\lambda_{0}\in\mathbb{C}$. Given any $\lambda\in\mathbb{C}$, let
$\Phi(\cdot,\lambda)$ be the fundamental matrix solution of (1.3)λ, that is
$\Phi(\cdot,\lambda)$ solves (1.3)λ, or (2.1)λ, respectively, and satisfies
$\Phi(\rho(t_{0}),\lambda)=I_{2d}$; note that (7.3) then holds. As
$\Phi(t,\lambda)$ and $\Phi\left(t,\lambda_{0}\right)$ are both invertible,
there exists an invertible matrix $X(t,\lambda)$ such that
(7.6) $\Phi(t,\lambda)=\Phi\left(t,\lambda_{0}\right)X(t,\lambda).$
We will show that $X(t,\lambda)$ is bounded for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. Using (7.6), the nabla product rule,
and the simple useful time-scale formula $X^{\rho}=X-\nu X^{\nabla}$, we see
that
$\displaystyle\Phi^{\nabla}(t,\lambda)=\Phi^{\rho}\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda)+\Phi^{\nabla}\left(t,\lambda_{0}\right)X(t,\lambda),$
(7.7)
$\displaystyle(\Upsilon\Phi)(t,\lambda)=(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)-\nu(t)\operatorname{diag}\\{0,I_{d}\\}\Phi^{\rho}\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda),$
for $\Upsilon$ in (1.4). From (7.7) and the fact that $\Phi(\cdot,\lambda)$
and $\Phi(\cdot,\lambda_{0})$ are fundamental solution matrices for (1.3)λ and
(1.3)${}_{\lambda_{0}}$, respectively, we arrive at
(7.8) $X^{\nabla}(t,\lambda)=Q(t,\lambda)X(t,\lambda),$
where we have taken
(7.9) $\displaystyle Q(t,\lambda)$ $\displaystyle=$
$\displaystyle(\lambda-\lambda_{0})Z^{-1}(t,\lambda)(\Upsilon\Phi)^{*}(t,\lambda_{0})W(t)(\Upsilon\Phi)(t,\lambda_{0}),$
(7.10) $\displaystyle Z(t,\lambda)$ $\displaystyle=$
$\displaystyle(\Upsilon\Phi)^{*}(t,\lambda_{0})\begin{pmatrix}0&-I_{d}+\nu(t)A^{*}(t)\\\
I_{d}&\nu(t)(B(t)+\lambda W_{2}(t))\end{pmatrix}\Phi^{\rho}(t,\lambda_{0}).$
The multiplier $(\Upsilon\Phi)^{*}(t,\lambda_{0})$ appears in (7.9) and (7.10)
via (1.8), and will help in the sequel with the analysis on $Q(t,\lambda)$.
First we focus on $Z(t,\lambda)$. From (1.3)${}_{\lambda_{0}}$ we have that
(7.11)
$\Phi^{\rho}(t,\lambda_{0})=\begin{pmatrix}I_{d}-\nu(t)A(t)&-\nu(t)(B(t)+\lambda_{0}W_{2}(t))\\\
0&I_{d}\end{pmatrix}(\Upsilon\Phi)(t,\lambda_{0}).$
If we substitute (7.11) into (7.10), we see that
(7.12) $\displaystyle Z(t,\lambda)$ $\displaystyle=$
$\displaystyle(\lambda-\lambda_{0})\int_{\rho(t)}^{t}(\Upsilon\Phi)^{*}(s,\lambda_{0})\operatorname{diag}\\{0,W_{2}(s)\\}(\Upsilon\Phi)(s,\lambda_{0})\nabla
s$
$\displaystyle+(\Upsilon\Phi)^{*}(t,\lambda_{0})\begin{pmatrix}0&-I_{d}+\nu(t)A^{*}(t)\\\
I-\nu(t)A(t)&0\end{pmatrix}(\Upsilon\Phi)(t,\lambda_{0}),$
where we have used the time-scale formula
$\nu(t)f(t)=\int_{\rho(t)}^{t}f(s)\nabla s$ in the first line of (7.12). As
all solutions of (1.3)${}_{\lambda_{0}}$ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$,
$\Phi(\cdot,\lambda_{0})\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, whence
(7.13)
$V(\lambda_{0}):=\int_{\rho(t_{0})}^{\infty}(\Upsilon\Phi)^{*}(s,\lambda_{0})W(s)(\Upsilon\Phi)(s,\lambda_{0})\nabla
s<\infty.$
Consequently the first term on the right-hand side of (7.12) tends to zero as
$t\rightarrow\infty$ in the time scale. Let us denote by $\Psi$ the second
term on the right-hand side of (7.12). From (7.11) we have
(7.14)
$\Psi(t)=\Phi^{\rho*}(t,\lambda_{0})J\Phi^{\rho}(t,\lambda_{0})+2i\operatorname{Im}\lambda_{0}\int_{\rho(t)}^{t}(\Upsilon\Phi)^{*}(s,\lambda_{0})\operatorname{diag}\\{0,W_{2}(s)\\}(\Upsilon\Phi)(s,\lambda_{0})\nabla
s,$
where we have used the time-scale formula
$\nu(t)f(t)=\int_{\rho(t)}^{t}f(s)\nabla s$ again; the second term on the
right-hand side of (7.14) goes to 0 as $t\rightarrow\infty$ in the time scale
since $\Phi(\cdot,\lambda_{0})\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. By Lemma 2.3 and the
initial condition for $\Phi(\cdot,\lambda_{0})$ we see that
$\Phi^{\rho*}(t,\lambda_{0})J\Phi^{\rho}(t,\lambda_{0})=J+2i\operatorname{Im}\lambda_{0}\int_{\rho(t_{0})}^{\rho(t)}(\Upsilon\Phi)^{*}(s,\lambda_{0})W(s)(\Upsilon\Phi)(s,\lambda_{0})\nabla
s.$
It then follows from (7.13) and (7.14) that
$\lim_{t\rightarrow\infty}\Psi(t)=J+2i\operatorname{Im}\lambda_{0}V(\lambda_{0}),$
so that
(7.15)
$\lim_{t\rightarrow\infty}Z(t,\lambda)=J+2i\operatorname{Im}\lambda_{0}V(\lambda_{0}).$
From (7.10) and (7.11) we see that $Z(t,\lambda)$ is invertible for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$; we need to show that
$Z^{-1}(t,\lambda)$ is bounded for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. From (7.11) we have that
$\det(\Upsilon\Phi)^{*}(t,\lambda_{0})=\det\Phi^{\rho*}(t,\lambda_{0})\det
E^{*}(t),$
so that from (7.10) we obtain
$\displaystyle\det Z(t,\lambda)$ $\displaystyle=$
$\displaystyle\det(\Upsilon\Phi)^{*}(t,\lambda_{0})\det\left(I_{d}-\nu(t)A^{*}(t)\right)\det\Phi^{\rho}(t,\lambda_{0})$
$\displaystyle=$
$\displaystyle\det\Phi^{\rho*}(t,\lambda_{0})\det\Phi^{\rho}(t,\lambda_{0})=1$
since $\Phi(\cdot,\lambda_{0})$ satisfies (7.3). As a result,
(7.16) $Z^{-1}(t,\lambda)=(\det
Z(t,\lambda))^{-1}\operatorname{adj}Z(t,\lambda),$
where $\operatorname{adj}Z(t,\lambda)$ is the adjugate matrix of
$Z(t,\lambda)$. Moreover, from (7.15) we see that
$\operatorname{adj}Z(t,\lambda)$ is bounded on
$[\rho(t_{0}),\infty)_{\mathbb{T}}$, whence $Z^{-1}(t,\lambda)$ is as well by
(7.16). Let $c\in\mathbb{R}$ be a positive constant such that
(7.17)
$\|Z^{-1}(t,\lambda)\|:=\left(\sum_{k=1}^{2d}\sum_{j=1}^{2d}|z_{jk}(t)|\right)^{1/2}\leq
c,\quad t\in[\rho(t_{0}),\infty)_{\mathbb{T}}.$
Now we will show that
(7.18) $\int_{\rho(t_{0})}^{\infty}\|Q(t,\lambda)\|_{1}\nabla
t<\infty,\quad\|Q(t,\lambda)\|_{1}:=\sup_{\|\xi\|=1}\|Q(t,\lambda)\xi\|,$
where $Q(\cdot,\lambda)$ is given in (7.9). From (7.13) it follows that all
the diagonal entries of the expression
(7.19) $(\Upsilon\Phi)^{*}(t,\lambda_{0})W(t)(\Upsilon\Phi)(t,\lambda_{0})$
are nonnegative and absolutely summable over
$[\rho(t_{0}),\infty)_{\mathbb{T}}$. By referring to the nonnegativity of
(7.19), the absolute value of each non-diagonal entry of (7.19) is less than
or equal to the sum of the two diagonal entries that lie exactly in the same
column and row as the non-diagonal entry does. As a result, each non-diagonal
entry of (7.19) is also absolutely summable over
$[\rho(t_{0}),\infty)_{\mathbb{T}}$. Thus, it follows that
(7.20)
$\int_{\rho(t_{0})}^{\infty}\|(\Upsilon\Phi)^{*}(t,\lambda_{0})W(t)(\Upsilon\Phi)(t,\lambda_{0})\|\nabla
t<\infty.$
Consequently from (7.9), (7.17), and (7.20) we have that
$\int_{\rho(t_{0})}^{\infty}\|Q(t,\lambda)\|\nabla t<\infty$, so that (7.18)
follows.
We are ready to show that $X(t,\lambda)$ is bounded on
$[\rho(t_{0}),\infty)_{\mathbb{T}}$. For a solution of (7.8) to exist, we need
the coefficient matrix $Q$ to be $\nu-$regressive, in other words we need to
show that $I_{2d}-\nu(t)Q(t)$ is invertible for all
$t\in[t_{0},\infty)_{\mathbb{T}}$. From (2.2), (7.9), (7.10), and (7.11) we
have that
$I_{2d}-\nu(t)Q(t)=Z^{-1}(t,\lambda)(\Upsilon\Phi)^{*}(t,\lambda_{0})\begin{pmatrix}-(\lambda-\lambda_{0})\nu(t)W_{1}(t)&-E^{*-1}(t)\\\
E^{-1}(t)&0\end{pmatrix}(\Upsilon\Phi)(t,\lambda_{0});$
since $\Phi(\cdot,\lambda_{0})$ is a fundamental matrix, by (7.11) again we
see that every matrix on the right-hand side here is invertible, making
$I_{2d}-\nu(t)Q(t)$ invertible for all $t\in[t_{0},\infty)_{\mathbb{T}}$.
Therefore
$X(t,\lambda)=\hat{e}_{Q(\cdot,\lambda)}(t,\rho(t_{0}))X(\rho(t_{0}),\lambda)\stackrel{{\scriptstyle\eqref{shi5.4}}}{{=}}\hat{e}_{Q(\cdot,\lambda)}(t,\rho(t_{0}))$
is a well-defined matrix, and thus
$\|X(t,\lambda)\|_{1}\leq\exp\left\\{\int_{\rho(t_{0})}^{t}\|Q(s,\lambda)\|_{1}\nabla
s\right\\}.$
Combining this with (7.18) we conclude that $\|X(t,\lambda)\|_{1}$ is bounded
for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$.
Let us now show that all solutions of (1.3)λ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. From the second
line of (7.7) and (7.11) we have that
$\displaystyle(\Upsilon\Phi)^{*}(t,\lambda)W(t)(\Upsilon\Phi)(t,\lambda)$
$\displaystyle=$ $\displaystyle
X^{*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)W(t)(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)$
$\displaystyle-\nu(t)X^{*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda)$
$\displaystyle-\nu(t)X^{\nabla*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)$
$\displaystyle+(\nu(t))^{2}X^{\nabla*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda);$
using the simple formula $\nu X^{\nabla}=X-X^{\rho}$ again, we simplify this
to
$\displaystyle(\Upsilon\Phi)^{*}(t,\lambda)W(t)(\Upsilon\Phi)(t,\lambda)$
$\displaystyle=$ $\displaystyle
X^{*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{W_{1}(t),0\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)$
$\displaystyle+X^{*\rho}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X^{\rho}(t,\lambda).$
From the boundedness of $\|X(t,\lambda)\|_{1}$ and (7.20), we see that
$\int_{\rho(t_{0})}^{\infty}\|(\Upsilon\Phi)^{*}(t,\lambda)W(t)(\Upsilon\Phi)(t,\lambda)\|\nabla
t<\infty,$
putting $\Phi(\cdot,\lambda)\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. ∎
###### Remark 7.7.
Given the above results, we are able to generalize the remainder of [41,
Section 5] verbatim, as the proofs are unchanged. For completeness we include
the main results here.
###### Definition 7.8.
Let $d_{\pm}$ be the positive and negative defect indices of $H_{0}$, where
$H_{0}$ is the minimal operator corresponding to system (1.3) defined in
(4.7). Then the dynamic Hamiltonian nabla operator $\mathscr{L}$ given in
(2.7) is said to be in the limit $(d_{+},d_{-})$ case at $t=\infty$. In the
special case of $d_{+}=d_{-}=d$, $\mathscr{L}$ is said to be in the limit
point case $(l.p.c.)$ at $t=\infty$, and in the other special case of
$d_{+}=d_{-}=2d$, $\mathscr{L}$ is said to be in the limit circle case
$(l.c.c.)$ at $t=\infty$.
###### Remark 7.9.
It is clear that there may be at most $1+d^{2}$ cases for the singular dynamic
Hamiltonian nabla system (1.3) of degree $d$ by the largest defect index
theorem (Theorem 7.6) and by using the fact that $d\leq d_{\pm}\leq 2d$.
However, in the special case of $d=1$, the classification is simple just like
the formal self-adjoint second-order scalar difference operators; in other
words, $\mathscr{L}$ is either in $l.p.c.$ or in $l.c.c.$ at $t=\infty$ by
using the largest defect index theorem.
###### Theorem 7.10.
Assume (7.3). Then the following nine statements are equivalent.
1. (i)
$\mathscr{L}$ is in $l.c.c.$ at $t=\infty$;
2. (ii)
system (1.3)λ has $2d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;
3. (iii)
$\mathscr{R}_{0}(\lambda)$ is invertible for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;
4. (iv)
the limiting set $\overline{E}_{0}(\lambda)$ has nonempty interior for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;
5. (v)
$\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j}$ is finite and positive for
$1\leq j\leq d$, where $\mu_{j}(b)(1\leq j\leq d)$ are eigenvalues of
$F_{22}(b,\lambda)$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;
6. (vi)
system (1.3)${}_{\lambda_{0}}$ has $2d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
7. (vii)
$\mathscr{R}_{0}(\lambda_{0})$ is invertible for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
8. (viii)
the limiting set $\overline{E}_{0}(\lambda_{0})$ has nonempty interior for
some $\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
9. (ix)
$\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j}$ is finite and positive for
$1\leq j\leq d$, where $\mu_{j}(b)(1\leq j\leq d)$ are eigenvalues of
$F_{22}(b,\lambda_{0})$ for some $\lambda_{0}\in\mathbb{C}$ with
$\operatorname{Im}\lambda_{0}\neq 0$.
Similarly, the following equivalent conditions on the limit point case can be
concluded by Theorems 6.4 and 7.4.
###### Theorem 7.11.
Assume (7.3). Then the following seven statements are equivalent:
1. (i)
$\mathscr{L}$ is in $l.p.c.$ at $t=\infty$;
2. (ii)
system (1.3)λ has $d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;
3. (iii)
$\mathscr{R}_{0}(\lambda)=0$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;
4. (iv)
$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda)$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;
5. (v)
systems (1.3)${}_{\lambda_{0}}$ and (1.3)${}_{\overline{\lambda}_{0}}$ have
exactly $d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, respectively, for
some $\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
6. (vi)
$\mathscr{R}_{0}(\lambda_{0})=\mathscr{R}_{0}\left(\overline{\lambda}_{0}\right)=0$
for some $\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
7. (vii)
$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda_{0})$ and the smallest eigenvalue of
$F_{22}\left(b,\overline{\lambda}_{0}\right)$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$.
If all the coefficients of system (1.3)λ are real, we have the following
results by Theorems 7.6 and 7.9.
###### Corollary 7.12.
If $P(t)$ and $W(t)$ are real for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$,
then following nine statements are equivalent.
1. (i)
$\mathscr{L}$ is in $l.p.c.$ at $t=\infty$;
2. (ii)
system (1.3)λ has exactly $d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;
3. (iii)
$\mathscr{R}_{0}(\lambda)=0$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;
4. (iv)
the limiting set $E_{0}(\lambda)$ contains only one element for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;
5. (v)
$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda)$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda>0$;
6. (vi)
system (1.3)${}_{\lambda_{0}}$ has exactly $d$ linearly independent solutions
in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
7. (vii)
$\mathscr{R}_{0}(\lambda_{0})=0$ for some $\lambda_{0}\in\mathbb{C}$ with
$\operatorname{Im}\lambda_{0}\neq 0$;
8. (viii)
the limiting set $E_{0}(\lambda_{0})$ contains only one element for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;
9. (ix)
$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda_{0})$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$.
As a consequence of Theorems 6.6, 7.4, and 7.6, Corollary 7.12 holds in the
special case of $d=1$, no matter if the coefficients of (1.3)λ are real or
complex.
###### Corollary 7.13.
If $d=1$, then the equivalent statements $\rm{(i)}-\rm{(ix)}$ in Corollary
7.12 hold.
###### Remark 7.14.
Much of the theory of Weyl and Titchmarsh remains that can be extended to time
scales, such as $M(\lambda)$ theory in the limit point case [41, Section 6],
asymptotic expansion of Weyl-Titchmarsh matrices and Green’s matrices [16],
and so on, leaving the future of the subject open to interested researchers.
## 8\. alternative form
In this section we introduce a possible alternative form for this theory on
Sturmian time scales. For example, instead of system (1.3), consider the
alternative system
(8.1) $J(\Upsilon y)^{\Delta}(t)=\Big{(}\lambda W(t)+P(t)\Big{)}y(t),\quad
t\in[t_{0},\infty)_{\mathbb{T}},\quad
J=\left(\begin{smallmatrix}0_{n}&-I_{n}\\\
I_{n}&0_{n}\end{smallmatrix}\right),$
for the same block matrices $W$ and $P$, where we have the delta derivative
and $\Upsilon y$ on the left-hand side for $\Upsilon$ in (1.4), $y$ on the
right-hand side, and where this time we assume
(8.2) $E_{2}(t):=\Big{(}I_{n}+\mu(t)A^{*}(t)\Big{)}^{-1}$
exists instead of (2.2). System (8.1) may also be viewed as a generalization
of (1.1) and (1.2). For (8.1) we have the integration by parts formula
(compare with Theorem 2.2)
$\int_{a}^{b}\left[z^{*}J(\Upsilon y)^{\Delta}-\left(J(\Upsilon
z)^{\Delta}\right)^{*}y\right](t)\Delta t=(\Upsilon z)^{*}(b)J(\Upsilon
y)(b)-(\Upsilon z)^{*}(a)J(\Upsilon y)(a).$
Moreover, the scalar product in (4.1) is now replaced by a standard weighted
product without shifts given by
(8.3) $(x,y)_{W}:=\int_{t_{0}}^{\infty}y^{*}(t)W(t)x(t)\Delta t,\quad x,y\in
L_{W}^{2}([t_{0},\infty)_{\mathbb{T}}).$
Additionally, as in (2.3) we may rewrite (8.1) as the system
(8.4) $(\Upsilon y)^{\Delta}(t)=\mathcal{K}(t,\lambda)(\Upsilon
y)(t),\quad\mathcal{K}(\cdot,\lambda):=-J(\lambda W+P)\widehat{H},$
where on $[t_{0},\infty)_{\mathbb{T}}$ we use $E_{2}=(I_{n}+\mu A^{*})^{-1}$
and
(8.5) $\widehat{H}:=\begin{pmatrix}I_{n}&0_{n}\\\ \mu E_{2}(C-\lambda
W_{1})&E_{2}\end{pmatrix},\quad\text{with}\quad-J(\lambda
W+P)=\begin{pmatrix}A&\lambda W_{2}+B\\\ C-\lambda W_{1}&-A^{*}\end{pmatrix},$
since $E_{2}A^{*}=A^{*}E_{2}$ and $I-\mu A^{*}E_{2}=E_{2}$. Directly from the
definition of $\mathcal{K}(\cdot,\lambda)$ in (8.4) we have that
$I_{2n}+\mu(t)\mathcal{K}(t,\lambda)=\begin{pmatrix}I_{n}+\mu(t)A(t)&\mu(t)(\lambda
W_{2}(t)+B(t))\\\ 0_{n}&I_{n}\end{pmatrix}\widehat{H}(t),$
so that $I_{2n}+\mu\mathcal{K}(\cdot,\lambda)$ is invertible by (8.2),
$\mathcal{K}(\cdot,\lambda)$ is regressive, and the matrix equation $(\Upsilon
y)^{\Delta}=\mathcal{K}(\cdot,\lambda)\Upsilon y$ with initial condition
$(\Upsilon y)(t_{0})=y_{0}$ has a unique solution $\Upsilon y$ on
$[t_{0},\infty)_{\mathbb{T}}$. It follows that an initial value problem
involving (8.1) has a unique solution $y$ in
$\left\\{y=(y_{1},y_{2})^{\operatorname{T}}\Big{|}\;y_{1},y_{2}^{\rho}:[t_{0},\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{n}\;\text{are
delta differentiable}\right\\}.$
In summary, to unify (1.1) and (1.2) on Sturmian time scales, systems
equivalent to (1.3) or (8.1) must be used to account for the shifts in the
discrete case [41]. For such systems to admit the existence and uniqueness of
solutions to initial value problems, an integration by parts formula, and a
matrix weighted scalar product in a Hilbert space, the Sturmian assumption
(1.7) is essential.
## References
* [1] R.P. Agarwal, M. Bohner, and P.J.Y. Wong, Sturm-Liouville eigenvalue problems on time scales, _Appl. Math Computation_ 99 (1999) 153–166.
* [2] C.D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, _J. Math. Anal. Appl._ 180 (1993) 498–517.
* [3] C.D. Ahlbrandt, M. Bohner, and J. Ridenhour, Hamiltonian systems on time scales, _J. Math. Anal. Appl._ 250 (2000) 561–578.
* [4] C.D. Ahlbrandt, M. Bohner and T. Voepel, Variable change for Sturm-Liouville differential expressions on time scales, _J. Difference Eq. Appl._ 9(1) (2003) 93- 107.
* [5] C.D. Ahlbrandt and A.C. Peterson, _Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Ricatti Equations_ , Kluwer Academic, Boston, 1996.
* [6] D.R. Anderson, Dominant and recessive solutions of self-adjoint matrix systems on time scales, _Nonlinear Dyn. Sys. Theory_ , 9(3) (2009) 219- 238.
* [7] D.R. Anderson, Spectral theory for second-order vector equations on finite time-varying domains, submitted.
* [8] D.R. Anderson and B. Buchholz, Self-adjoint matrix equations on time scales, _PanAmerican Mathematical Journal_ , 17(2) (2007) 81–104.
* [9] D.R. Anderson, G.Sh. Guseinov and J. Hoffacker, Higher-order self adjoint boundary value problems on time scales, _J. Computational Appl. Math._ 194(2) (2006) 309–342.
* [10] F.M. Atici and G.Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, _J. Comput. Appl. Math._ 141 (2002), no. 1-2, 75–99.
* [11] F.V. Atkinson, _Discrete and Continuous Boundary Problems_ , Academic Press, New York, 1964.
* [12] E.D. Belokolos, F. Gesztesy, K.A. Makarov, L.A. Sakhnovich, Matrix-valued generalizations of the theorems of Borg and Hochstadt, in: G.Ruiz Goldstein, R.Nagel, S.Romanelli (Eds.), _Evolution Equations, Lecture Notes in Pure and Applied Mathematics_ , Vol. 234, Marcel Dekker, New York, 2003, pp.1–34.
* [13] M. Bohner, O. Došlý, and W. Kratz, Sturmian and spectral theory for discrete symplectic systems, _Trans. Amer. Math. Soc._ 361(6) (2009) 3109–3123.
* [14] M. Bohner and A. Peterson, _Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2001.
* [15] M. Bohner and A. Peterson, editors, _Advances in Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2003.
* [16] S. Clark and F. Gesztesy, On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems, _J. Comput. Appl. Math._ 171 (2004), no. 1-2, 151–184.
* [17] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators, _Proc. London Math. Soc._ 82 (2001) 701–724.
* [18] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators, _Trans. Amer. Math. Soc._ 354 (2002), no. 9, 3475–3534.
* [19] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics and Borg-type theorems for Dirac operators, _Trans. Amer. Math. Soc._ 354 (2002) 3475–3534.
* [20] S. Clark and F. Gesztesy, On Povzner-Wienholtz-type Self-Adjointness Results for Matrix-Valued Sturm-Liouville Operators, _Proc. Roy. Soc. Edinburgh_ 133A (2003) 747–758.
* [21] S. Clark, F. Gesztesy, H. Holden, and B.M. Levitan, Borg-type theorems for matrix-valued Schrödinger operators, _J. Differential Equations_ 167 (2000) 181–210.
* [22] V. Cormani, Liouville’s formula on time scales, _Dynamic Sys. Appl._ 12 (2003) 79–86.
* [23] W.N. Everitt, D.B. Hinton, and J.K. Shaw, The asymptotic form of the Titchmarsh-Weyl coeQcient for Dirac systems, _J. London Math. Soc._ 27 (2) (1983) 465-476.
* [24] W.N. Everitt and K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric di)erential expressions. I. The odd-order case, _Nieuw Arch. Wisk._ 24 (3) (1976) 109–145.
* [25] W.N. Everitt and K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric di)erential expressions. II. The general theory, _Nieuw Arch. Wisk._ 24 (3) (1976) 1–48.
* [26] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, _Results Math._ 18 (1990) 18–56.
* [27] R. Hilscher, Linear Hamiltonian systems on time scales: positivity of quadratic functionals, _Math. Comput. Modelling_ 32 (2000) 507–527.
* [28] D.B. Hinton and A. Schneider, On the Titchmarsh-Weyl coeffcients for singular S-Hermitian Systems I, _Math. Nachr._ 163 (1993) 323–342.
* [29] D.B. Hinton and A. Schneider, On the Titchmarsh-Weyl coeffcients for singular S-Hermitian Systems II, _Math. Nachr._ 185 (1997) 323–342.
* [30] D.B. Hinton and J.K. Shaw, On Titchmarsh-Weyl $M(\lambda)$-functions for linear Hamiltonian systems, _J.Differential Equations_ 40 (1981) 316–342.
* [31] D.B. Hinton and J.K. Shaw, On the spectrum of a singular Hamiltonian system, _Quaestiones Math._ 5 (1982) 29–81.
* [32] D.B. Hinton and J.K. Shaw, Hamiltonian systems of limit point or limit circle type with both endpoints singular, _J. Diferential Equations_ 50 (1983) 444–464.
* [33] J. Ji and B. Yang, Eigenvalue comparisons for a class of boundary value problems of second order difference equations, _Linear Algebra Appl._ 420(1) (2007) 218–227.
* [34] J. Ji and B. Yang, Eigenvalue comparisons for second order difference equations with Neumann boundary conditions, _Linear Algebra Appl._ 425(1) (2007) 171–183.
* [35] Q.K. Kong, Sturm-Liouville problems on time scales with separated boundary conditions, _Results Math._ 52(1-2) (2008) 111–121.
* [36] W. Kratz, _Quadratic Functionals in Variational Analysis and Control Theory_ , Akademie Verlag, Berlin, 1995.
* [37] K. Messer, A second-order self-adjoint equation on a time scale, _Dyn. Sys. Appl._ 12 (2003) 201–215.
* [38] W. T. Reid, _Sturmian Theory for Ordinary Differential Equations_ , Springer -Verlag, New York, 1980.
* [39] B. P. Rynne, $L^{2}$ spaces and boundary value problems on time-scales, _J. Math. Anal. Appl._ 328 (2007) 1217 -1236.
* [40] Y.M. Shi, Spectral theory of discrete linear Hamiltonian systems, _J. Math. Anal. Appl._ 289(2) (2004) 554–570.
* [41] Y.M. Shi, Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems, _Linear Algebra Appl._ 416(2-3) (2006) 452–519.
* [42] Y.M. Shi and S.Z. Chen, Spectral theory of second-order vector difference equations, _J. Math. Anal. Appl._ 239 (1999) 195–212.
* [43] Y.M. Shi and S.Z. Chen, Spectral theory of higher-order discrete vector Sturm-Liouville problems, _Linear Algebra Appl._ 323(1-3) (2001) 7–36.
* [44] G.L. Shi and H.Y. Wu, Spectral theory of Sturm-Liouville difference operators, _Linear Algebra Appl._ 430(2-3) (2009) 830–846.
* [45] G.L. Shi and R. Yan, Spectral theory of left definite difference operators, _J. Math. Anal. Appl._ 337(1) (2008) 116–122.
* [46] H.Q. Sun and Y.M. Shi, Eigenvalues of second-order difference equations with coupled boundary conditions, _Linear Algebra Appl._ 414(1) (2006) 361–372.
* [47] S.R. Sun, Y.M. Shi, and S.Z. Chen, The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems, _J. Math. Anal. Appl._ 327(2) (2007) 1360–1380.
* [48] E.C. Titchmarsh, _Eigenfunction Expansions_ , Oxford University Press, Oxford, 1962.
* [49] Y. Wang and Y.M. Shi, Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, _J. Math. Anal. Appl._ 309(1) (2005) 56–69.
* [50] J. Weidmann, _Linear Operators in Hilbert Spaces_ , Springer-Verlag, New York, 1980.
* [51] J. Weiss, Limit-point criteria for a second order dynamic equation on time scales, _Nonlinear Dyn. Syst. Theory_ 9 (2009), no. 1, 99–108.
* [52] H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, _Math. Ann._ 68 (1910) 220–269.
* [53] A. Zettl, _Sturm-Liouville Theory_ , American Mathematical Society, Providence, 2005.
|
arxiv-papers
| 2010-01-22T16:35:52 |
2024-09-04T02:49:07.948357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1001.4040"
}
|
1001.4119
|
201047-58Nancy, France 47 Xavier Allamigeon Stéphane Gaubert Éric Goubault
# The tropical double description method
Xavier Allamigeon Direction du Budget, 4ème sous-direction, Bureau des
transports, Paris, France , Stéphane Gaubert INRIA Saclay and CMAP, Ecole
Polytechnique, France and Éric Goubault CEA, LIST MeASI – Gif-sur-Yvette,
France firstname.lastname@polytechnique.org,inria.fr,cea.fr
###### Abstract.
We develop a tropical analogue of the classical double description method
allowing one to compute an internal representation (in terms of vertices) of a
polyhedron defined externally (by inequalities). The heart of the tropical
algorithm is a characterization of the extreme points of a polyhedron in terms
of a system of constraints which define it. We show that checking the
extremality of a point reduces to checking whether there is only one minimal
strongly connected component in an hypergraph. The latter problem can be
solved in almost linear time, which allows us to eliminate quickly redundant
generators. We report extensive tests (including benchmarks from an
application to static analysis) showing that the method outperforms
experimentally the previous ones by orders of magnitude. The present tools
also lead to worst case bounds which improve the ones provided by previous
methods.
###### Key words and phrases:
convexity in tropical algebra, algorithmics and combinatorics of tropical
polyhedra, computational geometry, discrete event systems, static analysis
###### 1991 Mathematics Subject Classification:
F.2.2.Geometrical problems and computations, G.2.2 Hypergraphs; Algorithms,
Verification
This work was performed when the first author was with EADS Innovation Works,
SE/IA – Suresnes, France and CEA, LIST MeASI – Gif-sur-Yvette, France
This work was partially supported by the Arpege programme of the French
National Agency of Research (ANR), project “ASOPT”, number ANR-08-SEGI-005 and
by the Digiteo project DIM08 “PASO” number 3389
## Introduction
Tropical polyhedra are the analogues of convex polyhedra in tropical algebra.
The latter deals with structures like the max-plus semiring
$\mathbb{R}_{\max}$ (also called _max-plus algebra_), which is the set
$\mathbb{R}\cup\\{-\infty\\}$, equipped with the addition $x\oplus
y:=\max(x,y)$ and the multiplication $x\otimes y:=x+y$.
The study of the analogues of convex sets in tropical or max-plus algebra is
an active research topic, and has been treated under various guises. It arose
in the work of Zimmerman [Zim77], following a way opened by Vorobyev [Vor67],
motivated by optimization theory. Max-plus cones were studied by Cuninghame-
Green [CG79]. Their theory was independently developed by Litvinov, Maslov and
Shpiz [LMS01] (see also [MS92]) with motivations from variations calculus and
asymptotic analysis, and by Cohen, Gaubert, and Quadrat [CGQ04] who initiated
a “geometric approach” of discrete event systems [CGQ99], further developed in
[Kat07, DLGKL09]. Other motivations arise from abstract convexity, see the
book by Singer [Sin97], and also the work of Briec and Horvath [BH04]. The
field has attracted recently more attention after the work of Develin and
Sturmfels [DS04], who pointed out connections with tropical geometry, leading
to several works by Joswig, Yu, and the same authors [Jos05, DY07, JSY07,
Jos09].
A tropical polyhedron can be represented in two different ways, either
internally, in terms of extreme points and rays, or externally, in terms of
linear inequalities (see Sect. 1 for details). As in the classical case,
passing from the external description of a polyhedron to its internal
description is a fundamental computational issue. This is the object of the
present paper.
Butkovič and Hegedus [BH84] gave an algorithm to compute the generators of a
tropical polyhedral cone described by linear inequalities. Gaubert gave a
similar one and derived the equivalence between the internal and external
representations [Gau92, Ch. III] (see [GK09] for a recent discussion). Both
algorithms rely on a successive elimination of inequalities, but have the
inconvenience of squaring at each step the number of candidate generators,
unless an elimination technique is used, as in the Maxplus toolbox of Scilab
[CGMQ]. Joswig developed a different approach, implemented in Polymake [GJ],
in which a tropical polytope is represented as a polyhedral complex [DS04,
Jos09].
The present work grew out from two applications: to discrete event systems
[Kat07, DLGKL09], and to software verification by static analysis [AGG08]. In
these applications, passing from the external to the internal representation
is a central difficulty. A further motivation originates from mean payoff
games [AGG09b]. These motivations are reviewed in Section 2.
Contributions. We develop a new algorithm which computes the extreme elements
of tropical polyhedra. It is based on a successive elimination of
inequalities, and a result (Th. 4.1) allowing one, given a polyhedron
$\mathcal{P}$ and a tropical halfspace $\mathcal{H}$, to construct a list of
candidates for the generators of $\mathcal{P}\cap\mathcal{H}$. The key
ingredient is a combinatorial characterization of the extreme generators of a
polyhedron defined externally (Th. 3.5 and 3.7): we reduce the verification of
the extremality of a candidate to the existence of a strongly connected
component reachable from any other in a directed hypergraph. We include a
complexity analysis and experimental results (Sect. 4), showing that the new
algorithm outperforms the earlier ones, allowing us to solve instances which
were previously by far inaccessible. Our result also leads to worst case
bounds improving the ones of previously known algorithms.
## 1\. Definitions: tropical polyhedra and polyhedral cones
The neutral elements for the addition $\oplus$ and multiplication $\otimes$,
i.e., the zero and the unit, will be denoted by $\mathbbb{0}:=-\infty$ and
$\mathbbb{1}:=0$, respectively. The tropical analogues of the operations on
vectors and matrices are defined naturally. The elements of
$\mathbb{R}_{\max}^{d}$, the $d$th fold Cartesian product of
$\mathbb{R}_{\max}$, will be thought of as vectors, and denoted by bold
symbols, like $\boldsymbol{x}=(\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{d})$.
A _tropical halfspace_ is a set of the vectors
$\boldsymbol{x}=(\boldsymbol{x}_{i})\in\mathbb{R}_{\max}^{d}$ verifying an
inequality constraint of the form
$\max_{1\leq i\leq d}a_{i}+\boldsymbol{x}_{i}\leq\max_{1\leq i\leq
d}b_{i}+\boldsymbol{x}_{i},\quad a_{i},b_{i}\in\mathbb{R}_{\max}.$
A _tropical polyhedral cone_ is defined as the intersection of $n$ halfspaces.
It can be equivalently written as the set of the solutions of a system of
inequality constraints $A\boldsymbol{x}\leq B\boldsymbol{x}$. Here,
$A=(a_{ij})$ and $B=(b_{ij})$ are $n\times d$ matrices with entries in
$\mathbb{R}_{\max}$, concatenation denotes the matrix product (with the laws
of $\mathbb{R}_{\max}$), and $\leq$ denotes the standard partial ordering of
vectors. For sake of readability, tropical polyhedral cones will be simply
referred to as _polyhedral cones_ or _cones_.
Tropical polyhedral cones are known to be generated by their extreme rays
[GK06, GK07, BSS07]. Recall that a ray is the set of scalar multiples of a
non-zero vector $\boldsymbol{u}$. It is extreme in a cone $\mathcal{C}$ if
$\boldsymbol{u}\in\mathcal{C}$ and if
$\boldsymbol{u}=\boldsymbol{v}\oplus\boldsymbol{w}$ with
$\boldsymbol{v},\boldsymbol{w}\in\mathcal{C}$ implies that
$\boldsymbol{u}=\boldsymbol{v}$ or $\boldsymbol{u}=\boldsymbol{w}$. A finite
set $G=(\boldsymbol{g}^{i})_{i\in I}$ of vectors is said to generate a
polyhedral cone $\mathcal{C}$ if each $\boldsymbol{g}^{i}$ belongs to
$\mathcal{C}$, and if every vector $\boldsymbol{x}$ of $\mathcal{C}$ can be
written as a _tropical linear combination_
$\mathop{\bigoplus}_{i}\lambda_{i}\boldsymbol{g}^{i}$ of the vectors of $G$
(with $\lambda_{i}\in\mathbb{R}_{\max}$). Note that in tropical linear
combinations, the requirement that $\lambda_{i}$ be nonnegative is omitted.
Indeed, $\mathbbb{0}=-\infty\leq\lambda$ holds for all scalar
$\lambda\in\mathbb{R}_{\max}$.
The tropical analogue of the Minkowski theorem [GK07, BSS07] shows in
particular that every generating set of a cone that is minimal for inclusion
is obtained by selecting precisely one (non-zero) element in each extreme ray.
A tropical polyhedron of $\mathbb{R}_{\max}^{d}$ is the affine analogue of a
tropical polyhedral cone. It is defined by a system of inequalities of the
form $A\boldsymbol{x}\oplus\boldsymbol{c}\leq
B\boldsymbol{x}\oplus\boldsymbol{d}$. It can be also expressed as the set of
the tropical affine combinations of its generators. The latter are of the form
$\mathop{\bigoplus}_{i\in
I}\lambda_{i}\boldsymbol{v}^{i}\,\,\oplus\,\,\mathop{\bigoplus}_{j\in
J}\mu_{j}\boldsymbol{r}^{j}$, where the $(\boldsymbol{v}^{i})_{i\in I}$ are
the extreme points, the $(\boldsymbol{r}^{j})_{j\in J}$ a set formed by one
element of each extreme ray, and
$\mathop{\bigoplus}_{i}\lambda_{i}=\mathbbb{1}$. It is known [CGQ04, GK07]
that every tropical polyhedron of $\mathbb{R}_{\max}^{d}$ can be represented
by a tropical polyhedral cone of $\mathbb{R}_{\max}^{d+1}$ thanks to an
analogue of the homogenization method used in the classical case (see [Zie98,
Sect. 1.5]). Then, the extreme rays of the cone are in one-to-one
correspondence with the extreme generators of the polyhedron. That is why, in
the present paper, we will only state the main results for cones, leaving to
the reader the derivation of the affine analogues, along the lines of [GK07].
In the sequel, we will illustrate our results on the polyhedral cone
$\mathcal{C}$ given in Fig. 2, defined by the system in the right side. The
left side is a representation of $\mathcal{C}$ in barycentric coordinates:
each element $(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\boldsymbol{x}_{3})$ is
represented as a barycenter with weights
$(e^{\boldsymbol{x}_{1}},e^{\boldsymbol{x}_{2}},e^{\boldsymbol{x}_{3}})$ of
the three vertices of the outermost triangle. Then two elements of a same ray
are represented by the same point. The cone $\mathcal{C}$ is depicted in solid
gray (the black border is included), and is generated by the extreme elements
$\boldsymbol{g}^{0}=(\mathbbb{0},0,\mathbbb{0})$,
$\boldsymbol{g}^{1}=(-2,1,0)$, $\boldsymbol{g}^{2}=(2,2,0)$, and
$\boldsymbol{g}^{3}=(0,\mathbbb{0},0)$.
$\boldsymbol{x}_{1}$$\boldsymbol{x}_{2}$$\boldsymbol{x}_{3}$$\boldsymbol{g}^{1}$$\boldsymbol{g}^{2}$$\boldsymbol{g}^{3}$$\boldsymbol{g}^{0}$$\left\\{\begin{aligned}
\boldsymbol{x}_{3}&\leq\boldsymbol{x}_{1}+2\\\
\boldsymbol{x}_{1}&\leq\max(\boldsymbol{x}_{2},\boldsymbol{x}_{3})\\\
\boldsymbol{x}_{1}&\leq\boldsymbol{x}_{3}+2\\\
\boldsymbol{x}_{3}&\leq\max(\boldsymbol{x}_{1},\boldsymbol{x}_{2}-1)\end{aligned}\right.$
Figure 1. A tropical polyhedral cone in $\mathbb{R}_{\max}^{3}$
$\mathtt{len\\_src}$$\mathtt{len\\_dst}$$\mathtt{n}$ Figure 2. memcpy
invariant
## 2\. Motivations from static analysis, discrete event systems, and mean
payoff games
Tropical polyhedra have been recently involved in static analysis by abstract
interpretation [AGG08]. It has been shown that they allow to automatically
compute complex invariants involving the operators $\min$ and $\max$ which
hold over the variables of a program. Such invariants are disjunctive, while
most existing techniques in abstract interpretation are only able to express
conjunctions of affine constraints, see in particular [CC77, CH78, Min01].
For instance, tropical polyhedra can handle notorious problems in verification
of memory manipulations. Consider the well-known memory string manipulating
function memcpy in C. A call to $\mathtt{memcpy(dst,src,n)}$ copies exactly
the first $\mathtt{n}$ characters of the string buffer $\mathtt{src}$ to
$\mathtt{dst}$. In program verification, precise invariants over the length of
the strings are needed to ensure the absence of string buffer overflows.
Recall that the length of a string is defined as the position of the first
null character in the string. To precisely analyze the function memcpy, two
cases have to be distinguished:
1. (i)
either $\mathtt{n}$ is strictly smaller than the source length
$\mathtt{len\\_src}$, so that only non-null characters are copied into
$\mathtt{dst}$, hence $\mathtt{len\\_dst}\geq\mathtt{n}$,
2. (ii)
or $\mathtt{n}\geq\mathtt{len\\_src}$ and the null terminal character of
$\mathtt{src}$ will be copied into $\mathtt{dst}$, thus
$\mathtt{len\\_dst}=\mathtt{len\\_src}$.
Thanks to tropical polyhedra, the invariant
$\min(\mathtt{len\\_src},\mathtt{n})=\min(\mathtt{len\\_dst},\mathtt{n})$, or
equivalently
$\max(-\mathtt{len\\_src},-\mathtt{n})=\max(-\mathtt{len\\_dst},-\mathtt{n})$,
can be automatically inferred. It is the _exact_ encoding of the disjunction
of the cases (i) and (ii). The invariant is represented by the non-convex set
of $\mathbb{R}^{3}$ depicted in Figure 2. In the application to static
analysis, the performance of the algorithm computing the extreme elements of
tropical polyhedra plays a crucial role in the scalability of the analyzer
(see [AGG08] for further details).
A second motivation arises from the “geometric approach” of max-plus linear
discrete event systems [CGQ99], in which the computation of feedbacks ensuring
that the state of the system meets a prescribed constraint (for instance that
certain waiting times remain bounded) reduces [Kat07] to computing the
greatest fixed point of an order preserving map on the set of tropical
polyhedra. Similar computations arise when solving dual observability problems
[DLGKL09]. Again, the effective handling of these polyhedra turns out to be
the bottleneck.
A third motivation arises from the study of mean payoff combinatorial games.
In particular, it is shown in [AGG09b] that checking whether a given initial
state of a mean payoff game is winning is equivalent to finding a vector in an
associated tropical polyhedral cone (with a prescribed finite coordinate).
This polyhedron consists of the super-fixed points of the dynamic programming
operator (potentials), which certify that the game is winning.
## 3\. Characterizing extremality from inequality constraints
### 3.1. Preliminaries on extremality
The following lemma, which is a variation on the proof of Th. 3.1 of [GK07]
and on Th. 14 of [BSS07], shows that extremality can be expressed as a
minimality property:
###### Proposition 3.1.
Given a polyhedral cone $\mathcal{C}\subset\mathbb{R}_{\max}^{d}$,
$\boldsymbol{g}$ is extreme if and only if there exists $1\leq t\leq d$ such
that $\boldsymbol{g}$ is a minimal element of the set
$\\{\,\boldsymbol{x}\in\mathcal{C}\mid\boldsymbol{x}_{t}=\boldsymbol{g}_{t}\,\\}$,
i.e. $\boldsymbol{g}\in\mathcal{C}$ and for each
$\boldsymbol{x}\in\mathcal{C}$, $\boldsymbol{x}\leq\boldsymbol{g}\text{ and
}\boldsymbol{x}_{t}=\boldsymbol{g}_{t}\text{ implies
}\boldsymbol{x}=\boldsymbol{g}.$ In that case, $\boldsymbol{g}$ is said to be
_extreme of type $t$_.
In Fig. 4, the light gray area represents the set of the elements
$(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\boldsymbol{x}_{3})$ of
$\mathbb{R}_{\max}^{3}$ such that
$(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\boldsymbol{x}_{3})\leq\boldsymbol{g}^{2}$
implies $\boldsymbol{x}_{1}<\boldsymbol{g}^{2}_{1}$. It clearly contains the
whole cone except $\boldsymbol{g}^{2}$, which shows that $\boldsymbol{g}^{2}$
is extreme of type $1$.
A tropical segment is the set of the tropical linear combinations of two
points. Using the fact that a tropical segment joining two points of a
polyhedral cone $\mathcal{C}$ yields a continuous path included in
$\mathcal{C}$, one can check that $\boldsymbol{g}$ is extreme of type $t$ in
$\mathcal{C}$ if and only if there is a neighborhood $N$ of $\boldsymbol{g}$
such that $\boldsymbol{g}$ is minimal in
$\\{\,\boldsymbol{x}\in\mathcal{C}\cap
N\mid\boldsymbol{x}_{t}=\boldsymbol{g}_{t}\,\\}$. Thus, extremality is a local
property.
Finally, the extremality of an element $\boldsymbol{g}$ in a cone
$\mathcal{C}$ can be equivalently established by considering the vector formed
by its non-$\mathbbb{0}$ coordinates. Formally, let
$\operatorname{\mathsf{supp}}(\boldsymbol{x}):=\\{i\mid\boldsymbol{x}_{i}\neq\mathbbb{0}\\}$
for any $\boldsymbol{x}\in\mathbb{R}_{\max}^{d}$. Then $\boldsymbol{g}$ is
extreme in $\mathcal{C}$ if and only if it is extreme in
$\\{\boldsymbol{x}\in\mathcal{C}\mid\operatorname{\mathsf{supp}}(\boldsymbol{x})\subset\operatorname{\mathsf{supp}}(\boldsymbol{g})\\}$.
This allows to assume that
$\operatorname{\mathsf{supp}}(\boldsymbol{g})=\\{1,\ldots,d\\}$ without loss
of generality.
### 3.2. Expressing extremality using the tangent cone
For now, the polyhedral cone $\mathcal{C}$ is supposed to be defined by a
system $A\boldsymbol{x}\leq B\boldsymbol{x}$ of $n$ inequalities.
Consider an element $\boldsymbol{g}$ of the cone $\mathcal{C}$, which we
assume, from the previous discussion, to satisfy
$\operatorname{\mathsf{supp}}(\boldsymbol{g})=\\{\,1,\ldots,d\,\\}$. In this
context, the _tangent cone_ of $\mathcal{C}$ at $\boldsymbol{g}$ is defined as
the tropical polyhedral cone $\mathcal{T}(\boldsymbol{g},\mathcal{C})$ of
$\mathbb{R}_{\max}^{d}$ given by the system of inequalities
$\max_{i\in\arg\max(A_{k}\boldsymbol{g})}\boldsymbol{x}_{i}\leq\max_{j\in\arg\max(B_{k}\boldsymbol{g})}\boldsymbol{x}_{j}\qquad\text{for
all }k\text{ such that }A_{k}\boldsymbol{g}=B_{k}\boldsymbol{g},$ (1)
where for each row vector $\boldsymbol{c}\in\mathbb{R}_{\max}^{1\times d}$,
$\arg\max(\boldsymbol{c}\boldsymbol{g})$ is defined as the argument of the
maximum $\boldsymbol{c}\boldsymbol{g}=\max_{1\leq i\leq
d}(\boldsymbol{c}_{i}+\boldsymbol{g}_{i})$, and where $A_{k}$ and $B_{k}$
denote the $k$th rows of $A$ and $B$, respectively.
The tangent cone $\mathcal{T}(\boldsymbol{g},\mathcal{C})$ provides a local
description of the cone $\mathcal{C}$ around $\boldsymbol{g}$:
###### Proposition 3.2.
There exists a neighborhood $N$ of $\boldsymbol{g}$ such that for all
$\boldsymbol{x}\in N$, $\boldsymbol{x}$ belongs to $\mathcal{C}$ if and only
if it is an element of
$\boldsymbol{g}+\mathcal{T}(\boldsymbol{g},\mathcal{C})$.
$\boldsymbol{x}_{1}$$\boldsymbol{x}_{2}$$\boldsymbol{x}_{3}$$\boldsymbol{g}^{2}$
Figure 3. Extremality of $\boldsymbol{g}^{2}$
$\boldsymbol{x}_{1}$$\boldsymbol{x}_{2}$$\boldsymbol{x}_{3}$$\boldsymbol{g}^{2}$
Figure 4. The set
$\boldsymbol{g}^{2}+\mathcal{T}(\boldsymbol{g}^{2},\mathcal{C})$
As an illustration, Fig. 4 depicts the set
$\boldsymbol{g}^{2}+\mathcal{T}(\boldsymbol{g}^{2},\mathcal{C})$ (in semi-
transparent light gray) when $\mathcal{C}$ is the cone given in Fig. 2. Both
clearly coincide in the neighborhood of $\boldsymbol{g}^{2}$.
Since extremality is a local property, it can be equivalently characterized in
terms of the tangent cone. Let $\boldsymbol{\mathbbb{1}}$ be the element of
$\mathbb{R}_{\max}^{d}$ whose all coordinates are equal to $\mathbbb{1}$.
###### Proposition 3.3.
The element $\boldsymbol{g}$ is extreme in $\mathcal{C}$ iff the vector
$\boldsymbol{\mathbbb{1}}$ is extreme in
$\mathcal{T}(\boldsymbol{g},\mathcal{C})$.
The problem is now reduced to the characterization of the extremality of the
vector $\boldsymbol{\mathbbb{1}}$ in a _$\\{\,\mathbbb{0},\mathbbb{1}\,\\}$
-cone_, i.e. a polyhedral cone defined by a system of the form
$C\boldsymbol{x}\leq D\boldsymbol{x}$ where
$C,D\in\\{\,\mathbbb{0},\mathbbb{1}\,\\}^{n\times d}$. The following
proposition states that only $\\{\,\mathbbb{0},\mathbbb{1}\,\\}$-vectors, i.e.
elements of the tropical regular cube $\\{\,\mathbbb{0},\mathbbb{1}\,\\}^{d}$,
have to be considered:
###### Proposition 3.4.
Let $\mathcal{D}\subset\mathbb{R}_{\max}^{d}$ be a
$\\{\,\mathbbb{0},\mathbbb{1}\,\\}$-cone. Then $\boldsymbol{\mathbbb{1}}$ is
extreme of type $t$ if and only if it is the unique element $\boldsymbol{x}$
of $\mathcal{D}\cap\\{\,\mathbbb{0},\mathbbb{1}\,\\}^{d}$ satisfying
$\boldsymbol{x}_{t}=\mathbbb{1}$.
The following criterion of extremality is a direct consequence of Prop. 3.3
and 3.4:
###### Theorem 3.5.
Let $\mathcal{C}\subset\mathbb{R}_{\max}^{d}$ be a polyhedral cone. Then
$\boldsymbol{g}\in\mathcal{C}$ is extreme of type $t$ if and only if the
vector $\boldsymbol{\mathbbb{1}}$ is the unique
$\\{\,\mathbbb{0},\mathbbb{1}\,\\}$-element of the tangent cone
$\mathcal{T}(\boldsymbol{g},\mathcal{C})$ whose $t$-th coordinate is
$\mathbbb{1}$.
Figure 6 shows that in our running example, the
$\\{\,\mathbbb{0},\mathbbb{1}\,\\}$-elements of
$\mathcal{T}(\boldsymbol{g}^{2},\mathcal{C})$ distinct from
$\boldsymbol{\mathbbb{1}}$ (in squares) all satisfy
$\boldsymbol{x}_{1}=\mathbbb{0}$. Naturally, testing, by exploration, whether
the set of $2^{d-1}$ $\\{\,\mathbbb{0},\mathbbb{1}\,\\}$-elements
$\boldsymbol{x}$ verifying $\boldsymbol{x}_{t}=\mathbbb{1}$ belonging to
$\mathcal{T}(\boldsymbol{g},\mathcal{C})$ consists only of
$\boldsymbol{\mathbbb{1}}$ does not have an acceptable complexity. Instead,
the approach of the next section will rely on the equivalent formulation of
the criterion of Th. 3.5:
$\forall
l\in\\{\,1,\dots,d\,\\},\;\bigl{[}\forall\boldsymbol{x}\in\mathcal{T}(\boldsymbol{g},\mathcal{C})\cap\\{\,\mathbbb{0},\mathbbb{1}\,\\}^{d},\;\boldsymbol{x}_{l}=\mathbbb{0}\implies\boldsymbol{x}_{t}=\mathbbb{0}\bigr{]}.$
(2)
$\boldsymbol{x}_{1}$$\boldsymbol{x}_{2}$$\boldsymbol{x}_{3}$$\boldsymbol{\mathbbb{1}}$$(\mathbbb{0},\mathbbb{1},\mathbbb{0})$$(\mathbbb{0},\mathbbb{0},\mathbbb{1})$$(\mathbbb{0},\mathbbb{1},\mathbbb{1})$
Figure 5. The $\\{\,\mathbbb{0},\mathbbb{1}\,\\}$-elements of
$\mathcal{T}(\boldsymbol{g}^{2},\mathcal{C})$
$u$$v$$w$$x$$y$$t$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$e_{5}$ Figure 6. A directed
hypergraph
### 3.3. Characterizing extremality with directed hypergraphs
A _directed hypergraph_ is a couple $(N,E)$ such that each element of $E$ is
of the form $(T,H)$ with $T,H\subset N$.
The elements of $N$ and $E$ are respectively called _nodes_ and _hyperedges_.
Given a hyperedge $e=(T,H)\in E$, the sets $T$ and $H$ represent the _tail_
and the _head_ of $e$ respectively, and are also denoted by $T(e)$ and $H(e)$.
Figure 6 depicts an example of hypergraph whose nodes are $u,v,w,x,y,t$, and
of hyperedges $e_{1}=(\\{u\\},\\{v\\})$, $e_{2}=(\\{v\\},\\{w\\})$,
$e_{3}=(\\{w\\},\\{u\\})$, $e_{4}=(\\{v,w\\},\\{x,y\\})$, and
$e_{5}=(\\{w,y\\},\\{t\\})$.
Reachability is extended from digraphs to directed hypergraphs by the
following recursive definition: given $u,v\in N$, then $v$ is _reachable from
$u$ in $\mathcal{H}$_, which is denoted $u\rightsquigarrow_{\mathcal{H}}v$, if
one of the two conditions holds: $u=v$, or there exists $e\in E$ such that
$v\in H(e)$ and all the elements of $T(e)$ are reachable from $u$. In our
example, $t$ is reachable from $u$.
The size $\mathsf{size}(\mathcal{H})$ of a hypergraph $\mathcal{H}=(N,E)$ is
defined as $\left\lvert N\right\rvert+\sum_{e\in E}(\left\lvert
T(e)\right\rvert+\left\lvert H(e)\right\rvert)$. In the rest of the paper,
directed hypergraphs will be simply referred to as hypergraphs.
We associate to the tangent cone $\mathcal{T}(\boldsymbol{g},\mathcal{C})$ the
hypergraph $\mathcal{H}(\boldsymbol{g},\mathcal{C})=(N,E)$ defined by:
$N=\\{\,1,\dots,d\,\\}\qquad
E=\left\\{\,(\arg\max(B_{k}\boldsymbol{g}),\arg\max(A_{k}\boldsymbol{g}))\mid
A_{k}\boldsymbol{g}=B_{k}\boldsymbol{g},\;1\leq k\leq n\,\right\\}.$
The extremality criterion of Eq. (2) suggests to evaluate, given an element of
$\mathcal{T}(\boldsymbol{g},\mathcal{C})\cap\\{\,\mathbbb{0},\mathbbb{1}\,\\}^{d}$,
the effect of setting its $l$-th coordinate to the other coordinates. Suppose
that it has been discovered that $\boldsymbol{x}_{l}=\mathbbb{0}$ implies
$\boldsymbol{x}_{j_{1}}=\dots=\boldsymbol{x}_{j_{n}}=\mathbbb{0}$. For any
hyperedge $e$ of $\mathcal{H}(\boldsymbol{g},\mathcal{C})$ such that
$T(e)\subset\\{\,l,j_{1},\dots,j_{n}\,\\}$, $\boldsymbol{x}$ satisfies:
$\max_{i\in H(e)}\boldsymbol{x}_{i}\leq\max_{j\in
T(e)}\boldsymbol{x}_{j}=\mathbbb{0}$, so that $\boldsymbol{x}_{i}=\mathbbb{0}$
for all $i\in H(e)$. Thus, the propagation of the value $\mathbbb{0}$ from the
$l$-th coordinate to other coordinates mimicks the inductive definition of the
reachability relation from the node $l$ in
$\mathcal{H}(\boldsymbol{g},\mathcal{C})$:
###### Proposition 3.6.
For all $l\in\\{\,1,\dots,d\,\\}$, the statement given between brackets in Eq.
(2) holds if and only if $t$ is reachable from $l$ in the hypergraph
$\mathcal{H}(\boldsymbol{g},\mathcal{C})$.
Hence, the extremality criterion can be restated thanks to some considerations
on the strongly connected components of
$\mathcal{H}(\boldsymbol{g},\mathcal{C})$. The _strongly connected components_
(Sccs for short) of a hypergraph $\mathcal{H}$ are the equivalence classes of
the equivalence relation $\equiv_{\mathcal{H}}$, defined by
$u\equiv_{\mathcal{H}}v$ if $u\rightsquigarrow_{\mathcal{H}}v$ and
$v\rightsquigarrow_{\mathcal{H}}u$. They form a partition of the set of nodes
of $\mathcal{H}$. They can be partially ordered by the relation
$\preceq_{\mathcal{H}}$, defined by $C_{1}\preceq_{\mathcal{H}}C_{2}$ if
$C_{1}$ and $C_{2}$ admit a representative $u$ and $v$ respectively such that
$v\rightsquigarrow_{\mathcal{H}}u$ (_beware of the order of $v$ and $u$ in
$v\rightsquigarrow_{\mathcal{H}}u$_). Then Prop. 3.6 and Th. 3.5 imply the
following statement:
###### Theorem 3.7.
Let $\mathcal{C}\subset\mathbb{R}_{\max}^{d}$ be a polyhedral cone, and
$\boldsymbol{g}\in\mathcal{C}$. Then $\boldsymbol{g}$ is extreme if and only
if the set of the Sccs of the hypergraph
$\mathcal{H}(\boldsymbol{g},\mathcal{C})$, partially ordered by
$\preceq_{\mathcal{H}(\boldsymbol{g},\mathcal{C})}$, admits a least element.
This theorem is reminiscent of a classical result, showing that a point of a
polyhedron defined by inequalities is extreme if and only if the family of
gradients of active inequalities at this point is of full rank. Here, the
hypergraph encodes precisely the subdifferentials (set of generalized
gradients) of the active inequalities but a major difference is that the rank
condition must be replaced by the above minimality condition, which is
essentially stronger. Indeed, using this theorem, it is shown in [AGK09] that
an important class of tropical polyhedra has fewer extreme rays than its
classical analogue.
An algorithm due to Gallo et al. [GLPN93] shows that one can compute the set
of nodes that are reachable from a given node in linear time in an hypergraph.
The following result shows that one can in fact compute the minimal Sccs with
almost the same complexity. The algorithm is included in the extended version
of the present paper [AGG09c]. Although it shows some analogy with the
classical Tarjan algorithm, the hypergraph case differs critically from the
graph case in that one cannot compute all the Sccs using the same technique.
###### Theorem 3.8.
The set of minimal Sccs of a hypergraph $\mathcal{H}=(N,E)$ can be computed in
time $O(\mathsf{size}(\mathcal{H})\times\alpha(\left\lvert N\right\rvert))$,
where $\alpha$ denotes the inverse of the Ackermann function.
## 4\. The tropical double description method
$\boldsymbol{x}_{1}$$\boldsymbol{x}_{2}$$\boldsymbol{x}_{3}$$\boldsymbol{g}^{0}$$\boldsymbol{g}^{1}$$\boldsymbol{g}^{2}$$\boldsymbol{g}^{3}$$\boldsymbol{h}^{2,0}$$\boldsymbol{h}^{3,0}$$\boldsymbol{h}^{1,0}$
Figure 7. Intersecting a cone with a halfspace
Figure 8. Intersecting $10$ affine hyperplanes in dimension $3$
Our algorithm is based on a successive elimination of inequalities. Given a
polyhedral cone $\mathcal{C}$ defined by a system of $n$ constraints, the
algorithm computes by induction on $k$ ($0\leq k\leq n$) a generating set
$G_{k}$ of the intermediate cone defined by the first $k$ constraints. Then
$G_{n}$ forms a generating set of the cone $\mathcal{C}$. Passing from the set
$G_{k}$ to the set $G_{k+1}$ relies on a result which, given a polyhedral cone
$\mathcal{K}$ and a tropical halfspace
$\mathcal{H}=\\{\,\boldsymbol{x}\mid\boldsymbol{a}\boldsymbol{x}\leq\boldsymbol{b}\boldsymbol{x}\,\\}$,
allows to build a generating set $G^{\prime}$ of $\mathcal{K}\cap\mathcal{H}$
from a generating set $G$ of $\mathcal{K}$:
###### Theorem 4.1.
Let $\mathcal{K}$ be a polyhedral cone generated by a set
$G\subset\mathbb{R}_{\max}^{d}$, and
$\mathcal{H}=\\{\,\boldsymbol{x}\mid\boldsymbol{a}\boldsymbol{x}\leq\boldsymbol{b}\boldsymbol{x}\,\\}$
a tropical halfspace
($\boldsymbol{a},\boldsymbol{b}\in\mathbb{R}_{\max}^{1\times d}$). Then the
polyhedral cone $\mathcal{K}\cap\mathcal{H}$ is generated by the set
$\\{\,\boldsymbol{g}\in
G\mid\boldsymbol{a}\boldsymbol{g}\leq\boldsymbol{b}\boldsymbol{g}\,\\}\ \cup\
\\{\,(\boldsymbol{a}\boldsymbol{h})\boldsymbol{g}\oplus(\boldsymbol{b}\boldsymbol{g})\boldsymbol{h}\mid\boldsymbol{g},\boldsymbol{h}\in
G\text{, }\boldsymbol{a}\boldsymbol{g}\leq\boldsymbol{b}\boldsymbol{g}\text{,
and }\boldsymbol{a}\boldsymbol{h}>\boldsymbol{b}\boldsymbol{h}\,\\}.$
For instance, consider the cone defined in Fig. 2 and the constraint
$\boldsymbol{x}_{2}\leq\boldsymbol{x}_{3}+2.5$ (depicted in semi-transparent
gray in Fig. 8). The three generators $\boldsymbol{g}^{1}$,
$\boldsymbol{g}^{2}$, and $\boldsymbol{g}^{3}$ satisfy the constraint, while
$\boldsymbol{g}^{0}$ does not. Their combinations are the elements
$\boldsymbol{h}^{1,0}$, $\boldsymbol{h}^{2,0}$, and $\boldsymbol{h}^{3,0}$
respectively. The resulting algorithm is given in Figure 9. As in the
classical case, this inductive approach produces redundant generators, hence,
the heart of the algorithm is the extremality test in Line 10. We use here the
hypergraph characterization (Theorems 3.7 and 3.8).
1:procedure ComputeExtreme($A,B,n$) $\quad\triangleright$
$A,B\in\mathbb{R}_{\max}^{n\times d}$
2: if $n=0$ then $\quad\triangleright$ Base case
3: return the tropical canonical basis $(\boldsymbol{\epsilon}_{i})_{1\leq
i\leq d}$
4: else $\quad\triangleright$ Inductive case
5: split $A\boldsymbol{x}\leq B\boldsymbol{x}$ into $C\boldsymbol{x}\leq
D\boldsymbol{x}$ and
$\boldsymbol{a}\boldsymbol{x}\leq\boldsymbol{b}\boldsymbol{x}$, with
$C,D\in\mathbb{R}_{\max}^{(n-1)\times d}$ and
$\boldsymbol{a},\boldsymbol{b}\in\mathbb{R}_{\max}^{1\times d}$
6: $G:=\textsc{ComputeExtreme}{\textnormal{(}}C,D,n-1{\textnormal{)}}$
7: $G^{\leq}:=\\{\,\boldsymbol{g}^{i}\in
G\mid\boldsymbol{a}\boldsymbol{g}^{i}\leq\boldsymbol{b}\boldsymbol{g}^{i}\,\\}$,
$G^{>}:=\\{\,\boldsymbol{g}^{j}\in
G\mid\boldsymbol{a}\boldsymbol{g}^{j}>\boldsymbol{b}\boldsymbol{g}^{j}\,\\}$,
$H:=G^{\leq}$
8: for all $\boldsymbol{g}^{i}\in G^{\leq}$ and $\boldsymbol{g}^{j}\in G^{>}$
do
9:
$\boldsymbol{h}:=(\boldsymbol{a}\boldsymbol{g}^{j})\boldsymbol{g}^{i}\oplus(\boldsymbol{b}\boldsymbol{g}^{i})\boldsymbol{g}^{j}$
10: if $\boldsymbol{h}$ is extreme in $\\{\,\boldsymbol{x}\mid
A\boldsymbol{x}\leq B\boldsymbol{x}\,\\}$ then
11: append $\kappa\boldsymbol{h}$ to $H$, where $\kappa$ is the opposite of
the first non-$\mathbbb{0}$ coefficient of $\boldsymbol{h}$
12: end
13: done
14: end
15: return $H$
16:end
Figure 9. Our main algorithm computing the extreme rays of tropical cones
#### Complexity analysis.
The complexity of the elementary step of ComputeExtreme, i.e. the computation
of the elements provided by Th. 4.1 and the elimination of non-extreme ones
(Lines 7 to 13), can be precisely characterized to $O(nd\alpha(d)\left\lvert
G\right\rvert^{2})$, where $G$ is the generating set of the last intermediate
cone. By comparison, for classical polyhedra, the same step in the refined
double description method by Fukuda and Prodon [FP96] takes a time
$O(n\left\lvert G\right\rvert^{3})$. Note that $\left\lvert G\right\rvert$ can
take values much larger that $d$.
The overall complexity of the algorithm ComputeExtreme depends on the size of
the sets returned in the intermediate steps. In classical geometry, the upper
bound theorem of McMullen [McM70] shows that the maximal number of extreme
points of a convex polytope in $\mathbb{R}^{d}$ defined by $n$ inequality
constraints is equal to
$U(n,d):=\binom{n-\lfloor(d+1)/2\rfloor}{n-d}+\binom{n-\lfloor(d+2)/2\rfloor}{n-d}\enspace.$
The polars of the _cyclic polytopes_ (see [Zie98]) are known to reach this
bound. Allamigeon, Gaubert, and Katz [AGK09] showed that a similar bound is
valid in the tropical setting.
###### Theorem 4.2 ([AGK09]).
The number of extreme rays of a tropical cone in
$(\mathbb{R}\cup\\{-\infty\\})^{d}$ defined as the intersection of $n$
tropical half-spaces cannot exceed
$U(n+d,d-1)=O((n+d)^{\lfloor(d-1)/2\rfloor}$.
The bound is asymptotically tight for a fixed $n$, as $d$ tends to infinity,
being approached by a tropical generalization of the (polar of) the cyclic
polytope [AGK09]. The bound is believed not to be tight for a fixed $d$, as
$n$ tends to infinity. Finding the optimal bound is an open problem. By
combining Theorem 4.2, Theorem 3.8, and Theorem 3.7, we readily get the
following complexity result, showing that the execution time is smaller in the
tropical case than in the classical case, even with the refinements of [FP96].
###### Proposition 4.3.
The hypergraph implementation of the tropical double description method
returns the set of extreme rays of a polyhedral cone defined by $n$
inequalities in dimension $d$ in time $O(n^{2}d\alpha(d)G_{\max}^{2})$, where
$G_{\max}$ is the maximal number of extreme rays of a cone defined by a
subsystem of inequalities taken from $A\boldsymbol{x}\leq B\boldsymbol{x}$. In
particular, the time can be bounded by $O(n^{2}d\alpha(d)(n+d)^{d-1})$.
#### Alternative approaches.
The existing approachs discussed in the introduction have a structure which is
similar to ComputeExtreme. However, their implementation in the Maxplus
toolbox of Scilab [CGMQ] and in our previous work [AGG08] relies on a much
less efficient elimination of redundant generators. Its principle is the
following: an element $\boldsymbol{h}$ is extreme in the cone generated by a
given set $H$ if and only if $\boldsymbol{h}$ can not be expressed as the
tropical linear combination of the elements of $H$ which are not proportional
to it. This property can be checked in $O(d\times\left\lvert H\right\rvert)$
time using residuation (see [BSS07] for algorithmic details). In the context
of our algorithm, the worst case complexity of the redundandy test is
therefore $O(d\left\lvert G\right\rvert^{2})$, where $G$ is the set of the
extreme rays of the last intermediary cone. This is much worse that our method
in $O(nd\alpha(d))$ based on directed hypergraphs, since the cardinality of
the set $G$ may be exponential in $d$ (see Theorem 4.2). This is also
confirmed by our experiments (see below).
We next sketch a different method relying on arrangement of tropical
hyperplanes (arrangements of classical hyperplanes yield naive bounds).
Indeed, Theorem 3.7 implies that every extreme ray belongs to the intersection
of $d-1$ tropical hyperplanes, obtained by saturating $d-1$ inequalities among
the $n+d$ taken from $A\boldsymbol{x}\leq B\boldsymbol{x}$ and
$\boldsymbol{x}_{i}\geq-\infty$, for $i\in[d]$. The max-plus Cramer theorem
(see [AGG09a] and the references therein) implies that for generic values of
the matrices $A,B$, every choice of $d-1$ saturated inequalities yields at
most one candidate to be an extreme ray, which can be computed in $O(d^{3})$
time. This yields a list of $O((n+d)^{d-1})$ candidates, from which the
extreme rays can be extracted by using the present hypergraph characterization
(Theorems 3.7 and 3.8), leading to a $O((nd\alpha(d)+d^{3})(n+d)^{d-1})$
execution time, which is better than the one of Proposition 4.3 by a factor
$n/\alpha(d)$ when $n\gg d$. However, the resulting algorithm is of little
practical use, since the worst case execution time is essentially always
achieved, whereas the double description method takes advantage of the fact
that $G_{\max}$ is in general much smaller than the upper bound of Theorem 4.2
(which is probably not optimal in the case $n\gg d$).
A third approach, along the lines of [DS04, Jos09], would consist in
representing tropical polyhedra by polyhedral complexes in the usual sense.
However, an inconvenient of polyhedral complexes is that their number of
vertices (called “pseudo-vertices” to avoid ambiguities) is exponential in the
number of extreme rays [DS04]. Hence, the representations used here are more
concise. This is illustrated in Figure 8 (generated using Polymake), which
shows an intersection of 10 signed tropical hyperplanes, corresponding to the
“natural” pattern studied in [AGK09]. There are only 24 extreme rays, but 1215
pseudo-vertices.
#### Experiments.
Table 1. Benchmarks on a single core of a $3\,\,\mathrm{GHz}$ Intel Xeon with
$3\,\,\mathrm{Gb}$ RAM
| $d$ | $n$ | # final | # inter. | $T$ (s) | $T^{\prime}$ (s) | $T/T^{\prime}$
---|---|---|---|---|---|---|---
rnd$100$ | $12$ | $15$ | $32$ | $59$ | $0.24$ | $6.72$ | $0.035$
rnd$100$ | $15$ | $10$ | $555$ | $292$ | $2.87$ | $321.78$ | $8.9\cdot 10^{-3}$
rnd$100$ | $15$ | $18$ | $152$ | $211$ | $6.26$ | $899.21$ | $7.0\cdot 10^{-3}$
rnd$30$ | $17$ | $10$ | $1484$ | $627$ | $15.2$ | $4667.9$ | $3.3\cdot 10^{-3}$
rnd$10$ | $20$ | $8$ | $5153$ | $1273$ | $49.8$ | $50941.9$ | $9.7\cdot 10^{-4}$
rnd$10$ | $25$ | $5$ | $3999$ | $808$ | $9.9$ | $12177.0$ | $8.1\cdot 10^{-4}$
rnd$10$ | $25$ | $10$ | $32699$ | $6670$ | $3015.7$ | — | —
cyclic | $10$ | $20$ | $3296$ | $887$ | $25.8$ | $4957.1$ | $5.2\cdot 10^{-3}$
cyclic | $15$ | $7$ | $2640$ | $740$ | $8.1$ | $1672.2$ | $5.2\cdot 10^{-3}$
cyclic | $17$ | $8$ | $4895$ | $1589$ | $44.8$ | $25861.1$ | $1.7\cdot 10^{-3}$
cyclic | $20$ | $8$ | $28028$ | $5101$ | $690$ | $45\text{ days}$ | $1.8\cdot 10^{-4}$
cyclic | $25$ | $5$ | $25025$ | $1983$ | $62.6$ | $8\text{ days}$ | $9.1\cdot 10^{-5}$
cyclic | $30$ | $5$ | $61880$ | $3804$ | $261$ | — | —
cyclic | $35$ | $5$ | $155040$ | $7695$ | $1232.6$ | — | —
| # var | # lines | $T$ (s) | $T^{\prime}$ (s) | $T/T^{\prime}$
---|---|---|---|---|---
oddeven8 | $17$ | $118$ | $7.6$ | $152.1$ | $0.050$
oddeven9 | $19$ | $214$ | $128.0$ | $22101.2$ | $5.8\cdot 10^{-3}$
oddeven10 | $21$ | $240$ | $1049.0$ | — | —
Allamigeon has implemented Algorithm ComputeExtreme in OCaml, as part of the
“Tropical polyhedral library” (TPLib), http://penjili.org/tplib.html. Table 1
reports some experiments for different classes of tropical cones: samples
formed by several cones chosen randomly (referred to as rnd$x$ where $x$ is
the size of the sample), and signed cyclic cones which are known to have a
very large number of extreme elements [AGK09]. The successive columns
respectively report the dimension $d$, the number of constraints $n$, the size
of the final set of extreme rays, the mean size of the intermediary sets, and
the execution time $T$ (for samples of “random” cones, we give average
results).
The result provided by ComputeExtreme does not depend on the order of the
inequalities in the initial system. This order may impact the size of the
intermediary sets and subsequently the execution time. In our experiments,
inequalities are dynamically ordered during the execution: at each step of the
induction, the inequality
$\boldsymbol{a}\boldsymbol{x}\leq\boldsymbol{b}\boldsymbol{x}$ is chosen so as
to minimize the number of combinations
$(\boldsymbol{a}\boldsymbol{g}^{j})\boldsymbol{g}^{i}\oplus(\boldsymbol{b}\boldsymbol{g}^{i})\boldsymbol{g}^{j}$.
This strategy reports better results than without ordering.
We compare our algorithm with a variant using the alternative extremality
criterion which is discussed in Sect. 4 and used in the other existing
implementations [CGMQ, AGG08]. Its execution time $T^{\prime}$ is given in the
seventh column. The ratio $T/T^{\prime}$ shows that our algorithm brings a
huge breakthrough in terms of execution time. When the number of extreme rays
is of order of $10^{4}$, the second algorithm needs several days to terminate.
Therefore, the comparison could not be made in practice for some cases.
Table 1 also reports some benchmarks from applications to static analysis. The
experiments oddeven$i$ correspond to the static analysis of the odd-even
sorting algorithm of $i$ elements. It is a sort of worst case for our
analysis. The number of variables and lines in each program is given in the
first columns. The analyzer automatically shows that the sorting program
returns an array in which the last (resp. first) element is the maximum
(minimum) of the array given as input. It clearly benefits from the
improvements of ComputeExtreme, as shown by the ratio with the execution time
$T^{\prime}$ of the previous implementation of the static analyzer [AGG08].
## References
* [AGG08] X. Allamigeon, S. Gaubert, and É. Goubault. Inferring min and max invariants using max-plus polyhedra. In SAS’08, volume 5079 of LNCS, pages 189–204. Springer, Valencia, Spain, 2008.
* [AGG09a] M. Akian, S. Gaubert, and A. Guterman. Linear independence over tropical semirings and beyond. In G.L. Litvinov and S.N. Sergeev, editors, Proc. of the International Conference on Tropical and Idempotent Mathematics, volume 495 of Contemp. Math., pages 1–38. AMS, 2009.
* [AGG09b] M. Akian, S. Gaubert, and A. Guterman. Tropical polyhedra are equivalent to mean payoff games. arXiv:0912.2462, 2009.
* [AGG09c] X. Allamigeon, S. Gaubert, and E. Goubault. Computing the extreme points of tropical polyhedra. Eprint arXiv:math/0904.3436v2, 2009.
* [AGK09] X. Allamigeon, S. Gaubert, and R. D. Katz. The number of extreme points of tropical polyhedra. Eprint arXiv:math/0906.3492, accepted for publication in JCTA, 2009.
* [BH84] P. Butkovič and G. Hegedüs. An elimination method for finding all solutions of the system of linear equations over an extremal algebra. Ekonomicko-matematicky Obzor, 20, 1984.
* [BH04] W. Briec and C. Horvath. $\mathbb{B}$-convexity. Optimization, 53:103–127, 2004.
* [BSS07] P. Butkovič, H. Schneider, and S. Sergeev. Generators, extremals and bases of max cones. Linear Algebra Appl., 421(2-3):394–406, 2007.
* [CC77] P. Cousot and R. Cousot. Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In Conference Record of the Fourth Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 238–252, Los Angeles, California, 1977. ACM Press, New York, NY.
* [CG79] R. A. Cuninghame-Green. Minimax algebra, volume 166 of Lecture Notes in Economics and Mathematical Systems. Springer, 1979.
* [CGMQ] G. Cohen, S. Gaubert, M. McGettrick, and J.-P. Quadrat. Maxplus toolbox of scilab. Available at http://minimal.inria.fr/gaubert/maxplustoolbox/; now integrated in ScicosLab. http://www.scicoslab.org.
* [CGQ99] G. Cohen, S. Gaubert, and J.P. Quadrat. Max-plus algebra and system theory: where we are and where to go now. Annual Reviews in Control, 23:207–219, 1999.
* [CGQ04] G. Cohen, S. Gaubert, and J. P. Quadrat. Duality and separation theorem in idempotent semimodules. Linear Algebra and Appl., 379:395–422, 2004.
* [CH78] P. Cousot and N. Halbwachs. Automatic discovery of linear restraints among variables of a program. In Conference Record of the Fifth Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 84–97, Tucson, Arizona, 1978. ACM Press.
* [DLGKL09] M. Di Loreto, S. Gaubert, R. D. Katz, and J.-J. Loiseau. Duality between invariant spaces for max-plus linear discrete event systems. Eprint arXiv:0901.2915., 2009.
* [DS04] M. Develin and B. Sturmfels. Tropical convexity. Doc. Math., 9:1–27 (electronic), 2004.
* [DY07] M. Develin and J. Yu. Tropical polytopes and cellular resolutions. Experimental Mathematics, 16(3):277–292, 2007.
* [FP96] K. Fukuda and A. Prodon. Double description method revisited. In Selected papers from the 8th Franco-Japanese and 4th Franco-Chinese Conference on Combinatorics and Computer Science, pages 91–111, London, UK, 1996. Springer.
* [Gau92] S. Gaubert. Théorie des systèmes linéaires dans les dioïdes. Thèse, École des Mines de Paris, July 1992.
* [GJ] E. Gawrilow and M. Joswig. Polymake. http://www.math.tu-berlin.de/polymake/.
* [GK06] S. Gaubert and R. Katz. Max-plus convex geometry. In R. A. Schmidt, editor, RelMiCS/AKA 2006, volume 4136 of Lecture Notes in Comput. Sci., pages 192–206. Springer, 2006.
* [GK07] S. Gaubert and R. Katz. The Minkowski theorem for max-plus convex sets. Linear Algebra and Appl., 421:356–369, 2007.
* [GK09] S. Gaubert and R. Katz. Minimal half-spaces and external representation of tropical polyhedra. Eprint arXiv:math/0908.1586, 2009.
* [GLPN93] G. Gallo, G. Longo, S. Pallottino, and S. Nguyen. Directed hypergraphs and applications. Discrete Appl. Math., 42(2-3):177–201, 1993.
* [Jos05] M. Joswig. Tropical halfspaces. In Combinatorial and computational geometry, volume 52 of Math. Sci. Res. Inst. Publ., pages 409–431. Cambridge Univ. Press, Cambridge, 2005.
* [Jos09] M. Joswig. Tropical convex hull computations. In G.L. Litvinov and S.N. Sergeev, editors, Proc. of the International Conference on Tropical and Idempotent Mathematics, volume 495 of Contemp. Math. AMS, 2009.
* [JSY07] M. Joswig, B. Sturmfels, and J. Yu. Affine buildings and tropical convexity. Albanian J. Math., 1(4):187–211, 2007.
* [Kat07] R. D. Katz. Max-plus $(A,B)$-invariant spaces and control of timed discrete event systems. IEEE Trans. Aut. Control, 52(2):229–241, 2007.
* [LMS01] G.L. Litvinov, V.P. Maslov, and G.B. Shpiz. Idempotent functional analysis: an algebraic approach. Math. Notes, 69(5):696–729, 2001.
* [McM70] P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179–184, 1970.
* [Min01] A. Miné. The octagon abstract domain. In AST 2001 in WCRE 2001, IEEE, pages 310–319. IEEE CS Press, October 2001. http://www.di.ens.fr/~mine/publi/article-mine-ast01.pdf.
* [MS92] V. Maslov and S. Samborskiĭ, editors. Idempotent analysis, volume 13 of Adv. in Sov. Math. AMS, RI, 1992.
* [Sin97] I. Singer. Abstract convex analysis. Wiley, 1997.
* [Vor67] N.N. Vorobyev. Extremal algebra of positive matrices. Elektron. Informationsverarbeitung und Kybernetik, 3:39–71, 1967\. in Russian.
* [Zie98] G. M. Ziegler. Lectures on Polytopes. Springer, 1998.
* [Zim77] K. Zimmermann. A general separation theorem in extremal algebras. Ekonom.-Mat. Obzor, 13(2):179–201, 1977.
|
arxiv-papers
| 2010-01-23T02:01:06 |
2024-09-04T02:49:07.959776
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xavier Allamigeon, Stephane Gaubert, Eric Goubault",
"submitter": "Stephane Gaubert",
"url": "https://arxiv.org/abs/1001.4119"
}
|
1001.4126
|
# Bihamiltonian Structure of the Two-component Kadomtsev-Petviashvili
Hierarchy of type B
Chao-Zhong Wu, Dingdian Xu
Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, P. R. China
wucz05@mails.tsinghua.edu.cnxudd06@mails.tsinghua.edu.cn
###### Abstract
We employ a Lax pair representation of the two-component BKP hierarchy and
construct its bihamiltonian structure with $R$-matrix techniques.
Key words: BKP hierarchy, Hamiltonian structure, $R$-matrix
## 1 Introduction
The Kadomtsev-Petviashvili (KP) hierarchy of type B (BKP for short) was
introduced in [6, 7], and generalized to multi-component cases by Date, Jimbo,
Kashiwara, Miwa [4] in the form of bilinear equations. Among these multi-
component integrable systems, the two-component BKP hierarchy is of special
interest.
In the original definition of the two-component BKP hierarchy, the solution
space of tau functions can be regarded as the vacuum orbit in the two-
component neutral free fermionic Fock representation of the infinite
dimensional Lie algebra $D_{\infty}$ [5, 14], which corresponds to the
infinite Dynkin diagram of type D [15]. The Lie algebra $D_{\infty}$ can be
reduced to the affine Lie algebra $D_{n}^{(1)}$ under the so-called
$(2n-2,2)$-reduction in [5], see also [14, 17]. This reduction reduces the
two-component BKP hierarchy to a hierarchy that is equivalent with the Kac-
Wakimoto hierarchy corresponding to the principal vertex operator realization
of the basic representation of $D^{(1)}_{n}$, the Drinfeld-Sokolov hierarchy
associated to the Lie algebra $D^{(1)}_{n}$ and the zeroth vertex $c_{0}$ of
its Dynkin diagram, as well as the Givental-Milanov hierarchy satisfied by the
total descendant for the $D_{n}$ singularity, see [9, 12, 13, 16, 19, 26] and
references therein. Such a reduction is analogous to the one that reduces the
KP hierarchy to the $n$th Gelfand-Dickey hierarchy (see e.g. [8]) that
corresponds to the reduction of Lie algebras: $A_{\infty}\mapsto A^{(1)}_{n}$.
So in this sense to compare the two-component BKP hierarchy with the KP
hierarchy would deepen our understanding of integrable hierarchies and
relevant theories, such as Jacobi/Prym varieties in algebraic geometry and
Landau-Ginzburg Models of topological strings, see e.g. [22, 23, 24].
In this article our aim is to study the two-component BKP hierarchy from the
view point of Hamiltonian structures. To our best knowledge, this topic has
not been considered in the literature, possibly for the reason that the KP-
analogue Lax pair representation of the two-component BKP hierarchy was
unknown. Recall that the two-component BKP hierarchy was defined to be the
bilinear equation of a single tau function:
$\displaystyle\mathrm{res}_{z}z^{-1}X(\mathbf{t};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\mathbf{t}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime})$
$\displaystyle=\mathrm{res}_{z}z^{-1}X(\hat{\mathbf{t}};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\hat{\mathbf{t}}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime}),$
(1.1)
where $\mathbf{t}=(t_{1},t_{3},t_{5},\cdots)$,
$\hat{\mathbf{t}}=(\hat{t}_{1},\hat{t}_{3},\hat{t}_{5},\cdots)$, and $X$ is a
vertex operator given by
$X(\mathbf{t};z)=\exp\left(\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}t_{k}z^{k}\right)\,\exp\left(-\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}\frac{2}{k\,z^{k}}\frac{\partial}{\partial
t_{k}}\right).$
Here the residue of a Laurent series is taken as
$\mathrm{res}_{z}(\sum_{i\in\mathbb{Z}}f_{i}z^{i})=f_{-1}$. In [22] Shiota
proposed a scalar Lax representation of the hierarchy (1), though this did not
attract much attention as it contains pseudo-differential operators with
derivations of two spatial variables. Recently, a Lax pair representation of
the two-component BKP hierarchy was found by Liu, Zhang and one of the authors
[19]. It was shown that the hierarchy (1) can be redefined by certain
extension of the following Lax equations (see Section 3 below):
$\displaystyle\frac{\partial P}{\partial
t_{k}}=[(P^{k})_{+},P],\quad\frac{\partial\hat{P}}{\partial
t_{k}}=[(P^{k})_{+},\hat{P}],$ (1.2) $\displaystyle\frac{\partial
P}{\partial\hat{t}_{k}}=[-(\hat{P}^{k})_{-},P],\quad\frac{\partial\hat{P}}{\partial\hat{t}_{k}}=[-(\hat{P}^{k})_{-},\hat{P}]$
(1.3)
with $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$, where
$P=D+\sum_{i\geq 1}u_{i}D^{-i},\quad\hat{P}=D^{-1}\hat{u}_{-1}+\sum_{i\geq
1}\hat{u}_{i}D^{i}\hbox{ with }D=\frac{\mathrm{d}}{\mathrm{d}x}$
are pseudo-differential operators such that $P^{*}=-DPD^{-1}$,
$\hat{P}^{*}=-D\hat{P}D^{-1}$. Note that the first equation in (1.2) is just
the Lax formulation of the BKP hierarchy appearing in [6]. Our arguments will
be based on the Lax pair representation (1.2), (1.3) of the two-component BKP
hierarchy.
Observe that the expression (1.2), (1.3) is similar to the Lax pair
representation of the two-dimensional Toda hierarchy [25], which carries a
tri-Hamiltonian structure [1]. Following the idea of [1], we want to use the
$R$-matrix theory to construct Hamiltonian structures of the two-component BKP
hierarchy (1.2), (1.3).
We are also motivated by the recent work [2], in which Carlet, Dubrovin and
Mertens constructed an infinite-dimensional Frobenius manifold underlying the
two-dimensional Toda hierarchy. Due to the similarity of the Lax
representations mentioned above, we expect that there also exists an infinite
dimensional Frobenius manifold that underlies the two-component BKP hierarchy.
A hint is that the potential $F$ (in the notion of [23], namely the
dispersionless limit of the logarithm of the tau function, see Section 3
below) of the dispersionless two-component BKP hierarchy was discovered to
satisfy certain infinite-dimensional WDVV-type associativity equation [3].
While in the finite-dimensional case, the concept of Frobenius manifolds [10]
is known as a geometric description of the WDVV equations, and associated to
certain nondegenerate Frobenius manifold there lies a Poisson pencil so that a
bihamiltonian hierarchy can be constructed [11]. We hope that this article and
follow-up work might help to understand the theory of infinite-dimensional
manifolds.
This article is arranged as follows. In next section we recall the definition
and some properties of pseudo-differential operators introduced in [19], and
in Section 3 we recall the Lax pair representation of the two-component BKP
hierarchy. In Sections 4 and 5, an $R$-matrix will be used to construct
Poisson brackets on an algebra of pseudo-differential operators, and then
after appropriate reductions of the Poisson brackets we obtain a bihamiltonian
structure of the two-component BKP hierarchy. In Section 6 we compute the
dispersionless limit of this bihamiltonian structure. Finally some remarks are
given in Section 7.
## 2 Pseudo-differential operators
For preparation we recall the notion of pseudo-differential operators over a
ring with certain gradation as introduced in [19].
Let $\mathcal{A}$ be a ring, and $D:\mathcal{A}\to\mathcal{A}$ be a
derivation. The algebra of usual pseudo-differential operators is
$\mathcal{D}^{-}=\left\\{\sum_{i<\infty}f_{i}D^{i}\mid
f_{i}\in\mathcal{A}\right\\}.$ (2.1)
This algebra is topologically complete with a topological basis given by the
following filtration:
$\cdots\subset\mathcal{D}^{-}_{(d-1)}\subset\mathcal{D}^{-}_{(d)}\subset\mathcal{D}^{-}_{(d+1)}\subset\cdots,\quad\mathcal{D}^{-}_{(d)}=\left\\{\sum_{i\leq
d}f_{i}D^{i}\mid f_{i}\in\mathcal{A}\right\\},$
and in this algebra two elements are multiplied as series of the following
product of monomials:
$fD^{i}\cdot gD^{j}=\sum_{r\geq 0}\binom{i}{r}f\,D^{r}(g)\,D^{i+j-r},\quad
f,g\in\mathcal{A}.$ (2.2)
Assume there is a gradation on $\mathcal{A}$ such that
$\mathcal{A}=\prod_{i\geq 0}\mathcal{A}_{i},\quad
D:\mathcal{A}_{i}\to\mathcal{A}_{i+1},\quad\mathcal{A}_{i}\cdot\mathcal{A}_{j}\subset\mathcal{A}_{i+j},$
and consider the linear space
$\mathcal{D}=\left\\{\sum_{i\in\mathbb{Z}}f_{i}D^{i}\mid
f_{i}\in\mathcal{A}\right\\}.$
Obviously $\mathcal{D}^{-}\subset\mathcal{D}$.
For any $k\in\mathbb{Z}$, denote by $\mathcal{D}_{k}$ the set of homogeneous
operators with degree $k$ in $\mathcal{D}^{-}$, i.e.,
$\mathcal{D}_{k}=\left\\{\sum_{i\leq k}f_{i}D^{i}\mid
f_{i}\in\mathcal{A}_{k-i}\right\\}.$
Let $\mathcal{D}^{+}$ be a subspace of $\mathcal{D}$ that reads
$\mathcal{D}^{+}=\bigcup_{d\in\mathbb{Z}}\mathcal{D}^{+}_{(d)},\quad\mathcal{D}^{+}_{(d)}=\prod_{k\geq
d}\mathcal{D}_{k},$ (2.3)
and $\mathcal{D}^{+}$ have a topological basis given by the filtration
$\cdots\supset\mathcal{D}^{+}_{(d-1)}\supset\mathcal{D}^{+}_{(d)}\supset\mathcal{D}^{+}_{(d+1)}\supset\cdots.$
In fact, every element $A\in\mathcal{D}^{+}$ has the following normal
expansion [19]
$A=\sum_{i\in\mathbb{Z}}\left(\sum_{j\geq\max\\{0,m-i\\}}a_{i,j}\right)D^{i},\quad
a_{i,j}\in\mathcal{A}_{j}$
with some integer $m$. Note that
$\mathcal{D}_{k}\cdot\mathcal{D}_{l}\subset\mathcal{D}_{k+l}$ according to the
multiplication defined by (2.2), then this multiplication can be naturally
extended to $\mathcal{D}^{+}$ such that $\mathcal{D}^{+}$ becomes an
associative algebra.
###### Definition 2.1 ([19])
Elements of $\mathcal{D}^{-}$ (resp. $\mathcal{D}^{+}$) are called pseudo-
differential operators of the first type (resp. the second type) over
$\mathcal{A}$. The intersection of $\mathcal{D}^{-}$ and $\mathcal{D}^{+}$ in
$\mathcal{D}$ is denoted by
$\mathcal{D}^{b}=\mathcal{D}^{-}\cap\mathcal{D}^{+},$
and its elements are called bounded pseudo-differential operators.
Sometimes to indicate the ring $\mathcal{A}$ and the derivation $D$, we will
use the notations $\mathcal{D}^{\pm}(\mathcal{A},D)$ instead of
$\mathcal{D}^{\pm}$.
Pseudo-differential operators of the second type have similar properties to
those of the operators in $\mathcal{D}^{-}$. For any operator
$A=\sum_{i\in\mathbb{Z}}f_{i}D^{i}\in\mathcal{D}^{\pm},$ (2.4)
its positive part, negative part, residue and adjoint operator are defined to
be respectively
$\displaystyle A_{+}=\sum_{i\geq 0}f_{i}D^{i},\quad
A_{-}=\sum_{i<0}f_{i}D^{i},$ (2.5) $\displaystyle\mathrm{res}\,A=f_{-1},\quad
A^{*}=\sum_{i\in\mathbb{Z}}(-D)^{i}\cdot f_{i}.$ (2.6)
Note that the formulae (2.5) give two projections of $\mathcal{D}$, and they
induce the following decompositions of spaces
$\mathcal{D}^{\pm}=(\mathcal{D}^{\pm})_{+}\oplus(\mathcal{D}^{\pm})_{-}.$
(2.7)
Particularly one sees that
$(\mathcal{D}^{-})_{+}\subset\mathcal{D}^{b},\quad(\mathcal{D}^{+})_{-}\subset\mathcal{D}^{b}.$
(2.8)
An element $A$ of $(\mathcal{D}^{\pm})_{+}$ is called a _differential
operator_. Let $A(f)$ denote the action of a differential operator $A$ on
$f\in\mathcal{A}$.
Elements of the quotient space
$\mathcal{F}=\mathcal{A}/(D(\mathcal{A})\oplus\mathbb{C})$ are called _local
functionals_ , which are denoted as
$\int f\,\mathrm{d}x=f+D(\mathcal{A}),\quad f\in\mathcal{A}.$
Introduce a map
$\langle\,\,\rangle:\ \mathcal{D}\to\mathcal{F},\quad A\mapsto\langle
A\rangle=\int\mathrm{res}A\,\mathrm{d}x.$ (2.9)
Then the pairing
$\langle A,B\rangle=\langle AB\rangle$ (2.10)
defines an inner product on each of $\mathcal{D}^{\pm}$.
Given any subspace $\mathcal{S}\subset\mathcal{D}^{\pm}$, we denote by
$\mathcal{S}^{*}$ the dual space of $\mathcal{S}$ (c.f. the notation of
adjoint operators). Via the above inner product, we have the following
identification of dual spaces
$(\mathcal{D}^{\pm})^{*}=\mathcal{D}^{\pm}.$ (2.11)
Consider the decompositions (2.7), it is easy to see that
$\big{(}(\mathcal{D}^{\pm})_{\pm}\big{)}^{*}=(\mathcal{D}^{\pm})_{\mp}.$
We also decompose $\mathcal{D}^{\pm}$ as
$\mathcal{D}^{\pm}=\mathcal{D}^{\pm}_{0}\oplus\mathcal{D}^{\pm}_{1},$ (2.12)
where
$\mathcal{D}^{\pm}_{\nu}=\left\\{A\in\mathcal{D}^{\pm}\mid
A^{*}=(-1)^{\nu}A\right\\},\quad\nu=0,1.$
Since $\langle A\rangle=-\langle A^{*}\rangle$ for any
$A\in\mathcal{D}^{\pm}$, then the dual subspaces of $\mathcal{D}^{\pm}_{\nu}$
read
$(\mathcal{D}^{\pm}_{\nu})^{*}=\mathcal{D}^{\pm}_{1-\nu},\quad\nu=0,1.$ (2.13)
For more details on properties of pseudo-differential operators one can refer
to [8, 19].
## 3 The two-component BKP hierarchy
The two types of pseudo-differential operators serve in [19] to give a scalar
Lax pair representation of the two-component BKP hierarchy, which is reviewed
as follows.
Let $\tilde{M}$ be an infinite-dimensional manifold with local coordinates
$(a_{1},a_{3},a_{5},\dots,b_{1},b_{3},b_{5},\dots),$
and $\tilde{\mathcal{A}}$ be the algebra of differential polynomials on
$\tilde{M}$:
$\tilde{\mathcal{A}}=C^{\infty}(\tilde{M})[[a_{k}^{(s)},b_{k}^{(s)}\mid
k\in\mathbb{Z^{\mathrm{odd}}_{+}},s\geq 1]].$
We assign a gradation on $\tilde{\mathcal{A}}$ by
$\deg f=0\hbox{ for }f\in C^{\infty}(\tilde{M}),\quad\deg a_{k}^{(s)}=\deg
b_{k}^{(s)}=s$
which make $\tilde{\mathcal{A}}$ a topologically complete algebra:
$\tilde{\mathcal{A}}=\prod_{i\geq
0}\tilde{\mathcal{A}}_{i},\quad\tilde{\mathcal{A}}_{i}\cdot\tilde{\mathcal{A}}_{j}\subset\tilde{\mathcal{A}}_{i+j}.$
Note that this gradation is induced from the derivation
$D:\tilde{\mathcal{A}}\to\tilde{\mathcal{A}},\quad D=\sum_{s\geq
0}\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}\left(a_{k}^{(s+1)}\frac{\partial}{\partial
a_{k}^{(s)}}+b_{k}^{(s+1)}\frac{\partial}{\partial b_{k}^{(s)}}\right)$
with $a_{k}^{(0)}=a_{k}$, $b_{k}^{(0)}=b_{k}$. So one can define the algebras
$\tilde{\mathcal{D}}^{\pm}=\mathcal{D}^{\pm}(\tilde{\mathcal{A}},D)$ of
pseudo-differential operators as was done in last section.
Introduce two operators
$\displaystyle\Phi=1+\sum_{i\geq
1}a_{i}D^{-i}\in\tilde{\mathcal{D}}^{-},\quad\Psi=1+\sum_{i\geq
1}b_{i}D^{i}\in\tilde{\mathcal{D}}^{+},$ (3.1)
where $a_{2},a_{4},a_{6},\dots,b_{2},b_{4},b_{6},\dots\in\tilde{\mathcal{A}}$
are determined by the following conditions
$\Phi^{*}=D\Phi^{-1}D^{-1},\quad\Psi^{*}=D\Psi^{-1}D^{-1}.$ (3.2)
Then the two-component BKP hierarchy (1) can be redefined to be
$\displaystyle\frac{\partial\Phi}{\partial
t_{k}}=-(P^{k})_{-}\Phi,\quad\frac{\partial\Psi}{\partial
t_{k}}=\bigl{(}(P^{k})_{+}-\delta_{k1}\hat{P}^{-1}\bigr{)}\Psi,$ (3.3)
$\displaystyle\frac{\partial\Phi}{\partial\hat{t}_{k}}=-(\hat{P}^{k})_{-}\Phi,\quad\frac{\partial\Psi}{\partial\hat{t}_{k}}=(\hat{P}^{k})_{+}\Psi,$
(3.4)
where $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$, and the operators $P$, $\hat{P}$
read
$P=\Phi D\Phi^{-1}\in\tilde{\mathcal{D}}^{-},\quad\hat{P}=\Psi
D^{-1}\Psi^{-1}\in\tilde{\mathcal{D}}^{+}.$ (3.5)
The operators $P$, $\hat{P}$ have the following expressions:
$P=D+\sum_{i\geq 1}u_{i}D^{-i},\quad\hat{P}=D^{-1}\hat{u}_{-1}+\sum_{i\geq
1}\hat{u}_{i}D^{i},$ (3.6)
with $\hat{u}_{-1}=(\Psi^{-1})^{*}(1)$, and they satisfy
$P^{*}=-DPD^{-1},\quad\hat{P}^{*}=-D\hat{P}D^{-1},$ (3.7)
which implies
$(P^{k})_{+}(1)=0,\quad(\hat{P}^{k})_{+}(1)=0,\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (3.8)
Observe that the coefficients of $P$ and $\hat{P}$ are elements of the algebra
$\tilde{\mathcal{A}}$, and among these coefficients the ones with odd
subscript are independent, while the others are determined by the conditions
(3.7). Assume that
$\mathbf{u}=(u_{1},u_{3},\dots,\hat{u}_{-1},\hat{u}_{1},\hat{u}_{3},\dots)$
(3.9)
serves as a coordinate of some infinite-dimensional manifold $M$, then the
algebra $\mathcal{A}$ of differential polynomials on $M$ reads
$\mathcal{A}=C^{\infty}(M)[[\mathbf{u}^{(s)}\mid s\geq 1]],$
which is a subalgebra of $\tilde{\mathcal{A}}$. Similarly as above, one can
assign a gradation to $\mathcal{A}$ that is induced from the derivation
$D:\mathcal{A}\to\mathcal{A},\quad D=\sum_{s\geq
0}\mathbf{u}^{(s+1)}\cdot\frac{\partial}{\partial\mathbf{u}^{(s)}}$
with $\mathbf{u}^{(0)}=\mathbf{u}$, and then define the algebras
$\mathcal{D}^{\pm}=\mathcal{D}^{\pm}(\mathcal{A},D)$ of pseudo-differential
operators over $\mathcal{A}$.
Clearly $P\in\mathcal{D}^{-}$, $\hat{P}\in\mathcal{D}^{+}$. When the two-
component BKP hierarchy (3.3), (3.4) is restricted from $\tilde{\mathcal{A}}$
to $\mathcal{A}$, it becomes
$\displaystyle\frac{\partial P}{\partial
t_{k}}=[(P^{k})_{+},P],\quad\frac{\partial\hat{P}}{\partial
t_{k}}=[(P^{k})_{+},\hat{P}],$ (3.10) $\displaystyle\frac{\partial
P}{\partial\hat{t}_{k}}=[-(\hat{P}^{k})_{-},P],\quad\frac{\partial\hat{P}}{\partial\hat{t}_{k}}=[-(\hat{P}^{k})_{-},\hat{P}]$
(3.11)
with $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$. In the present article we regard the
two-component BKP hierarchy as the evolutionary equations (3.10), (3.11)
defined on the algebra $\mathcal{A}$.
In fact, the hierarchy (3.10), (3.11) possesses a tau function
$\tau=\tau(\mathbf{t},\hat{\mathbf{t}})$ defined by
$\displaystyle\omega=\mathrm{d}(2\,\partial_{x}\log\tau)~{}~{}\hbox{ with
}~{}~{}x=t_{1},$ (3.12)
where $\omega$ is the following closed $1$-form:
$\omega=\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}(\mathrm{res}\,P^{k}\,\mathrm{d}t_{k}+\mathrm{res}\,\hat{P}^{k}\,\mathrm{d}\hat{t}_{k}).$
This tau function solves the bilinear equation (1), which is the original
definition of the two-component BKP hierarchy.
###### Remark 3.1
The dispersionless limit of the flows (3.10), (3.11) first exists in [23],
where Takasiki also considered the dispersionless limit of the logarithm of
the tau function as given in (3.12). Inspired by [23], Chen and Tu [3]
discovered that the leading term of $\log\tau$ solves an infinite-dimensional
associativity equation of WDVV type.
## 4 $R$-matrix and pseudo-differential operators
To show that the two-component BKP hierarchy (3.10), (3.11) possesses a
bihamiltonian structure, we need to construct a Poisson pencil for it. The
method is to use the standard $R$-matrix theory and introduce Poisson brackets
on a Lie algebra (see [21, 18, 20] and references therein), then restrict the
Poisson brackets to certain submanifold of the Lie algebra. Our approach is
similar with that used by Carlet [1] for the two-dimensional Toda hierarchy.
We first recall the $R$-matrix formalism. Let $\mathfrak{g}$ be a Lie algebra,
and $R:\mathfrak{g}\to\mathfrak{g}$ be a linear transformation. Then $R$ is
called an $R$-matrix [21] on $\mathfrak{g}$ if it defines a Lie bracket by
$[X,Y]_{R}=[R(X),Y]+[X,R(Y)],\quad X,Y\in\mathfrak{g}.$ (4.1)
A sufficient condition for a transformation $R$ being an $R$-matrix is that
$R$ solves the modified Yang-Baxter equation [21]
$[R(X),R(Y)]-R([X,Y]_{R})=-[X,Y]$ (4.2)
for all $X$, $Y\in\mathfrak{g}$.
Assume that $\mathfrak{g}$ is an associative algebra, with the Lie bracket
defined naturally by commutators, and there is a map
$\langle~{}\rangle:\mathfrak{g}\to\mathbb{C}$ that defines a non-degenerate
symmetric invariant bilinear form (inner product) $\langle\,,\rangle$ by
$\langle X,Y\rangle=\langle XY\rangle=\langle YX\rangle,\quad
X,Y\in\mathfrak{g}.$
Via this inner product one can identify $\mathfrak{g}$ with its dual space
$\mathfrak{g}^{*}$. The tangent and the cotangent bundles of $\mathfrak{g}$
are denoted by $T\mathfrak{g}$ and $T^{*}\mathfrak{g}$ respectively, with
fibers $T_{A}\mathfrak{g}=\mathfrak{g}$ and
$T^{*}_{A}\mathfrak{g}=\mathfrak{g}^{*}$ at every point $A\in\mathfrak{g}$.
Let $R^{*}$ be the adjoint transformation of $R$ with respect to the above
inner product. We introduce the notations of the symmetric and the anti-
symmetric parts of $R$ respectively as
$R_{s}=\frac{1}{2}(R+R^{*}),\quad R_{a}=\frac{1}{2}(R-R^{*}).$
The $R$-matrix formalism is briefly stated as follows. Given an $R$-matrix
$R:\mathfrak{g}\to\mathfrak{g}$ that satisfies certain conditions, there
define three compatible Poisson brackets on $\mathfrak{g}$, say, the linear,
the quadratic and the cubic brackets in the notion of [18, 20].
In particular, let us recall the quadratic bracket, which will be used to
construct a Poisson pencil for the two-component BKP hierarchy.
###### Lemma 4.1 ([18, 20])
Let $f$, $g$ be two arbitrary smooth functions on $\mathfrak{g}$, and $\nabla
f,\nabla g\in T_{A}^{*}\mathfrak{g}$ be their gradients at any point
$A\in\mathfrak{g}$. Given a linear transformation
$R:\mathfrak{g}\to\mathfrak{g}$, if both $R$ and its anti-symmetric part
$R_{a}$ satisfy the modified Yang-Baxter equation (4.2), then the quadratic
bracket
$\\{f,g\\}(A)=\frac{1}{4}\big{(}\langle[A,\nabla f],R(A\nabla g+\nabla g\cdot
A)\rangle-\langle[A,\nabla g],R(A\nabla f+\nabla f\cdot A)\rangle\big{)}$
(4.3)
defines a Poisson bracket on $\mathfrak{g}$.
Note that the bracket (4.3) can be rewritten as
$\\{f,g\\}(A)=\langle\nabla f,\mathcal{P}_{A}(\nabla g)\rangle,$
where $\mathcal{P}:T^{*}\mathfrak{g}\to T\mathfrak{g}$ is a Poisson tensor
given by
$\displaystyle\mathcal{P}_{A}(\nabla g)=$
$\displaystyle-\frac{1}{4}[A,R(A\nabla g+\nabla g\cdot
A)]-\frac{1}{4}AR^{*}([A,\nabla g])-\frac{1}{4}R^{*}([A,\nabla g])A,$
namely,
$\displaystyle\mathcal{P}_{A}(\nabla g)=$
$\displaystyle-\frac{1}{2}A\big{(}R_{s}(A\nabla g)+R_{a}(\nabla g\cdot
A)\big{)}+\frac{1}{2}\big{(}R_{a}(A\nabla g)+R_{s}(\nabla g\cdot A)\big{)}A.$
(4.4)
Henceforth we take $\mathfrak{g}$ to be the algebra
$\mathfrak{D}=\mathcal{D}^{-}\times\mathcal{D}^{+},$
where $\mathcal{D}^{-}$ and $\mathcal{D}^{+}$ are the sets of pseudo-
differential operators of the first type and the second type over some
differential algebra $\mathcal{A}$ as defined in Section 2. In $\mathfrak{D}$
the elements read $\mathbf{X}=(X,\hat{X})$, and the operations are defined
diagonally as
$(X,\hat{X})+(Y,\hat{Y})=(X+Y,\hat{X}+\hat{Y}),\quad(X,\hat{X})(Y,\hat{Y})=(XY,\hat{X}\hat{Y}).$
So $\mathfrak{D}$ is indeed an associative algebra. Moreover, the algebra
$\mathfrak{D}$ is equipped with an inner product define by
$\langle(X,\hat{X}),(Y,\hat{Y})\rangle=\langle(X,\hat{X})(Y,\hat{Y})\rangle=\langle
X,Y\rangle+\langle\hat{X},\hat{Y}\rangle,$
see (2.9), (2.10). Via this inner product we have the identification of dual
spaces as above:
$\mathfrak{D}^{*}=(\mathcal{D}^{-})^{*}\times(\mathcal{D}^{+})^{*}=\mathcal{D}^{-}\times\mathcal{D}^{+}=\mathfrak{D}.$
Inspired by [1], we introduce a linear transformation of $\mathfrak{D}$ as
follows
$R:\mathfrak{D}\to\mathfrak{D},\quad(X,\hat{X})\mapsto(X_{+}-X_{-}+2\hat{X}_{-},\hat{X}_{-}-\hat{X}_{+}+2X_{+}).$
(4.5)
Since $R=\Pi-\tilde{\Pi}$, where
$\Pi(X,\hat{X})=(X_{+}+\hat{X}_{-},\hat{X}_{-}+X_{+}),\quad\tilde{\Pi}(X,\hat{X})=(X_{-}-\hat{X}_{-},\hat{X}_{+}-X_{+})$
are two projections of $\mathfrak{D}$ onto its subalgebras, more exactly,
$\displaystyle\Pi\mathfrak{D}=\\{(X,X)\mid
X\in\mathcal{D}^{b}\\},\quad\tilde{\Pi}\mathfrak{D}=(\mathcal{D}^{-})_{-}\times(\mathcal{D}^{+})_{+},$
$\displaystyle\Pi^{2}=\Pi,~{}~{}\tilde{\Pi}^{2}=\tilde{\Pi},~{}~{}\tilde{\Pi}\,\Pi=0=\Pi\,\tilde{\Pi},~{}~{}\Pi+\tilde{\Pi}=\mathrm{id},$
then transformation $R$ satisfies the modified Yang-Baxter equation (4.2).
Hence $R$ is an $R$-matrix on $\mathfrak{D}$.
On the other hand, with respect to the inner product on $\mathfrak{D}$ the
adjoint transformation of $R$ reads
$R^{*}:\mathfrak{D}\to\mathfrak{D},\quad(X,\hat{X})\mapsto(X_{-}-X_{+}+2\hat{X}_{-},\hat{X}_{+}-\hat{X}_{-}+2X_{+}).$
Then the symmetric and anti-symmetric parts of $R$ are given by
$R_{s}(X,\hat{X})=2(\hat{X}_{-},X_{+}),\quad
R_{a}(X,\hat{X})=(X_{+}-X_{-},\hat{X}_{-}-\hat{X}_{+}).$ (4.6)
Observe that $R_{a}$ can be expressed as the difference of two projections
onto subalgebras of $\mathfrak{D}$, hence $R_{a}$ also solves the Yang-Baxter
equation (4.2). Thus the $R$-matrix given in (4.5) fulfills the condition of
Lemma 4.1.
We regard $\mathfrak{D}$ as an infinite-dimensional manifold, whose coordinate
is given by the coefficients of the general expression of its elements
$\mathbf{A}=\left(\sum_{i\in\mathbb{Z}}w_{i}D^{i},\,\sum_{i\in\mathbb{Z}}\hat{w}_{i}D^{i}\right)\in\mathfrak{D}.$
(4.7)
The set $\mathcal{F}$ of local functionals over the differential algebra
$\mathcal{A}$ (see Section 2) plays the role of $C^{\infty}(\mathfrak{g})$.
For any $F=\int f\,\mathrm{d}x\in\mathcal{F}$, the variational gradient of $F$
at $\mathbf{A}$ given in (4.7) is defined to be
$\frac{\delta
F}{\delta\mathbf{A}}=\left(\sum_{i\in\mathbb{Z}}D^{-i-1}\frac{\delta F}{\delta
w_{i}(x)},\,\sum_{i\in\mathbb{Z}}D^{-i-1}\frac{\delta
F}{\delta\hat{w}_{i}(x)}\right),$
where $\delta F/\delta w(x)=\sum_{j\geq 0}(-D)^{j}\left(\partial f/\partial
w^{(j)}\right)$. Note that $\delta F/\delta\mathbf{A}$ is not contained in
$\mathfrak{D}^{*}=\mathfrak{D}$ in general, so to go forward we need to do
some restriction.
It shall be indicated that, in this paper we only consider functionals with
variational gradients lying in $\mathfrak{D}$. Let $\mathcal{F}_{0}$ denote
the set of such functionals.
Now we can use Lemma 4.1 and the formulae (4.4), (4.6) to obtain the following
result.
###### Lemma 4.2
Let $F$ and $G$ be two arbitrary functionals in $\mathcal{F}_{0}$. On the
algebra $\mathfrak{D}$ there is a quadratic Poisson bracket
$\\{F,G\\}(\mathbf{A})=\left\langle\frac{\delta
F}{\delta\mathbf{A}},\mathcal{P}_{\mathbf{A}}\left(\frac{\delta
G}{\delta\mathbf{A}}\right)\right\rangle,\quad\mathbf{A}=(A,\hat{A})\in\mathfrak{D},$
(4.8)
where the Poisson tensor $\mathcal{P}:T\mathfrak{D}^{*}\to T\mathfrak{D}$ is
defined by
$\displaystyle\mathcal{P}_{(A,\hat{A})}(X,\hat{X})=$
$\displaystyle\big{(}A(XA)_{-}-(AX)_{-}A-A(\hat{A}\hat{X})_{-}+(\hat{X}\hat{A})_{-}A,$
$\displaystyle~{}~{}\hat{A}(\hat{X}\hat{A})_{+}-(\hat{A}\hat{X})_{+}\hat{A}-\hat{A}(AX)_{+}+(XA)_{+}\hat{A}\big{)}.$
(4.9)
Aiming at Hamiltonian structures of the two-component BKP hierarchy, we need
to reduce the Poisson structure (4.2) to an appropriate submanifold of
$\mathfrak{D}$. Recall the decompositions (2.12), let us decompose the space
$\mathfrak{D}$ as
$\mathfrak{D}=\mathfrak{D}_{0}\oplus\mathfrak{D}_{1},$ (4.10)
where $\mathfrak{D}_{\nu}=\mathcal{D}^{-}_{\nu}\times\mathcal{D}^{+}_{\nu}$
for $\nu=0,1$. Since the subspaces $\mathfrak{D}_{0}$ and $\mathfrak{D}_{1}$
are dual to each other with respect to the inner product on $\mathfrak{D}$,
then for any $\mathbf{A}\in\mathfrak{D}_{\nu}$ we have
$T_{\mathbf{A}}^{*}\mathfrak{D}_{\nu}=(\mathfrak{D}_{\nu})^{*}=\mathfrak{D}_{1-\nu}$
for $\nu=0,1$. It is straightforward to verify the following lemma.
###### Lemma 4.3
The Poisson structure (4.2) on $\mathfrak{D}$ can be properly restricted to
each of its subspaces $\mathfrak{D}_{0}$ and $\mathfrak{D}_{1}$.
## 5 Bihamiltonian representation of the two-component BKP hierarchy
In this section, we are to find a submanifold of $\mathfrak{D}$ where the
Poisson pencil for the two-component BKP hierarchy lies, then after a further
reduction of the Poisson structure constructed in last section we will express
the hierarchy (3.10), (3.11) to the form of Hamiltonian equations.
Recall the operators $P\in\mathcal{D}^{-}$, $\hat{P}\in\mathcal{D}^{+}$ given
in (3.5), we let
$\mathbf{A}=(P^{2}D^{-1},D\hat{P}^{2}).$ (5.1)
It is easy to see that $\mathbf{A}\in\mathfrak{D}_{1}$ (see (4.10)), and
$\mathbf{A}=(A,\hat{A})$ has the following expression:
$\displaystyle A=P^{2}D^{-1}=D+\sum_{i\geq
0}(v_{-i}D^{-2i-1}+f_{-i}D^{-2i-2}),$ (5.2)
$\displaystyle\hat{A}=D\hat{P}^{2}=\rho D^{-1}\rho+\sum_{i\geq
1}(\hat{v}_{i}D^{2i-1}+\hat{f}_{i}D^{2i-2}),\quad\rho=\hat{u}_{-1}.$ (5.3)
Denote $\mathbf{v}=(v_{0},v_{-1},\dots,\hat{v}_{0},\hat{v}_{1},\dots)$ with
$\hat{v}_{0}=\rho^{2}$. Then the coordinate $\mathbf{v}$ is related to
$\mathbf{u}$ given in (3.9) by a Miura-type transformation, while $f_{-i}$ and
$\hat{f}_{i}$ are linear functions of derivatives of $\mathbf{v}$ determined
by the symmetry property $(A^{*},\hat{A}^{*})=-(A,\hat{A})$. Hence the flows
of the hierarchy (3.10), (3.11) can be described in the coordinate
$\mathbf{v}$.
Given any local functional $F\in\mathcal{F}_{0}$ (remind the notation
$\mathcal{F}_{0}$ in last section), its variational gradient with respect to
$\mathbf{A}$, say $\delta F/\delta\mathbf{A}$, is defined to be
$\mathbf{X}=(X,\hat{X})\in\mathfrak{D}$ with
$\displaystyle X=$ $\displaystyle\frac{1}{2}\sum_{i\geq 0}\left(\frac{\delta
F}{\delta v_{-i}(x)}D^{2i}+D^{2i}\frac{\delta F}{\delta v_{-i}(x)}\right),$
(5.4) $\displaystyle\hat{X}=$ $\displaystyle\frac{1}{2}\sum_{i\geq
0}\left(\frac{\delta F}{\delta\hat{v}_{i}(x)}D^{-2i}+D^{-2i}\frac{\delta
F}{\delta\hat{v}_{i}(x)}\right).$ (5.5)
In a coordinate-free way, $\delta F/\delta\mathbf{A}=\mathbf{X}$ can be
defined by
$\displaystyle\delta
F=\langle\mathbf{X},\delta\mathbf{A}\rangle,\quad\mathbf{X}\in\mathfrak{D}_{0}.$
(5.6)
Note that in the latter definition, the variational gradient is determined up
to a kernel part $\mathbf{Z}=(Z,\hat{Z})\in\mathfrak{D}_{0}$ such that
$Z_{+}=0,\quad\hat{Z}_{-}=0,\quad\hat{Z}_{+}(\rho)=0.$ (5.7)
Let us consider the coset $(D,0)+\mathcal{U}$ consisting of operators of the
form (5.1), namely,
$\mathcal{U}=(\mathcal{D}^{-}_{1})_{-}\times\big{(}(\mathcal{D}^{+}_{1})_{+}\times\mathcal{M}\big{)},\quad\mathcal{M}=\\{\rho
D^{-1}\rho\mid\rho\in\mathcal{A}\\}.$ (5.8)
Here $\mathcal{M}$ is regarded as a $1$-dimensional manifold with coordinate
$\rho$, and this manifold has tangent spaces of the form
$T_{\rho}\mathcal{M}=\\{\rho D^{-1}f+fD^{-1}\rho\mid f\in\mathcal{A}\\}.$
So the tangent bundle, denoted by $T\mathcal{U}$, of the coset
$(D,0)+\mathcal{U}$ has fibers
$T_{\mathbf{A}}\mathcal{U}=(\mathcal{D}^{-}_{1})_{-}\times\big{(}(\mathcal{D}^{+}_{1})_{+}\times
T_{\rho}\mathcal{M}\big{)},\quad\mathbf{A}\in(D,0)+\mathcal{U},$ (5.9)
while the cotangent bundle $T^{*}\mathcal{U}$ of $(D,0)+\mathcal{U}$ is
composed of
$T_{\mathbf{A}}^{*}\mathcal{U}=(\mathcal{D}^{-}_{0})_{+}\times\big{(}(\mathcal{D}^{+}_{0})_{-}\times
T_{\rho}^{*}\mathcal{M}\big{)},\quad T^{*}_{\rho}\mathcal{M}=\mathcal{A}.$
(5.10)
From (5.4), (5.5) one sees that $\delta F/\delta\mathbf{A}\in
T_{\mathbf{A}}^{*}\mathcal{U}$ for any $F\in\mathcal{F}_{0}$.
Now we are ready to do the desired reduction of the Poisson structure.
###### Lemma 5.1
The map
$\mathcal{P}:T^{*}\mathcal{U}\to T\mathcal{U}$ (5.11)
defined by the formula (4.2) is a Poisson tensor on the coset
$(D,0)+\mathcal{U}$ that consists of operators of the form (5.1).
Proof. We only need to show that the map defined by (4.2) admits the
restriction to the coset $(D,0)+\mathcal{U}$, i.e., the following map is well
defined:
$\mathcal{P}_{\mathbf{A}}:T^{*}_{\mathbf{A}}\mathcal{U}\to
T_{\mathbf{A}}\mathcal{U},\quad\mathbf{A}\in(D,0)+\mathcal{U}.$ (5.12)
Assume $\mathbf{X}=(X,\hat{X})\in
T^{*}_{\mathbf{A}}\mathcal{U}\subset\mathfrak{D}_{0}$. It follows from Lemma
4.3 that $\mathcal{P}_{\mathbf{A}}(X)\in\mathfrak{D}_{1}$. More precisely, the
first component of $\mathcal{P}_{\mathbf{A}}(X)$ belongs to
$(\mathcal{D}^{-}_{1})_{-}$. On the other hand, for any
$\hat{Y}\in(\mathcal{D}^{+})_{+}$ we have
$\displaystyle(\hat{A}\hat{Y}+\hat{Y}^{*}\hat{A})_{-}=$ $\displaystyle(\rho
D^{-1}\rho\hat{Y}+\hat{Y}^{*}\rho D^{-1}\rho)_{-}$ $\displaystyle=$
$\displaystyle-(\hat{Y}^{*}\rho
D^{-1}\rho)^{*}_{-}+\hat{Y}^{*}(\rho)D^{-1}\rho$ $\displaystyle=$
$\displaystyle\rho D^{-1}\hat{Y}^{*}(\rho)+\hat{Y}^{*}(\rho)D^{-1}\rho\in
T_{\rho}\mathcal{M},$
then by taking $\hat{Y}=(\hat{X}\hat{A})_{+},(AX)_{+}$ it follows that the
second component of $\mathcal{P}_{\mathbf{A}}(\mathbf{X})$ lies in
$(\mathcal{D}^{+}_{1})_{+}\times T_{\rho}\mathcal{M}$. Thus
$\mathcal{P}_{\mathbf{A}}(\mathbf{X})\in T_{\mathbf{A}}\mathcal{U}$, i.e., the
map (5.12) is well defined. The lemma is proved. $\Box$
###### Remark 5.2
The proof of this lemma is the simplest case of the Dirac reduction procedure
for Poisson tensors, see e.g. [20]. In fact, one can express the manifolds
$\mathfrak{D}_{1}$ and $\mathfrak{D}_{1}^{*}$ as
$\displaystyle\mathfrak{D}_{1}=\mathcal{U}\times\mathcal{V}=T_{\mathbf{A}}\mathcal{U}\times\mathcal{V}_{\mathbf{A}},\quad\mathfrak{D}_{1}^{*}=\mathfrak{D}_{0}=T_{\mathbf{A}}^{*}\mathcal{U}\times\mathcal{V}^{*}_{\mathbf{A}},$
(5.13)
where
$\displaystyle\mathcal{V}=\mathcal{V}_{\mathbf{A}}=(\mathcal{D}^{-}_{1})_{+}\times\mathcal{N},\quad\mathcal{N}=\\{X\in(\mathcal{D}^{+}_{1})_{-}\mid\mathrm{res}X=0\\},$
$\displaystyle\mathcal{V}^{*}_{\mathbf{A}}=(\mathcal{D}^{-}_{0})_{-}\times(T_{\rho}^{*})^{\bot}\mathcal{M},\quad(T_{\rho}^{*})^{\bot}\mathcal{M}=\\{\hat{Y}\in(\mathcal{D}^{+}_{0})_{+}\mid\hat{Y}(\rho)=0\\}.$
Similar as the proof of Lemma 5.1, one can show that the map
$\mathcal{P}_{\mathbf{A}}=\left(\begin{array}[]{cc}\mathcal{P}_{\mathbf{A}}^{\mathcal{U}\mathcal{U}}&\mathcal{P}_{\mathbf{A}}^{\mathcal{U}\mathcal{V}}\\\
\mathcal{P}_{\mathbf{A}}^{\mathcal{V}\mathcal{U}}&\mathcal{P}_{\mathbf{A}}^{\mathcal{V}\mathcal{V}}\\\
\end{array}\right):T_{\mathbf{A}}^{*}\mathcal{U}\times\mathcal{V}^{*}_{\mathbf{A}}\to
T_{\mathbf{A}}\mathcal{U}\times\mathcal{V}_{\mathbf{A}}$
defined by (4.2) is diagonal. Hence from Lemma 4.3 it follows that the map
(4.2) gives a Poisson tensor on the coset
$(D,0)+\mathcal{U}\subset\mathcal{D}_{1}$.
###### Lemma 5.3
On the coset $(D,0)+\mathcal{U}$ there are two compatible Poisson tensors
defined by the following formulae:
$\displaystyle\mathcal{P}_{1}(X,\hat{X})=$
$\displaystyle\big{(}A(XD^{-1})_{-}+D^{-1}(XA)_{-}-(D^{-1}X)_{-}A-(AX)_{-}D^{-1}$
$\displaystyle~{}~{}-A(D\hat{X})_{-}-D^{-1}(\hat{A}\hat{X})_{-}+(\hat{X}D)_{-}A+(\hat{X}\hat{A})_{-}D^{-1},$
$\displaystyle~{}~{}\hat{A}(\hat{X}D)_{+}+D(\hat{X}\hat{A})_{+}-(D\hat{X})_{+}\hat{A}-(\hat{A}\hat{X})_{+}D$
$\displaystyle~{}~{}-\hat{A}(D^{-1}X)_{+}-D(AX)_{+}+(XD^{-1})_{+}\hat{A}+(XA)_{+}D\big{)},$
(5.14) $\displaystyle\mathcal{P}_{2}(X,\hat{X})=$
$\displaystyle\big{(}A(XA)_{-}-(AX)_{-}A-A(\hat{A}\hat{X})_{-}+(\hat{X}\hat{A})_{-}A,$
$\displaystyle~{}~{}\hat{A}(\hat{X}\hat{A})_{+}-(\hat{A}\hat{X})_{+}\hat{A}-\hat{A}(AX)_{+}+(XA)_{+}\hat{A}\big{)}$
(5.15)
with $(X,\hat{X})\in T^{*}_{\mathbf{A}}\mathcal{U}$ at any point
$\mathbf{A}=(A,\hat{A})\in(D,0)+\mathcal{U}$.
Proof. Lemma 5.1 shows that $\mathcal{P}_{2}$ is a Poisson tensor on the coset
$(D,0)+\mathcal{U}$.
Introduce a shift transformation on $(D,0)+\mathcal{U}$ as
$\mathscr{S}:(A,\hat{A})\mapsto(A+sD^{-1},\hat{A}+sD)$
with $s$ being a parameter. Then the push-forward of the Poisson tensor
$\mathcal{P}_{2}$ reads
$(\mathscr{S}_{*}\mathcal{P}_{2})(X,\hat{X})=\mathcal{P}_{2}(X,\hat{X})+s\mathcal{P}_{1}(X,\hat{X})+s^{2}\mathcal{P}_{0}(X,\hat{X}),$
(5.16)
where
$\displaystyle\mathcal{P}_{0}(X,\hat{X})=$
$\displaystyle\big{(}D^{-1}(XD^{-1})_{-}-(D^{-1}X)_{-}D^{-1}-D^{-1}(D\hat{X})_{-}+(\hat{X}D)_{-}D^{-1},$
$\displaystyle~{}~{}D(\hat{X}D)_{+}-(D\hat{X})_{+}D-D(D^{-1}X)_{+}+(XD^{-1})_{+}D\big{)}.$
By virtue of the symmetry property $(X^{*},\hat{X}^{*})=(X,\hat{X})$ that
yields the formulae
$\displaystyle(XD^{-1})_{\pm}=X_{\pm}D^{-1}\mp X_{+}(1)D^{-1},$
$\displaystyle(D^{-1}X)_{\pm}=D^{-1}X_{\pm}\mp D^{-1}\cdot X_{+}(1),$
$\displaystyle(D\hat{X})_{\pm}=D\hat{X}_{\pm},\quad(\hat{X}D)_{\pm}=\hat{X}_{\pm}D,$
one can check $\mathcal{P}_{0}(X,\hat{X})=0$. Hence the expansion (5.16)
implies that $\mathcal{P}_{1}$ is a Poisson tensor that is compatible with
$\mathcal{P}_{2}$. The lemma is proved. $\Box$
Let $\\{\cdot,\cdot\\}_{1,2}$ denote the Poisson brackets given in (4.8) with
Poisson tensors being $\mathcal{P}_{1,2}$ respectively. We arrive at the main
result of this article.
###### Theorem 5.4
The two-component BKP hierarchy (3.10), (3.11) can be expressed in the
following bihamiltonian recursion form
$\displaystyle\frac{\partial F}{\partial
t_{k}}=\\{F,H_{k+2}\\}_{1}(\mathbf{A})=\\{F,H_{k}\\}_{2}(\mathbf{A}),$ (5.17)
$\displaystyle\frac{\partial
F}{\partial\hat{t}_{k}}=\\{F,\hat{H}_{k+2}\\}_{1}(\mathbf{A})=\\{F,\hat{H}_{k}\\}_{2}(\mathbf{A})$
(5.18)
with $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$, where $F\in\mathcal{F}_{0}$,
$\mathbf{A}=(P^{2}D^{-1},D\hat{P}^{2})$ as given in (5.1), and the
Hamiltonians are
$\displaystyle H_{k}=\frac{2}{k}\langle
P^{k}\rangle,~{}~{}\hat{H}_{k}=-\frac{2}{k}\langle\hat{P}^{k}\rangle,\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (5.19)
Proof. First let us compute the variational gradients of the Hamiltonian
functionals. Since
$\delta H_{k}=\langle P^{k-2},\delta P^{2}\rangle=\langle
DP^{k-2},\delta(P^{2}D^{-1})\rangle=\langle(DP^{k-2},0),\delta\mathbf{A}\rangle$
and similarly
$\delta\hat{H}_{k}=\langle(0,-\hat{P}^{k-2}D^{-1}),\delta\mathbf{A}\rangle,$
then up to kernel parts given in (5.7) we have the variational gradients of
the Hamiltonians:
$\frac{\delta
H_{k}}{\delta\mathbf{A}}=(DP^{k-2},0),\quad\frac{\delta\hat{H}_{k}}{\delta\mathbf{A}}=(0,-\hat{P}^{k-2}D^{-1})$
(5.20)
One can easily see that different choices of the kernel parts do not change
the definition of the Poisson tensors $\mathcal{P}_{1,2}$.
According to the flows (3.10), (3.11) one has
$\frac{\partial\mathbf{A}}{\partial
t_{k}}=\left([(P^{k})_{+},P^{2}]D^{-1},D[(P^{k})_{+},\hat{P}^{2}]\right).$
Note that
$\frac{\partial F}{\partial t_{k}}=\left\langle\frac{\delta
F}{\delta\mathbf{A}},\frac{\partial\mathbf{A}}{\partial t_{k}}\right\rangle,$
then to show (5.17) we only need to verify the equations
$\displaystyle\frac{\partial\mathbf{A}}{\partial
t_{k}}=\mathcal{P}_{1}\left(\frac{\delta
H_{k+2}}{\delta\mathbf{A}}\right)=\mathcal{P}_{2}\left(\frac{\delta
H_{k}}{\delta\mathbf{A}}\right).$ (5.21)
The verification is straightforward by substituting (5.20) into (5.3), (5.15)
with the help of the following formulae induced from (3.8):
$(DP^{k}D^{-1})_{\pm}=D(P^{k})_{\pm}D^{-1},\quad(D\hat{P}^{k}D^{-1})_{\pm}=D(\hat{P}^{k})_{\pm}D^{-1},\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$
The equations (5.18) can be checked similarly. The theorem is proved. $\Box$
This theorem implies that the tau function (3.12) of the two-component BKP
hierarchy is defined from the tau-symmetry of Hamiltonian densities [11] (up
to the signs of $\hat{H}_{k}$).
###### Remark 5.5
One can also construct Hamiltonian structures of the two-component BKP
hierarchy by reducing the linear and the cubic Poisson brackets induced from
the $R$-matrix mentioned in last section. However, from these brackets we have
not found bihamiltonian recursion relations like (5.17), (5.18).
## 6 Dispersionless limit of the bihamiltonian structure
Let us compute the leading term of the bihamiltonian structure in (5.17),
(5.18) of the two-component BKP hierarchy, which would make sense in studying
the corresponding Frobenius manifold if there be.
First we replace the pseudo-differential operators by Laurent series of
symbols. In the dispersionless case, the operator
$\mathbf{A}=(P^{2}D^{-1},D\hat{P}^{2})$ becomes
$\displaystyle(a(z),\hat{a}(z))=\left(z+\sum_{i\geq
0}v_{-i}z^{-2i-1},\sum_{i\geq 0}\hat{v}_{i}z^{2i-1}\right),$ (6.1)
and the coordinate-type local functionals $v_{-i}(y)$, $\hat{v}_{j}(y)$ have
variational gradients $(z^{2i}\delta(x-y),0)$, $(0,z^{-2j}\delta(x-y))$
respectively. Substituting these Laurent series into the Poisson brackets
defined by the formulae (4.8), (5.3), (5.15), we obtain the following result.
For the convenience of expression we set $v_{1}=1$, $v_{i}=0$ when $i\geq 2$,
and $\hat{v}_{j}=0$ when $j\leq-1$.
* i)
The first bracket: for $i,j\geq 0$,
$\displaystyle\\{v_{-i}(x),v_{-j}(y)\\}_{1}^{[0]}=(1-\delta_{i0}-\delta_{j0})\big{(}2(i+j-1)v_{-i-j+1}(x)\,\delta^{\prime}(x-y)$
$\displaystyle\qquad\qquad\qquad\qquad+(2j-1)v_{-i-j+1}^{\prime}(x)\,\delta(x-y)\big{)},$
(6.2)
$\displaystyle\\{\hat{v}_{i}(x),\hat{v}_{j}(y)\\}_{1}^{[0]}=-(1-\delta_{i0}-\delta_{j0})\big{(}2(i+j-1)\hat{v}_{i+j}(x)\,\delta^{\prime}(x-y)$
$\displaystyle\qquad\qquad\qquad\qquad+(2j-1)\hat{v}_{i+j}^{\prime}(x)\,\delta(x-y)\big{)},$
(6.3) $\displaystyle\\{v_{-i}(x),\hat{v}_{j}(y)\\}_{1}^{[0]}$
$\displaystyle\qquad=2(i-j)\big{(}(1-\delta_{j0})v_{j-i}(x)+(1-\delta_{i0})\hat{v}_{j-i+1}(x)\big{)}\delta^{\prime}(x-y)$
$\displaystyle\qquad\quad-(2j-1)\big{(}(1-\delta_{j0})v_{j-i}^{\prime}(x)+(1-\delta_{i0})\hat{v}_{j-i+1}^{\prime}(x)\big{)}\delta(x-y).$
(6.4)
* ii)
The second bracket: for $i,j\geq 0$,
$\displaystyle\\{v_{-i}(x),v_{-j}(y)\\}_{2}^{[0]}$
$\displaystyle\qquad=\sum_{r=-1}^{i-1}\Big{(}2(i+j-2r-1)v_{-r}(x)\,v_{-i-j+r+1}(x)\,\delta^{\prime}(x-y)$
$\displaystyle\qquad\quad+(2j-2r-1)v_{-r}(x)\,v_{-i-j+r+1}^{\prime}(x)\,\delta(x-y)$
$\displaystyle\qquad\quad+(2i-2r-1)v_{-r}^{\prime}(x)\,v_{-i-j+r+1}(x)\,\delta(x-y)\Big{)},$
(6.5)
$\displaystyle\\{\hat{v}_{i}(x),\hat{v}_{j}(y)\\}_{2}^{[0]}=-\sum_{r=0}^{i}\Big{(}2(i+j-2r+1)\hat{v}_{r}(x)\,\hat{v}_{i+j-r+1}(x)\,\delta^{\prime}(x-y)$
$\displaystyle\qquad\quad+(2j-2r+1)\hat{v}_{r}(x)\,\hat{v}_{i+j-r+1}^{\prime}(x)\,\delta(x-y)$
$\displaystyle\qquad\quad+(2i-2r+1)\hat{v}_{r}^{\prime}(x)\,\hat{v}_{i+j-r+1}(x)\,\delta(x-y)\Big{)},$
(6.6) $\displaystyle\\{v_{-i}(x),\hat{v}_{j}(y)\\}_{2}^{[0]}$
$\displaystyle\qquad=\sum_{r=\max\\{-1,i-j-1\\}}^{i-1}\Big{(}2(i-j)v_{-r}(x)\,\hat{v}_{-i+j+r+1}(x)\,\delta^{\prime}(x-y)$
$\displaystyle\qquad\quad+(2r-2j+1)v_{-r}(x)\,\hat{v}_{-i+j+r+1}^{\prime}(x)\,\delta(x-y)$
$\displaystyle\qquad\quad+(2r-2i+1)v_{-r}^{\prime}(x)\,\hat{v}_{-i+j+r+1}(x)\,\delta(x-y)\Big{)}.$
(6.7)
## 7 Concluding remarks
Based on the Lax pair representation (3.10), (3.11) of the two-component BKP
hierarchy, we obtain a bihamiltonian structure of this hierarchy. Our method
in the construction of the Poisson brackets is to employ the standard
$R$-matrix formalism, which is analogous to that for the two-dimensional Toda
hierarchy [1]. In comparison with the two-dimensional Toda hierarchy, we
expect that there would be an infinite-dimensional Frobenius manifold
underlying the two-component BKP hierarchy.
As shown in [19], the two-component BKP hierarchy (3.10), (3.11) is reduced to
the Drinfeld-Sokolov hierarchy of type $(D_{n}^{(1)},c_{0})$ under the
constraint $P^{2n-2}=\hat{P}^{2}$. Whether such a constraint induces a
reduction of the bihamiltonian structure is unclear yet. We hope that
considering this example would help to understand the relations between
Frobenius manifolds of infinite and finite dimensions.
Acknowledgments. The authors thank Si-Qi Liu and Youjin Zhang for their
advice, and thank Yang Shi for her comment. They are also grateful to the
referee for the helpful suggestions. This work is partially supported by the
National Basic Research Program of China (973 Program) No.2007CB814800 and the
NSFC No.10801084.
## References
* [1] Carlet, G. The Hamiltonian structures of the two-dimensional Toda lattice and $R$-matrices. Lett. Math. Phys. 71 (2005), no. 3, 209–226.
* [2] Carlet, G.; Dubrovin, B.; Mertens, L.P. Infinite-dimensional Frobenius manifolds for 2+1 integrable systems, preprint arXiv: math-ph/0902.1245v1.
* [3] Chen, Y.T.; Tu, M.H. On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy. J. Math. Phys. 47 (2006), no. 10, 102702, 19 pp.
* [4] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type. Phys. D 4 (1981/82), no. 3, 343–365.
* [5] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy. Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1077–1110.
* [6] Date, E.; Kashiwara, M.; Jimbo, M.; Miwa, T. Transformation groups for soliton equations. Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981), 39–119, World Sci. Publishing, Singapore, 1983.
* [7] Date, E.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. II. Vertex operators and $\tau$ functions. Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no. 8, 387–392.
* [8] Dickey, L.A. Soliton equations and Hamiltonian systems. Second edition. Advanced Series in Mathematical Physics, 26. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
* [9] Drinfeld, V.G.; Sokolov, V.V. Lie algebras and equations of Korteweg-de Vries type. (Russian) Current problems in mathematics, Vol. 24, 81–180, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
* [10] Dubrovin, B. Geometry of $2$D topological field theories. Integrable systems and quantum groups (Montecatini Terme, 1993), 120–348, Lecture Notes in Math., 1620, Springer, Berlin, 1996.
* [11] Dubrovin, B.; Zhang, Y. Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, preprint arXiv: math.DG/0108160.
* [12] Frenkel, E.; Givental, A.; Milanov, T. Soliton equations, vertex operators, and simple singularities, preprint arXiv: math.QA/0909.4032v1.
* [13] Givental, A.; Milanov, T. Simple singularities and integrable hierarchies. The breadth of symplectic and Poisson geometry, 173–201, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005.
* [14] Jimbo, M.; Miwa, T. Solitons and infinite-dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 943–1001.
* [15] Kac, V.G. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990.
* [16] Kac, V.G.; Wakimoto, M. Exceptional hierarchies of soliton equations. Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987), 191–237, Proc. Sympos. Pure Math., 49, Part 1, Amer. Math. Soc., Providence, RI, 1989.
* [17] ten Kroode, F.; van de Leur, J. Bosonic and fermionic realizations of the affine algebra $\widehat{\rm so}_{2n}$. Comm. Algebra 20 (1992), no. 11, 3119–3162.
* [18] Li, L.C.; Parmentier, S. Nonlinear Poisson structures and $r$-matrices. Comm. Math. Phys. 125 (1989), no. 4, 545–563.
* [19] Liu, S.Q.; Wu C.Z.; Zhang Y. On the Drinfeld-Sokolov hierarchies of D type, preprint arXiv: nlin.SI/0912.5273v1.
* [20] Oevel, W.; Ragnisco, O. $R$-matrices and higher Poisson brackets for integrable systems. Phys. A 161 (1989), no. 1, 181–220.
* [21] Semenov-Tyan-Shanskii, M.A. What a classical $r$-matrix is. (Russian) Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, 17–33. English translation: Functional Anal. Appl. 17 (1983), no. 4, 259–272.
* [22] Shiota, T. Prym varieties and soliton equations. Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), 407–448, Adv. Ser. Math. Phys., 7, World Sci. Publ., Teaneck, NJ, 1989.
* [23] Takasaki, K. Integrable hierarchy underlying topological Landau-Ginzburg models of $D$-type. Lett. Math. Phys. 29 (1993), no. 2, 111–121.
* [24] Takasaki, K. Dispersionless Hirota equations of two-component BKP hierarchy. SIGMA Symmetry Integrability Geom. Methods Appl. 2 (2006), Paper 057, 22 pp.
* [25] Ueno, K.; Takasaki, K. Toda lattice hierarchy. Group representations and systems of differential equations (Tokyo, 1982), 1–95, Adv. Stud. Pure Math., 4, North-Holland, Amsterdam, 1984.
* [26] Wu, C.Z. A remark on Kac-Wakimoto hierarchies of D-type. J. Phys. A: Math. Theor. 43 (2010), 035201.
|
arxiv-papers
| 2010-01-25T11:27:55 |
2024-09-04T02:49:07.966600
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chao-Zhong Wu, Dingdian Xu",
"submitter": "Chao-Zhong Wu",
"url": "https://arxiv.org/abs/1001.4126"
}
|
1001.4221
|
# A Cosmological Model without Singularity and Dark Matter
Shi-Hao Chen Institute of Theoretical Physics, Northeast Normal University,
Changchun 130024, China.
shchen@nenu.edu.cn
###### Abstract
According to the cosmological model without singularity, there are s-matter
and v-matter which are symmetric and have oppose gravitational masses. In
V-breaking s-matter is similar to dark energy to cause expansion of the
universe with an acceleration now, and v-matter is composed of v-F-matter and
v-W-matter which are symmetric and have the same gravitational masses and
forms the world. The ratio of s-matter to v-matter is changeable. Based on the
cosmological model, we confirm that big bang nucleosynthesis is not spoiled by
that the average energy density of W-matter (mirror matter) is equal to that
of F-matter (ordinary matter). According to the present model, there are three
sorts of dark matter which are v-W-baryon matter (4/27), unknown v-F-matter
(9.5/27) and v-W-matter (9.5/27). Given v-F-baryon matter (4/27) and v-W-
baryon matter can cluster and respectively form the visible galaxies and dark
galaxies. Unknown v-F-matter and v-W-matter cannot cluster to form any
celestial body, loosely distribute in space, are equivalent to cold dark
matter, and their compositions are unknown. The number in a bracket is the
ratio of the density of a sort of matter to total density of v-matter. The
decisive predict is that there are dark celestial bodies and dark galaxies.
The energy of F-matter can transform into the energy of W-matter by such a
process in which the reaction energy is high enough.
Primordial nucleosynthesis, Dark matter, Cosmology of theories beyond the SM,
Dark energy, Physics of early universe
###### pacs:
95.30.Cq, 98.80.Es, 98.80.Cq, 95.35.+d, 11.10Wx
LABEL:FirstPage1 LABEL:LastPage#146
###### Contents
1. I Introduction
2. II The average density of W-matter is equal to that of F-matter
3. III The sorts and average density of dark matter
4. IV The interaction of F-scalar fields and W-scalar fields
5. V Features and observation of dark matter in present model
6. VI Conclusions
## I Introduction
Mirror matter as dark matter has been presented${}^{[1]}.$ In order that the
model is consistent with the standard cosmological framework, the model
ascribes the macroscopic asymmetry of the universe to asymmetric initial
conditions of matter and mirror matter[2]. However, when temperature is high
enough, the expectation values of all Higgs fields are zero so that the static
masses of all particles are zero, and all particles can transform from one
into other. Consequently the energy densities of two sorts of matter must be
equal if both are symmetric so that their degrees of freedom are equal to each
other. Thus the asymmetric initial conditions make one to be uncomfortable.
A quantum field theory without divergence has been constructed which can
obtain all results of the given quantum field theory and in which there is no
divergence of loop-corrections and the energy of the vacuum must be zero
without normal product[3]. There must be two sorts of matter in this quantum
field theory. The two sorts of matter are called $F-matter$ and $W-matter$,
respectively. $F-matter$ and $W-matter$ are symmetric, and there are only
gravitation and very weak interaction by Higgs bosons in low energy between
both. $F-matter$ forms the given world and $W-matter$ is dark matter for a
$F-observer^{[4]}$. This model of dark matter is equivalent to the model of
mirror matter as dark matter.
A cosmological model without singularity has been constructed${}^{[5]}.$ The
model can well explain evolution of the universe, formation of large scale
structure and the features of huge voids, naturally determines the
cosmological constant to be zero, has no singularity and gives some
predictions. Based on the cosmological model, we will see in the present paper
that the average density of $W-matter$ may be equal to that of $F-matter$. The
average density of dark matter is the $23/4$ times of that of visible matter,
because dark matter is composed of $W-matter$ and a part of $F-matter$.
Second 2 discusses that the average density of W-matter is equal to that of
F-matter. Second 3 discusses the sorts and average density of dark matter.
Second 4 the interaction of F-scalar fields and W-scalar fields. Section 5 is
the conclusions.
## II The average density of W-matter is equal to that of F-matter
According to the cosmological model without singularity, the space evolving
equations are[5]
$\overset{\cdot}{R}^{2}(t)+K=\eta\left[\rho_{v}+V_{v}(\varpi_{v})+V_{0}-\rho_{s}-V_{s}(\varpi_{s})\right]R^{2}\left(t\right)=\eta\left(\rho_{g}+V_{g}\right)R^{2}\left(t\right),$
(1)
where $\omega=\Omega,$ $\Phi,$ and $\chi,$ $\eta\equiv 8\pi G/3;$ $\rho_{v}$
and $\rho_{s}$ are the mass densities of $v-matter$ and $s-matter$
$\left(\text{here }c=1\right)$, respectively; $V_{v}$ and $V_{s}$ are the
densities of $v-Higgs$ potential energy and $v-Higgs$ potential energy,
respectively. the curvature factor $K$ is a function of the gravitational mass
density, and
$\rho_{g}\equiv\rho_{v}-\rho_{s},\text{ \ }p_{g}=p_{v}-p_{s},\text{ \
}V_{g}=V_{v}(\varpi_{v})+V_{0}-V_{s}(\varpi_{s}).$ (3) $1\geq K>0\text{ \ for
}\rho_{g}>0,\text{ }K=0\text{ \ for }\rho_{g}=0,\text{ \ }0>K\geq-1\text{ \
for }\rho_{g}<0.$ (4)
In $V-breaking$ in which the expectation values of $v-Higgs$ fields
$\langle\omega_{v}\rangle\equiv\varpi_{v}=0$, and the expectation values of
$s-Higgs$ fields
$\langle\omega_{s}\rangle^{\prime}\equiv\varpi_{s}^{\prime}\neq 0,$ the
gravitational mass of $v-matter$ is positive and the gravitational mass of
$s-matter$ is negative.
When temperature rises to the critical temperature $T_{cr}$ because space
contracts${}^{[5]},$ $\varpi_{v}=\varpi_{s}=0,$ $\rho_{v}=\rho_{s},$
$V_{v}(\varpi_{v})=V_{s}(\varpi_{s})=0$ so that
$\displaystyle\overset{\cdot}{R}^{2}(t)$ $\displaystyle=$
$\displaystyle-K+\eta V_{0}R^{2}\left(t\right),\TCItag{5}$ (2)
$\displaystyle\overset{\cdot\cdot}{R}(t)$ $\displaystyle=$ $\displaystyle-\eta
V_{0}R\left(t\right)<0.\TCItag{6}$ (3)
There is the highest temperature $T_{\max}$ at which $\overset{\cdot}{R}=0,$
$R\left(t\right)=R_{\min}\equiv\sqrt{K/\eta V_{0}}$ and space inflation must
occur.
$V-matter$ is composed of $v-F-matter$ and $v-W-matter$, $s-matter$ is
composed of $s-F-matter$ and $s-W-matter$. $F-matter$ and $W-matter$ are
symmetric so that divergence of loop corrections is eliminated and the energy
of the vacuum is determined to be zero. There is only the gravitation and the
interaction by Higgs bosons between $F-particles$ and $W-particles.$ The
interaction by Higgs bosons can be ignored when temperature is low because the
masses of the Higgs bosons are all very large. Consequently $W-matter$ is
regarded dark matter relative to $F-matter$, and vice versa. These are the
necessary inferences of the quantum field theory without divergence${}^{[3]}.$
$F-matter$ and $W-matter$ correspond to ordinary matter and mirror matter in
Ref${}^{[1,2]},$ respectively.
The state with $T\geq T_{cr}$ is such a state with the highest symmetry.
$\rho_{sF}=\rho_{sW}=\rho_{vF}=\rho_{vW}$ and $T_{sF}=T_{sW}=T_{vF}=T_{vW}=T$
in this state because $F-matter$ and $W-matter$ are symmetric and can
transform from one into another, $s-matter$ and $v-matter$ are symmetric and
can transform from one into another, and the thermal equilibrium can realized
due to $\overset{\cdot}{R}\sim 0$ when $T\geq T_{cr}.$
After space inflation, reheating process occurs. After reheating process, the
potential energy density transforms into $\rho_{v}^{\prime}=xV_{0}$ and
$\rho_{s}^{\prime}=\left(1-x\right)V_{0}.$
$\rho_{v}^{\prime}>\rho_{s}^{\prime}$ because
$\langle\omega_{v}\rangle=0\longrightarrow\langle\omega_{v}\rangle_{0}\neq 0,$
$\langle\omega_{s}\rangle=0\longrightarrow\langle\omega_{s}\rangle_{0}=0,$
$V_{v}=0\longrightarrow-V_{0}$ and $V_{s}=0\longrightarrow 0^{[5]}.$ Thus,
after reheating process, the evolving equations become
$\overset{\cdot}{R}^{2}(t)+K=\eta\left[\rho_{vF}+\rho_{vW}-\rho_{sF}-\rho_{sW}\right]R^{2}\left(t\right)=\eta\left[2\rho_{vF}-2\rho_{sF}\right]R^{2}\left(t\right),$
(7)
$\overset{\cdot\cdot}{R}(t)=-\eta\left[\left(\rho_{vF}+3p_{vF}\right)-\left(\rho_{sF}+3p_{sF}\right)\right]R\left(t\right),$
(8)
Here we still denote $\rho_{v}^{\prime}+\rho_{v}$ and
$\rho_{s}^{\prime}+\rho_{s}$ by $\rho_{v}$ and $\rho_{s},$ respectively, for
convenience.
It is seen from $\left(4\right)$ and $\left(7\right)$ that in contrast with
the conventional cosmological models, although $\rho_{vF}=\rho_{vW},$ the
$\overset{\cdot}{R}^{2}(t)+K$ cannot doubled because of
$\rho_{s}=\rho_{sF}+\rho_{sW}.$ Consequently the big bang nucleosynthesis
cannot be spoiled by $\rho_{vF}=\rho_{vW}$ provided $\rho_{s}$ is suitably
chosen. For example, if $\rho_{s}=\rho_{vW}$ in the stage of the big bang
nucleosynthesis, the evolving equations $\left(7\right)-\left(8\right)$ become
$\overset{\cdot}{R}^{2}(t)+K=\eta\rho_{vF}R^{2}\left(t\right)$
$\overset{\cdot\cdot}{R}(t)=-\frac{1}{2}\eta\left(\rho_{vF}+3p_{vF}\right)R\left(t\right),$
This is consistent with the conventional theory.
## III The sorts and average density of dark matter
Recent astronomical observations show that the universe expanded with a
deceleration early and is expanding with an acceleration now. This implies
that there is dark energy[6]. $\rho_{de}/\rho_{tot}=0.73,$
$\rho_{M}/\rho_{tot}=0.27$, $\rho_{M}=\rho_{VM}+\rho_{DM}$,
$\rho_{VM}\sim\rho_{B},$ $\rho_{DM}/\rho_{tot}=0.23$ and
$\rho_{B}/\rho_{tot}=0.04$, here $\rho_{de}$ is the density of dark energy,
$\rho_{tot}$ is the density of the total energy of the universe
$\left(c=1\right)$, $\rho_{VM}$ is the energy density of visible matter,
$\rho_{DM}$ is the energy density of dark matter, and $\rho_{B}$ is the energy
density of visible baryon matter. According to the cosmological model without
singularity[5], in the $V-breaking$, the effects of $s-matter$ are equivalent
to that of the so-called dark energy, and $\rho_{v}=\rho_{M}$. According to
this dark-matter model $[3,4]$, because of the symmetry of $F-matter$ and
$W-matter$, we have
$\displaystyle\rho_{M}$ $\displaystyle=$
$\displaystyle\rho_{v}=\rho_{vF}+\rho_{vW}=2\rho_{vF},\text{ \
}\rho_{B}=\rho_{vFB},$ $\displaystyle\rho_{vF}$ $\displaystyle=$
$\displaystyle\rho_{vFB}+\rho_{vFu},\text{ \
}\rho_{vW}=\rho_{vWB}+\rho_{vWu},$ $\displaystyle\rho_{vFB}$ $\displaystyle=$
$\displaystyle\rho_{vWB},\text{ \ \ }\rho_{vFu}=\rho_{vWu},\text{ \
}\rho_{vD}=\rho_{vFu}+\rho_{vW}=\rho_{DM},\TCItag{9}$ (4)
where $\rho_{v}$ is the total energy density of $v-matter$, $\rho_{vF}$ and
$\rho_{vW}$ are respectively the energy density of $v-F-matter$ and the energy
density of $v-W-matter,$ $\rho_{vFB}$ and $\rho_{vWB}$ are respectively the
energy density of $v-F-baryon$ matter $(v-FBM)$ and the energy density of
$v-W-baryon$ matter $(v-WBM)$, $\rho_{vFu}$ is the energy density of unknown
$v-F-matter$ $(v-UFM),$ $\rho_{vWu}$ is the energy density of $v-W-matter$
$(v-UWM)$ corresponding to $v-UFM,$ and $\rho_{vD}$ is the total energy
density of invisible $v-matter$. Here $v-FBM$ is the given and visible matter
which contains given baryon matter, black holes and neutrinos etc., $F-matter$
contains $v-FBM$ and invisible and unknown $v-UFM$. Considering
$\rho_{vF}=\rho_{vW}$ because $F-matter$ and $W-matter$ are symmetric and can
transform from one into another when temperature is high enough, we can
determine the ratios of a density to another.
$\displaystyle\frac{\rho_{vF}}{\rho_{vW}}$ $\displaystyle=$
$\displaystyle\frac{0.27/2}{0.27/2}=1=\frac{\rho_{vFB}}{\rho_{vW}B}=\frac{\rho_{vFu}}{\rho_{vW}u},\TCItag{10a}$
(5) $\displaystyle\frac{\rho_{vFB}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{vWB}}{\rho_{v}}=\frac{0.04}{0.27}=\frac{4}{27},\TCItag{10b}$
(6) $\displaystyle\frac{\rho_{vFu}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{vWu}}{\rho_{v}}=\frac{0.27/2-0.04}{0.27}=\frac{9.5}{27},\TCItag{10c}$
(7) $\displaystyle\frac{\rho_{vD}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{vW}+\rho_{vFu}}{\rho_{v}}=\frac{23}{27},\TCItag{10d}$
(8) $\displaystyle\frac{\rho_{vD}}{\rho_{vFB}}$ $\displaystyle=$
$\displaystyle\frac{0.27/2+0.095}{0.04}=\frac{23}{4}.\TCItag{10e}$ (9)
Thus, according to the present model${}^{[3]},$ there are three sorts of dark
matter which are $v-UFM,$ $v-WBM$ and $v-UWM.$ Given $v-FBM$ can cluster and
form visible galaxies. $v-WBM$ can cluster and form dark galaxies. $v-UFM$ and
$v-UWM$ cannot cluster to form any celestial body, loosely distribute in
space, and their compositions are unknown.
$\rho_{vWu}/\rho_{vFu}=\rho_{vFB}/\rho_{vWB}=1$ is invariant because of the
symmetry of $W-matter$ and $F-matter$.
According to the cosmological model without singularity${}^{[5]},$ there are
$s-matter$ and $v-matter$ which are symmetric and repulsive each other. In
$V-breaking$, $v-SU(5)$ breaks into $v-SU(3)\times U(1),$ $v-FBM$ forms the
visible world, $v-WBM,$ $v-UFM$ and $v-UWM$ form the dark-matter world;
$s-SU(5)$ does not break, all $s-particles$ form $s-SU(5)$ color single states
which loosely distribute in whole space and cause space to expand with an
acceleration in the present stage. This is because there is no interaction
similar to the given electroweak interaction among the $s-SU(5)$ color single
states[5].
$\rho_{s}/\rho_{v}$ is changeable as space expands[5] because
$\rho_{vM}\propto R^{-3},$ $\rho_{v\gamma}\propto R^{-4},$ and
$\rho_{s}=\rho_{sM}\propto R^{-3}.$ Here $\rho_{vM}$ and $\rho_{v\gamma}$ are
respectively the energy density of massive $v-particles$ and the energy
density of massless $v-particles$, $\rho_{s}$ is the energy density of
$s-SU(5)$ color single states. The masses of all color single states are not
zero, hence $\rho_{s}\propto R^{-3}.$
## IV The interaction of F-scalar fields and W-scalar fields
The Ref. $[7]$ given the interaction of $F-scalar$ fields and $W-scalar$
fields
$V=-\frac{2A}{225}Tr\Phi_{f}^{2}Tr\Phi_{w}^{2},$ (11)
where $\Phi_{f}$ and $\Phi_{w}$ are the $\underline{24}$ representation of
$SU(5)$ group. The breaking component is
$\Phi_{i}=Diagonal\left(1,1,1,-\frac{3}{2},-\frac{3}{2}\right)\left(\sigma_{i}+\varphi_{i}\right),$
(12)
where the subscript $i=f,$ $w.$ From $\left(11\right)-\left(12\right)$ we
obtain
$V_{fw}=-A\left(2\sigma_{f}\sigma_{w}\varphi_{f}\varphi_{w}+\sigma_{f}\varphi_{f}\varphi_{w}^{2}+\sigma_{w}\varphi_{w}\varphi_{f}^{2}+\frac{1}{2}\varphi_{f}^{2}\varphi_{w}^{2}\right).$
(13)
$\mid\sigma_{w}\mid=\mid\sigma_{f}\mid$ because of the symmetry of $s-matter$
and $f-matter$. Both $\sigma_{i}$ and $m\left(\varphi_{i}\right)$ are
functions of temperature $T$. When $T\geq T_{cr},$
$\sigma_{i}=m\left(\varphi_{i}\right)=0$. Consequently $f-particles$ and
$w-particles$ can easily transform from one to another so that
$\rho_{F}=\rho_{W}.$ When $T\sim 0,$ both $\mid\sigma_{i}\mid$and
$m\left(\varphi_{i}\right)$ are large enough. Consequently interaction between
$f-particles$ and $w-particles$ by the scalar bosons may be ignored. Thus
there is only the gravitation between $f-matter$ and $w-matter$ when
temperature is low.
There are the couplings of fermions (and gauge particles) and scalar
bosons${}^{[7]}.$ Hence there are the interactions of $f-fermions$ and
$w-fermions$ shown in figures 1-3 and the interactions of $f-gauge$ bosons and
$w-gauge$ bosons via the scalar bosons $\varphi_{f}$ and $\varphi_{w}.$ In the
figures the dotted lines with arrows denote $W-fermion$ field $\psi_{w}$, the
dotted lines without arrow denote $W-scalar$ field $\varphi_{w},$ the lines
with arrows denote $F-fermion$ field $\psi_{f}$, the lines without arrow
denote $F-scalar$ field $\varphi_{f},$ $M^{2}=-2A\sigma_{f}\sigma_{w}$,
$R_{f}=-A\sigma_{f}$, $R_{w}=-A\sigma_{w}$ and $S=-A/2.$
It can be seen from figure 1 and $\left(13\right)$ that when
$-A\sigma_{f}\sigma_{w}>0$ and $k^{2}-m^{2}<0$ or $-A\sigma_{f}\sigma_{w}<0$
and $k^{2}-m^{2}>0,$ $f-fermions$ and $w-fermions$ are repulsive each other;
when $-A\sigma_{f}\sigma_{w}>0$ and $k^{2}-m^{2}>0$ or
$-A\sigma_{f}\sigma_{w}<0$ and $k^{2}-m^{2}<0,$ $f-fermions$ and $w-fermions$
are attractive each other.
Figure 1: $f+\overline{f}\longrightarrow w+\overline{w}$ realized by the tree
digram.
Figure 2: $f+\overline{f}\longrightarrow w+\overline{w}$ realized by the two
one-loop digrams.
Figure 3: $f+\overline{f}\longrightarrow w+\overline{w}$ realized by the two-
loop diagram.
## V Features and observation of dark matter in present model
According to the present model[3,4], $v-UFM$ and $v-UWM$ cannot form cluster,
loosely distribute in space and have positive gravitational masses, hence both
should be identified as cold dark matter. From $\left(10\right)$ we have
$\displaystyle\frac{\rho_{vFu}+\rho_{vWu}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{0.095\times 2}{0.27}=\frac{19}{27},$
$\displaystyle\frac{\rho_{vFu}+\rho_{vWu}}{\rho_{vD}}$ $\displaystyle=$
$\displaystyle\frac{0.095\times 2}{0.23}=\frac{19}{23}.\TCItag{14}$ (10)
It is seen that the present model cannot differ from the cold dark matter
model by $v-UFM$ and $v-UWM.$ $W-baryon$ matter $v-WBM$ can form clusters of
dark matter and dark galaxies[3,4]. There possibly are such celestial bodies
which are composed of mixture of $F-matter$ and $W-matter$ because both have
positive gravitational masses. This is a decisive predict of the present model
and mirror dark matter model. Dark celestial bodies flying to the earth are
possibly detected by probing gravity-meters of clustering dark matter[4].
It can be seen from $\left(13\right)$ and figures $1-3$ that the energy of
$F-matter$ can transform into the energy of $W-matter$ by such a process in
which the reaction energy is high enough.
## VI Conclusions
According to the cosmological model without singularity, there are $s-matter$
and $v-matter$ which are symmetric and have oppose gravitational masses. In
$V-breaking$ $s-matter$ is similar to dark energy to cause expansion of the
universe with an acceleration now, and $v-matter$ is composed of $v-F-matterv$
and $v-W-matterv$ which are symmetric and have the same gravitational masses.
The ratio of $s-matter$ to $v-matter$ is changeable.
Based on the cosmological model, we confirm that big bang nucleosynthesis is
not spoiled by that the average energy density of $W-matter$ (mirror matter)
is equal to that of $F-matter$ (ordinary matter).
According to the present model, there are three sorts of dark matter which are
$v-W-baryon$ matter ($4/27)$ and unknown $v-F-matterv$ ($9.5/27)$ and $v-W-
matter$ ($9.5/27).$ Given $v-F-bayon$ matter ($4/27)$ can cluster to form the
visible galaxies. $V-W-bayon$ matter can cluster to form dark celestial bodies
and dark galaxies. Unknown $v-F-matterv$ and $v-W-matter$ cannot cluster to
form any celestial body, loosely distribute in space, are equivalent to cold
matter, and their compositions are unknown. The number in a bracket is the
ratio of the density of a sort of matter to total density of $v-matter$.
The decisive predict is that there are dark celestial bodies and dark
galaxies. The energy of $F-matter$ can transform into the energy of $W-matter$
by such a process in which the reaction energy is high enough.
Acknowledgement
I am very grateful to professor Zhao Zhan-yue and professor Wu Zhao-yan for
their helpful discussions and best support.
## References
* (1) Z. Berezhiani, D. Comelli and F. L. Villante, Phys. Lett. B 503 (2001) 362-375.
* (2) Y. Ignatiev and R. R. Volkas, Phys. Rev. D 68 (2003) 023518.
* (3) S-H. Chen, ‘Quantum Field Theory Without Divergence A’ hep-th/0203220, ‘Quantum Field Theory :New Research’, O. Kovras Editor, Nova Science Publishers, Inc. (2005), p103-170. ‘Significance of Negative Energy State in Quantum Field Theory A’ hep-th/0203230.
* (4) S-H. Chen, ‘A Possible Candidate for Dark Matter’, hep-th/0103234; ‘Progress in Dark Matter Research’ Editor: J. Val Blain, pp.65-72. 2005 Nova Science Publishers, Inc.
* (5) S-H. Chen, ‘A Possible Universal Model without Singularity and its Explanation for Evolution of the Universe’, hep-ph/0611283; ‘Discussion of a Possible Universal Model without Singularity’, arXiv. 0908.1495.
* (6) R.R.Caldwell, Phys. World 17, 37 (2004); T.padmanabhan, Phys. Rep. 380. 325 (2003); P.J.E. Peebles and B. Ratra; Rev. Mod. Phys. 75,559 (2003).
* (7) S-H. Chen, ‘$SU(5)$ Grand Unified Model and Dark Matter’, arXiv. 0912.2427.
|
arxiv-papers
| 2010-01-25T07:22:25 |
2024-09-04T02:49:07.992798
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shi-Hao Chen",
"submitter": "Shihao Chen",
"url": "https://arxiv.org/abs/1001.4221"
}
|
1001.4829
|
201071-82Nancy, France 71
László Babai
Anandam Banerjee
Raghav Kulkarni
Vipul Naik
# Evasiveness and the Distribution of Prime Numbers
L. Babai University of Chicago, Chicago, IL, USA. , A. Banerjee
Northeastern University, Boston, MA, USA. , R. Kulkarni and V. Naik
###### Abstract.
A Boolean function on $N$ variables is called _evasive_ if its decision-tree
complexity is $N$. A sequence $B_{n}$ of Boolean functions is _eventually
evasive_ if $B_{n}$ is evasive for all sufficiently large $n$.
We confirm the eventual evasiveness of several classes of monotone graph
properties under widely accepted number theoretic hypotheses. In particular we
show that Chowla’s conjecture on Dirichlet primes implies that (a) for any
graph $H$, “forbidden subgraph $H$” is eventually evasive and (b) all
nontrivial monotone properties of graphs with $\leq n^{3/2-\epsilon}$ edges
are eventually evasive. ($n$ is the number of vertices.)
While Chowla’s conjecture is not known to follow from the Extended Riemann
Hypothesis (ERH, the Riemann Hypothesis for Dirichlet’s $L$ functions), we
show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH.
We also prove unconditional results: (a′) for any graph $H$, the query
complexity of “forbidden subgraph $H$” is $\binom{n}{2}-O(1)$; (b′) for some
constant $c>0$, all nontrivial monotone properties of graphs with $\leq cn\log
n+O(1)$ edges are eventually evasive.
Even these weaker, unconditional results rely on deep results from number
theory such as Vinogradov’s theorem on the Goldbach conjecture.
Our technical contribution consists in connecting the topological framework of
Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot,
and Shi (2002), with a deeper analysis of the orbital structure of permutation
groups and their connection to the distribution of prime numbers. Our
unconditional results include stronger versions and generalizations of some
result of Chakrabarti et al.
###### Key words and phrases:
Decision tree complexity, evasiveness, graph property, group action, Dirichlet
primes, Extended Riemann Hypothesis
###### 1991 Mathematics Subject Classification:
F.2.2, F.1.1, F.1.3
22footnotetext: Partially supported by NSF Grant CCF-0830370.
## 1\. Introduction
### 1.1. The framework
A graph property $P_{n}$ of $n$-vertex graphs is a collection of graphs on the
vertex set $[n]=\\{1,\dots,n\\}$ that is invariant under relabeling of the
vertices. A property $P_{n}$ is called monotone (decreasing) if it is
preserved under the deletion of edges. The trivial graph properties are the
empty set and the set of all graphs. A class of examples are the forbidden
subgraph properties: for a fixed graph $H$, let $Q_{n}^{H}$ denote the class
of $n$-vertex graphs that do not contain a (not necessarily induced) subgraph
isomorphic to $H$.
We view a set of labeled graphs on $n$ vertices as a Boolean function on the
$N=\binom{n}{2}$ variables describing adjacency. A Boolean function on $N$
variables is evasive if its deterministic query (decision-tree) complexity is
$N$.
The long-standing Aanderaa-Rosenberg-Karp conjecture asserts that every
nontrivial monotone graph property is evasive. The problem remains open even
for important special classes of monotone properties, such as the forbidden
subgraph properties.
### 1.2. History
In this note, $n$ always denotes the number of vertices of the graphs under
consideration.
Aanderaa and Rosenberg (1973) [17] conjectured a lower bound of
$\Omega(n^{2})$ on the query complexity of monotone graph properties. Rivest
and Vuillemin (1976) [19] verified this conjecture, proving an $n^{2}/16$
lower bound. Kleitman and Kwiatkowski (1980) [10] improved this to $n^{2}/9.$
Karp conjectured that nontrivial monotone graph properties were in fact
evasive. We refer to this statement as the Aanderaa-Rosenberg-Karp (ARK)
conjecture.
In their seminal paper, Kahn, Saks, and Sturtevant [11] observe that non-
evasiveness of monotone Boolean functions has strong topological consequences
(contracibility of the associated simplicial complex). They then use results
of R. Oliver about fixed points of group actions on such complexes to verify
the ARK conjecture when $n$ is a prime-power. As a by-product, they improve
the lower bound for general $n$ to $n^{2}/4.$
Since then, the topological approach of [11] has been influential in solving
various interesting special cases of the ARK conjecture. Yao (1988) [25]
proves that non-trivial monotone properties of bipartite graphs with a given
partition $(U,V)$ are evasive (require $|U||V|$ queries). Triesch (1996) [22]
shows (in the original model) that any monotone property of bipartite graphs
(all the graphs satisfying the property are bipartite) is evasive.
Chakrabarti, Khot, and Shi (2002) [3] introduce important new techniques which
we use; we improve over several of their results (see Section 1.4).
### 1.3. Prime numbers in arithmetic progressions
Dirichlet’s Theorem (1837) (cf. [5]) asserts that if $\gcd(a,m)=1$ then there
exist infinitely many primes $p\equiv a\pmod{m}$. Let $p(m,a)$ denote the
smallest such prime $p$. Let $p(m)=\max\\{p(m,a)\mid\gcd(a,m)=1\\}$. Linnik’s
celebrated theorem (1947) asserts that $p(m)=O(m^{L})$ for some absolute
constant $L$ (cf. [16, Chap. V.]). Heath-Brown [9] shows that $L\leq 5.5$.
Chowla [4] observes that under the Extended Riemann Hypothesis (ERH) we have
$L\leq 2+\epsilon$ for all $\epsilon>0$ and conjectures that $L\leq
1+\epsilon$ suffices:
###### Conjecture 1.1 (S. Chowla [4]).
For every $\epsilon>0$ and every $m$ we have $p(m)=O(m^{1+\epsilon})$.
This conjecture is widely believed; in fact, number theorists suggest as
plausible the stronger form $p(m)=O(m(\log m)^{2})$ [8]. Turán [23] proves the
tantalizing result that for almost all $a$ we have $p(m,a)=O(m\log m)$ .
Let us call a prime $p$ an $\epsilon$-near Fermat prime if there exists an
$s\geq 0$ such that $2^{s}\mid p-1$ and $\frac{p-1}{2^{s}}\leq p^{\epsilon}$.
We need the following weak form of Chowla’s conjecture:
###### Conjecture 1.2 (Weak Chowla Conjecture).
For every $\epsilon>0$ there exist infinitely many $\epsilon$-near Fermat
primes.
In other words, the weak conjecture says that for every $\epsilon$, for
infinitely many values of $s$ we have $p(2^{s},1)<(2^{s})^{1+\epsilon}$.
### 1.4. Main results
For a graph property $P$ we use $P_{n}$ to denote the set of graphs on vertex
set $[n]$ with property $P$. We say that $P$ is eventually evasive if $P_{n}$
is evasive for all sufficiently large $n$.
Our first set of results states that the “forbidden subgraph” property is
“almost evasive” under three different interpretations of this phrase.
###### Theorem 1.3 (Forbidden subgraphs).
For all graphs $H$, the forbidden subgraph property $Q_{n}^{H}$ (a) is
eventually evasive, assuming the Weak Chowla Conjecture; (b) is evasive for
almost all $n$ (unconditionally); and (c) has query complexity
$\binom{n}{2}-O(1)$ for all $n$ (unconditionally).
Part (b) says the asymptotic density of values of $n$ for which the problem is
not evasive is zero. Part (c) improves the bound $\binom{n}{2}-O(n)$ given in
[3]. Parts (a) and (c) will be proved in Section 3. We defer the proof of part
(b) to the journal version.
The term “monotone property of graphs with $\leq m$ edges” describes a
monotone property that fails for all graphs with more than $m$ edges.
###### Theorem 1.4 (Sparse graphs).
All nontrivial monotone properties of graphs with at most $f(n)$ edges are
eventually evasive, where (a) under Chowla’s Conjecture,
$f(n)=n^{3/2-\epsilon}$ for any $\epsilon>0$; (b) under ERH,
$f(n)=n^{5/4-\epsilon}$; and (c) unconditionally, $f(n)=cn\log n$ for some
constant $c>0$. (d) Unconditionally, all nontrivial monotone properties of
graphs with no cycle of length greater than $(n/4)(1-\epsilon)$ are eventually
evasive (for all $\epsilon>0$).
Part (c) of Theorem 1.4 will be proved in Section 4. Parts (a) and (b) follow
in Section 5. The proof of part (d) follows along the lines of part (c); we
defer the details to the journal version of this paper.
We note that the proofs of the unconditional results (c) and (d) in Theorem
1.4 rely on Haselgrove’s version [7] of Vinogradov’s Theorem on Goldbach’s
Conjecture (cf. Sec. 4.2).
Recall that a _topological subgraph_ of a graph $G$ is obtained by taking a
subgraph and replacing any induced path $u-\dots-v$ in the subgraph by an edge
$\\{u,v\\}$ (repeatedly) and deleting parallel edges. A minor of a graph is
obtained by taking a subgraph and contracting edges (repeatedly). If a class
of graphs is closed under taking minors then it is also closed under taking
topological subgraphs but not conversely; for instance, graphs with maximum
degree $\leq 3$ are closed under taking toopological subgraphs but every graph
is a minor of a regular graph of degree 3.
###### Corollary 1.5 (Excluded topological subgraphs).
Let $P$ be a nontrivial class of graphs closed under taking topological
subgraphs. Then $P$ is eventually evasive.
This unconditional result extends one of the results of Chakrabarti et al.
[3], namely, that nontrival classes of graphs closed under taking minors is
eventually evasive.
Corollary 1.5 follows from part (c) of Theorem 1.4 in the light of Mader’s
Theorem which states that if the average degree of a graph $G$ is greater than
$2^{\binom{k+1}{2}}$ then it contains a topological $K_{k}$ [13, 14].
Theorem 1.4 suggests a new stratification of the ARK Conjecture. For a
monotone (decreasing) graph property $P_{n}$, let
$\dim(P_{n}):=\max\\{|E(G)|-1\ |\ G\in P_{n}\\}.$
We can now restate the ARK Conjecture:
###### Conjecture 1.6.
If $P_{n}$ is a non-evasive, non-empty, monotone decreasing graph property
then $\dim(P_{n})=\binom{n}{2}-1.$
## 2\. Preliminaries
### 2.1. Group action
For the basics of group theory we refer to [18]. All groups in this paper are
finite. For groups $\Gamma_{1},\Gamma_{2}$ we use $\Gamma_{1}\leq\Gamma_{2}$
to denote that $\Gamma_{1}$ is a subgroup; and $\Gamma_{1}\lhd\Gamma_{2}$ to
denote that $\Gamma_{1}$ is a (not necessarily proper) normal subgroup. We say
that $\Gamma$ is a $p$-group if $|\Gamma|$ is a power of the prime $p$.
For a set $\Omega$ called the “permutation domain,” let
$\operatorname{Sym}(\Omega)$ denote the symmetric group on $\Omega$,
consisting of the $|\Omega|!$ permutations of $\Omega$. For
$\Omega=[n]=\\{1,\dots,n\\}$, we set $\Sigma_{n}=\operatorname{Sym}([n])$. For
a group $\Gamma$, a homomorphism
$\varphi\,:\,\Gamma\to\operatorname{Sym}(\Omega)$ is called a $\Gamma$-action
on $\Omega$. The action is faithful if $\ker(\varphi)=\\{1\\}$. For
$x\in\Omega$ and $\gamma\in\Gamma$ we denote by $x^{\gamma}$ the image of $x$
under $\varphi(\gamma)$. For $x\in\Omega$ we write
$x^{\Gamma}=\\{x^{\gamma}\,:\,\gamma\in\Gamma\\}$ and call it the orbit of $x$
under the $\Gamma$-action. The orbits partition $\Omega$.
Let $\binom{\Omega}{t}$ denote the set of $t$-subsets of $\Omega$. There is a
natural induced action
$\operatorname{Sym}(\Omega)\to\operatorname{Sym}(\binom{\Omega}{t})$ which
also defines a natural $\Gamma$-action on $\binom{\Omega}{t}$. We denote this
action by $\Gamma^{(t)}$. Similarly, there is a natural induced
$\Gamma$-action on $\Omega\times\Omega$. The orbits of this action are called
the orbitals of $\Gamma$. We shall need the undirected version of this
concept; we shall call the orbits of the $\Gamma$-action on
$\binom{\Omega}{2}$ the u-orbitals (undirected orbitals) of the
$\Gamma$-action.
By an action of the group $\Gamma$ on a structure $\mathfrak{X}$ such as a
group or a graph or a simplicial complex we mean a homomorphism
$\Gamma\to\operatorname{Aut}({\mathfrak{X}})$ where
$\operatorname{Aut}({\mathfrak{X}})$ denotes the automorphism group of
$\mathfrak{X}$.
Let $\Gamma$ and $\Delta$ be groups and let
$\psi\,:\,\Delta\to\operatorname{Aut}(\Gamma)$ be a $\Delta$-action on
$\Gamma$. These data uniquely define a group $\Theta=\Gamma\rtimes\Delta$, the
_semidirect product_ of $\Gamma$ and $\Delta$ with respect to $\psi$. This
group has order $|\Theta|=|\Gamma||\Delta|$ and has the following properites:
$\Theta$ has two subgroups $\Gamma^{*}\cong\Gamma$ and $\Delta^{*}\cong\Delta$
such that $\Gamma^{*}\lhd\Theta$; $\Gamma^{*}\cap\Delta^{*}=\\{1\\}$; and
$\Theta=\Gamma^{*}\Delta^{*}=\\{\gamma\delta\mid\gamma\in\Gamma^{*},\delta\in\Delta^{*}\\}$.
Moreover, identifying $\Gamma$ with $\Gamma^{*}$ and $\Delta$ with
$\Delta^{*}$, for all $\gamma\in\Gamma$ and $\delta\in\Delta$ we have
$\gamma^{\psi(\delta)}=\delta^{-1}\gamma\delta$.
$\Theta$ can be defined as the set $\Delta\times\Gamma$ under the group
operation
$(\delta_{1},\gamma_{1})(\delta_{2},\gamma_{2})=(\delta_{1}\delta_{2},\gamma_{1}^{\psi(\delta_{2})}\gamma_{2})\quad\quad(\delta_{i}\in\Delta,\gamma_{i}\in\Gamma).$
For more on semidirect products, which we use extensively, see [18, Chap. 7].
The group $\operatorname{AGL}(1,q)$ of affine transformations $x\mapsto ax+b$
of $\mathbb{F}_{q}$ ($a\in\mathbb{F}_{q}^{\times}$, $b\in\mathbb{F}_{q}$) acts
on $\mathbb{F}_{q}$. For each $d\mid q-1$, $\operatorname{AGL}(1,q)$ has a
unique subgroup of order $qd$; we call this subgroup $\Gamma(q,d)$. We note
that $\mathbb{F}_{q}^{+}\lhd\Gamma(q,d)$ and $\Gamma(q,d)/\mathbb{F}_{q}^{+}$
is cyclic of order $d$ and is isomorphic to a subgroup $\Delta$ of
$\operatorname{AGL}(1,q)$; $\Gamma(q,d)$ can be described as a semidirect
product $(\mathbb{F}_{q}^{+})\rtimes\Delta$.
### 2.2. Simplicial complexes and monotone graph properties
An abstract simplicial complex ${\mathcal{K}}$ on the set $\Omega$ is a subset
of the power-set of $\Omega$, closed under subsets: if $B\subset
A\in{{\mathcal{K}}}$ then $B\in{{\mathcal{K}}}$. The elements of
${\mathcal{K}}$ are called its faces. The dimension of $A\in{{\mathcal{K}}}$
is $\dim(A)=|A|-1$; the dimension of ${\mathcal{K}}$ is
$\dim({{\mathcal{K}}})=\max\\{\dim(A)\mid A\in{{\mathcal{K}}}\\}$. The Euler
characteristic of ${{\mathcal{K}}}$ is defined as
$\chi({{\mathcal{K}}}):=\sum_{A\in{{\mathcal{K}}},A\neq\emptyset}{(-1)^{\dim(A)}}.$
Let $[n]:=\\{1,2,\ldots,n\\}$ and $\Omega=\binom{[n]}{2}$. Let $P_{n}$ be a
subset of the power-set of $\Omega$, i. e., a set of graphs on the vertex set
$[n]$. We call $P_{n}$ a graph property if it is invariant under the induced
action $\Sigma_{n}^{(2)}$. We call this graph property monotone decreasing if
it is closed under subgraphs, i. e., it is a simplicial complex. We shall omit
the adjective “decreasing.”
### 2.3. Oliver’s Fixed Point Theorem
Let ${\mathcal{K}}\subseteq 2^{\Omega}$ be an abstract simplicial complex with
a $\Gamma$-action. The fixed point complex ${\mathcal{K}}_{\Gamma}$ action is
defined as follows. Let $\Omega_{1},\dots,\Omega_{k}$ be the $\Gamma$-orbits
on $\Omega$. Set
${\mathcal{K}}_{\Gamma}:=\\{S\subseteq[k]\mid\bigcup_{i\in
S}\Omega_{i}\in{\mathcal{K}}\\}.$
We say that a group $\Gamma$ satisfies Oliver’s condition if there exist (not
necessarily distinct) primes $p,q$ such that $\Gamma$ has a (not necessarily
proper) chain of subgroups $\Gamma_{2}\lhd\Gamma_{1}\lhd\Gamma$ such that
$\Gamma_{2}$ is a $p$-group, $\Gamma_{1}/\Gamma_{2}$ is cyclic, and
$\Gamma/\Gamma_{1}$ is a $q$-group.
###### Theorem 2.1 (Oliver [15]).
Assume the group $\Gamma$ satisfies Oliver’s condition. If $\Gamma$ acts on a
nonempty contractible simplicial complex ${\mathcal{K}}$ then
$\chi({\mathcal{K}}_{\Gamma})\equiv 1\pmod{q}.$ (2.1)
In particular, such an action must always have a nonempty invariant face.
### 2.4. The KSS approach and the general strategy
The topological approach to evasiveness, initiated by Kahn, Saks, and
Sturtevant, is based on the following key observation.
###### Lemma 2.2 (Kahn-Saks-Sturtevant [11]).
If $P_{n}$ is a non-evasive graph property then $P_{n}$ is contractible.
Kahn, Saks, and Sturtevant recognized that Lemma 2.2 brought Oliver’s Theorem
to bear on evasiveness. The combination of Lemma 2.2 and Theorem 2.1 suggests
the following general strategy, used by all authors in the area who have
employed the topological method, including this paper: We find primes $p,q$, a
group $\Gamma$ satisfying Oliver’s condition with these primes, and a
$\Gamma$-action on $P_{n}$, such that $\chi(P_{n})\equiv 0\pmod{q}$. By
Oliver’s Theorem and the KSS Lemma this implies that $P_{n}$ is evasive. The
novelty is in finding the right $\Gamma$.
KSS [11] made the assumption that $n$ is a prime power and used as
$\Gamma=\operatorname{AGL}(1,n)$, the group of affine transformations
$x\mapsto ax+b$ over the field of order $n$. While we use subgroups of such
groups as our building blocks, the attempt to combine these leads to hard
problems on the distribution of prime numbers.
Regarding the “forbidden subgraph” property, Chakrabarti, Khot, and Shi [3]
built considerable machinery which we use. Our conclusions are considerably
stronger than theirs; the additional techniques involved include a study of
the orbitals of certain metacyclic groups, a universality property of
cyclotomic graphs derivable using Weil’s character sum estimates, plus the
number theoretic reductions indicated.
For the “sparse graphs” result (Theorem 1.4) we need $\Gamma$ such that all
u-orbitals of $\Gamma$ are large and therefore
$(P_{n})_{\Gamma}=\\{\emptyset\\}$.
In both cases, we are forced to use rather large building blocks of size $q$,
say, where $q$ is a prime such that $q-1$ has a large divisor which is a prime
for Theorem 1.4 and a power of 2 for Theorem 1.3.
## 3\. Forbidden subgraphs
In this section we prove parts (a) and (c) of Theorem 1.3.
### 3.1. The CKS condition
A homomorphism of a graph $H$ to a graph $H^{\prime}$ is a map $f\,:\,V(H)\to
V(H^{\prime})$ such that $(\forall x,y\in V(H))(\\{x,y\\}\in
E(H)\Rightarrow\\{f(x),f(y)\\}\in E(H^{\prime}))$. (In particular,
$f^{-1}(x^{\prime})$ is an independent set in $H$ for all $x^{\prime}\in
V(H^{\prime})$.) Let $Q_{r}^{[[H]]}$ be the set of those $H^{\prime}$ with
$V(H^{\prime})=[r]$ that do not admit an $H\to H^{\prime}$ homomorphism. Let
further $T_{H}:=\min\\{2^{2^{t}}-1\ \mid\ 2^{2^{t}}\geq h\\}$ where $h$
denotes the number of vertices of $H$. The following is the main lemma of
Chakrabarti, Khot, and Shi [3].
###### Lemma 3.1 (Chakrabarti et al. [3]).
If $r\equiv 1\pmod{T_{H}}$ then $\chi(Q_{r}^{[[H]]})\equiv 0\pmod{2}$.
### 3.2. Cliques in generalized Paley graphs
Let $q$ be an odd prime power and $d$ an even divisor of $q-1.$ Consider the
graph $P(q,d)$ whose vertex set is $\mathbb{F}_{q}$ and the adjacency between
the vertices is defined as follows: $i\sim j\iff(i-j)^{d}=1.$ $P(q,d)$ is
called a generalized Paley graph.
###### Lemma 3.2.
If $(q-1)/d\leq q^{1/(2h)}$ then $P(q,d)$ contains a clique on $h$ vertices.
This follows from the following lemma which in turn can be proved by a routine
application of Weil’s character sum estimates (cf. [1]).
###### Lemma 3.3.
Let $a_{1},\ldots,a_{t}$ be distinct elements of the finite field
$\mathbb{F}_{q}.$ Assume $\ell\mid q-1$. Then the number of solutions
$x\in\mathbb{F}_{q}$ to the system of equations $(a_{i}+x)^{(q-1)/\ell}=1$ is
$\frac{q}{\ell^{t}}\pm t\sqrt{q}.$ ∎
Let $\Gamma(q,d)$ be the subgroup of order $qd$ of $\operatorname{AGL}(1,q)$
defined in Section 2.1. Each u-orbital of $\Gamma(q,d)$ is isomorphic to
$P(q,d)$. ∎
###### Corollary 3.4.
If ${\frac{q-1}{d}\leq q^{1/(2h)}}$ then each u-orbital of $\Gamma(q,d)$
contains a clique of size $h.$
### 3.3. $\epsilon$-near-Fermat primes
The numbers in the title were defined in Section 1.3. In this section we prove
Theorem 1.3, part (a).
###### Theorem 3.5.
Let $H$ be a graph on $h$ vertices. If there are infinitely many
$\frac{1}{2h}$-near-Fermat primes then $Q_{n}^{H}$ is eventually evasive.
Proof. Fix an odd prime $p\equiv 2\pmod{T_{H}}$ such that $p\geq|H|.$ If there
are infinitely many $\frac{1}{2h}$-near-Fermat primes then infinitely many of
them belong to the same residue class mod $p$, say $a+\mathbb{Z}p$. Let
$q_{i}$ be the $i$-th $\frac{1}{2h}$-near-Fermat prime such that $q_{i}\geq p$
and $q_{i}\equiv a\pmod{p}.$ Let $r^{\prime}=na^{-1}\pmod{p}$ and
$k^{\prime}=\sum_{i=1}^{r^{\prime}}q_{i}.$ Then $k^{\prime}\equiv n\pmod{p}$
and therefore $n=pk+k^{\prime}$ for some $k$.
Now in order to use Lemma 3.1, we need to write $n$ as a sum of $r$ terms
where $r\equiv 1\pmod{T_{H}}$. We already have $r^{\prime}$ of these terms; we
shall choose each of the remaining $r-r^{\prime}$ terms to be $p$ or $p^{2}$.
If there are $t$ terms equal to $p^{2}$ then this gives us a total of
$r=t+(k-tp)+r^{\prime}$ terms, so we need $t(p-1)\equiv
k+r^{\prime}\pmod{T_{H}}$. By assumption, $p-1\equiv 1\pmod{T_{H}}$; therefore
such a $t$ exists; for large enough $n$, it will also satisfy the constraints
$0\leq t\leq k/p$,
Let now
$\Lambda_{1}:=\left((\mathbb{F}^{+}_{p^{2}})^{t}\times(\mathbb{F}^{+}_{p})^{k-tp}\right)\rtimes\mathbb{F}_{p^{2}}^{\times}$
acting on $[pk]$ with $t$ orbits of size $p^{2}$ and $k-pt$ orbits of size $p$
as follows: on an orbit of size $p^{i}$ ($i=1,2$) the action is
$\operatorname{AGL}(1,p^{i})$. The additive groups act independently, with a
single multiplicative action on top. $\mathbb{F}_{p^{2}}^{\times}$ acts on
$\mathbb{F}_{p}^{+}$ through the group homomorphism
$\mathbb{F}_{p^{2}}^{\times}\to\mathbb{F}_{p}^{\times}$ defined by the map
$x\mapsto x^{p-1}$. Let $B_{j}$ denote an orbit of $\Lambda_{1}$ on $[kp]$.
Now the orbit of any pair $\\{u,v\\}\in{B_{j}\choose 2}$ is a clique of size
$|B_{j}|\geq p\geq h$, therefore a $\Lambda_{1}$-invariant graph cannot
contain an intra-cluster edge.
Let $d_{i}$ be the largest power of 2 that divides $q_{i}-1.$ Let $C_{i}$ be
the subgroup of $\mathbb{F}_{q_{i}}^{\times}$ of order $d_{i}.$ Let
$\displaystyle{\Lambda_{2}:=\prod_{i=1}^{r^{\prime}}\Gamma(q_{i},d_{i}),}$
acting on $[k^{\prime}]$ with $r^{\prime}$ orbits of sizes
$q_{1},\dots,q_{r^{\prime}}$ in the obvious manner.
From Lemma 3.2 we know that the orbit of any $\\{u,v\\}\in{[q_{i}]\choose 2}$
must contain a clique of size $h.$ Hence, an invariant graph cannot contain
any intra-cluster edge.
Overall, let $\Gamma:=\Lambda_{1}\times\Lambda_{2}$, acting on $[n].$ Since
$q_{i}\geq p,$ we have $\gcd(q_{i},p^{2}-1)=1.$ Thus, $\Gamma$ is a “$2$-group
extension of a cyclic extension of a $p$-group” and therefore satisfies
Oliver’s Condition (stated before Theorem 2.1). Hence, assuming $Q_{n}^{H}$ is
non-evasive, Lemma 2.2 and Theorem 2.1 imply
$\chi((Q_{n}^{H})_{\Gamma})\equiv 1\pmod{2}.$
On the other hand, we claim that the fixed-point complex
$(Q_{n}^{H})_{\Gamma}$ is isomorphic to $Q_{r}^{[[H]]}$. The (simple) proof
goes along the lines of Lemma 4.2 of [3]. Thus, by Lemma 3.1 we have
$\chi(Q_{r}^{[[H]]})\equiv 0\pmod{2},$ a contradiction. ∎
### 3.4. Unconditionally, $Q_{n}^{H}$ is only $O(1)$ away from being evasive
In this section, we prove part (c) of Theorem 1.3.
###### Theorem 3.6.
For every graph $H$ there exists a number $C_{H}$ such that the query
complexity of $Q_{n}^{H}$ is $\geq\binom{n}{2}-C_{H}.$
Proof. Let $h$ be the number of vertices of $H$. Let $p$ be the smallest prime
such that $p\geq h$ and $p\equiv 2\pmod{T_{H}}$. So $p<f(H)$ for some function
$f$ by Dirichlet’s Theorem (we don’t need any specific estimates here). Since
$p-1\equiv 1\pmod{T_{H}},$ we have $\gcd(p-1,T_{H})=1$ and therefore
$\gcd(p-1,pT_{H})=1$. Now, by the Chinese Remainder Theorem, select the
smallest positive integer $k^{\prime}$ satisfying $k^{\prime}\equiv
n\pmod{pT_{H}}$ and $k^{\prime}\equiv 1\pmod{p-1}$. Note that
$k^{\prime}<p^{2}T_{H}$. Let $k=(n-k^{\prime})/(pT_{H})$; so we have
$n=kpT_{H}+k^{\prime}$.
Let $N^{\prime}=\binom{n}{2}-\binom{k^{\prime}}{2}$. Consider the following
Boolean function $B_{n}^{H}$ on $N^{\prime}$ variables. Consider graphs $X$ on
the vertex set $[n]$ with the property that they have no edges among their
last $k^{\prime}$ vertices. These graphs can be viewed as Boolean functions of
the remaining $N^{\prime}$ variables. Now we say that such a graph has
property $B_{n}^{H}$ if it does not contain $H$ as a subgraph.
Claim. The function $B_{n}^{H}$ is evasive.
The Claim immediately implies that the query complexity of $Q_{n}^{H}$ is at
least $N^{\prime}$, proving the Theorem with
$C_{H}=\binom{k^{\prime}}{2}<p^{4}T_{H}^{2}<f(H)^{4}T_{H}^{2}$.
To prove the Claim, consider the groups
$\Lambda:=(\mathbb{F}_{p}^{+})^{kT_{H}}\rtimes\mathbb{F}_{p}^{\times}$ and
$\Gamma:=\Lambda\times\mathbb{Z}_{k^{\prime}}$. Here $\Lambda$ acts on
$[pkT_{H}]$ in the obvious way: we divide $[pkT_{H}]$ into $kT_{H}$ blocks of
size $p$; $\mathbb{F}_{p}^{+}$ acts on each block independently and
$\mathbb{F}_{p}^{\times}$ acts on the blocks simultaneously (diagonal action)
so on each block they combine to an $\operatorname{AGL}(1,p)$-action.
$\mathbb{Z}_{k^{\prime}}$ acts as a $k^{\prime}$-cycle on the remaining
$k^{\prime}$ vertices. So $\Gamma$ is a cyclic extension of a $p$-group
(because $\gcd(p-1,k^{\prime})=1$).
If $B_{n}^{H}$ is not evasive then from Theorem 2.1 and Lemma 2.2, we have
$\chi\left((B_{n}^{H})_{\Gamma}\right)=1$.
On the other hand we claim that, $(B_{n}^{H})_{\Gamma}\cong Q_{r}^{[[H]]},$
where $r=kT_{H}+1.$ The proof of this claim is exactly the same as the proof
of Lemma 4.2 of [3]. Thus, from Lemma 3.1, we conclude that
$\chi(Q_{r}^{[[H]]})$ is even. This contradicts the previous conclusion that
$\chi(Q_{r}^{[[H]]})=1.$ ∎
###### Remark 3.7.
Specific estimates on the smallest Dirichlet prime can be used to estimate
$C_{H}$. Linnik’s theorem implies $C_{H}<h^{O(1)}$, extending Theorem 3.6 to
strong lower bounds for variable $H$ up to $h=n^{c}$ for some positive
constant $c$.
## 4\. Sparse graphs: unconditional results
We prove part (c) of Theorem 1.4.
###### Theorem 4.1.
If the non-empty monotone graph property $P_{n}$ is not evasive then
$\dim(P_{n})=\Omega(n\log n).$
### 4.1. The basic group construction
Assume in this section that $n=p^{\alpha}k$ where $p$ is prime. Let
$\Delta_{k}\leq\Sigma_{k}$. We construct the group
$\Gamma_{0}(p^{\alpha},\Delta_{k})$ acting on $[n].$
Let $\Delta=(\mathbb{F}_{p^{\alpha}}^{\times}\times\Delta_{k})$. Let
$\Gamma_{0}(p^{\alpha},\Delta_{k})$ be the semidirect product
$(\mathbb{F}_{p^{\alpha}}^{+})^{k}\rtimes\Delta$ with respect to the
$\Delta$-action on $(\mathbb{F}_{p^{\alpha}}^{+})^{k}$ defined by
$(a,\sigma):(b_{1},\ldots,b_{k})\mapsto(ab_{\sigma^{-1}(1)},\ldots,ab_{\sigma^{-1}(k)}).$
We describe the action of $\Gamma_{0}(p^{\alpha},\Delta_{k})$ on $[n]$.
Partition $[n]$ into $k$ clusters of size $p^{\alpha}$ each. Identify each
cluster with the field of order $p^{\alpha},$ i.e., as a set,
$[n]=[k]\times\mathbb{F}_{p^{\alpha}}.$ The action of
$\gamma=(b_{1},\ldots,b_{k},a,\sigma)$ is described by
$\gamma:(x,y)\mapsto(\sigma(x),ay+b_{\sigma(x)}).$
An unordered pair $(i,j)\in[n]$ is termed an intra-cluster edge if both $i$
and $j$ are in the same cluster, otherwise it is termed an inter-cluster edge.
Note that every u-orbital under $\Gamma$ has only intra-cluster edges or only
inter-cluster edges. Denote by $m_{\operatorname{intra}}$ and
$m_{\operatorname{inter}}$ the minimum sizes of u-orbitals of intra-cluster
and inter-cluster edges respectively.
We denote by $m^{\prime}_{k}$ the minimum size of an orbit in $[k]$ under
$\Delta_{k}$ and by $m^{\prime\prime}_{k}$ the minimum size of a u-orbital in
$[k].$ We then have:
$m_{\operatorname{intra}}\geq\binom{p^{\alpha}}{2}\times m^{\prime}_{k},\qquad
m_{\operatorname{inter}}\geq(p^{\alpha})^{2}\times m^{\prime\prime}_{k}$
Let $m_{k}`:=\min\\{m^{\prime}_{k},m^{\prime\prime}_{k}\\}$ and define $m^{*}$
as the minimum size of a u-orbital in $[n].$ Then
$m^{*}=\min\\{m_{\operatorname{intra}},m_{\operatorname{inter}}\\}=\Omega(p^{2\alpha}m_{k})$
(4.1)
### 4.2. Vinogradov’s Theorem
The Goldbach Conjecture asserts that every even integer can be written as the
sum of two primes. Vinogradov’s Theorem [24] says that every sufficiently
large odd integer $k$ is the sum of three primes $k=p_{1}+p_{2}+p_{3}$. We use
here Haselgrove’s version [7] of Vinogradov’s theorem which states that we can
require the primes to be roughly equal: $p_{i}\sim k/3$. This can be combined
with the Prime Number Theorem to conclude that every sufficiently large even
integer $k$ is a sum of four roughly equal primes.
### 4.3. Construction of the group
Let $n=p^{\alpha}k$ where $p$ is prime. Assume $k$ is not bounded. Write $k$
as a sum of $t\leq 4$ roughly equal primes $p_{i}$. Let
$\Delta_{k}:=\prod_{i}\mathbb{Z}_{p_{i}}$ where $\mathbb{Z}_{p_{i}}$ denotes
the cyclic group of order $p_{i}$ and the direct product is taken over the
distinct $p_{i}$.
$\Delta_{k}$ acts on $[k]$ as follows: partition $k$ into parts of sizes
$p_{1},\dots,p_{t}$ and call these parts $[p_{i}].$ The group
$\mathbb{Z}_{p_{i}}$ acts as a cyclic group on the part $[p_{i}].$ In case of
repetitions, the same factor $\mathbb{Z}_{p_{i}}$ acts on all the parts of
size $p_{i}.$
We follow the notation of Section 4.1 and consider the group
$\Gamma_{0}(p^{\alpha},\Delta_{k})$ with our specific $\Delta_{k}$. We have
$m_{k}=\Omega(k)$ and hence we get, from equation (4.1):
###### Lemma 4.2.
Let $n=p^{\alpha}k$ where $p$ is a prime. For the group
$\Gamma_{0}(p^{\alpha},\Delta_{k})$, we have
$m^{*}=\Omega(p^{2\alpha}k)=\Omega(p^{\alpha}n),$ where $m^{*}$ denotes the
minimum size of a u-orbital.
### 4.4. Proof for the superlinear bound
Let $n=p^{\alpha}k$ where $p^{\alpha}$ is the largest prime power dividing
$n$; so $p^{\alpha}=\Omega(\log n)$; this will be a lower bound on the size of
u-orbitals. Our group $\Gamma$ will be of the general form discussed in
Section 4.1.
Case 1. $p^{\alpha}=\Omega(n^{2/3}).$
Let $\Gamma=\Gamma_{0}(p^{\alpha},\\{1\\})$. Following the notation of Section
4.1, we get $m_{k}^{\prime}=m_{k}^{\prime\prime}=1,$ and this yields that
$m^{*}=\Omega((p^{\alpha})^{2})=\Omega(n^{4/3})=\Omega(n\log n).$ Oliver’s
condition is easily verified for $\Gamma$.
Case 2. $k=\Omega(n^{1/3}).$
Consider the $\Gamma:=\Gamma_{0}(p^{\alpha},\Delta_{k})$ acting on $[n]$ where
$\Delta_{k}$ is as described in Section 4.3. The minimum possible size $m^{*}$
of a u-orbital is $\Omega(np^{\alpha})$ by Lemma 4.2. Finally, since
$p^{\alpha}=\Omega(\log n)$, we obtain $m^{*}=\Omega(n\log n).$
If all $p_{i}$ are co-prime to $p^{\alpha}-1$ then
$\mathbb{F}_{p^{\alpha}}^{\times}\times\Delta_{k}$ becomes a cyclic group and
$\Gamma$ becomes a cyclic extension of a $p$-group.
Since $p_{i}=\Omega(k)=\Omega(n^{1/3})$ for all $i$ and
$p^{\alpha}=O(n^{2/3})$, size considerations yield that at most one $p_{i}$
divides $p^{\alpha}-1$ and $p_{i}^{2}$ does not. Suppose, without loss of
generality, $p_{1}$ divides $p^{\alpha}-1.$ Let $p^{\alpha}-1=p_{1}d,$ then
$d$ must be co-prime to each $p_{i}.$ Thus,
$\Delta=(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{d})\times(\mathbb{Z}_{p_{1}}\times\ldots\times\mathbb{Z}_{p_{t}})=(\mathbb{Z}_{d}\times\mathbb{Z}_{p_{2}}\times\ldots\times\mathbb{Z}_{p_{r}})\times(\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}}).$
Thus, $\Delta$ is a $p_{1}$-group extension of a cyclic group. Hence, $\Gamma$
satisfies Oliver’s Condition (cf. Theorem 2.1). ∎
###### Remark 4.3.
For almost all $n,$ our proof gives a better dimension lower bound of
$\Omega(n^{1+\frac{1+o(1)}{\ln\ln n}}).$
## 5\. Sparse graphs: conditional improvements
In this section we prove parts (a) and (b) of Theorem 1.4.
### 5.1. General Setup
Let $n=pk+r,$ where $p$ and $r$ are prime numbers. Let $q$ be a prime divisor
of $(r-1).$ We partition $[n]$ into two parts of size $pk$ and $r$, denoted by
$[pk]$ and $[r]$ respectively. We now construct a group $\Gamma(p,q,r)$ acting
on $[n]$ as a direct product of a group acting on $[pk]$ and a group acting on
$[r],$ as follows:
$\Gamma=\Gamma(p,q,r):=\Gamma_{0}(p,\Delta_{k})\times\Gamma(r,q)$
Here, $\Gamma_{0}(p,\Delta_{k})$ acts on $[pk]$ and is as defined in Section
4.3, and involves choosing a partition of $k$ into upto four primes that are
all $\Omega(k).$
$\Gamma(r,q)$ is defined as the semidirect product $\mathbb{F}_{r}^{+}\rtimes
C_{q},$ with $C_{q}$ viewed as a subgroup of the group
$\mathbb{F}_{r}^{\times}.$ It acts on $[r]$ as follows: We identify $[r]$ with
the field of size $r.$ Let $(b,a)$ be a typical element of $\Gamma_{r}$ where
$b\in\mathbb{F}_{r}$ and $a\in C_{q}.$ Then, $(b,a):x\mapsto ax+b.$
Thus, $\Gamma=\Gamma(p,q,r)$ acts on $[n].$ Let $m^{*}$ be the minimum size of
the orbit of any edge $(i,j)\in{[n]\choose 2}$ under the action of $\Gamma.$
One can show that
$m^{*}=\Omega(\min\\{p^{2}k,pkr,qr\\}).$ (5.1)
We shall choose $p,q,r$ carefully such that (a) the value of $m^{*}$ is large,
and (b) Oliver’s condition holds for $\Gamma(p,q,r)$.
### 5.2. ERH and Dirichlet primes
The Extended Riemann Hypothesis (ERH) implies the following strong version of
the Prime Number Theorem for arithmetic progressions. Let $\pi(n,D,a)$ denote
the numer of primes $p\leq n$, $p\equiv a\pmod{D}$. Then for $D<n$ we have
$\pi(n,D,a)=\frac{\operatorname{li}(n)}{\varphi(D)}+O(\sqrt{x}\ln x)$ (5.2)
where $\operatorname{li}(n)=\int_{2}^{n}dt/t$ and the constant implied by the
big-Oh notation is absolute (cf. [16, Ch. 7, eqn. (5.12)] or [2, Thm. 8.4.5]).
This result immediately implies “Bertrand’s Postulate for Dirichlet primes:”
###### Lemma 5.1 (Bertrand’s Postulate for Dirichlet primes).
Assume ERH. Suppose the sequence $D_{n}$ satisfies
$D_{n}=o(\sqrt{n}/\log^{2}n)$. Then for all sufficiently large $n$ and for any
$a_{n}$ relatively prime to $D_{n}$ there exists a prime $p\equiv
a_{n}\pmod{D_{n}}$ such that $\frac{n}{2}\leq p\leq n.$
### 5.3. With ERH but without Chowla
We want to write $n=pk+r,$ where $p$ and $r$ are primes, and with $q$ a prime
divisor of $r-1,$ as described in Section 5.1. Specifically, we try for:
$p=\Theta(n^{1/4}),\quad\frac{n}{4}\leq r\leq\frac{n}{2},\quad
q=\Theta(n^{1/4-\epsilon})$
We claim that under ERH, such a partition of $n$ is possible.
To see this, fix some $p=\Theta(n^{1/4})$ such that $\gcd(p,n)=1.$ Fix some
$q=\Theta(n^{1/4-\epsilon}).$ Now, $r\equiv 1\pmod{q}$ and $r\equiv n\pmod{p}$
solves to $r\equiv a\pmod{pq}$ for some $a$ such that $\gcd(a,pq)=1.$ Since
$pq=\Theta(n^{1/2-\epsilon}),$ we can conclude under ERH (using Lemma 5.1)
that there exists a prime $r\equiv a\pmod{pq}$ such that $\frac{n}{4}\leq
r\leq\frac{n}{2}.$ This gives us the desired partition. One can verify that
our $\Gamma$ satisfies Oliver’s Condition. Equation (5.1) gives
$m^{*}=\Omega(n^{5/4-\epsilon}).$ This completes the proof of part (b) of
Theorem 1.4. ∎
### 5.4. Stronger bound using Chowla’s conjecture
Let $a$ and $D$ be relatively prime. Let $p$ be the first prime such that
$p\equiv a\pmod{D}.$ Chowla’s conjecture tells us that $p=O(D^{1+\epsilon})$
for every $\epsilon>0.$ Using this, we show $m^{*}=\Omega(n^{3/2-\epsilon}).$
We can use Chowla’s conjecture, along with the general setup of Section 5.1,
to obtain a stronger lower bound on $m^{*}.$ The new bounds we hope to achieve
are:
$p=\Theta(\sqrt{n}),\quad n^{1-2.5\delta}\leq r\leq n^{1-0.5\delta},\quad
q=\Theta(n^{1/2-\delta})$
Such a partition is always possible assuming Chowla’s conjecture. To see this,
first fix $p=\Theta(n^{1/2}),$ then fix $q=\Theta(n^{1/2-2\delta})$ and find
the least solution for $r\equiv 1\pmod{q}$ and $r\equiv n\pmod{p},$ which is
equivalent to solving for $r\equiv a\pmod{pq}$ for some $a<pq.$ The least
solution will be greater than $pq$ unless $a$ happens to be a prime. In this
case, we add another constraint, say $r\equiv a+1\pmod{3}$ and resolve to get
the least solution greater than $pq.$ Note that $n^{1-2.5\delta}\leq r\leq
n^{1-0.5\delta}.$ Now, from Equation (5.1), we get the lower bound of
$m^{*}=\Omega(n^{3/2-4\delta}).$ This completes the proof of part (a) of
Theorem 1.4. ∎
### Acknowledgment.
Raghav Kulkarni expresses his gratitude to Sasha Razborov for bringing the
subject to his attention and for helpful initial discussions.
## References
* [1] Babai, L., Gál, A., Wigderson, A.: Superpolynomial lower bounds for monotone span programs. Combinatorica 19 (1999), 301–320.
* [2] Bach, E., Shallit, J.: Algorithmic Number Theory, Vol. 1. The MIT Press 1996.
* [3] Chakrabarti, A., Khot, S., Shi, Y.: Evasiveness of Subgraph Containment and Related Properties. SIAM J. Comput. 31(3) (2001), 866-875.
* [4] Chowla, S. On the least prime in the arithmetical progression. J. Indian Math. Soc. 1(2) (1934), 1–3.
* [5] Davenport, H.: Multiplicative Number Theory. (2nd Edn) Springer Verlag, New York, 1980.
* [6] Granville, A., Pomerance, C.: On the least prime in certain arithmetic progressions. J. London Math. Soc. 41(2) (1990), 193–200.
* [7] Haselgrove, C. B.: Some theorems on the analytic theory of numbers. _J. London Math. Soc._ 36 (1951) 273–277
* [8] Heath-Brown, D. R.: Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambr. Phil. Soc. 83 (1978) 357–376.
* [9] Heath-Brown, D. R.: Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. 64(3) (1992) 265–338.
* [10] Kleitman, D. J., Kwiatkowski, D. J.: Further results on the Aanderaa-Rosenberg Conjecture J. Comb. Th. B 28 (1980), 85–90.
* [11] Kahn, J., Saks, M., Sturtevant, D.: A topological approach to evasiveness. Combinatorica 4 (1984), 297–306.
* [12] Lutz, F. H.: Examples of $\mathbb{Z}$-acyclic and contractible vertex-homogeneous simplicial complexes.. Discrete Comput. Geom. 27 (2002), No. 1, 137–154.
* [13] Mader, W.: Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Math. Ann. 174 (1967), 265–268.
* [14] Mader, W.: Homomorphiesätze für Graphen. Math. Ann. 175 (1968), 154–168.
* [15] Oliver, R.: Fixed-point sets of group actions on finite acyclic complexes. Comment. Math. Helv. 50 (1975), 155–177.
* [16] Prachar, K.: Primzahlverteilung. Springer, 1957.
* [17] Rosenberg A. L.: On the time required to recognize properties of graphs: A problem. SIGACT News 5 (4) (1973), 15–16.
* [18] Rotman, J.: An Introduction to the Theory of Groups. Springer Verlag, 1994.
* [19] Rivest, R.L., Vuillemin, J.: On recognizing graph properties from adjacency matrices. Theoret. Comp. Sci. 3 (1976), 371–384.
* [20] Smith P. A.: Fixed point theorems for periodic transformations. Amer. J. of Math. 63 (1941), 1–8.
* [21] Titchmarsh, E. C.: A divisor problem. Rend. Circ. Mat. Palermo 54 (1930), 419–429.
* [22] Triesch, E.: On the recognition complexity of some graph properties. Combinatorica 16 (2) (1996) 259–268.
* [23] Turán, P.: Über die Primzahlen der arithmetischen Progression. Acta Sci. Math. (Szeged) 8 (1936/37) 226–235.
* [24] Vinogradov, I. M.: The Method of Trigonometrical Sums in the Theory of Numbers (Russian). Trav. Inst. Math. Stekloff 10, 1937.
* [25] Yao, A. C.: Monotone bipartite properties are evasive. SIAM J. Comput. 17 (1988), 517–520.
|
arxiv-papers
| 2010-01-27T00:31:30 |
2024-09-04T02:49:08.008421
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Laszlo Babai, Anandam Banerjee, Raghav Kulkarni, Vipul Naik",
"submitter": "Raghav Kulkarni",
"url": "https://arxiv.org/abs/1001.4829"
}
|
1001.4856
|
21-11-2010
The probability that x and y commute
in a compact group
Karl H. Hofmann and Francesco G. Russo
Abstract. We show that a compact group $G$ has finite conjugacy classes, i.e.,
is an FC-group if and only if its center $Z(G)$ is open if and only if its
commutator subgroup $G^{\prime}$ is finite. Let $d(G)$ denote the Haar measure
of the set of all pairs $(x,y)$ in $G\times G$ for which $[x,y]=1$; this,
formally, is the probability that two randomly picked elements commute. We
prove that $d(G)$ is always rational and that it is positive if and only if
$G$ is an extension of an FC-group by a finite group. This entails that $G$ is
abelian by finite. The proofs involve measure theory, transformation groups,
Lie theory of arbitrary compact groups, and representation theory of compact
groups. Finally, for an arbitrary pair of positive natural numbers $m$ and $n$
we investigate the effect on the group structure of the positivity of the
probability that the powers $x^{m}$ and $y^{n}$ commute for two randomly
picked elements $x,y\in G$. This involves additional additional complications
which are studied in the case of compact Lie groups. Examples and references
to the history of the discussion are given at the end of the paper.
MSC 2010: Primary 20C05, 20P05; Secondary 43A05.
Key words: Probability of commuting pairs, commutativity degree, FC-groups,
compact groups, Haar measure.
1\. Introducing the problem
We fix a compact group $G$ and natural numbers $m,n=1,2,3,\dots$, and we let
$\nu$ denote Haar measure of $G$.
Question. What is the probability that a randomly picked pair $(x,y)\in
G\times G$ of elements of $G$ has the property that $x$ and $y$ commute?
Technically speaking: Derive useful information from possible knowledge on the
real number
$(\nu\times\nu)(\\{(x,y)\in G\times G:[x,y]=1\\}).$
We shall develop a theory which will show that in a compact group $G$ the
probability of the commuting of two randomly picked elements $x$ and $y$ is
positive only in very special groups, namely, those that have a certain
characteristic open abelian normal subgroup. Our theory will allow us to
compute this probability explicitly and to establish that it is always a
rational number.
However, we make efforts to present the theory in a frame work that is general
enough to cover, to a certain extent, the following, more general question:
For fixed natural numbers $m$ and $n$, assume that the probability of the
commuting of the powers $x^{m}$ and $y^{n}$ for randomly selected elements $x$
and $y$ from a compact group $G$ is positive. What are the consequences for
the structure of $G$?
The question applies, in particular, to Lie groups as well as to profinite
groups. In both of these cases, the issue is somewhat precarious. It is
therefore clear that for an answer to this more general question, some
conditions depending on $m$ and $n$ will be needed.
So let us say that a compact group $G$ is $n$-straight for a natural number
$n$ whenever for a closed nowhere dense subgroup $H$ the set $\\{g\in
G:g^{n}\in H\\}$ has Haar measure $0$. Every compact group is 1-straight, and
every finite group is $n$-straight for every $n$, but the profinite group
$({Z}/2{Z})^{N}$ is not $2$-straight since the squaring map is constant, nor
is the Lie group ${R}/{Z}\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt$}\\{1,-1\\}$ (the “continuous dihedral
group”) $2$-straight, since all elements of ${R}/{Z}\times\\{-1\\}$ have order
2.
The FC-center of a group is the set of elements whose conjugacy class is
finite; a group is an FC-group if it agrees with its FC-center. We shall prove
the following Theorems:
Theorem 1.1. (The structure of compact FC-groups) A compact group $G$ is an
FC-group if and only if its center is open, that is, $G$ is central by finite,
if and only if its commutator subgroup is finite, that is, $G$ is finite by
abelian. Theorem 1.2. (The structure of compact groups with frequent commuting
of elements) Let $G$ denote a compact group with Haar measure $\nu$ and FC-
center $F$. Then the following conditions are equivalent:
$(1)$ $(\nu\times\nu)(\\{(x,y)\in G\times G:[x,y]=1\\})>0$.
$(2)$ $F$ is open in $G$.
$(3)$ The center $Z(F)$ of $F$ is open in $G$.
Moroever, if $m$ and $n$ are arbitrary fixed natural numbers and $G$ is an
$m$\- and $n$-straight Lie group, then these conditions are equivalent to each
of the following:
$(4)$ $(\nu\times\nu)(\\{(x,y)\in G\times G:[x^{m},y^{n}]=1\\})>0$.
$(5)$ $G_{0}$ is abelian.
Our strategy in order to include (4) and (5) will be as follows: We write
$p_{n}\colon G\to G$ by $p_{n}(x)=x^{n}$.
The measures $\mu_{1}\buildrel\rm def\over{=}p_{m}(\nu)$ and
$\mu_{2}\buildrel\rm def\over{=}p_{n}(\nu)$ are defined by
$\mu_{1}(B)=p_{m}(\nu)(B)=\nu\big{(}p_{m}^{-1}(B)\big{)}$ for each Borel
subset $B$ of $G$, and analogously for $\mu_{2}$. Now
$D=\\{(x,y):[p_{m}(x),p_{n}(y)]=1\\}=(p_{m}\times
p_{n})^{-1}\\{(u,v):[u,v]=1\\}\\}$,
and thus, setting $E\buildrel\rm def\over{=}\\{(u,v)\in G\times G:[u,v]=1\\}$,
in the context of the theorem, we have
$0<(\nu\times\nu)(D)=(p_{m}\times p_{n})(\nu\times\nu)(E)=(p_{m}(\nu)\times
p_{n}(\nu))(E)=(\mu_{1}\times\mu_{2})(E)$.
We propose a proof based, in the end, on this formula, but we shall have to
draw from different branches of group theory such as the theory of compact
groups and transformation groups, Lie groups, and FC-groups.
2\. Actions and product measures
It is conceptually simpler to start by considering a more general situation.
Let $(g,x)\mapsto g{\cdot}x:G\times X\to X$ be a continuous action $\alpha$ of
a compact group $G$ on a compact space $X$. All spaces in sight are assumed to
be Hausdorff. We specify a Borel probability measure $P$ on $G\times X$ and
discuss the probability that a group element $g\in G$ fixes a phase space
element $x\in X$ for a pair $(g,x)$, randomly picked from $G\times X$, that
is, that $g{\cdot}x=x$. We define
$E\buildrel\rm def\over{=}\\{(g,x)\in G\times X:g{\cdot}x=x\\},$
that is, $E$ is the equalizer of the two functions $\alpha,\mathop{\rm
pr}\nolimits_{X}:G\times X\to X$ and is therefore a closed subset of $G\times
X$.
Let $G_{x}=\\{g\in G:g{\cdot}x=x\\}$ be the isotropy (or stability) group at
$x$ and let $X_{g}=\\{x\in X:g{\cdot}x=x\\}$ be the set of points fixed under
the action of $g$. We note that $G_{g{\cdot}x}=gG_{x}g^{-1}$. The function
$g\mapsto g{\cdot}x:G\to G{\cdot}x$ induces a continuous equivariant bijection
$G/G_{x}\to G{\cdot}x$ which, due to the compactness of $G$, is a
homeomorphism. We have
$\eqalign{E&=\\{(g,x):g\in G_{x},\,x\in X\\}=\bigcup_{x\in
X}G_{x}\times\\{x\\}\cr&=\\{(g,x):g\in G,\,x\in X_{g}\\}=\bigcup_{g\in
G}\\{g\\}\times X_{g}\subseteq G\times X{\cdot}\cr}$
Now we assume that $\mu$ and $\nu$ are Borel probability measures on $G$ and
$X$, respectively, and that $P=\mu\times\nu$ is the product measure. For
information on measure theory the reader may refer to [1]. Let $\chi_{E}\colon
G\times X\to{R}$ be the characteristic function of $E$. We define the function
$m\colon X\to{R}$, $m(x)=\mu(G_{x})$. Then by the Theorem of Fubini we compute
$\leqalignno{P(E)&=\int_{G\times
X}\chi_{E}(g,x)dP=\int_{X}\left(\int_{G}\chi_{E}(g,x)d\mu(g)\right)d\nu(x)&(*)\cr&=\int_{X}\mu(G_{x})d\nu(x)=\int
m\,d\nu.\cr}$
Likewise
$\leqalignno{P(E)&=\int_{G\times
X}\chi_{E}(g,x)dP=\int_{G}\left(\int_{X}\chi_{E}(g,x)d\nu(x)\right)d\mu(g)&(**)\cr&=\int_{G}\nu(X_{g})d\mu(g).\cr}$
We now see from $(*)$ that $P(E)>0$ implies, firstly, that there is at least
one $x\in X$ such that $\mu(G_{x})=m(x)>0$ holds, and that, secondly, the set
of all $x$ for which $m(x)>0$ has positive $\nu$-measure. Likewise, there is
at least one $g\in G$ such that $\nu(X_{g})>0$ and that the set of all of
these $g$ has positive $\mu$-measure. At this point we introduce a terminology
which we shall retain and use
Definition 2.1. Let $\cal C$ be a set of subgroups of a compact group $G$,
such as the set of all closed subgroups or all subgroups whose underlying set
is a Borel subset of $G$. We shall say that a Borel probability measure
$\sigma$ on $G$ respects $\cal C$-subgroups if every subgroup $H\in{\cal C}$
with $\sigma(H)>0$ is open. Recall that an open subgroup $H$ of a topological
group $G$, being the complement of all the cosets $gH$ for $g\notin H$, is
closed and that it contains the identity component $G_{0}$ of $G$. If $G$ is
compact, then $H$ has finite index in $G$.
We claim that
Haar measure $\mu$ on a compact group $G$ respects Borel subgroups.
Indeed, assume that $H$ is a Borel subgroup of $G$ with $\mu(H)>0$. Then by
[9], p. 296, Corollary 20.17, $H=HH$ contains a nonvoid open set and thus is
open.
If, in the cirumstances discussed here, $\mu$ respects closed subgroups, and
$\mu(G_{x})>0$ then the subgroup $G_{x}$ is open in $G$ hence contains
$G_{0}$. If $G$ is a compact Lie group, then the condition $G_{0}\subseteq
G_{x}$ is also sufficient for the openness of the subgroup $G_{x}$ of $G$,
since the identity component of a Lie group is open. If $G_{x}$ is open, then
$G{\cdot}x\cong G/G_{x}$ is discrete and compact, hence finite.
Let now
$F\buildrel\rm def\over{=}\\{x\in X:|G{\cdot}x|<\infty\\}=\\{x\in
X:G_{x}\hbox{ is open}\\}.$ $None$
These remarks require that henceforth whenever the measure $\mu$ occurs we
shall assume that $\mu$ respects closed subgroups.
Lemma 2.2. Let $G$ be a compact group acting on a compact space $X$.
(i) If $G$ is a Lie group, then for each $x\in X$ there is an open invariant
neighborhood $U_{x}$ of $x$ such that all isotropy groups of elements in $U$
are conjugate to a subgroup of the isotropy subgroup $G_{x}$.
(ii) Under these circumstances, $m$ takes its maximum on $U_{x}$ in $x$. That
is, $m^{-1}(\left]-\infty,m(x)\right])$ is a neighborhood of $x$. In
particular, $m$ is upper semicontinuous.
(iii) If $G$ is a Lie group, then the subspace $F$ of $X$ is compact.
(iv) If $G$ is an arbitrary compact group, then the subspace $F$ of $X$ is an
Fσ, that is, a countable union of closed subsets and thus is a Borel subset.
Proof. Assertion (i) is a consequence of the Tube Existence Theorem (see e.g.
[2], p. 86, Theorem 5.4, or [18], p. 40, Theorem 5.7).
(ii) Immediate from (i) and from $m(u)=\mu(G_{u})$.
(iii) We recall that $y\in F$ if and only if $m(y)>0$. Hence $F$ is the
complement of $m^{-1}(0)$. By conclusion (ii), however, $m^{-1}(0)$ is open.
Thus $F$ is closed and therefore compact.
(iv) For a natural number $n\in{N}$ we set $F(n)=\\{x\in X:|G{\cdot}x|\leq
n\\}$. We claim that $F(n)$ is closed in $X$ for all $n\in{N}$. Since
$F=\bigcup_{n=1}^{\infty}F(n)$, this claim will prove assertion (iv).
We prove the claim by contradiction and suppose that there is an $n\in{N}$
such that $\overline{F(n)}$ contains an $x^{\prime}\notin F(n)$. Then there
exist elements $g_{1},\dots,g_{n+1}\in G$ such that
$|\\{g_{1}{\cdot}x^{\prime},\dots,g_{n+1}{\cdot}x^{\prime}\\}|=n+1$. Now we
find a compact normal subgroup $N$ of $G$ such that $G/N$ is a Lie group and
that $Ng_{j}{\cdot}x^{\prime}\cap Ng_{k}{\cdot}x^{\prime}=\emptyset$ for all
$j\neq k$ in $\\{1,\dots,n+1\\}$. The Lie group $G/N$ acts on
$X/N=\\{N{\cdot}x:x\in X\\}$ via
$(gN){\bullet}(N{\cdot}x)=N{\cdot}(g{\cdot}x)$. By what we just saw
$|(G/N){\bullet}(N{\cdot}x_{0})|\geq n+1$. On the other hand,
$F_{N}(n)=\\{N{\cdot}x\in X/N:|(G/N){\bullet}(N{\cdot}x)|\leq n\\}$ is closed
by (ii) above. Since the orbit map $\pi_{N}\colon X\to X/N$ is continuous and
$\pi(F(n))\subseteq F_{N}(n)$ we have
$N{\cdot}x_{0}=\pi(x_{0})\subseteq\overline{F_{N}(n)}=F_{N}(n)$. Thus, by the
definition of $F_{N}(n)$ we have $|(G/N){\bullet}(N{\cdot}x_{0})|\leq n$. This
contradiction proves the claim. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Recall that we assume that $\mu$ respects closed subgroups. Now that we know
that $F\subseteq X$ is a Borel set, hence is $\nu$-measurable, we can state
that, regardless of any particular property of $\nu$, the function $m$
satisfies
$P(E)=\int_{X}md\nu=\int_{X}\chi_{H}{\cdot}md\nu=\int_{x\in F}m(x)d\nu(x).$
Here $x\in F$ implies $0<m(x)=\mu(G_{x})=1/|G/G_{x}|\leq 1$.
Lemma 2.3. If $\mu$ respects closed subgroups and $P(E)>0$, then
$0<P(E)\leq\nu(F)$. In particular, $F\neq\emptyset$.
Proof. We have seen that $P(E)=\int_{F}m\,d\nu$ since $F$ is Borel measurable.
The Lemma then follows from this fact and $m(x)\leq 1$ for $x\in F$. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
We shall say that the group $G$ acts automorphically on $X$ if $X$ is a
compact group and $x\mapsto g{\cdot}x:X\to X$ is an automorphism for all $g\in
G$.
Lemma 2.4. Assume that $G$ and $X$ are compact groups and assume the following
hypotheses:
(a) $G$ acts automorphically on $X$.
(b) $\mu$ respects closed subgroups.
(c) $\nu$ respects Borel subgroups or else $X$ is a Lie group and $\nu$
respects closed subgroups.
(d) $P(E)>0$.
Then $F$ is an open, hence closed subgroup of $X$. Proof. Let $x,y\in F$. Then
$G{\cdot}x$ and $G{\cdot}y$ are finite sets by the definition of $F$. Now
$G(xy^{-1})=\\{g{\cdot}(xy^{-1}):g\in G\\}=\\{(g{\cdot}x)(g{\cdot}y)^{-1}:g\in
G\\}\subseteq\\{(g{\cdot}x)(h{\cdot}y)^{-1}:g,h\in
G\\}=(G{\cdot}x)(G{\cdot}y)^{-1}$, and the last set is finite as a product of
two finite sets. Thus $xy^{-1}\in F$ and $F$ is a subgroup. By Lemma 2.3,
$\nu(F)>0$. Then by Lemma 2.2(iii),(iv) and the kind of subgroups respected by
$\nu$, we conclude that $F$ is an open subgroup. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
If $G$ acts automorphically on a compact group $X$, we let $\pi\colon
G\to\mathop{\rm Aut}\nolimits\ X$ be the representation given by
$\pi(g)(x)=g{\cdot}x$. Let $\mathop{\rm id}\nolimits_{X}$ denote the identity
function of $X$. Then the fixed point set $X_{g}$ is the equalizer of the
morphisms $\pi(g)$ and $\mathop{\rm id}\nolimits_{G}$ and is therefore a
closed subgroup of $X$. Let $I\subseteq G$ denote the set of all $g\in G$ for
which $X_{g}$ has inner points.
Lemma 2.5. Assume that $G$ and $X$ are compact groups and assume the following
hypotheses:
(a) $\mu$ and $\nu$ are the Haar measures on $G$ and $X$, respectively.
(b) $G$ acts automorphically on $X$.
(c) $G$ is finite.
Then $P(E)={1\over|G|}{\cdot}\sum_{g\in I}|X/X_{g}|^{-1}$. In particular,
$P(E)$ is a rational number. Proof. By $(**)$ above and the fact that Haar
measure on a finite group $G$ is counting measure with
$\mu(\\{g\\})=|G|^{-1}$, we have $P(E)={1\over|G|}{\cdot}\sum_{g\in
G}\nu(X_{g})$. If a closed subgroup $Y$ of the compact group $X$ has no inner
points, its Haar measure $\nu(Y)$ is zero. If it has inner points, it is open
and its measure $\nu(Y)$ is the reciprocal of its index, that is
$\nu(Y)=|X/Y|^{-1}$. Hence $P(E)={1\over|G|}{\cdot}\sum_{g\in
I}|X/X_{g}|^{-1}$ and the assertion follows. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
Our conclusions sum up to the following result: Proposition 2.6. Let $G$ and
$X$ be compact groups and assume that $G$ acts automorphically on $X$. Let
$\mu$ and $\nu$ be normalized positive Borel measures on $G$ and $X$,
respectively. Define
$E=\\{(g,x)\in G\times X:g{\cdot}x=x\\}\subseteq G\times X,$
whence $(\mu\times\nu)(E)=\int_{x\in F}\mu(G_{x})d\nu(x)$. Assume that $\mu$
respects closed subgroups and that $\nu$ either respects Borel subgroups of
$X$ or else $X$ is a Lie group and $\nu$ respects closed subgroups, then the
following statements are equivalent:
$(1)$ $(\mu\times\nu)(E)>0$.
$(2)$ The subgroup $F\leq X$ of all elements with finite $G$-orbits is open
and thus has finite index in $X$. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
3\. Action by inner automorphism
The main application of this general situation will be the case of a compact
group $G$ and the automorphic action of $G$ on $X=G$ via inner automorphisms:
$(g,x)\mapsto g{\cdot}x=gxg^{-1}:G\times X\to X.$
The orbit $G{\cdot}x$ of $x$ is the conjugacy class $C(x)$ of $x$, and the
isotropy group $G_{x}$ of the action at $x$ is the centralizer $Z(x,G)=\\{g\in
G:gx=xg\\}$ of $x$ in $G$. The set $E$ is the set $D=\\{(x,y)\in G\times
G:[x,y]=1\\}$, and $F$ is the union of all finite conjugacy classes. In
particular, $F$ is a characteristic Fσ subgroup of $G$ whose elements have
finite conjugacy classes, that is, the FC-center of $G$. Recall that a group
agreeing with its FC-center is called an FC-group.
In this setting, Proposition 2.6 has the following consequence:
Corollary 3.1. Let $G$ be a compact group and let $\mu$ and $\nu$ be Borel
probability measures on $G$ and assume that $\mu$ respects closed subgroups
and that $\nu$ respects Borel subgroups or, if $G$ is a Lie group, that $\nu$
respects closed subgroups.
Let $F$ be the FC-center of $G$. Then $F$ is an Fσ and we define
$D=\\{(g,x)\in G\times G:[g,x]=1\\}\subseteq G\times G.$
Then
$P(D)=\int_{x\in F}\mu(Z(x,G))d\nu(x),$
and the following statements are equivalent:
$(1)$ $(\mu\times\nu)(D)>0$.
$(2)$ $F$ is open in $G$ and thus has finite index in $G$. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
Moreover, under these conditions, $Z(F,G)$ contains the identity component
$G_{0}$, and the profinite group $G/Z(F,G)$ is acting effectively on $F$ with
orbits being exactly the finite conjugacy classes of $G$. We shall see later
in 3.10, that the center $Z(F)$ is an open subgroup, whence $Z(F,G)$,
containing $Z(F)$, is an open subgroup of $G$. Therefore $G/Z(F,G)$ will in
fact turn out to be finite.
In a nontechnical spirit, let us call a subgroup $U$ of $G$ large if it is
open, hence closed and with finite index in $G$. In 3.1 (2) we state that $F$
is a large characteristic subgroup which is itself an FC-group. Therefore, for
finding large abelian subgroups of $G$ under the equivalent hypotheses of 3.1,
it is sufficient to concentrate on compact FC-groups which we shall do now;
and while we are classifying compact FC-groups, measures will not play any
role. In a compact FC-group $G$, the identity component $G_{0}$ is abelian by
the structure theory of compact connected groups ([10], 9.23ff). Abbreviate
the centralizer $Z(G_{0},G)$ of $G_{0}$ by $C$. The action by inner
automorphisms of $G$ on $G_{0}$ induces an effective automorphic action of the
profinite group $G/C$ on the compact connected abelian group $G_{0}$ with
finite orbits. Let the Lie algebra ${g}={L}(G_{0})$ be defined as $\mathop{\rm
Hom}\nolimits({R},G_{0})\cong\mathop{\rm Hom}\nolimits(\widehat{G_{0}},{R})$
(see [10], Proposition 7.36, Theorem 7.66). The Lie algebra ${g}$ is a weakly
complete vector space, which is isomorphic as a topological vector space to
the complete locally convex space ${R}^{I}$ for a suitable set $I$ whose
cardinality is the rank of the torsionfree abelian group $\widehat{G_{0}}$.
There is a natural morphism $\mathop{\rm Aut}\nolimits G_{0}\to\mathop{\rm
Aut}\nolimits{g}$: Indeed each automorphism $\alpha$ of $G_{0}$ induces an
automorphism ${L}(\alpha)$ of ${g}$ as follows: Let $X\colon{R}\to G_{0}$,
$t\mapsto X(t)$, be a member of ${g}$, then ${L}(\alpha)(X)\colon{R}\to G_{0}$
is defined by ${L}(\alpha)(X)=\alpha\circ X$. An element $g\in G$ induces an
inner automorphism $I_{g}$ on $G_{0}$ via $I_{g}(x)=gxg^{-1}$ and the function
$g\mapsto I_{g}:G\to\mathop{\rm Aut}\nolimits{g}$ factors through the quotient
$G\to G/C$ giving us a chain of representations
$G\to G/C\to\mathop{\rm Aut}\nolimits G_{0}\to\mathop{\rm Aut}\nolimits{g},$
whose composition yields a continuous representation
$\pi\colon G/C\to\mathop{\rm Aut}\nolimits{g},\quad\pi(gC)(X)=I_{g}\circ
X,\hbox{ that is, }\pi(gC)(X)(t)=gX(t)g^{-1}.$
We claim that the fact, that every element of $G$ has finitely many
conjugates, implies for each $X\in{g}$ that $\pi(G/C)(X)\subseteq{g}$ is
contained in a finite dimensional vector subspace of ${g}$. That is, we
maintain that the $G/C$-module ${g}$ (see [10], Definition 2.2) satisfies
${g}_{\rm fin}={g}$. (See [10], Definition 3.1.) In order to prove this claim
in the next lemma we recall that a subgroup of a topological group is called
monothetic if it contains a dense cyclic subgroup, and solenoidal if it
contains a dense one-parameter subgroup. Lemma 3.2. Let $\Gamma$ be a compact
group acting automorphically on a compact group $G$. Assume that all orbits of
$\Gamma$ on $G$ are finite. Then the following conclusions hold:
(i) For each monothetic subgroup $M=\overline{\langle g\rangle}$ of $G$ there
is an open normal subgroup $\Omega$ of $\Gamma$ which fixes the elements of
$M$ elementwise and the finite group $\Gamma/\Omega$ acts on $M$ with the same
orbits as $\Gamma$.
(ii) The orbits of $\Gamma$ on ${g}={L}(G)$ for the induced automorphic action
of $\Gamma$ on ${L}(G)$ are finite.
Proof. (i) Assume that $A=\overline{\langle g\rangle}$ is a monothetic
subgroup. If $\alpha\in\Gamma$, then $\alpha\in\Gamma_{g}$ means
$\alpha{\cdot}g=g$, that is, $g$ belongs to the fixed point subgroup
Fix$(\alpha)$ of $\alpha$. This is equivalent to $A\subseteq{\rm
Fix}(\alpha)$, i.e., $\alpha\in\Gamma_{a}$ for all $a\in A$. So we have
$\Gamma_{g}\subseteq\Gamma_{a}$ for all $a\in A$. Then the normal finite index
subgroup $\Omega_{g}=\bigcap_{\gamma\in\Gamma}\gamma\Gamma_{g}\gamma^{-1}$ is
contained in all $\Gamma_{a}$, $a\in A$. This establishes (i) as the remainder
is clear. (ii) A compact abelian group $A$ is monothetic if and only if there
is a morphism $f\colon{Z}\to A$ with dense image exactly when (in view of
Pontryagin duality) there is an injective morphism
$\widehat{f}\colon\widehat{A}\to\widehat{Z}\cong{T}$. In the same spirit a
compact abelian group is solenoidal if and only if there is a morphism
$f\colon{R}\to A$ with dense image exactly when (in view of Pontryagin
duality) there is an injective morphism
$\widehat{f}\colon\widehat{A}\to\widehat{R}\cong{R}$. As $\widehat{A}$ is
discrete, this happens if and only if $\widehat{A}$ is algebraically a
subgroup of ${R}$. Now ${T}$ has a subgroup algebraically isomorphic to ${R}$
(see [10], Corollary A1.43). Thus if $\widehat{A}$ can be homomorphically
injected into ${R}$ it can be homomorphically injected into ${T}$. Therefore
every solenoidal compact group is monothetic.
Thus let $X\in{g}$ be a one-parameter subgroup of $G$. Then by (i), the image
$X({R})$ is contains in a monothetic subgroup. Hence by (i) above there is an
open normal subgroup $\Omega$ of $\Gamma$ such that each $\alpha\in\Omega$
satisfies $\alpha{\cdot}X(t)=X(t)$ for all $t\in{R}$. Hence $\alpha{\cdot}X=X$
in ${g}$ with respect to the action of $\Gamma$ induced on ${g}={L}(G)$. Hence
$\Gamma_{X}\supseteq\Omega$ has finite index for the action of $\Gamma$ on
${g}$. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Lemma 3.2(ii) completes the argument that the $G$-module ${g}$ in the case of
a compact FC-group $G$ satisfies ${g}_{\rm fin}={g}$.
In the following we refer to to the representation theory of compact groups as
presented in [10], Chapters 3 and 4.
Lemma 3.3. Let $V$ be a locally convex weakly complete vector space isomorphic
to ${R}^{I}$ and an effective $\Gamma$-module for a profinite group $\Gamma$.
In particular, the associated representation $\pi\colon\Gamma\to\mathop{\rm
Aut}\nolimits V$ is injective. Assume that $V_{\rm fin}=V$. Then
(i) $V$ is a finite direct sum (and product) $V_{1}\oplus\cdots\oplus V_{k}$
of isotypic components.
(ii) $\Gamma$ is finite.
Proof. (i) By [10], Theorem 4.22, $V_{\rm fin}$ is a direct sum of its
isotypic components $V_{\epsilon}$, where $\epsilon\in\widehat{\Gamma}$ is an
equivalence class of irreducible representations of $\Gamma$. Each
$V_{\epsilon}$ is a module retract of $V$ under a canonical projection
$P_{\epsilon}$ and is therefore complete, and thus as a closed vector subspace
of a weakly complete vector space is weakly complete. The universal property
of the product $W\buildrel\rm
def\over{=}\prod_{\epsilon\in\widehat{\Gamma}}V_{\epsilon}$ gives us an
equivariant $\Gamma$-module morphism $\phi\colon V\to W$ of weakly complete
$\Gamma$-modules such that $\mathop{\rm pr}\nolimits_{\epsilon}\ \circ\
\phi=P_{\epsilon}$. Since morphisms of weakly complete vector spaces have a
closed image (see [11], Theorem A2.12(b)) and
$\sum_{\epsilon\in\widehat{\Gamma}}V_{\epsilon}=\\{(v_{\epsilon})_{\epsilon\in\widehat{\Gamma}}\in
W:v_{\epsilon}=0\hbox{ for all but finitely many
}\epsilon\in\widehat{\Gamma}\\}\subseteq W$
is in the image of $\phi$ and is dense in $W$ we know that $\phi$ is
surjective. Now $V=V_{\rm fin}$ and $\phi(V_{\rm fin})\subseteq W_{\rm fin}$
whence $W_{\rm fin}=W$. It is readily verified that
$W_{\epsilon}=\\{(v_{\eta})_{\eta\in\widehat{\Gamma}}\in W:v_{\eta}=0\hbox{
for }\eta\neq\epsilon\\}$.
Therefore $W_{\rm fin}=\sum_{\epsilon\in\widehat{\Gamma}}V_{\epsilon}$, and
thus
$\sum_{\epsilon\in\widehat{\Gamma}}V_{\epsilon}=W=\prod_{\epsilon\in\widehat{\Gamma}}V_{\epsilon}.$
This equation, however, implies that the set $S\buildrel\rm
def\over{=}\\{\epsilon\in\widehat{\Gamma}:V_{\epsilon}\neq\\{0\\}\\}$ is
finite. List the members of $\\{V_{\epsilon}:\epsilon\in S\\}$ as
$V_{1},\dots,V_{k}$. Then assertion (i) follows. (ii) For a simple
$\Gamma$-module $F$ denote by $[F]\in\widehat{\Gamma}$ its equivalence class.
Let $F_{j}$, $j=1,\dots,k$ be simple $\Gamma$-modules such that
$S=\\{[F_{j}]:j=1,\dots,k\\}$, $V_{j}=V_{[F_{j}]}$. Then let
$\pi_{j}\colon\Gamma\to{\rm GL}(F_{j})$, $j=1,\dots,k$ denote the associated
representations defined by $\pi_{j}(g)(v)=g{\cdot}v$. By [10], Theorem 4.22,
$V_{j}\cong\mathop{\rm Hom}\nolimits_{\Gamma}(F_{j},V)\otimes F_{j}$, where
$g\in\Gamma$, $f\in\mathop{\rm Hom}\nolimits(F_{j},V)$, and $v\in V$ imply
$g{\cdot}(f\otimes v)=f\otimes g{\cdot}v$. Since $V$ is an effective (or
faithful) $\Gamma$-module, (i) implies $\bigcap_{j=1}^{k}\ker\pi_{j}=\\{1\\}$.
However, a simple $\Gamma$-module is finite-dimensional (see [10], Theorem
3.51), and so GL$(F_{j})$ is a Lie group. Hence
$\Gamma/\ker\pi_{j}\cong\mathop{\rm im}\nolimits\pi_{j}$ is a compact
profinite Lie group and is therefore finite. Hence $\ker\pi_{j}$ is open in
$\Gamma$. Thus $\\{1\\}$, being a finite intersection of open subgroups, is
open in $\Gamma$. Therefore, $\Gamma$ is discrete and compact, hence finite.
$\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Thus we get the following result: Lemma 3.4. Let $G$ be a compact FC-group.
Then the centralizer $Z(G_{0},G)$ of the identity component is open. Proof. By
Lemma 3.2 we can apply Lemma 3.3 with $\Gamma=G/Z(G_{0},G)$ and the
$\Gamma$-module ${g}={L}(G)={L}(G_{0})$. We conclude that $\Gamma$ is finite.
This proves the assertion. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Let us explain the structure of a central extension of a compact abelian group
by a profinite group:
Proposition 3.5. Let $G$ be a compact group such that $G_{0}$ is central. Then
there is a profinite normal subgroup $\Delta$ such that $G=G_{0}\Delta$. In
particular,
$G\cong{G_{0}\times\Delta\over D},\quad D=\\{(g,g^{-1}):g\in
G_{0}\cap\Delta\\}.$
Proof. See [10], Theorem 9.41 and Corollary 9.42. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
In order to pursue the structure of compact FC-groups further we cite Lemma
2.6, p. 1281 from the paper by Shalev [17], proved with the aid of the Baire
Category Theorem: Lemma 3.6. If $G$ is a profinite FC-group then its
commutator subgroup $G^{\prime}$ is finite. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
In order to exploit this information, we need a further lemma: Lemma 3.7. If
$N$ is a compact nilpotent group of class $\leq 2$ and $N^{\prime}$ is
discrete, then the center $Z(N)$ has finite index in $N$. Proof. Since $N$ is
nilpotent of class at most 2, we have $N^{\prime}\subseteq Z(N)$. Hence so for
each $y\in N$ the function $x\mapsto[x,y]:N\to N^{\prime}$ is a morphism.
Therefore, if we set $A=N/N^{\prime}$ we have a continuous ${Z}$-bilinear map
of abelian groups $b\colon A\times A\to N^{\prime}$, where
$b(xN^{\prime},yN^{\prime})=[x,y]$. Since $N^{\prime}$ is discrete, $\\{1\\}$
is a neighborhood of the identity in $N^{\prime}$. On the other hand,
$b(\\{1_{A}\\}\times A)=\\{1\\}$. So for each $a\in A$ we have open
neighborhoods $U_{a}$ of $1_{A}$ and $V_{a}$ of $a$, respectively, such that
$b(U_{a}\times V_{a})=\\{1\\}$ As $A$ is compact, we find a finite set
$F\subseteq A$ of elements such that $A=\bigcup_{a\in F}V_{a}$. Let
$U=\bigcap_{a\in F}U_{a}$; then $U$ is an identity neighborhood of $A$ and
$b(U\times A)=\\{1\\}$. Since $b$ is bilinear, this implies $b(\langle
U\rangle\times A)=\\{1\\}$. Then the full inverse image $M$ of $\langle
U\rangle$ in $N$ under the quotient morphism $N\to A$ is an open subgroup of
$N$ satisfying $[M,N]=\\{1\\}$ and is, therefore, central. Thus the center
$Z(N)$ of $N$ is open and so has finite index in $N$. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
We remark that the proof of this lemma resembles that of Proposition 13.11, p.
574 of [11]. From the preceding two pieces of information we derive
Proposition 3.8. Let $G$ be a compact group whose commutator subgroup
$G^{\prime}$ is finite, and let $Z(G^{\prime},G)$ be the centralizer of
$G^{\prime}$ in $G$. Then the center of $Z(G^{\prime},G)$ is a characteristic
abelian open subgroup $A_{G}\buildrel\rm def\over{=}Z(Z(G^{\prime},G))$ of
$G$.
Proof. Let $f\colon G\to\mathop{\rm Aut}\nolimits(G^{\prime})$ denote the
morphism defined by $f(g)(x)=gxg^{-1}$ for $x\in G^{\prime}$. Since
$G^{\prime}$ is finite by Lemma 3.6, so is $\mathop{\rm Aut}\nolimits\
G^{\prime}$. Hence $f(G)$ is finite as well and so $G/\ker f\cong f(G)$ is
finite. If follows that the centralizer $C\buildrel\rm
def\over{=}Z(G^{\prime},G)$ of $G^{\prime}$ in $G$, being equal to $\ker f$,
has finite index in $G$ and thus is open. The center
$Z(G^{\prime})=G^{\prime}\cap C$ of $G^{\prime}$ is finite abelian, and so the
isomorphism $C/Z(G^{\prime})\to G^{\prime}C/G^{\prime}\subseteq G/G^{\prime}$
shows that the commutator subgroup $C^{\prime}\subseteq Z(G^{\prime})$ is a
finite subgroup of $C$. Since $G^{\prime}\subseteq
Z(Z(G^{\prime},G),G)=Z(C,G)$, the subgroup $C^{\prime}\subseteq G^{\prime}\cap
C$ is central in $C$. Hence $C$ is nilpotent of class at most two with a
finite commutator subgroup. Now Lemma 3.7 shows that $Z(C)$ is open in $C$,
and since $C$ is open in $G$ we know that $Z(C)$, a characteristic subgroup of
$G$, is open in $G$. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
By Lemma 3.6, this applies to all profinite FC-groups. Collecting our
information on compact FC-groups, we can establish the following main
structure theorem on the structure of compact FC-groups: Theorem 3.9. (The
Structure of Compact FC-groups) Let $G$ be a compact group. Then the following
two statements are equivalent:
$(1)$ $G$ is an FC-group.
$(2)$ $G/Z(G)$ is finite, that is, $G$ is center by finite.
$(3)$ The commutator subgroup $G^{\prime}$ of $G$ is finite.
Proof. (1)$\Rightarrow$(2): By Lemma 3.4, $Z(G_{0},G)$ is open and so by
Proposition 3.5 there is a profinite FC-group $\Delta$ commuting elementwise
with $G_{0}$ such that $Z(G_{0},G)=G_{0}\Delta$. By Lemma 3.6 and Proposition
3.8, $A_{\Delta}$ is an open characteristic abelian subgroup of $\Delta$
whence $A\buildrel\rm def\over{=}G_{0}A_{\Delta}$ is an open characteristic
abelian subgroup of $G_{0}\Delta=Z(G_{0},G)$. Since $Z(G_{0},G)$ is normal in
$G$, the subgroup $A$ is normal in $G$. Since $A$ is open in $Z(G_{0},G)$ and
this latter group is open in $G$ by Lemma 3.4, $A$ is an open normal subgroup
of $G$. Thus $G$ is an abelian by finite FC-group. We claim that $G$ is
therefore a center by finite group: Indeed, let $f\colon G/A\to\mathop{\rm
Hom}\nolimits(A,A)$ be defined by $f(gA)(a)=gag^{-1}a^{-1}$. For each coset
$\gamma=gA$ the image of $f(\gamma)$ is finite since $G$ is an FC-group, and
so $Z(g,G)=\ker f(\gamma)$ has finite index. Therefore
$Z(G)=\bigcap_{\gamma\in G/A}\ker f(\gamma)$ has finite index, and this proves
the claim.
(2)$\Rightarrow$(1): Since $Z(G)\subseteq Z(g,G)$ for each $g$, by (2), the
quotient $G/Z(g,G)$ is finite for all $g$. Hence $G$ is an FC-group.
(1)$\Rightarrow$(3): Let $G$ be an FC-group. By the equivalence of (1) and (2)
we know that $G_{0}$, being contained in the open subgroup $Z(G)$, is central.
By Proposition 3.5, there is a profinite subgroup $\Delta$ such that
$G=G_{0}\Delta$. Then $[G,G]=[\Delta,\Delta]$. Now by Lemma 3.6,
$[\Delta,\Delta]$ is finite.
(3)$\Rightarrow$(1): Set $C(g)=\\{xgx^{-1}:x\in G\\}$; then
$C(g)g^{-1}\subseteq G^{\prime}$, whence $C(g)\in G^{\prime}g$. Thus the
finiteness of $G^{\prime}$ implies that of $C(g)$ for all $g\in G$. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
In fact, a group $G$ satisfying the equivalent conditions of Theorem 3.9 above
is what has been called a BFC-group, that is, an FC-group with all conjugacy
classes of elements having bounded length. Let us remark that center by finite
groups are subject to classical central extension theory. For instance, if
$Z(G)$ happens to be divisible by $q=|G/Z(G)|$, then $G=Z(G)E$ for a finite
subgroup $E$ such that $Z(G)\cap E$ is contained in $\\{z\in Z(G):z^{q}=1\\}$.
(Cf. method of proof of Theorem 6.10(i) of [10].)
At this point we return to the issue of probability of commuting elements and
thus to Borel probability measures on $G$. Recalling Definition 2.1 we now
summarize our discussion as follows:
Theorem 3.10. Let $G$ be a compact group and $F$ its FC-center. Further let
$\mu_{1}$ and $\mu_{2}$ two Borel probability measures on $G$ and set
$P=\mu_{1}\times\mu_{2}$ and $D\buildrel\rm def\over{=}\\{(g,h)\in G\times
G:[g,h]=1\\}$. Assume that $\mu_{j}$ respect closed subgroups for $j=1,2$ and
that, if $G$ is not a Lie group, $\mu_{2}$ respects even Borel subgroups. Then
the following conditions are equivalent:
$(1)$ $P(D)>0$.
$(2)$ $F$ is open in $G$.
$(3)$ The characteristic abelian subgroup $Z(F)$ is open in $G$.
Under these conditions, the centralizer $Z(F,G)$ of $F$ in $G$ is open, and
the finite group $\Gamma\buildrel\rm def\over{=}G/Z(F,G)$ is finite and acts
effectively on $F$ with the same orbits as $G$ under the well defined action
$\gamma{\cdot}x=gxg^{-1}$ for $(\gamma,x)\in\Gamma\times F$, $g\in\gamma$. The
isotropy group $\Gamma_{x}$ at $x\in F$ is $Z(x,G)/Z(F,G)$, and the set
$F_{\gamma}$ of fixed points under the action of $\gamma$ is $Z(g,F)$ for any
$g\in\gamma$. Proof. (1)$\Leftrightarrow$(2): This is a part of Corollary 3.1.
(2)$\Rightarrow$(3): The FC-center $F$ of $G$ is an FC-group in its own right.
Then Theorem 3.9 shows that $Z(F)$ is open in $F$. By (2), $F$ is open in $G$.
Then $Z(F)$ is open, and this establishes (3).
The implication (3)$\Rightarrow$(2) is trivial.
Now assume that these conditions are satisfied. Then the open subgroup $Z(F)$
is contained in the the centralizer $Z(F,G)=\bigcap_{x\in F}Z(x,G)$, which is
the kernel of the morphism $G\to\mathop{\rm Aut}\nolimits F$ sending $g\in G$
to $x\mapsto gxg^{-1}$. The homomorphism
$\pi\colon\Gamma=G/Z(F,G)\to\mathop{\rm Aut}\nolimits F$ is therefore well-
defined by $\pi(\gamma)(x)=gxg^{-1}$, $\gamma=gZ(F,G)$, independently of the
choice of the representative $g\in\gamma$. If $x\in F$ and $\gamma\in\Gamma$,
say $\gamma=gZ(F,G)$, then $\gamma\in\Gamma_{x}$ iff $\gamma{\cdot}x=x$ iff
$gxg^{-1}=x$ iff $g\in Z(x,G)$ regardless of the choice of $g\in\gamma$. Thus
$\Gamma_{\gamma}=Z(x,G)/Z(F,G)$. Similarly, we have $x\in F_{\gamma}$ iff
$\gamma{\cdot}x=x$ iff $gxg^{-1}=x$ for $g\in\gamma$ iff $x\in Z(g,F)$.
$\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Later we shall specialize the measures $\mu_{1}$ and $\mu_{2}$ in a reasonable
way and then search for useful measures $\mu_{1}$ and $\mu_{2}$ respecting
appropriate classes of subgroups.
Meanwhile, we shall specialize $\mu_{1}$ and $\mu_{2}$ both to Haar measure on
$G$ and derive the main theorem on commuting elements in a compact group.
This case represents the special class of examples in which $P(E)$ is the
probability that two randomly picked elements commute. In this situation one
has called $P(\\{(x,y)\in G\times G:[x,y]=1\\})$ the commutativity degree
$d(G)$ of $G$.
Here, by the Main Theorem 3.10, the finite group $\Gamma=G/Z(F,G)$ acts
effectively on $F$ so that its orbits are the $G$-conjugacy clases of elements
of $F$ and that for the isotropy groups and fixed point sets we have
$\Gamma_{x}=Z(x,G)/Z(F,G)\quad\hbox{and}\quad(\forall
g\in\gamma)\,F_{\gamma}=Z(g,F).$
In particular, if $\nu_{F}$ is Haar measure of $F$, for the closed subgroup
$Z(g,F)$ of $F$ and $g\in\gamma\in\Gamma$ we conclude
$\nu_{F}(F_{\gamma})=\cases{0&if $F_{\gamma}$ has no inner points in
$F$,\cr|F/Z(g,F)|^{-1}&if $F_{\gamma}$ is open in $F$.\cr}$
We now formulate and prove the following fundamental result: Main Theorem
3.11. Let $G$ be any compact group and denote by $d(G)$ its commutativity
degree. Then we have the following conclusions: Part(i) The following
conditions are equivalent:
$(1)$ $d(G)>0$.
$(2)$ The center $Z(F)$ of the FC-center $F$ of $G$ is open in $G$.
Part(ii) Assume that these conditions of (i) are satisfied. Then there is a
finite set of elements $g_{1},\dots,g_{n}\in G$, $n\leq|G/Z(F,G)|$, such that
$d(G)={1\over|G/F|{\cdot}|G/Z(F,G)|}{\cdot}\sum_{j=1}^{n}|F/Z(g_{j},F)|^{-1}.$
Part(iii) $d(G)$ is always a rational number.
Proof. (i) follows directly from 3.10. For a proof of (ii) we let $\nu_{G}$
and $\nu_{F}$ be the Haar measures of $G$ and $F$, respectively, and recall
from 3.1 that
$d(G)=P(D)=\int_{x\in F}\nu_{G}(Z(x,G))d\nu_{G}(x)=\int_{x\in
F}|G/Z(x,G)|^{-1}d\nu_{G}(x).$
We note that the $\nu_{G}$-measure of $F$ is $|G/F|^{-1}$; thus
$\nu_{F}=|G/F|{\cdot}(\nu_{G}|F)$. Hence
$d(G)={1\over|G/F|}{\cdot}\int_{F}|G/Z(x,G)|^{-1}d\nu_{F}(x).$
Now
$G/Z(x,G)\cong(G/Z(F,G))/(Z(x,G)/Z(F,G))=\Gamma/\Gamma_{x},$
and so, letting
$E_{F}=\\{(\gamma,x)\in\Gamma\times F:\gamma{\cdot}x=x\\}$
and $P=\nu_{\Gamma}\times\nu_{F}$, by $(*)$ above, we get
$d(G)={1\over|G/F|}{\cdot}\int_{F}|\Gamma/\Gamma_{x}|^{-1}d\nu_{F}={1\over|G/F|}{\cdot}\int_{F}\nu_{\Gamma}(\Gamma_{x})d\nu_{F}={P(E_{F})\over|G/F|}.$
Next we select a set $\\{g_{1},\dots,g_{n}\\}$ of elements of $G$ such that
the cosets $g_{j}Z(F,G)$ are exactly those elements $\gamma\in\Gamma$ whose
fixed point set $F_{\gamma}$ is open in $F$. Then
$\Gamma_{g_{j}Z(F,G)}=Z(g_{j},F)$ for all $j=1,\dots,n$. We also recall
$\Gamma=G/Z(F,G)$ and apply Lemma 2.5 to see that
$P(E_{F})={1\over|G/Z(F,G)|}\sum_{j=1}^{n}|F/Z(g_{j},F)|^{-1}.$
Therefore
$d(G)={1\over|G/F|{\cdot}|G/Z(F,G)|}{\cdot}\sum_{j=1}^{n}|F/Z(g_{j},F)|^{-1}.$
This is what we claimed. Part (iii) is now an immediate consequence of (ii) as
$d(G)$ is trivially rational if it is $0$. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
The main result says: If the probablity that two randomly picked elements
commute in a compact group is positive, then, no matter how small it is, the
group is almost abelian.
We draw attention to the fact that in 3.11(ii) the commutativity degree of $G$
is expressed in purely arithmetic terms via the group theoretical data $F$,
$Z(F,G)$, and $Z(g_{j},F)$. In the much simpler case of finite groups this
aspect was discussed in [14].
4\. Compact Lie groups and measures respecting closed subgroups
Recall that for a continuous function $f\colon Y\to Z$ between compact spaces
and a Borel measure $\lambda$ on $Y$, the image measure $f(\lambda)$ on $Z$ is
given by $f(\lambda)(B)=\lambda(f^{-1}(B))$ for any Borel subset $B\subseteq
Z$.
On a smooth manifold $Y$ we denote by $T_{p}(Y)$ the tangent space at the
point $p\in Y$.
We say that a smooth function $f\colon Y\to Z$ between two real smooth
manifolds $Y$ and $Z$ of the same dimension is regular at a point $p$ if its
derivative $D_{p}f\colon T_{p}(Y)\to T_{f(p)}(Z)$ is invertible, that is, if
$\det(D_{p}f)\neq 0$. The complement of of the set of points at which $f$ is
regular is called the singular set of $f$. The singular set is the zero set of
the smooth function $\Delta\colon Y\to{R}$, $\Delta(p)=\det(D_{p}f)$ and thus
is closed. If $f$ is a real analytic function, then $\Delta$ is a real valued
real analytic function. Accordingly, it vanishes on a neighborhood of a point
$p$ if and only if it vanishes on the entire connected component of $p$ in
$Y$. Thus the singular set of a real analytic function $f$ is closed and
nowhere dense except for possibly containing entire components of $Y$.
Lemma 4.1. Let $H$ be a closed subgroup of a compact infinite Lie group $G$
and let $f\colon G\to G$ be a real analytic function whose singular set does
not contain a connected component of $G$. Assume that $f(\nu)(H)>0$ for the
normalized Haar measure $\nu$ of $G$. Then $H$ is open. In particular, the
measure $\mu\buildrel\rm def\over{=}f(\nu)$ respects closed subgroups. Proof.
We know that $0<\mu(H)=\nu\big{(}f^{-1}(H)\big{)}=\nu(\\{g\in G:f(g)\in
H\\})$. We claim that this implies that $\nu(H)>0$ which in turn will imply
that $H$ is open, as asserted.
Suppose that $H$ fails to be open. Then $H$ is a closed proper real analytic
submanifold of the real analytic manifold $G$. Since the singular set of $f$
is nowhere dense, it follows that $f^{-1}(H)$ is a closed nowhere dense real
analytic subset of $G$. Haar measure on $G$ is a real analytic $\dim(G)$-form
on $G$. Therefore $f(\nu)(H)=\nu(f^{-1}(H))=0$. This contradiction proves the
claim. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
As a next step we specialize $f_{1}$ and $f_{2}$ to power functions. We recall
that for a natural number $n\in\\{1,2,3,\dots\\}$ we define the power function
$p_{n}\colon G\to G$ by $p_{n}(g)=g^{n}$ and $f_{j}$ to be $p_{n_{j}}$.
Therefore we need a precise understanding of the singularities of $p_{n}$ on a
Lie group. Clearly $p_{n}$ is a real analytic self-map of the not necessarily
connected compact real manifold $G$. Firstly, let us say that a real analytic
function $f\colon G\to G$ is totally singular at $g\in G$ if $f$ takes a
constant value on a neighborhood of $g$ and therefore on the entire connected
component $gG_{0}$ of $g$ in $G$. This happens for $f=p_{2}$ on
$G={T}\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt$}\\{1,-1\\}$ (see Example 6.4 below):
Here $p_{2}$ is totally singular at each point of ${T}\times\\{-1\\}$. Let
$\exp\colon{g}\to G$ be the exponential function of $G$ and $U$ an open
neighborhood of $0$ in ${g}$ so small that $\exp|U:U\to\exp U$ is a
diffeomorphism onto an open neighborhood of 1 in $G$. The function $p_{n}$ is
singular, or totally singular, at $g$ if and only if the function
$x\mapsto((\exp x)g)^{n}:U\to G$ is singular, respectively, totally singular
at $0$. For the following we recall that each element $g\in G$ yields a
canonical Lie algebra automorphism $\mathop{\rm Ad}\nolimits g\colon{g}\to{g}$
such that $g(\exp x)g^{-1}=\exp(\mathop{\rm Ad}\nolimits g)x$.
Proposition 4.2. (The singularities of the power function) Let $G$ be a
compact Lie group and $n=1,2,\dots$ a natural number. Then the following
statements are equivalent for an element $g\in G$:
(i) $p_{n}$ is singular at $g\in G$.
(ii) At least one eigenvalue of $\mathop{\rm Ad}\nolimits g$ is an $n$-th root
of unity different from $1$.
Moreover, the following conditions are also equivalent:
(a) $p_{n}$ is totally singular in $g\in G$.
(b) $g^{n}=1$ and every eigenvalue of $\mathop{\rm Ad}\nolimits g$ is an
$n$-th root of unity different from 1.
Proof. From $g(\exp y)g^{-1}=\exp(\mathop{\rm Ad}\nolimits g)y$ for all
$y\in{g}$ it follows by induction that
$\big{(}(\exp x)g\big{)}^{n}=(\exp x)(\exp(\mathop{\rm Ad}\nolimits
g)x)(\exp(\mathop{\rm Ad}\nolimits g)^{2}x)\cdots(\exp(\mathop{\rm
Ad}\nolimits g)^{n-1}x)g^{n}.$ $None$
For fixed $g$ and $n$ we may assume that $U$ is so small that the Campbell-
Hausdorff multiplication $*$ is defined where needed for $x\in U$ such that
$\eqalign{&(\exp x)(\exp(\mathop{\rm Ad}\nolimits g)x)(\exp(\mathop{\rm
Ad}\nolimits g)^{2}x)\cdots(\exp(\mathop{\rm Ad}\nolimits
g)^{n-1}x)\cr=&\exp(x*(\mathop{\rm Ad}\nolimits g)x*(\mathop{\rm Ad}\nolimits
g)^{2}x*\cdots*(\mathop{\rm Ad}\nolimits g)^{n-1}x).\cr}$
We note that
$x*\mathop{\rm Ad}\nolimits(g)x*(\mathop{\rm Ad}\nolimits
g)^{2}x*\cdots*(\mathop{\rm Ad}\nolimits g)^{n-1}x=({\bf 1}+\mathop{\rm
Ad}\nolimits g+(\mathop{\rm Ad}\nolimits g)^{2}+\cdots+(\mathop{\rm
Ad}\nolimits g)^{n-1})x+r(x)$
where $r(x)\in{g}$ is a function $r\colon U\to{g}$ satisfying $\lim_{x\to
0}\|x\|^{-1}{\cdot}r(x)=0$ for one, hence all norms on ${g}$. Such a function
we shall call a remainder function. Thus for $x\in U$ we have
$p_{n}\big{(}(\exp x)g)\big{)}=\big{(}\exp\big{(}{\bf 1}+\mathop{\rm
Ad}\nolimits g+(\mathop{\rm Ad}\nolimits g)^{2}+\cdots+(\mathop{\rm
Ad}\nolimits g)^{n-1})x+r(x)\big{)}\big{)}g^{n}.$
Since $\exp$ is regular on $U$, and since right translation by $g^{n}$ is a
diffeomorphism on $G$, the function $x\mapsto\big{(}(\exp x)g\big{)}^{n}:U\to
G$ is regular at 0 if and only if
$\alpha_{g}\buildrel\rm def\over{=}\sum_{m=0}^{n-1}(\mathop{\rm Ad}\nolimits
g)^{m}:{g}\to{g}$
is an isomorphism. Let $\lambda$ be an eigenvalue of $\mathop{\rm Ad}\nolimits
g$, then $\lambda\neq 0$ since $\mathop{\rm Ad}\nolimits g$ is an automorphism
of ${g}$. Using the semisimplicity of $\mathop{\rm Ad}\nolimits g$, which due
to the fact that $G$ is compact, we see that
$\rho\buildrel\rm def\over{=}\sum_{m=0}^{n-1}\lambda^{m}=\cases{n&if
$\lambda=1$,\cr{\lambda^{n}-1\over\lambda-1}&if $\lambda\neq 1$,\cr}$
is an eigenvalue of $\alpha_{g}$, and all eigenvalues $\rho$ of $\alpha_{g}$
are so obtained. Thus $p_{n}$ is singular in $g$ if and only if one of the
eigenvalues $\rho$ vanishes, and that is the case if and only if
$\lambda^{n}=1$, $\lambda\neq 1$. This completes the proof of the equivalence
of (i) and (ii). Next we observe that (a) happens if and only if $(\exp
x)g)^{n}=1$ for all $x\in{g}$, including, of course, $x=0$. By $(\dag)$ this
is equivalent to $g^{n}=1$ and
$(\exp x)\exp((\mathop{\rm Ad}\nolimits g)x)\exp((\mathop{\rm Ad}\nolimits
g)^{2}x)\cdots\exp((\mathop{\rm Ad}\nolimits g)^{n-1}x)=1\hbox{ for all
}x\in{g}.$
In particular, due to analyticiy, this is equivalent to the fact that for all
sufficiently small $x$ for which the required Campbell-Hausdorff products
exist we have
$x*\mathop{\rm Ad}\nolimits(g)x*(\mathop{\rm Ad}\nolimits
g)^{2}x*\cdots*(\mathop{\rm Ad}\nolimits g)^{n-1}x=0.$ $None$
By the Campbell-Hausdorff formalism, there is a zero-neighborhood of ${g}$ and
a remainder function $r$ such that, for
$\alpha_{g}={\bf 1}+\mathop{\rm Ad}\nolimits g+(\mathop{\rm Ad}\nolimits
g)^{2}+\cdots+(\mathop{\rm Ad}\nolimits g)^{n-1},$
we have
$(\forall x\in U)\,x*\mathop{\rm Ad}\nolimits(g)x*(\mathop{\rm Ad}\nolimits
g)^{2}x*\cdots*(\mathop{\rm Ad}\nolimits g)^{n-1}x=\alpha_{g}(x)+r(x).$
Thus by (u) above, condition (a) is equivalent to
(a′) $g^{n}=1$ and there is a sufficiently small neighborhood of $0$ in ${g}$
and a remainder function $r$ such that $(\forall x\in
U)\,\alpha_{g}(x)+r(x)=0$.
Let $0\neq y\in{g}$. Setting $x=t{\cdot}y$ with $t>0$ we have
$\|x\|=t{\cdot}\|y\|$ and $x\in U$ if $t$ is sufficiently small. Then (a′)
implies $0=\alpha(t{\cdot}x)+r(t{\cdot}y)$ and thus $0=\alpha_{g}(y)+{1\over
t}{\cdot}r(t{\cdot}y)\to\alpha_{g}(y)$ for $t\to 0$ by the definition of a
remainder function. Hence $\alpha_{g}=0$. In view of the semisimplicity of
$\mathop{\rm Ad}\nolimits g$, no eigenvalue of $\mathop{\rm Ad}\nolimits g$
can then be $1$, while $g^{n}-1$ implies $(\mathop{\rm Ad}\nolimits
g)^{n}={\bf 1}$. Thus all eigenvalues of $\mathop{\rm Ad}\nolimits g$ are
$n$-th roots of unity different from $1$. This establishes
(a)$\Rightarrow$(b).
Conversely, assume (b) and let $\lambda$ be an eigenvalue of $\mathop{\rm
Ad}\nolimits g$. Then $\lambda$ is an $n$-th root of unity and $\lambda\neq
1$. Then
$1+\lambda+\lambda^{2}+\cdots+\lambda^{n-1}={\lambda^{n}-1\over\lambda-1}=0.$
Since $\mathop{\rm Ad}\nolimits g$ is semisimple (and thus diagonalizable over
${C}$) we conclude that $\alpha_{g}=0$. Then (a′) holds with $r\equiv 0$.
Hence (a) and (b) are equivalent. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Assume that the equivalent conditions (a) and (b) are satisfied and assume,
without essential loss of generality, that $n$ is the order of the element
$g$, then $\langle G\cup\\{g\\}\rangle=G_{0}\langle g\rangle$ is an open
subgroup of $G$ which is isomorphic to the semidirect product
$G_{g}\buildrel\rm def\over{=}G_{0}\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt_{\iota}$}{Z}/n{Z}$ where the morphism
$\iota\colon{Z}/n{Z}\to\mathop{\rm Aut}\nolimits G_{0}$ is defined by by
$\iota(m+n{Z})(h)=g^{m}hg^{-m}$. Recall that the multiplication on $G_{g}$ is
given by
$(h,m+n{Z})(h^{\prime},m^{\prime}+n{Z})=(hg^{m}h^{\prime}g^{-m},m+m^{\prime}+n{Z}).$
The power function $p_{n}$ is strongly singular in $G_{g}$ in each point of
$G_{0}\times\\{1+n{Z}\\}$. In this sense these semidirect products (such as
the continuous dihedral group ${T}\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt$}\\{1,-1\\}$) are typical for the
presence of totally singular points.
Lemma 4.3. Let $G$ be an infinite compact Lie group with normalized Haar
measure $\nu$, and let $n\in\\{2,3,\dots\\}$. Then the following conditions
are equivalent.
(i) $p_{n}(\nu)$ respects closed subgroups of $G$.
(ii) $p_{n}$ is nowhere totally singular.
Proof. Proposition 4.2 shows that $p_{n}$ is a real analytic self-map of $G$
whose set of singular points is contained in the union of the sets
$S_{m}\buildrel\rm def\over{=}\\{g\in G:\det(\mathop{\rm Ad}\nolimits
g-e^{2\pi im/n}{\cdot}\mathop{\rm id}\nolimits)=0\\},\quad m=1,2,\dots,n-1.$
On each of the finitely many connected components $G_{0}c$, $c\in G$, the set
$S_{m}\cap G_{0}c$ is a real algebraic variety and thus is a closed nowhere
dense analytic subset or else contains all of $G_{0}c$. Hence (ii) is
equivalent to
(ii′) $p_{n}$ is a real analytic function whose singular set does not contain
a connected component of $G$
Then by Lemma 4.1, (ii′)$\Rightarrow$(i).
Conversly, assume (i) and suppose that (ii) is false. Then we have a component
$G_{0}c$ such that $p_{n}(G_{0}c)=\\{1\\}$ and so
$p_{n}(\nu)(\\{1\\})=\nu(p_{n}^{-1}\\{1\\})\geq\nu(G_{0}c)=\nu(G_{0})=|G/G_{0}|>0$.
Since $p_{n}(\nu)$ respects closed subgroups by (ii), we conclude that
$\\{1\\}$ is open. Hence the compact group $G$ is discrete and thus finite.
This is a contradiction to the hypothesis that $G$ is infinite. Hence
(i)$\Rightarrow$(ii) and the proof is complete. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
For the simplicity of notation, let us adopt the following notation Definition
4.4. Let $m$ be a natural number. A compact Lie group $G$ is called to be
$m$-straight if no element $g\in G$ satisfies $g^{m}$ and every eigenvalue of
$\mathop{\rm Ad}\nolimits g$ is an $m$-th root of unity different from $1$.
By Proposition 4.2 this is the same as saying that the power function $p_{m}$
is totally singular at no $g\in G$.
With this notation we have Theorem 4.5. Let $n_{1}$ and $n_{2}$ be natural
numbers and $G$ a compact Lie group. Assume that $G$ is $n_{j}$-straight for
$j=1,2$. Then the following statements are equivalent
$(1)$ The probability that for a randomly picked pair $(x,y)$ of $G$ the
powers $x^{n_{1}}$ and $y^{n_{2}}$ commute is positive.
$(2)$ $G_{0}$ is abelian.
Proof. By Lemmas 4.2 and 4.3 we know that the measures
$\mu_{1}=p_{n_{1}}(\nu)$ and $\mu_{2}=p_{n_{2}}(\nu)$ respect closed subgroups
of the Lie group $G$. Then (i) says that
$(\mu_{1}\times\mu_{2})(\\{(x,z)\in G\times G:[x,y]=1\\})>0.$
Now by Theorem 3.10, this is equivalent to
($2^{\prime}$) $Z(F)$ is open.
then $G_{0}$, being contained in the abelian group $Z(H)$ is commutative. Now
assume (2). Since $G/G_{0}$ is finite for a compact Lie group and $G_{0}$ is
abelian each conjugacy class of $G_{0}$ is finite, that is, $G_{0}\subseteq
F$, where $F$ is the FC-center of $G$. Hence $F$ is open and so by 3.10, this
implies (1). $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
For a compact group $G$ we write $\cal N$ for the set of all closed normal
subgroups $N$ of $G$ for which $G/N$ is a Lie group. Then
$G\cong\lim_{N\in{\cal N}}G/N$. (See [10], Lemma 9.1, p. 448.)
We shall now keep $n_{1}$ and $n_{2}$ fixed throughout the remainder of the
section and show that Condition 4.5(1) implies the commutativity of the
identity component $G_{o}$ of an arbitrary compact group $G$ provided all Lie
group quotients are $n_{j}$-straight for $j=1,2$. First a general remark. We
have $G\cong\lim_{N\in{\cal N}}G/N$. For $N\in{\cal N}$ and for a fixed
$n=1,2,\dots$ we set
$\eqalign{D(G)&=\\{(x,y)\in G\times G:[x^{n_{1}},y^{n_{2}}]=1\\},\cr
D_{N}(G)&=\\{(x,y)\in G\times G:[x^{n_{1}},y^{n_{2}}]\in N\\}.\cr}$
Obviously $N\subseteq M$ implies
$D_{N}(G)\subseteq D_{M}(G);\quad\hbox{also}\quad D(G)=\bigcap_{P\in{\cal
N}}D_{P}(G).$ $None$
Now we define the generalized commutativity degree (depending on $n_{1}$ and
$n_{2}$:
$d(G)\buildrel\rm def\over{=}(\nu_{G}\times\nu_{G})(D(G)).$
Note that with the quotient morphism $\pi_{N}\colon G\to G_{N}$ we have
$D(G/N)=\\{(xN,yN)\in G/N\times G/N:[x^{n_{1}},y^{n_{2}}]\in
N\\}=(\pi_{N}\times\pi_{N})(D_{N}(G)).$
For a quotient map $\rho\colon\Gamma\to\Omega$ of compact groups and a Borel
set $B\subseteq\Omega$ we have $\nu_{\Gamma}(\rho^{-1}(B))=\nu_{\Omega}(B)$.
Hence
$d(G/N)=(\nu_{G}\times\nu_{G})(D_{N}(G))\geq(\nu_{G}\times\nu_{G})(D(G))=d(G)$
In particular, $\\{d(G/N):N\in{\cal N}\\}$ is a decreasing family of numbers
in $[0,1]$ with $d(G)$ as a lower bound. So by the Monotone Convergence
Theorem of nets of real numbers, we have
$\inf\\{d(G/N):N\in{\cal N}\\}=\lim_{N\in{\cal N}}d(G/N).$
Proposition 4.6. $d(G)=\lim_{N\in{\cal N}}d(G/N)$. Proof. In view of
observation $(\\#)$, this is a consequence of the outer regularity of Haar
measure. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Theorem 4.7. Let $n_{1}$ and $n_{2}$ be natural numbers and $G$ a compact
group such that all Lie groups $G/N$, $N\in{\cal N}$ are $n_{1}$\- and
$n_{2}$-straight. Assume that the probability that $g_{1}^{n_{1}}$ and
$g_{2}^{n_{2}}$ commute for two randomly picked elements $g_{1},g_{2}\in G$ is
positive. Then $G_{0}$ is abelian.
Proof. Let $N\in{\cal N}$. We know $0<d(G)\leq d(G/N)$. Thus, from Theorem 4.4
for Lie groups we know that $(G/N)_{0}$ is abelian. On the other hand,
$(G/N)_{0}=G_{0}N/N$, and so $[G_{0},G_{0}]\subseteq N$. Since the
intersection of all normal subgroups $N$ such that $G/N$ is a Lie group is
singleton, we obtain $[G_{0},G_{0}]=\\{1\\}$. $\hbox
to0.0pt{$\sqcap$\hss}\sqcup$
It may be useful to recall that $G=G_{0}D$ for a profinite group $D$ such that
$G_{0}\cap D$ is normal in $G$. This follows from Dong Hoon Lee’s Supplement
Theorem for Compact Groups (see [10], Theorem 9.41). However, the issue of
commuting powers in the case of a profinite group $G$ is not discussed in this
paper.
5\. Examples
Finally, we record some examples for the simplest case that
$D(G)=\\{(x,y)\in G\times G:[x,y]=1\\},$
where $P=\nu\times\nu$ with normalized Haar measure $\nu$ on $G$. Recall that
in this case $d(G)\buildrel\rm def\over{=}P(D(G))$ is the commutativity
degree. All of the examples in our list are metabelian, and the first two are
in fact nilpotent. The Main Theorem 3.11 explains why this is not too far from
the most general situation.
The first example is trivial and arises from finite groups:
Example 5.1. Let $H$ be the 8-element quaternion group and $P=\nu\times\nu$.
Then $d(H)=5/8$. Let $A$ be any compact connected group, e.g., the circle
group ${T}$. Let $G=A\times H$. The random selection of pairs of commuting
elements in $A$ and the random selection of pairs of commuting elements in $H$
are independent events, and thus
$d(G)=P(D(A))\times P(D(H))=d(A){\cdot}d(H)=5/8.$
In view of Theorem 3.9, this example is a compact FC-group.
Example 5.2. Let $H$ be the class 2 nilpotent compact group of all $3\times
3$-matrices
$M(a,b;z)\buildrel\rm def\over{=}\pmatrix{1&a&z\cr 0&1&b\cr 0&0&1\cr},$
where $a,b,z$ range through the ring ${Z}_{p}$ of $p$-adic integers. Let $Z$
be the closed central subgroup of $H$ of all $M(0,0;pz)$ , $z\in{Z}_{p}$. Then
$G=H/Z$ is a compact nilpotent $p$-group of class 2 whose commutator group
$[H,H]/Z$ contains all elements $M(0,0;z)Z$ and thus is isomorphic to
${Z}(p)={Z}/p{Z}$. Its center $Z(G)$ consists of all elements $M(a,b;z)Z$ with
$a,b\in p{Z}_{p}$, $z\in{Z}_{p}$, whence $G/Z(G)\cong{Z}(p)^{2}$. The factor
group $G/[G,G]$ is isomorphic to ${Z}_{p}^{2}$. The subgroup of all
$M(a,0;0)$, $a\in{Z}_{p}$, is isomorphic to ${Z}_{p}$; it is topologically
generated by $M(1,0;0)$. By Theorem 3.9 again, this example yields a compact
FC-group. which is not the product of a central group with a finite group.
$\hbox to0.0pt{$\sqcap$\hss}\sqcup$
Erfanian and the second author have shown the following proposition, whose
proof draws on ideas of Lescot [14] on finite groups.
Proposition 5.3. ([7], Theorem A) Assume $G$ is an nonabelian compact FC-group
such that $G/Z(G)\cong({Z}/p{Z})^{2}$ for a prime $p$. Then
$d(G)=(p^{2}+p-1)/p^{3}.$
The group $G$ in Example 5.2 therefore has the commutativity degree
$(p^{2}+p-1)/p^{3}$. All examples of compact groups $G$ in the existing
literature having commutativity degree $d(G)=(p^{2}+p-1)/p^{3}$ arise from
direct products of finite groups such as e.g. Example 5.1 above. In this sense
$G=M(a,b;z)$ exhibits a first “nontrival” explicit compact example
illustrating Proposition 5.3.
Similar formulae as in 5.3 can be found in [5, 7, 8].
The following examples are nonnilpotent metabelian groups. Example 5.4. Let
$A$ be a compact abelian group and set $A_{a}=\\{a\in A:2{\cdot}a=0\\}$.
Define $G=A\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt$}\\{1,-1\\}$ with multiplication
$(a,\epsilon)(a^{\prime},\epsilon^{\prime})=(a+\epsilon{\cdot}a^{\prime},\epsilon\epsilon^{\prime}).$
Then $Z((a,1),G)=A\times\\{1\\}$ for $2a\neq 0$, and
$Z((0,-1),G)=A_{2}\times\\{-1,1\\}$. The orbits are $G{\cdot}(t,1)=\\{(\pm
a,1)\\}$, $G{\cdot}(0,-1)=2{\cdot}A\times\\{-1\\}$. Then
$\eqalign{D(G)&=(A\times\\{1\\})^{2}\cup(A\times\\{-1\\})\times(A_{2}\times\\{1\\})\cr&\cup(A_{2}\times\\{1\\})\times(A\times\\{-1\\})\cup(A_{2}\times\\{-1\\})^{2},\cr}$
and so, setting $t=\nu_{A}(A_{2})$, we get
$d(G)={1\over 4}+{1\over 2}t+{1\over 4}t^{2}=\left({1+t\over 2}\right)^{2}.$
If $A_{2}$ is without inner points in $A$ we have $t=0$ and so $d(G)={1\over
4}$. For $A={T}$ we get the “continuous dihedral group” which shows, among
other things, that under the circumstances of Theorem 3.10, we may have
$Z(F)=F\neq G$ and that $G$ may not be center by finite. This example also
provides a power function $p_{2}$ on a Lie group which is constant equal to 1
and hence is totally singular on the component ${T}\times\\{-1\\}$. In support
of the next Example we prove the following lemma: Lemma 5.5. Let $G_{n}$,
$n=1,2,\dots$ be a sequence of finite groups with the commutativity degrees
$d(G_{n})=d_{n}$. Then the product $G\buildrel\rm
def\over{=}\prod_{n=1}^{\infty}G_{n}$ has the commutativity degree
$\prod_{n=1}^{\infty}d_{n}\buildrel\rm
def\over{=}\lim_{n\to\infty}d_{1}d_{2}\cdots d_{n}$. Proof. Define
$E_{n}=D(G_{1})\times\cdots\times D(G_{n})\times
G_{n+1}^{2}\times\cdots\subseteq\prod_{n=1}^{\infty}(G_{n}\times G_{n})\cong
G\times G.$
Let $P=\nu\times\nu$ for Haar measure on $G$. Then $P(E_{n})=d_{1}\cdots
d_{n}$. If $\prod_{n=1}^{\infty}G_{n}\times G_{n}$ is identified with $G\times
G$, then $\bigcap_{n=1}^{\infty}E_{n}=D(G)$. Since $\nu$ is $\sigma$-additive,
so is $P$ and therefore
$P(\bigcap_{n=1}^{\infty}E_{n})=\lim_{n\to\infty}d_{1}\cdots d_{n}$. The
assertion follows. $\hbox to0.0pt{$\sqcap$\hss}\sqcup$
This example, among other things, illustrates Proposition 4.6. Example 5.6.
Let $G=({T}\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt$}\\{1,-1\\})^{N}$. Then $d(G)=0$ and
$G_{0}$ is abelian and not open.
Indeed $d({T}\hbox{$\mathrel{\times}{\hskip-4.6pt{\vrule
height=4.7pt,depth=0.5pt}}\hskip 2.0pt$}\\{1,-1\\})=1/4$ by Example 5.4. Then
by Lemma 5.5 we have $d(G)=\lim_{n\to\infty}1/4^{n}=0$. In the context of
Lemma 2.1(iv), the following Example is relevant:
Example 5.7. Let ${I}$ denote the unit interval $[0,1]$ under ordinary
multiplication, a compact connected topological monoid. Set
${I}_{0}=[0,1-1/p]$, ${I}_{1}=]1-1/p,1-1/p^{2}]$,
$\dots,\quad{I}_{n}=]1-1/p^{n},1-1/p^{n+1}],\dots,{I}_{\infty}=\\{1\\}$. We
form the compact topological product monoid $S={I}\times{Z}_{p}$ for the
additive group ${Z}_{p}$ of $p$-adic integers. For $r\in{I}$ set
$J_{r}=\cases{{Z}_{p},&if $r\in{I}_{0}$,\cr p^{n}{Z},&if $r\in{I}_{n}$,
$n=1,2,\dots$,\cr\\{0\\},&if $\in{I}_{\infty}$.\cr}$
The binary relation $R$, whose cosets are $R(t,z)=\\{t\\}\times(z+J_{t})$ is a
closed congruence relation. Therefore, $X\buildrel\rm def\over{=}S/R$ is a
compact connected abelian monoid with zero $R(0,0)$ whose group of units is
$G\buildrel\rm def\over{=}(\\{1\\}\times{Z}_{p})/R\cong{Z}_{p}$. For $t<1$ in
${I}$ and $x=R(t,z)$ we have
$G{\cdot}x=(\\{t\\}\times(z+{Z}_{p}/I_{p})\cong{Z}_{p}/p^{n}{Z}_{p}\cong{Z}/p^{n}{Z}$
for $t\in{I}_{n}$, $n=0,1,\dots$.
In particular, considering $X$ as a $G$-space under multiplication, the space
of finite orbits $F$ is $([0,1[\times G)/R$ is not closed in $X$. Thus Lemma
2.2(iv), saying that $F$ is an Fσ cannot be improved to read that $F$ is
closed.
For $p=2$, the space $X$ is the standard binary tree with $G$ as the Cantor
set of leaves. Compact monoids like $X$ above were considered in [13] rather
generally under the name cylindrical semigroups; for our construction see in
particular D-2.3.3ff on p. 241.
6\. On the history and background of the problem
A study of the probability that two randomly picked elements $x$ and $y$ of a
compact group $G$ commute was initiated by W. H. Gustafson in [8]. Thereby he
extended to the infinite case an idea which was put forward a few years
earlier by P. Erdős and P. Turán in the context of finite groups (s. [8] again
and [4]). Perhaps Erdős and Turán attempted to reformulate in a statistical
way a famous problem, posed by Paul Erdős himself and solved partially by
Bernhard Neumann in [16]:
For any class of groups $\cal{X}$, let $\cal{X}^{*}$ denote the class of all
groups $G$ such that every infinite subset of $G$ contains a pair of distinct
elements which generate an $\cal{X}$-subgroup of $G$. Is it true
$\cal{X}^{*}\subseteq\cal{X}$?
In [16], Bernhard Neumann answered this problem in the negative by showing
that for the class $\cal{A}$ of all abelian groups, the class $\cal{A}^{*}$
turns out to be the class of finite central extensions of abelian groups. His
methods dealt with combinatorial techniques, coverings of suitable sets and
remarks on the size of centralizers. Exactly the same methods can be found in
[4, 8]. The approach of P. X. Gallagher for computing the probability that two
randomly picked elements of a finite group commute is completely different in
so far as he used character theory (s. [14]) to express this probability in
terms of the number of conjugates of a given element. More details can be
found in [3] and [14]. In particular, Diaconis’ survey [3] illustrates the
fact that considerable information on the structure of groups may be obtained
along this route, and it helped to motivate the growing interest in these
issues that has emerged in the literature in recent years. Regarding infinite
groups, compact topological groups were the natural ones to which the results
on finite groups could be extended as soon as appropriate methods suitable for
the topological context were found. We have seen that the concepts of measure
theory serve as a substitute for an approach through character theory. However
no visible contributions so far deal with this point, even though the
literature on the representation theory of compact groups is considerable.
In [10], one approaches the subject of this paper from the other end,
following up on a classical theme. In [10], Proposition 6.86 and Exercise
E6,18 is is shown that in a nonabelian connected compact Lie group $G$ the set
of all pairs $(x,y)\in G\times G$ such that $\langle x,y\rangle$ is free
nonabelian and dense in $G$ is (i) dense in $G\times G$, (ii) has a meager
complement, and (iii) has Haar measure 1 in $G\times G$. Thus one has no
chance that the commutativity degree of such groups is positive, as is amply
confirmed through the principal results of this article.
In [15], Levai and Pyber discuss profinite, that is totally disconnected
profinite groups and, for this special case prove results similar to the ones
presented in the present paper.
The papers [5, 6, 7] illustrate that it is possible to extend W. H.
Gustafson’s initial idea to more sophisticated notions of probability on
compact groups. In principle, for any finite sequence of words
$w_{\alpha}(x_{1},x_{2},\ldots,x_{n})$, $\alpha=1,\dots,N$ in $n$ variables
and for any compact group $G$ the probability of the set
$\\{(x_{1},x_{2},\ldots,x_{n})\in
G^{n}:w_{1}(x_{1},x_{2},\ldots,x_{n})=\cdots=w_{N}(x_{1},\dots,x_{n})=1\\}$
in the group $G^{n}$ becomes a reasonable object of study. For instance, in
[5] and [7] the authors consider the words
$w_{ij}(x_{1},\ldots,x_{n})=[x_{i},x_{j}]$ for $1\leq i<j\leq n$ and the
measure of the set
$\\{(x_{1},\ldots,x_{n})\in G^{n}:\ [x_{i},x_{j}]=1,1\leq i<j\leq n\\}.$
Some simple observations concerning the relation $w=1$ on compact Lie groups
are made in [10], Lemma 6.83.
Acknowledgements. We express our appreciation to Karl-Hermann Neeb; during a
seminar lecture by the second author at TU Darmstadt in January 2010, Neeb
posed the question whether $P(\\{(x,y)\in G\times G:[x,y]=1\\})>0$ on a
compact group implies the existence of an open normal abelian subgroup. This
question motivated and inspired many of our arguments which finally led us to
an affirmative answer. We thank Georg W. Hofmann of Dalhousie University in
Halifax for motivating that portion of Theorem 4.4 resulting in a proof that
the commutativity degree $d(G)$ of a compact group is always rational. We are
grateful to an anonymous referee of an earlier version of this paper who
detected several flaws in our original presentation which could be eliminated
from the main body of our results. The paper [15] by L. Levai and L. Pyber
came to our attention after we completed our manuscript through a lecture by
Aleksander Ivanov at the conference on Automorphism groups of Topological
Structure in Eilat in June 2010. While they treat profinite groups only, in
the area of positive probability for commuting pairs there is a good deal of
overlap between their results and ours.
Literature
[1] Bourbaki, N., Intégration, Chap. 7 et 8, Hermann, Paris, 1963.
[2] Bredon, G., Introduction to Compact Transformation Groups, Academic Press,
New York, 1972.
[3] Diaconis, P., Random walks on groups: characters and geometry, in: Groups
St. Andrews 2001 in Oxford, Vol. I, London Math. Soc. Lecture Note Ser. 304,
Cambridge Univ. Press, Cambridge, 2003, pp. 120–142.
[4] Erdős, P., and P. Túran, On some problems of statistical group theory,
Acta Math. Acad. Sci. Hung. 19 (1968), 413–435.
[5] Erfanian, A., and R. Kamyabi–Gol, On the mutually commuting $n$-tuples in
compact groups, Int. J. Algebra 1 (2007), 251–262.
[6] Erfanian, A., and R. Rezaei, On the commutativity degree of compact
groups, Arch. Math. (Basel) 93 (2009), 201–212.
[7] Erfanian, A., and F. Russo, Probability of mutually commuting $n$-tuples
in some classes of compact groups, Bull. Iran. Math. Soc. 34 (2008), 27–37.
[8] Gustafson, W. H., What is the probability that two group elements commute?
Amer. Math. Monthly 80 (1973), 1031–1304.
[9] Hewitt, E., and K. Ross, Abstract Harmonic Analysis, Vol.I, Springer,
Berlin, 1963.
[10] Hofmann, K. H., and S. A. Morris, The Structure of Compact Groups, de
Gruyter, Berlin, Second Edition 2006.
[11] —, The Lie Theory of connected Pro-Lie Groups, Eur. Math. Soc. Publ.
House, 2007. [12] Hofmann, K. H., and P. S. Mostert, Splitting in Topological
Groups, Memoirs of the Amer. Math. Soc. 43 (1963), 75 pp. Third Printing 1993.
[13] —, Elements of Compact Semigroups, Charles E. Marrill, Columbus, Ohio,
1966, xii+384 pp.
[14] Lescot, P., Isoclinism classes and commutativity degrees of finite
groups, J. Algebra 177 (1985), 847–869.
[15] Lévai, L., and L. Pyber, Profinite groups with many commuting pairs or
involutions, Arch. Math. (Basel) 75 (2000), 1–7.
[16] Neumann, B. H., On a problem of P. Erdős, J. Aust. Math. Soc. Ser. A 21
(1976), 467–472.
[17] Shalev, A., Profinite groups with restricted centralizers, Proc. Amer.
Math. Soc. 122 (1994), 1279–1284.
[18] tom Dieck, T., Transformation Groups, de Gruyter, Berlin, 1987.
Authors’ addresses:
Karl H. Hofmann,
Fachbereich Mathematik
Technische Universität
Schlossgartenstr. 7
64289 Darmstadt, Germany
hofmann@mathematik.tu-darmstadt.de
Francesco G. Russo
Department of Mathematics
University of Palermo
via Archirafi 14
90123 Palermo, Italy
francescog.russo@yahoo.com
|
arxiv-papers
| 2010-01-27T05:41:04 |
2024-09-04T02:49:08.016430
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Karl H. Hofmann (Technische Universitaet Darmstadt, Darmstadt,\n Germany) and Francesco G. Russo (Universita' degli Studi di Palermo, Palermo,\n Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1001.4856"
}
|
1001.4974
|
# Infrared Spectral Energy Distributions of Seyfert Galaxies: Spitzer Space
Telescope Observations of the 12 $\mu$m Sample of Active Galaxies
J. F. Gallimore11affiliation: Department of Physics and Astronomy, Bucknell
University, Lewisburg, PA 17837 22affiliation: Currently on leave at NRAO, 520
Edgemont Rd., Charlottesville, VA 22903 , A. Yzaguirre11affiliation:
Department of Physics and Astronomy, Bucknell University, Lewisburg, PA 17837
33affiliation: Department of Physics, California State University, Fullerton,
P.O. Box 6866, Fullerton, CA 92834-6866 , J. Jakoboski11affiliation:
Department of Physics and Astronomy, Bucknell University, Lewisburg, PA 17837
44affiliation: Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA
91190 , M. J. Stevenosky11affiliation: Department of Physics and Astronomy,
Bucknell University, Lewisburg, PA 17837 55affiliation: Franklin Pierce Law
Center, Two White Street, Concord, NH 03301 , D. J. Axon66affiliation:
Department of Physics, Rochester Institute of Technology, 84 Lomb Memorial
Drive, Rochester, NY 14623 77affiliation: School of Mathematical and Physical
Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK , S. A.
Baum88affiliation: Chester F. Carlson Center for Imaging Science, Rochester
Institute of Technology, 54 Lomb Memorial Drive, Rochester, NY 14623 , C. L.
Buchanan99affiliation: School of Physics, University of Melbourne, Parkville,
Victoria, 3010 Australia , M. Elitzur1010affiliation: Department of Physics
and Astronomy, University of Kentucky, Lexington, KY 40506 , M.
Elvis1111affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138 , C. P. O’Dea66affiliation: Department of Physics,
Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623
, A. Robinson66affiliation: Department of Physics, Rochester Institute of
Technology, 84 Lomb Memorial Drive, Rochester, NY 14623 draft
###### Abstract
The mid-infrared spectral energy distributions (SEDs) of 83 active galaxies,
mostly Seyfert galaxies, selected from the extended 12 $\mu$m sample are
presented. The data were collected using all three instruments, IRAC, IRS, and
MIPS, aboard the Spitzer Space Telescope. The IRS data were obtained in
spectral mapping mode, and the photometric data from IRAC and IRS were
extracted from matched, 20″ diameter circular apertures. The MIPS data were
obtained in SED mode, providing very low resolution spectroscopy ($R\sim 20$)
between $\sim 55$ and 90 $\mu$m in a larger, 20″ $\times$ 30″ synthetic
aperture. We further present the data from a spectral decomposition of the
SEDs, including equivalent widths and fluxes of key emission lines; silicate
10 $\mu$m and 18 $\mu$m emission and absorption strengths; IRAC magnitudes;
and mid-far infrared spectral indices. Finally, we examine the SEDs averaged
within optical classifications of activity. We find that the infrared SEDs of
Seyfert 1s and Seyfert 2s with hidden broad line regions (HBLR, as revealed by
spectropolarimetry or other technique) are qualitatively similar, except that
Seyfert 1s show silicate emission and HBLR Seyfert 2s show silicate
absorption. The infrared SEDs of other classes within the 12 $\mu$m sample,
including Seyfert 1.8-1.9, non-HBLR Seyfert 2 (not yet shown to hide a type 1
nucleus), LINER, and HII galaxies, appear to be dominated by star-formation,
as evidenced by blue IRAC colors, strong PAH emission, and strong far-infrared
continuum emission, measured relative to mid-infrared continuum emission.
galaxies: active — galaxies: Seyfert — galaxies: spiral — infrared: galaxies
## 1 Introduction
Three components dominate the mid-infrared spectrum of an active galaxy
(Genzel & Cesarsky, 2000): (1) thermal radiation from a dusty, compact medium
that surrounds the active nucleus (AGN) and can obscure direct sight-lines to
it; (2) PAH features and thermal dust continuum associated with star-formation
or perhaps a powerful starburst; and (3) line features arising from molecular,
atomic, and ionic species.
The dusty medium surrounding the AGN is commonly referred to as “the dusty
torus.” Its presence is inferred for many AGNs based, for example, on
spectropolarimetry (Antonucci & Miller, 1985; Antonucci, 1993; Urry &
Padovani, 1995; Smith et al., 2004), X-ray spectroscopy (Risaliti et al.,
2007, 2002), and infrared aperture synthesis measurements (Jaffe et al., 2004;
Tristram et al., 2007). Observations further indicate that the dusty medium
must be axisymmetric, permitting low extinction sight-lines to type 1 AGNs,
where the broad-line region (BLR) is apparent in total intensity (Stokes $I$)
spectra (the “pole-on” view), but high-extinction sight-lines to type 2 AGNs,
which show suppressed broad line emission and AGN continuum (the “edge-on”
view).
Since the dusty torus re-radiates incident AGN continuum in the mid-infrared,
its SED provides an indirect measure of the AGN luminosity (Nenkova et al.,
2008b), important especially for heavily obscured AGNs where other indirect
diagnostics may not be available. Mid-infrared fine-structure lines can also
constrain the intrinsic shape of the AGN SED, because particular line ratios
are sensitive to the shape of the SED but less sensitive to extinction
compared to optical/UV lines of species with similar ionization energies
(Alexander et al., 2000).
The torus SED depends on the geometry and clumpiness of the torus, among other
properties (Nenkova et al., 2002, 2008b). For example, smooth, cylindrical
tori produce very strong 10 $\mu$m silicate (Sil) features, whether in
emission when viewed more nearly pole-on, or deep absorption when viewed edge-
on (Pier & Krolik, 1992). Clumpy tori, somewhat independent of the assumed
geometry, instead produce much weaker Sil features, because inter-clump sight-
lines can provide a view of hotter dust on the far side of the torus; our view
of clumpy tori includes a mix of cold and hot clump surfaces that dilutes the
Sil features (Nenkova et al., 2008b). Individual clumps are heated from
outside and cannot produce the strong absorption that a centrally heated dust
shell can have (Sirocky et al., 2008; Nenkova et al., 2008a; Levenson et al.,
2007).
The mid-infrared SED is also an important diagnostic of star-formation. Young
star clusters embedded in GMCs are predicted to produce weaker PAH equivalent
width in comparison to older clusters, owing to photodestruction and continuum
dilution by hot dust grains (Efstathiou et al., 2000). In global models, PAH
features are sensitive to the fractional luminosity of OB associations and the
gas density of their surrounding ISM (Siebenmorgen & Krügel, 2007). Since the
dusty medium re-radiates incident stellar radiation, the mid-far infrared
luminosity constrains the luminous contribution of star-formation.
The mid-infrared SED of active galaxies therefore contains diagnostics of the
AGN, its surrounding dusty torus, and any surrounding star-formation. The
described tracers can, in principle, be disentangled by spectral decomposition
of the global SED (Farrah et al., 2003; Marshall et al., 2007). At the
present, available infrared instruments, at least in their default mode of
use, suffer from mismatched apertures leading to discontinuities in the
observed SEDs or mismatched coverage of spectral lines that bias line ratios.
To remedy this problem of aperture matching, we have used the Spitzer Space
Telescope to observe a sample of active galaxies, mostly Seyferts, in
synthetically matched, 20″ diameter apertures spanning $\lambda$3.6-36 $\mu$m
and a larger aperture, $\sim 20\arcsec\times 30\arcsec$, covering
$\lambda$55-90 $\mu$m. These moderate to low resolution SEDs are well-suited
for spectral decomposition, studies of the PAH and Sil features, as well as
global constraints on the AGN and star-formation contributions to the
luminosity.
The observations and data obtained for this survey are presented in this work.
Section 2 describes the sample selection. Section 3 provides a detailed
account of the observations and data reduction, with attention to artifact
correction, synthetic aperture matching, and corrections for extended
emission. Section 4 summarizes the extraction of spectral features using a
modified version of the PAHFIT tool (Smith et al., 2007b) and measurements
based on the line-subtracted SEDs. We conclude in Section 5 with a summary of
the properties of the sample and avenues for future analysis.
## 2 Sample Selection
The sample, listed in Table 1, comprises a subset of the extended 12 $\mu$m
sample of AGNs (Rush et al., 1993). This parent sample was defined by an IRAS
detection at 12 $\mu$m, $F_{12}({\rm IRAS})>0.22$ Jy, and color selection
$F_{60}({\rm IRAS})>0.5F_{12}({\rm IRAS})$ or $F_{100}({\rm IRAS})>F_{12}({\rm
IRAS})$ to remove stars but few galaxies. AGNs were identified based on prior
(usually optical) classification. We restricted the sample to include (1) only
those objects categorized by Rush et al. (1993) as Seyferts or LINERs, and (2)
only those sources with $cz<10,000$ km s-1. Three sources were removed
subsequent to observations owing either to pointing errors or saturation in
the Spitzer observations: NGC 1068, M-3-34-64, and NGC 4922. Our sample
ultimately includes 83 Seyfert and LINER nuclei.
The main advantage of this sample over other AGN catalogs is its large
collection of published and archival multiwavelength observations (Rush et
al., 1996; Hunt & Malkan, 1999; Hunt et al., 1999; Thean et al., 2001, 2000;
Gallimore et al., 2006; Spinoglio et al., 2002). In addition, there are
comparable numbers of Seyfert 1 (S1) and Seyfert 2 (S2) nuclei, and their
redshift distributions are statistically indistinguishable (Thean et al.,
2001, 2000).
The original survey paper of Rush et al. (1993) broadly assigned optical
classifications of type 1 (broad-line AGN) and type 2 (narrow-line AGN), but
Tran (2003) pointed out that, even given such coarse binning of activity type,
there were many misclassifications. To aid in a more sophisticated comparison
of infrared properties to optical classifications, we endeavored to collect
updated and more precise classifications from the literature. The revised
classifications are included in Table 1 and illustrated in Figure 1. We find
that 20% (7 / 35) of the Rush et al. Type 1s are re-classified as hidden broad
line region (HBLR) Seyfert 2s (S1h or S1i), LINER, or HII (star-forming
galaxy), and 28% (13 / 46) of the Rush et al. Type 2s are re-classified as
S1.n, LINER, or HII. Note that HBLRs have been sought in all but three of the
20 S2s in our survey: NGC 1125, E33-G2, and NGC 4968 (Tran, 2003).
## 3 Observations and Data Reduction
The sample galaxies were observed using all of the instruments of the Spitzer
Space Telescope (Program ID 3269, Gallimore, P.I.): the four broadband
channels (3.6 $\mu$m, 4.5 $\mu$m, 5.8 $\mu$m, and 8.0 $\mu$m) of the Infrared
Array Camera (IRAC; Fazio et al., 2004); the low resolution gratings of the
Infrared Spectrograph (IRS; Houck et al., 2004), operating in spectral mapping
mode; and the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al.,
2004), operating in SED mode. The resulting SEDs are provided in Figure 2.
Several sample galaxies were observed as part of other Spitzer programs that
made use of different observing strategies; for example, in some cases only
single-pointing (“Staring Mode”) IRS spectra are available. We summarize the
observations and data reduction techniques both for our observing program and
archival Spitzer data.
### 3.1 Spitzer IRAC Observations
IRAC observations were centered on the NED coordinates of the sample galaxies
based on catalog names listed in Rush et al. (1993). A beamsplitter and
supporting optics centers the target on two detectors simultaneously (Fazio et
al., 2004), either at 3.6 and 5.8 $\mu$m or 4.5 and 8.0 $\mu$m, and so each
observation consisted of two pointings at a common orientation of the focal
plane relative to sky. These observations were taken as snapshots with no
attempt to mosaic or dither. To guard against saturation, we used the high-
dynamic range (HDR) mode which provides 0.6 s and 12 s integrations at each
pointing.
The data were initially processed and calibrated by the IRAC basic calibration
data (BCD) pipeline, version S14.0. The pipeline performs basic processing
tasks, including bias and dark current subtraction; response linearization for
pixels near saturation; flat-fielding based on observations of high-zodiacal
background regions; saturation and cosmic-ray flagging, the latter indicated
by signal detections more compact than the PSF; and finally flux calibration
based on observations of standard stars111Details are provided in the IRAC
Data Handbook, available at http://ssc.spitzer.caltech.edu/irac/dh/.. The
nominal photometric stability for compact sources is better than 3% in all
detectors (Reach et al., 2005). Corrections for extended
sources222http://ssc.spitzer.caltech.edu/irac/calib are accurate to $\sim
10$%, but the contribution of extended emission and the resulting correction
is small for most of the sample galaxies. Color-corrections are typically much
greater, especially in the 8.0 $\mu$m channel where in-band PAH emission can
result in factors of two or greater corrections.
The data were further processed for remaining artifacts, including cosmic-rays
and detector artifacts not corrected by the BCD pipeline. The steps performed
for artifact removal and photometric extraction are detailed below in Sections
3.1.1–3.1.4. Figure 3 illustrates the effect of our artifact removal
techniques.
The IRAC photometry is listed in Table 2. Detailed below, the photometry is
presented in the IRAC magnitude system with zero-point flux densities 280.9,
179.7, 115.0, and 64.13 Jy for the 3.6, 4.5, 5.8, and 8.0 $\mu$m channels,
respectively (Reach et al., 2005). The photometry includes corrections for
extended emission and color-corrections.
#### 3.1.1 Cosmic-ray and bandwidth-effect mitigation
All of the images were affected to varying degrees by cosmic-ray and solar
proton hits. Moreover, 5.8 and 8.0 $\mu$m images near saturation suffered from
the bandwidth effect, which manifests as a row-wise trail of fading source
images repeating every four pixels from the affected source. These multiple
images do not conserve flux but artificially add signal to the sources; for
our purposes, these bandwidth effect artifacts behave like cosmic ray hits
near our target galaxies.
The BCD pipeline attempts to identify cosmic-ray hits by locating detections
that are narrower than the PSF. Extended cosmic ray tracks are however missed
by this procedure. The v. S14.0 BCD pipeline further has no means of
mitigating the bandwidth effect.
We corrected these artifacts in three, post-BCD steps. Firstly, we generated
difference maps between the long and short exposure images. Difference signal
that exceeded 5$\sigma$ on the long exposure images was flagged as an
artifact, and these artifacts were replaced with signal from the short
exposure image. The matching mask and uncertainty images were updated
accordingly. This technique was particularly successful at removing bandwidth
artifacts with little image degradation; the affected image regions are at
relatively high signal-to-noise on the short exposure images.
We next applied van Dokkum’s (2001) algorithm to flag cosmic rays not picked
up on difference images. Images are convolved with a Laplacian filter, which
enhances sharp edges on image features and effectively identifies cosmic ray
tracks. The IRAC point response function (PRF) is however undersampled in all
four detectors, and real, compact sources such as field stars or the AGN
appear as false positives. These false positives can be filtered by measuring
the asymmetry of the detected source, parameter $f_{lim}$ in van Dokkum’s
notation. Cosmic ray hits tend to be more asymmetric than the PRF,
corresponding to a larger value of $f_{lim}$. Using a few sample images and
trial-and-error, we determined lower threshold values for $f_{lim}$ that flag
obvious cosmic rays but pass field stars and galaxy images; specifically, we
found $f_{lim}=8$ worked well for the 4.5, 5.8, and 8.0 $\mu$m images, and
$f_{lim}=10$ for the 3.6 $\mu$m images.
Finally, the few remaining cosmic rays were flagged interactively by
inspection of four-color images. Residual cosmic rays appear as color-
saturated pixels in this representation and so were easily identified,
flagged, and replaced by bilinear interpolation of neighboring pixels.
#### 3.1.2 Bias artifacts
Residual bias artifacts affect the pipeline-processed data. Software is
currently available at the Spitzer Science Center to mitigate these artifacts
where the image comprises mainly compact sources, but the algorithm breaks
down in the presence of extended or diffuse emission. The host galaxy is
detected for many of the sample AGNs, and so we had to develop new techniques
to eliminate these bias artifacts.
The 3.6 and 4.5 $\mu$m images show the effects of column pull-down and
multiplexer bleed (“muxbleed”). Column pull-down is evident as a depressed
bias level along columns that run through pixels near saturation. The bias
adjustment is nearly constant along a column, but there may be slightly
different bias offsets above and below saturated pixels.
One approach is to evaluate the bias depression in source-free regions, but,
for some sources, the presence of extended emission over a majority fraction
of the array reduces or eliminates valid background regions. We suppressed the
diffuse emission by performing row-wise median filtering across pull-down
columns and used the median difference to determine the bias correction.
Muxbleed also affects rows containing pixels near saturation, most evident as
a row pull-up, but also impacting the bias level on neighboring rows. The
images are read as four separate readout channels that are interlaced every
four columns. The bias on each readout channel is affected differently,
resulting in a vertical pinstripe pattern over some region of the array near
saturated pixels.
To mitigate the muxbleed artifact in the presence of extended emission, we
first median-filtered the image using an $8\times 8$ kernel to smooth out the
pinstripe pattern over two readout cycles, subtracted the median-filtered
image to remove diffuse emission, and generated residual images of each
readout channel. Muxbleed appears on each channel readout image as decaying,
horizontal stripes along rows containing pixels near saturation with
surrounding bands of weaker, constant bias offset. We determined a final
muxbleed model by fitting the brighter muxbleed stripes with a cubic
polynomial and determining the median DC-level offset in the surrounding
bands.
The 5.8 $\mu$m images were affected by residual dark current, appearing as a
slowly varying surface brightness gradient of the background. This “first
frame” artifact results from the sensitivity of the dark current to properties
of the previous observation and the time elapsed since that observation. To
remove this dark current gradient, we first masked bright stars and diffuse
emission from galaxies and then fit a bilinear surface brightness model to the
background.
Several 5.8 and 8.0 $\mu$m images were also affected by banding, which is a
decaying signal along rows or columns containing saturated pixels. These bands
were reduced by fitting separate row-wise and column-wise polynomials to the
surface brightness of off-source regions of the array.
#### 3.1.3 Saturation and Distortion
Final corrections were performed using custom IDL scripts and the MOPEX
software package333http://ssc.spitzer.caltech.edu/postbcd/mopex.html (Makovoz
& Khan, 2005; Makovoz et al., 2006). The short exposure images were used to
replace saturated pixels on the long-exposure images. The images were then
corrected for distortion and registered to a 1$\farcs$22 $\times$ 1$\farcs$22
grid. The resulting data products are the science image, calibrated in surface
brightness units MJy sr-1; an uncertainty image based on the propagation of
statistical uncertainties through the pipeline and post-BCD processing, but
which does not include systematic uncertainties associated with calibration;
and a coverage image, which, for these snapshot exposures, marks good pixels
as “1,” bad pixels (i.e., known bad pixels or cosmic ray hits) as “0,” and
intermediate values indicating that good and bad pixels were used in the
distortion correction for that pixel.
#### 3.1.4 Photometric extraction
We extracted flux density measurements using a synthetic, 20″ diameter
circular aperture centered on the brightest infrared source associated with
the active galaxy. The exception is NGC 1097, which has an off-nucleus star-
forming region that is brighter than the central point source; in that case,
the aperture was centered on the central point source. To determine the point-
source contribution to the aperture, each image was convolved with a 2-D,
rotationally symmetric Ricker wavelet (González-Nuevo et al., 2006). The width
of the central peak of the wavelet was tuned to match the width of the nominal
IRAC PRF, effectively subtracting extended emission and enhancing point
sources. We used field stars to calibrate the point source response of the
wavelet-convolved image; this calibration includes a correction for aperture
losses. We next applied extended source
corrections444http://ssc.spitzer.caltech.edu/irac/calib/ to the residual
signal (total $-$ point source).
Finally, the resulting photometry was color-corrected following the
prescription described in the IRAC Data Handbook. The uncorrected calibration
provides a flux density measurement at a nominal wavelength assuming a nominal
spectral shape $\nu F_{\nu}=$ constant over the broadband response. The color-
correction adjusts the calibration based on the true shape of the spectrum.
The result is the true flux density at a nominal wavelength rather than a
broadband average; for example, the color-corrected flux reported for the 8
$\mu$m camera will be closer to peak of the 7.7 $\mu$m PAH feature plus any
underlying continuum at that wavelength rather than bandpass-weighted average.
For reference, the nominal wavelengths for the IRAC cameras are 3.550, 4.493,
5.731, and 7.872 $\mu$m.
Color-correction involves integrating the infrared spectrum weighted by the
IRAC filter bandpasses. To perform this integration, we used our IRS spectra
(Section 3.3, below) with extrapolation to shorter wavelengths assuming a
power law slope matching the observed, uncorrected 3.6 $\mu$m flux density.
Note that this color-correction technique does not force a match between the
IRS and IRAC photometry; the flux scale of the spectrum is normalized so that
only the shape of the IRS spectrum influences the color correction.
Photometric corrections for extended sources are not accurately known for the
IRS spectra, and so the accuracy of the IRS spectral extractions precluded
separating the color corrections for extended and point source contributions
for a given source; rather, the color correction was determined based on the
integrated signal in the aperture.
### 3.2 Archival Spitzer IRAC Observations
IRAC observations of 16 sample galaxies were obtained through public release
to the Spitzer archive. Data processing largely followed the techniques
described above for our observations, except that, where our observations
comprise snapshots, the archival observations all employed dithering or
mosaicking techniques. The archival data were initially mosaicked using
standard procedures with MOPEX, including background matching based on
overlaps.
Artifacts, particularly banding, seriously affected the matching of mosaic
overlap regions and so we developed a modified mosaicking procedure building
on the algorithm of Regan & Gruendl (1995). We modified their least-squares
approach to employ least absolute deviations as a more robust measure of the
quality of overlap matching, and we further masked bright source and artifact
regions prior to overlap matching. The final images were assembled using a
median stack of astrometrically aligned images, background subtracted, and
finally sub-imaged to roughly 5′ square to match our survey observations.
Residual bias artifacts were removed by computing column or row biases where
sufficient background was available for bias determination. Cosmic rays were
largely mitigated by the median stack, but residual bad pixels were identified
and flagged interactively, as described in Section 3.1.1.
### 3.3 Spitzer IRS Spectral Mapping
We observed the sample galaxies using the Spitzer IRS in spectral mapping mode
and using the Short-Low (SL) and Long-Low (LL) modules. The modules cover a
wavelength range of $\sim 5$ – 36 $\mu$m with resolution
$R=\lambda/\Delta\lambda$ ranging from 64 to 128\. The integration times were
6 seconds per slit pointing.
The spectral maps were constructed to span $>20\arcsec$ in the cross-slit
direction and centered on the target source. The observations were stepped
perpendicular to the slit by half slit-width spacings. For the SL data, the
mapping involved 13 observations stepped by 1$\farcs$8 perpendicular to the
slit, and, for the LL data, 5 observations stepped by 5$\farcs$25
perpendicular to the slit. The resulting spectral cubes span roughly $25\farcs
2\times 54\farcs 6\times$ (5.3 – 14.2 $\mu$m) for the SL data and $29\farcs
1\times 151\arcsec\times$ (14.2 – 36 $\mu$m) for the LL data.
The raw data were processed through the Spitzer BCD pipeline, version S15.3.0.
The pipeline handled primary processing tasks including identification of
saturated pixels; detection of cosmic ray hits; correction for “droop,” in
which charge stored in an individual pixel is affected by the total flux
received by the detector array; dark current subtraction; and flat-fielding
and response linearization. Details are provided in the IRS Data
Handbook555http://ssc.spitzer.caltech.edu/mips/dh.
Sky subtraction used off-source orders. In SL and LL observations, the source
is centered in the first or second order separately, with the “off-order”
observing the sky at a position offset parallel to the slit: 79″ away for SL
and 192″ away for LL. Sky frames were constructed using median combinations of
the off-source data and subtracted from the on-source data of matching order.
No detectable contamination of the sky frame appeared in our observations
based on inspection of the sky frame data. In addition, a portion of the first
order slit spectrum appears on the second order observation and is used as an
additional check of spectral features that appear near the first / second
order spectral boundary.
Data cubes were constructed by first registering the individual, single-slit
spectra onto a uniform grid (slit-position vs. wavelength). The IRS slits do
not align with the detector grid, and the detector pixels undersample the
spatial resolution. Interpolating the undersampled slit onto a uniform grid
produces resampling noise (Allington-Smith et al., 1989), which significantly
impacts the spectra of unresolved sources. We instead used the pixel re-
gridding algorithm described by Smith et al. (2007a) that minimizes resampling
noise. Pixels from the original image are effectively placed atop the new,
registered pixel grid. Uniform surface brightness is assumed across the
original pixel, and that original signal is weighted and distributed based on
the fractional overlap with pixels on the new grid. After registering the
single-slit frames, the data were re-gridded by bilinear interpolation, one
wavelength plane at a time, into the final data cube. Flux uncertainty images
were similarly processed, with modifications to accommodate variance
propagation, to produce an uncertainty cube.
The spectra were extracted from synthetic, 20″ diameter circular apertures
centered on the brightest, compact IR source nearest the target coordinates.
Fractional pixels at the edge of the aperture were accounted for by assuming
uniform surface brightness across the pixel and weighting by the area of
intersection between the pixel and aperture mask.
The cube-extracted spectra were optimally weighted based on a three-
dimensional modification of Horne’s (1986) two-dimensional slit extraction
algorithm. Successful application of this algorithm requires estimation of
spatial profiles $P_{x\lambda}$, the probability that a detected photon falls
in a given pixel $x$ rather than some other pixel on the spectral map at
wavelength $\lambda$. Optimal extraction requires that the fractional
uncertainty of the estimator for $P_{x\lambda}$ is less than the fractional
uncertainty of original spectral image. Generation of $P_{x\lambda}$ therefore
requires some smoothing along the wavelength axis. To help preserve real
variations of $P_{x\lambda}$ with wavelength, we employed Savitzky-Golay
(1964) polynomial smoothing. In this smoothing scheme, $P_{x\lambda}$ is
calculated by a polynomial fit to the spectrum at pixel $x$ over the fitting
window $\delta\lambda$ either centered on $\lambda$ or limited by the ends of
the spectrum. We performed trial-and-error smoothing experiments on
CGCG381-051, which shows relatively weak continuum at short wavelengths but
high eqw PAH features, to decide on the polynomial order and window smoothing
parameters; ultimately we selected quadratic polynomials and 5-pixel smoothing
windows to balance improved signal-to-noise and the tracking of real
variations of $P_{x\lambda}$ with wavelength.
The effect of optimal extraction is illustrated in Figures 4 and 5, which
compare the optimally-extracted and non-weighted extraction of the IRS SL
spectrum of NGC 3079 and F01475-0740, and Table 3, which compares the
measurements derived from these extractions (see Section 4.1 for a description
of the measurement technique). NGC 3079 presents a challenging case because it
is edge-on and shows bright, extended PAH emission. The spatial profile is
therefore a complex function of wavelength, and coarse smoothing in the
wavelength direction could potentially affect the measurement of the PAH
fluxes and equivalent widths. We find however that the fractional difference
between the optimally-weighted spectrum and the unweighted spectrum is
typically only a few %, comparable to the statistical uncertainties in the
unweighted spectrum. Furthermore, the measured fluxes and equivalent widths
agree to within the measurement uncertainties; in fact, the measurement
uncertainties are dominated by the systematics of the measurement technique.
Compared to NGC 3079, F01475-0740 is a compact source at relatively low
signal-to-noise. Figure 5 clearly illustrates the advantage of optimum
weighting. The continuum shape and PAH spectral features are preserved, but
the formal statistical uncertainties are reduced by factor of 2–3 over the SL
spectral range. Again, the fluxes and equivalent widths of lines agree to
within the measurement uncertainties, but the optimally weighted spectrum
produced signficant ($>3\sigma$) detections of PAH 6.2 $\mu$m, Ne V 14.3
$\mu$m, and S III 18.7 $\mu$m that are too faint for the unweighted spectrum.
Fringing at the 5% – 10% level was apparent in the LL observations,
particularly those of sample objects with a strong point source contribution.
The spectra undersample the fringing, and so the fringe pattern could not be
removed using a conventional filtering in Fourier space. We employed instead
the technique described by Kester et al. (2003), which involves fitting
sinusoids in wavenumber to line-free sections of the spectrum. Successive
fringe components are added to the model until the next model fringe amplitude
falls below the noise level of the spectrum. The effect of this technique is
illustrated in Figure 6.
The flux scale was calibrated against archival staring and spectral mapping
observations of the stars HR7341 (SL) and HR6606 (LL). The staring mode
spectra were extracted using
SPICE666http://ssc.spitzer.caltech.edu/postbcd/spice.html., and the spectral
mapping observations were processed into cubes as described above for our
sample galaxies. The flux calibration curve was determined by the ratio of the
flux-calibrated staring mode spectra to the uncalibrated, cube-extracted
spectra and smoothed by a polynomial fit.
No correction was produced for resolved, extended emission. The flux
calibration includes an aperture loss correction factor appropriate for
compact sources, but extended sources do not suffer as much aperture loss.
Therefore, wavelength bands containing extended emission relative to the PRF
will be calibrated systematically high. Discussed below in Section 3.6, the
systematic error introduced is of order 10%, particularly near the 8 $\mu$m
PAH features.
We note that these IRS observations have been independently and differently
processed and recently presented in Wu et al. (2009). The main differences in
data processing are: (1) Wu et al. used the CUBISM tool and its calibrations
for spectral cube construction (Smith et al., 2007a); (2) it is not clear that
Wu et al. extracted the spectra optimally; (3) a rectangular aperture, which
covers a similar solid angle to our extraction aperture, was commonly used
(20$\farcs$4 $\times$ 15$\farcs$3; larger for SINGS galaxies); (4) there
appears to have been no attempt to de-fringe the long wavelength (LL) data;
(5) for an unknown number of sources, Wu et al. had to scale the SL data to
match up with the flux calibration of the LL data. We address these
differences in turn, excepting (3), the minor difference of extraction
aperture geometry.
We found that (1) CUBISM forced the data onto a new grid in which the original
pointing center was shifted, commonly resulting in nuclei that were centered
between pixels rather than on a pixel. This shift apparently introduced a
systematic error in the resampling of the surface brightness, because we
noticed artifacts and a few percent reduced signal in the spectrum extracted
from cubes produced by CUBISM. Our cubing technique was designed to require
minimal resampling of the observational grid and mitigated these artifacts.
We also found that (2) non-optimal extractions decreased the signal-to-noise
of the extracted spectra of fainter sources by factors of $\sim 50\%$ or more
(see Figure 5 and the discussion above). Although it varies source to source,
(4) fringing can produce $\sim 10\%$ artifacts between 14 and 36 $\mu$m (cf.
their extraction of MRK 1239 and M-6-30-15, among others). Finally, (5) we did
not need to apply any additional scaling to the spectra extracted from SL
cubes; rather our calibrations of the SL module show excellent agreement with
both the IRAC measurements and LL measurements, even though the calibrations
for each respective Spitzer module were based on different calibration stars;
this result suggests that the calibration against spectrally mapped stars, as
presented here, was more robust for calibration and systematic corrections,
accepting the $\sim 10\%$ calibration uncertainty for sources dominated by
extended emission.
### 3.4 Archival IRS Data
Archival data of our sample galaxies, such as obtained by the SINGS
collaboration, were processed in the same manner as for our program data.
Spectral mapping data were available in LL for all sources, but there are
however a few sources for which only SL staring mode observations are
available: NGC 526A, NGC 4941, NGC 3227, IC 5063, NGC 7172, and NGC 7314. The
staring mode spectra were extracted using SPICE and subjectively scaled to
achieve the best match to align with LL measurements and, where possible, IRAC
5.8 and 8.0 $\mu$m measurements. Not surprisingly, there result disagreements
between the IRAC measurements and scaled, IRS staring observations owing to
the contribution of the host galaxy in the 20″ aperture.
### 3.5 Spitzer MIPS-SED Observations
The sample galaxies were observed using the 70 $\mu$m low-resolution
spectrometer (SED mode) of the MIPS instrument. The spectrometer provides
spectral resolution $R\sim 15$–25 between $\lambda$55 and 95 $\mu$m. The slit
dimensions are $20\arcsec\times 120\arcsec$.
Each observation consisted of three pairs of on-source and off-source
measurements, where the off-position was located 1′ – 3′ from the on-source
position. Integration times were 3 or 10 seconds depending on the IRAS 60
$\mu$m flux density of the source. The observations include measurements of
built-in calibration light sources, called stimulators, for flux calibration.
We used post-BCD products of the S14.4.0 pipeline; the pipeline processing
includes flux calibration, background subtraction, co-addition of the on-
source pointings, and uncertainty images. Details are provided in the MIPS
data handbook 777http://ssc.spitzer.caltech.edu/mips/dh.
The MIPS spectra were extracted by summing across a 29″-wide synthetic
aperture centered on the target source (three-column extraction). The spectra
were further corrected for aperture losses assuming a point source model; the
present extractions include no corrections for spatially extended emission.
The photometric accuracy is expected to be 10% for compact sources and $\sim
15$% for extended sources (Lu et al., 2008).
Note that the MIPS aperture covers a solid angle $\sim 1.8\times$ larger than
the IRAC and IRS apertures. Modeling SEDs of sources with extended
($>20\arcsec$) far-infrared emission will therefore require an (unknown)
aperture correction that is potentially much greater than the reported
calibration uncertainty. Candidates for MIPS aperture corrections will show a
jump in the MIPS flux relative to a suitable extrapolation of the IRS
spectrum, although a real spectral peak in the 40–50 $\mu$m range might
similarly result in an observed MIPS SED that appears too blue even in the
absence of aperture effects.
### 3.6 Comparison of IRS and IRAC Photometry
The IRS spectra overlap with the 5.8 and 8.0 $\mu$m IRAC channels. To compare
the relative spectrophotometry, we interpolated the IRS spectra to find the
flux densities at the effective wavelengths of the color-corrected IRAC data,
5.731 $\mu$m and 7.872 $\mu$m, respectively. The average relative difference
of the overlapping flux densities, $F_{\nu}({\rm IRS})/F_{\nu}({\rm IRAC})-1$,
are $4\pm 1$% at 5.8 $\mu$m and $6\pm 1$% at 8.0 $\mu$m. The frequency
distributions are shown in Figure 7. For comparison, the average standard
scores, $(F_{\nu}[{\rm IRS}]-F_{\nu}[{\rm IRAC}])/\sigma$, are $0.00\pm 0.03$
and $0.18\pm 0.05$.
These results indicate that the IRS flux densities are, on average,
systematically higher than the IRAC measurements by a few percent. However,
the excess at 5.8 $\mu$m is dominated by statistical uncertainties, as the
average standard score is consistent with zero.
Strong, extended PAH emission however contributes significantly to the 8.0
$\mu$m IRAC channel. This spectral feature is difficult to color-correct
accurately owing to numerical integration errors that arise at sharp spectral
features. In addition, PAH emission is often spatially resolved in this
sample, and the present IRS calibration includes no correction for extended
emission. The systematic offset of 6% may result from the overcorrection of
aperture losses for extended PAH sources. The IRAC photometry of extended
sources is moreover accurate to only $\sim 10$%.
Figure 8 shows the 8 $\mu$m IRS / IRAC flux ratio as a function of the
fractional contribution of the central point source to the 20″ nuclear
aperture. The fainter sources provide appreciable scatter, but the brighter
sources reveal a trend in which point-source dominated objects show better
agreement, but more extended objects present $\sim 10$% IRS excesses. This
trend supports the interpretation that residual extended source calibration
uncertainties are the primary cause for the IRS – IRAC discrepancies at 8
$\mu$m.
## 4 Analysis
### 4.1 Spectrum Decomposition
In addition to continuum radiation from dust grains and associated Sil
emission and absorption bands, the infrared spectrum of Seyfert galaxies
includes diagnostics such as fine structure lines tracing a range of
ionization states, H2 lines, and PAH emission. We used the spectrum fitting
tool PAHFIT (Smith et al., 2007b), which is tailored to low resolution IRS
spectroscopy. The fits to the SEDs of NGC 4151 and NGC 7213 are provided in
Figure 9 as examples of the decomposition. The results are summarized in the
following tables: integrated PAH fluxes are given in Table 4; PAH equivalent
widths (EQWs) in Table 5; H2 line fluxes (mostly upper limits) in Table 6 and
EQWs in Table 7; ionic fine structure line fluxes in Table 8, and their EQWs
in Table 9. The line measurements are not corrected for model extinction, but
for completeness we list the best-fit model dust opacity (normalized to 10
$\mu$m) in Table 10.
As provided, PAHFIT is best suited for nearly normal or star-forming galaxies;
its model includes thermal continuum radiation from dust grains, PAH features,
fine-structure lines from lower ionization state species, H2, and Sil
absorption, whether by assumed mixed or foreground screen extinction. We added
two components to the PAHFIT model to fit the Seyfert SEDs: (1) fine-structure
lines from high ionization states, such as [Ne V] and [Ne VI]; and (2), to fit
silicate emission features, a simple model for warm dust clouds that are
optically thin at infrared wavelengths.
By default, PAHFIT models extinction based on the dust opacity law of Kemper
et al. (2004). Sirocky et al. (2008) found that the cold dust model of
Ossenkopf et al. (1992) better matches the high 18 $\mu$m Sil / 10 $\mu$m Sil
absorption found in active ultraluminous infrared galaxies. From inspection,
the present spectra similarly show relatively strong 18 $\mu$m features,
whether in emission or absorption, and so we further modifed PAHFIT to use the
cold dust model of Ossenkopf et al. (1992).
The thin, warm dust model assumes clouds with simple, slab geometry and
opacity at 10 $\mu$m, $\tau_{10}<1$. These warm clouds are further assumed to
be partially covered by cold, absorbing dust clouds. The model spectrum is
then,
$F_{\nu}=\left(1-C_{f}\right)B_{\nu}(T_{W})\left(1-e^{-\tau_{W}(\nu)}\right)+C_{f}B_{\nu}(T_{W})\left(1-e^{-\tau_{W}(\nu)}\right)e^{-\tau_{C}(\nu)},$
(1)
where $C_{f}$ is the covering fraction of cold, foreground clouds, modeled
independent of the galaxy extinction; $\tau_{W}$ is the opacity through the
warm clouds; $\tau_{C}$ is the opacity through the cold, foreground clouds,
and $B_{\nu}(T_{w})$ is the source function, for which we adopt a scaled
Planck spectrum at (fitted) temperature $T_{W}$ for simplicity. Note that the
cold dust opacities described by Eq. 1 are taken to be independent of the
global PAHFIT dust opacity model; the opacity values listed in Table 10 are
determined by the global dust opacity fit. The intention of including this
additional model component is to provide a realistic continuum baseline for
emission line fits and Sil strength determination rather than an
interpretable, radiative transfer model, and details of the opacity law or
more realistic source functions are beyond the scope of the present spectral
decomposition.
We further modified PAHFIT better to accommodate the IRAC broadband and MIPS-
SED measurements. Both instruments provide much lower sampling density in
wavelength than IRS data, and consequently they receive lower weight in the
$\chi^{2}$ minimization procedure. We therefore employed the sampling weight
correction described by Marshall et al. (2007), which effectively re-weights
individual data points based on the sampling density local to that data point;
regions of low sampling density receive increased weight. The weighting is
normalized so that each data point carries, on average, unity sampling weight.
We also introduced as a fitted parameter an aperture correction factor for the
MIPS-SED wavelength range that boosts the model SED by a factor up to 1.81,
which is the areal aperture ratio of the MIPS-SED extraction to the nominal
20″ diameter extraction aperture. Table 10 includes the best-fit aperture
corrections.
There are a few caveats to the results of this decomposition. These low
spectral resolution data are not best suited for measuring line fluxes,
particularly in spectrally crowded regions. PAHFIT takes a conservative
approach, severely restricting the centroid wavelengths and widths of the
fitted lines; for example, the fine-structure lines are assumed to be
unresolved and their widths are fixed at the instrumental resolution. Even so,
the fine-structure lines near 35 $\mu$m are crowded, and furthermore the IRS
measurements are very noisy at that region of the spectrum, with sensitivity
$\sim 10\times$ poorer than at 25 $\mu$m, depending on background
([http://ssc.spitzer.caltech.edu/documents/som/]Spitzer Observer’s Manual).
These limitations are reflected in the uncertainties and upper limits reported
in Tables 4–9.
Extracted line fluxes were however affected by occasional, residually poor
fits to the local continuum or PAH features. For example, if the PAHFIT model
placed the continuum too low locally to an emission line, PAHFIT would grow
the emission line to meet the data and therefore produce a line flux greater
than observed. Similarly, the PAHFIT model might produce a local continuum
that is too high and the line flux is reported too low. A good illustration of
this problem is the too-high continuum model surrounding the [Ne V]
$\lambda$14.3 $\mu$m line of NGC 4151 (Figure 9). To compensate for locally
poor continuum models, the mean and rms of the model-subtracted spectrum was
evaluated in spectral regions surrounding each line. Each fitted line peak was
accordingly adjusted by subtracting the local, mean residual continuum. Line
fluxes were similarly adjusted; line widths are unaffected as they were held
fixed to the instrumental resolution during the fit.
There is a weak PAH feature near 14.3 $\mu$m that potentially contaminates the
[Ne V] $\lambda$14.3 $\mu$m fine structure line (Sturm et al., 2000).
Deblending these features uses the fact that the PAH feature is somewhat
broader (FWHM $\sim 0.4$ $\mu$m) than the unresolved (FWHM $<0.1$ $\mu$m) [Ne
V] line (Smith et al., 2007b). In the worst-case scenario, the code may fail
to fit a weak but present PAH 14.3 feature resulting in an artificially
enhanced [Ne V] line strength. This effect is at least partially ameliorated
by the residuals analysis and is reflected in the large uncertainties of Table
8. The [Ne V] 14 / 24 ratio provides however a good check for contamination.
This ratio is a density diagnostic with lower limit $\sim 0.9$ (e.g.,
Alexander et al., 1999). We demonstrate in a companion paper (Baum et al.
2009, submitted) that for all of the sources where there is a [Ne V] 14.3
$\mu$m detection, the line ratio is consistent with the low density limit (cf.
Sturm et al., 2002); contamination from PAH 14.3 would push the ratio to a
(forbidden) value below the low density limit. We conclude that the PAH 14.3
$\mu$m feature does not significantly contaminate the [Ne V] $\lambda$14.3
$\mu$m line strengths in this study or that any contamination is within the
uncertainties of the line strength.
#### 4.1.1 Silicate Strengths
This spectrum decomposition technique provides a reasonable model for the 9 –
20 $\mu$m continuum, suitable to measure the relative strength of Sil
features. Spoon et al. (2007) defined the Sil strength as the log ratio of the
observed flux density at the center of the Sil feature, 10 $\mu$m or 18
$\mu$m, and the local continuum; e.g.,
$S_{10}=\ln\left(\frac{F_{10\mu{\rm m}}[{\rm observed}]}{F_{10\mu{\rm m}}[{\rm
continuum}]}\right).$ (2)
To measure the Sil strength for the 12 $\mu$m sample, we first subtracted PAH
and other emission line features as determined by PAHFIT to obtain
$S_{10\mu{\rm m}}[{\rm observed}]$. The continuum was derived from the
spectrum decomposition as the sum of the (optically thick) dust components,
stars, and the continuous part of the warm, thin dust component; the Sil
emission features of the warm, thin dust component were replaced by quadratic
interpolation between bracketing spectral regions. We used Monte Carlo
variation of the PAHFIT model parameters and the data uncertainties to
determine the Sil strength uncertainties.
The measured Sil strengths are provided in Table 11.
#### 4.1.2 Continuum Spectral Indices
We further used the PAHFIT spectral decomposition to produce line-free
continuum spectra over $\lambda$20–30 $\mu$m. The MIPS SED data are similarly
line-free, except for a few possible detections of O I ($\lambda$63 $\mu$m);
see, e.g., Figure 9.
The data show a range of continuum slopes, and we characterized the spectral
shape by fitting a powerlaw model, $F_{\lambda}\propto\lambda^{\alpha}$ where
$\alpha$ is the spectral index, to the rest wavelength ranges 20–30 $\mu$m and
55–90 $\mu$m. The results are listed in Table 12. Note that in this convention
for $\alpha$, the Rayleigh-Jeans tail of the Planck spectrum would give
$\alpha=-4$.
#### 4.1.3 Comparison with Measurements Employing Spline Approximations for
the IR Continuum
Wu et al. (2009) adopt a different but conventional approach to the
measurement of the PAH 6.2 $\mu$m and 11.2 $\mu$m features and the 10 $\mu$m
Sil strength, and we next consider systematic differences with our
measurements. Rather than decompose the spectrum with a dust and lines model
as PAHFIT does, their approach was to define a local continuum level based on
a spline fit to wavelength ranges narrowly bracketing PAH features. To measure
Sil strengths, they adopted the technique of Spoon et al. (2007), which
requires the identification of apparently feature-free continuum points to
anchor a broader spline interpolation across the Sil features.
Smith et al. (2007b) demonstrated that, for the nearly normal galaxies in the
SINGS sample, the PAH line strengths measured by PAHFIT are systematically
greater, by factors of 2–3, than line strengths that are based on a spline fit
to the neighboring pseudo-continuum. The reason is qualitatively illustrated
in the PAHFIT decomposition of NGC 7213 (Figure 9). Both the 6.2 $\mu$m and
11.3 $\mu$m features blend with weaker, overlapping PAH features. By defining
the continuum level based on neighboring spectral points without accounting
for PAH blending, the continuum level is overestimated, and the line strength
and eqw are underestimated. It is further evident from this decomposition that
Sil strengths will be systematically affected if the influence of PAH blends
and the underlying continuum shape are not reasonably accounted for; it would
not be surprising that the spline technique of produced very different Sil
strengths particularly for this source.
We illustrate the systematic differences between the present analysis and that
of Wu et al. (2009) in Figs. 10 and 11. As expected, PAHFIT measures, on
average, systematically higher values of PAH fluxes and eqws, because PAHFIT
removes the contamination of neighboring PAH features to measurement of the
local continuum. The measurements of Wu et al. (2009) fall somewhat below the
average ratio (PAHFIT / spline continuum) reported by Smith et al. (2007b),
but the spline measurements will be sensitive to systematic differences in how
the local continuum anchor points were defined in the analysis; such a
detailed reconciliation is beyond the scope of the present work.
Similarly, our PAHFIT-derived Sil strengths are systematically more positive,
by $\sim 0.1$–0.3 dex, indicating weaker Sil absorption or stronger Sil
emission depending on the sign of the Sil strength. Again, this result is
unsurprising, because PAH blends that are not accounted for by decomposition
can falsely mimic enhanced continuum surrounding the Sil 10 $\mu$m feature,
pushing the Sil strength to lower (more negative) values. Recall that we also
use an interpolation technique similar to that of Spoon et al. (2007) to
measure the strength of the Sil features; the difference is that we perform
the analysis on the line-subtracted continuum model produced by PAHFIT.
### 4.2 SEDs Averaged by Optical Classification
A key goal of this project is to identify and compare the infrared
characteristics of AGNs segregated by optical classification. Toward a
qualitative first look, we calculated average SEDs within the following
classification bins: (1) S1.0-1.5 & S1n; (2) S1.8-1.9; (3) HBLR Seyfert 2s
(S1h & S1i); (4) non-HBLR Seyfert 2s (S2); (5) LINERs; and (6) HII. The
results are presented in Figure 12. The separation of the non-HBLR and HBLR
S2s was motivated by inspection of Figure 2; non-HBLR S2s appear to have
higher-eqw PAH features, for example. Recall that the non-HBLR S2s may in fact
harbor a BLR that has not appeared in spectropolarimetric or infrared
measurements. All but three of the 20 non-HBLR S2s have been searched for an
HBLR, but the inclusion of these three sources in the non-HBLR subsample does
not appear to dilute the striking differences between the averaged spectra of
non-HBLR S2s and HBLR S2s.
To perform the averaging, all of the data were corrected for redshift and
interpolated to a common wavelength grid. The SEDs were converted to $\lambda
F_{\lambda}$ and normalized to the flux density integrated between 5 and 35
$\mu$m, $F$(5–35$\mu$m). Objects within a given classification bin were
averaged, and the median absolute deviation was computed as a robust estimator
of the characteristic spread of SEDs within a classification bin.
Tran (2003) demonstrated that S1s and HBLR S2s show similar IRAS broadband
colors, but non-HBLR S2s tend to show cooler IRAS colors (cf. Heisler et al.,
1997). The present study confirms this result in some finer detail based on
the averaged SEDs. From inspection of Figure 12, S1s (group 1) and HBLR S2s
(group 3) show the flattest infrared SEDs. They both present fine-structure
emission lines of high-ionization state species, such as [O IV], [Ne IV], and
[S IV], with similar equivalent width. Hidden S1s have a slightly redder SED,
on average, and also show evidence for Sil 10$\mu$m absorption.
Similarly, the average SEDs of S1.8-1.9s (group 2) and non-HBLR S2s (group 4)
are essentially indistinguishable. Both groups show strong PAH features, red
continuum, and blue IRAC colors suggesting significant contribution of stellar
photospheric emission at the short wavelength end. The SEDs of these groups
most closely resemble optically classified star-forming galaxies, or
starbursts (HII; group 6), except that the average HII SED for this sample
shows PAH features with somewhat greater equivalent widths. The [S III] and
[Si II] average equivalent widths appear comparable between S1.8-1.9, non-HBLR
S2, and HII galaxies.
The average LINER SED stands out by showing a bowl-shaped infrared SED. The
SEDs appear to be more strongly dominated by stellar photospheres, or perhaps
very hot dust, at shorter wavelengths compared to the other groups, including
HII galaxies where starlight appears to dominate the IRAC bands. In this way,
the LINERs in our survey are similar to the IR-faint LINERs in the larger
sample studied by Sturm et al. (2006). This result is somewhat tempered by the
broad range of IRAC colors observed among the LINERs in this survey; from
inspection of the six individual LINER SEDs, four show bowl-like SEDs
resembling the average (NGC 2639, NGC 4579, NGC 4594, & NGC 5005), and two
show HII-like SEDs (NGC 1097 & NGC 3079).
PAH features are commonly detected in this sample, and, even though such
features appear at reduced equivalent width in S1 and HBLR S2 objects (cf.
Clavel et al., 2000), they are sufficiently strong to obscure Sil features,
especially Sil emission. We therefore repeated the SED averaging after
subtracting PAH, H2, and fine-structure lines based on the results of the
PAHFIT decomposition (Section 4.1); the results are provided in Figure 13.
Here the average SED of S1 and HBLR S2 distinguish more clearly, with S1
showing clear 10 $\mu$m and 18 $\mu$m Sil emission features, similar to that
observed in QSOs (Hao et al., 2005a). In contrast, the averaged SED of known
HBLR S2s show Sil 10 $\mu$m in absorption.
The other classes again show broadly similar SEDs after line subtraction. The
notable exception is the non-HBLR S2 average, where weak 10 $\mu$m Sil
absorption appears underneath the subtraction of very strong PAH features. Sil
features are essentially absent among intermediate Seyferts (S1.8-1.9),
LINERs, and HII galaxies. It is further interesting to note that the IRAC
color [3.6] $-$ [4.5] of the averaged S2, LINER, and HII SEDs is consistent
with an undiluted Rayleigh-Jeans continuum. The averaged SEDs of the other
classes present redder [3.6] $-$ [4.5], indicating dilution from warm dust or
some flat-spectrum component.
## 5 Discussion
We have presented the data reduction and decomposition of Spitzer Space
Telescope 3.6 – 90 $\mu$m spectrophotometry of active galaxies from the
extended 12 $\mu$m survey. Careful attention was provided to matching 20″,
circular diameter apertures across the IRAC and IRS bands (3.6 – 36 $\mu$m)
with appropriate color and extended source corrections where possible or with
an evaluation of the systematic error where such corrections were not
available.
We further present SEDs averaged within groups defined by optical AGN
classification. We demonstrate that, within this sample, Seyfert 1s show Sil
emission on average, known HBLR Seyfert 2s show Sil absorption. This result is
broadly compatible with the obscuring torus interpretation, in which case
Seyfert 1s are viewed more nearly pole-on, affording a more direct view of
hot, Sil emitting dust. HBLR S2s are viewed more nearly edge-on,
preferentially through colder, Sil absorbing dust. That the Sil features are,
on average, weak is further compatible with the clumpy torus model (Nenkova et
al., 2008b; Sirocky et al., 2008; Nenkova et al., 2008a; Levenson et al.,
2007).
The other classes, Seyfert 1.8-1.9, non-HBLR S2, LINER, and HII galaxies,
produce very weak or absent Sil features. They further show stronger PAH
features, bluer IRAC colors, and stronger far-infrared emission (relative to
$F$[5–35$\mu$m]). Such SEDs appear to be more commonly dominated by stellar
photospheres and star-forming processes. Based on the present analysis,
however, we are unable to conclude whether Seyfert 1.8-1.9 and non-HBLR S2
galaxies are in fact more commonly dominated by star formation or whether this
result is peculiar to the 12 $\mu$m sample owing to selection effects; for
example, they may harbor less luminous AGNs, or more heavily absorbed AGNs,
but the contribution from star-formation enhanced the 12 $\mu$m flux density
sufficiently to be included in the 12 $\mu$m sample. On the other hand, our
results are consistent with the interpretation that the host galaxy dominates
the emission of non-HBLR S2s, diminishing our ability to detect the HBLR
(Alexander, 2001).
In companion work, we present a statistical analysis of the present
measurements with attention to differences and similarities between sources
grouped by optical classification (Baum et al. 2009, submitted). We are also
investigating a decomposition of the SEDs using gridded radiative transfer
models with the goal of measuring bolometric contributions of the AGN vs.
star-formation as well as constraints on clumpy torus parameters (Gallimore et
al. in prep.).
The authors gratefully acknowledge the anonymous referee for a careful reading
of the manuscript and very helpful comments. This work is based on
observations made with the Spitzer Space Telescope, which is operated by the
Jet Propulsion Laboratory, California Institute of Technology under a contract
with NASA. Support for this work at Bucknell University, the University of
Rochester and, the Rochester Institute of Technology was provided by NASA
through an award issued by JPL/Caltech. A. Yzaguirre received support from the
National Science Foundation REU Program, grant 0097424\. J. Jakoboski received
support as a Bucknell Presidential Fellow. Facility: Spitzer
## References
* Agüero et al. (2004) Agüero, E. L., Díaz, R. J., & Bajaja, E. 2004, A&A, 414, 453
* Alexander (2001) Alexander, D. M. 2001, MNRAS, 320, L15
* Alexander et al. (2000) Alexander, T., Lutz, D., Sturm, E., Genzel, R., Sternberg, A., & Netzer, H. 2000, ApJ, 536, 710
* Alexander et al. (1999) Alexander, T., Sturm, E., Lutz, D., Sternberg, A., Netzer, H., & Genzel, R. 1999, ApJ, 512, 204
* Allington-Smith et al. (1989) Allington-Smith, J. R., et al. 1989, MNRAS, 238, 603
* Alloin et al. (1981) Alloin, D., Edmunds, M. G., Lindblad, P. O., & Pagel, B. E. J. 1981, A&A, 101, 377
* Alloin et al. (1985) Alloin, D., Pelat, D., Phillips, M., & Whittle, M. 1985, ApJ, 288, 205
* Alonso-Herrero et al. (2000) Alonso-Herrero, A., Rieke, M. J., Rieke, G. H., & Shields, J. C. 2000, ApJ, 530, 688
* Antonucci (1993) Antonucci, R. 1993, ARA&A, 31, 473
* Antonucci & Miller (1985) Antonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621
* Aretxaga et al. (1999) Aretxaga, I., Joguet, B., Kunth, D., Melnick, J., & Terlevich, R. J. 1999, ApJ, 519, L123
* Clavel & Joly (1984) Clavel, J., & Joly, M. 1984, A&A, 131, 87
* Clavel et al. (2000) Clavel, J., et al. 2000, A&A, 357, 839
* Coziol et al. (1998) Coziol, R., Ribeiro, A. L. B., de Carvalho, R. R., & Capelato, H. V. 1998, ApJ, 493, 563
* Dahari & De Robertis (1988) Dahari, O., & De Robertis, M. M. 1988, ApJS, 67, 249
* de Grijp et al. (1992) de Grijp, M. H. K., Keel, W. C., Miley, G. K., Goudfrooij, P., & Lub, J. 1992, A&AS, 96, 389
* De Robertis & Osterbrock (1986) De Robertis, M. M., & Osterbrock, D. E. 1986, ApJ, 301, 98
* Dewangan & Griffiths (2005) Dewangan, G. C., & Griffiths, R. E. 2005, ApJ, 625, L31
* Efstathiou et al. (2000) Efstathiou, A., Rowan-Robinson, M., & Siebenmorgen, R. 2000, MNRAS, 313, 734
* Farrah et al. (2003) Farrah, D., Afonso, J., Efstathiou, A., Rowan-Robinson, M., Fox, M., & Clements, D. 2003, MNRAS, 343, 585
* Fazio et al. (2004) Fazio, G. G., et al. 2004, ApJS, 154, 10
* Feldmeier et al. (1999) Feldmeier, J. J., Brandt, W. N., Elvis, M., Fabian, A. C., Iwasawa, K., & Mathur, S. 1999, ApJ, 510, 167
* Filippenko & Halpern (1984) Filippenko, A. V., & Halpern, J. P. 1984, ApJ, 285, 458
* Filippenko & Sargent (1985) Filippenko, A. V., & Sargent, W. L. W. 1985, ApJS, 57, 503
* Gallimore et al. (2006) Gallimore, J. F., Axon, D. J., O’Dea, C. P., Baum, S. A., & Pedlar, A. 2006, AJ, 132, 546
* Genzel & Cesarsky (2000) Genzel, R., & Cesarsky, C. J. 2000, ARA&A, 38, 761
* Giannuzzo & Stirpe (1996) Giannuzzo, E. M., & Stirpe, G. M. 1996, A&A, 314, 419
* González Delgado et al. (1998) González Delgado, R. M., Heckman, T., Leitherer, C., Meurer, G., Krolik, J., Wilson, A. S., Kinney, A., & Koratkar, A. 1998, ApJ, 505, 174
* González Delgado & Perez (1996) González Delgado, R. M., & Perez, E. 1996, MNRAS, 280, 53
* González-Nuevo et al. (2006) González-Nuevo, J., Argüeso, F., López-Caniego, M., Toffolatti, L., Sanz, J. L., Vielva, P., & Herranz, D. 2006, MNRAS, 369, 1603
* Goodrich (1989) Goodrich, R. W. 1989, ApJ, 340, 190
* Goodrich et al. (1994) Goodrich, R. W., Veilleux, S., & Hill, G. J. 1994, ApJ, 422, 521
* Hao et al. (2005a) Hao, L., et al. 2005a, ApJ, 625, L75
* Hao et al. (2005b) —. 2005b, AJ, 129, 1783
* Heckman (1980) Heckman, T. M. 1980, A&A, 87, 152
* Heisler et al. (1997) Heisler, C. A., Lumsden, S. L., & Bailey, J. A. 1997, Nature, 385, 700
* Heisler et al. (1989) Heisler, C. A., Vader, J. P., & Frogel, J. A. 1989, AJ, 97, 986
* Ho et al. (1997) Ho, L. C., Filippenko, A. V., & Sargent, W. L. W. 1997, ApJS, 112, 315
* Horne (1986) Horne, K. 1986, PASP, 98, 609
* Houck et al. (2004) Houck, J. R., et al. 2004, ApJS, 154, 18
* Hunt & Malkan (1999) Hunt, L. K., & Malkan, M. A. 1999, ApJ, 516, 660
* Hunt et al. (1999) Hunt, L. K., Malkan, M. A., Moriondo, G., & Salvati, M. 1999, ApJ, 510, 637
* Inglis et al. (1993) Inglis, M. D., Brindle, C., Hough, J. H., Young, S., Axon, D. J., Bailey, J. A., & Ward, M. J. 1993, MNRAS, 263, 895
* Jaffe et al. (2004) Jaffe, W., et al. 2004, Nature, 429, 47
* Kay (1994) Kay, L. E. 1994, ApJ, 430, 196
* Keel (1983) Keel, W. C. 1983, ApJ, 269, 466
* Kemper et al. (2004) Kemper, F., Vriend, W. J., & Tielens, A. G. G. M. 2004, ApJ, 609, 826
* Kester et al. (2003) Kester, D. J. M., Beintema, D. A., & Lutz, D. 2003, in ESA Special Publication, Vol. 481, The Calibration Legacy of the ISO Mission, ed. L. Metcalfe, A. Salama, S. B. Peschke, & M. F. Kessler, 375
* Kewley et al. (2001) Kewley, L. J., Heisler, C. A., Dopita, M. A., & Lumsden, S. 2001, ApJS, 132, 37
* Kim et al. (1995) Kim, D.-C., Sanders, D. B., Veilleux, S., Mazzarella, J. M., & Soifer, B. T. 1995, ApJS, 98, 129
* Kirhakos & Steiner (1990) Kirhakos, S. D., & Steiner, J. E. 1990, AJ, 99, 1722
* Kollatschny et al. (1983) Kollatschny, W., Biermann, P., Fricke, K. J., Huchtmeier, W., & Witzel, A. 1983, A&A, 119, 80
* Kollatschny & Fricke (1985) Kollatschny, W., & Fricke, K. J. 1985, A&A, 143, 393
* Levenson et al. (2007) Levenson, N. A., Sirocky, M. M., Hao, L., Spoon, H. W. W., Marshall, J. A., Elitzur, M., & Houck, J. R. 2007, ApJ, 654, L45
* Lu et al. (2008) Lu, N., et al. 2008, PASP, 120, 328
* Lumsden et al. (2001) Lumsden, S. L., Heisler, C. A., Bailey, J. A., Hough, J. H., & Young, S. 2001, MNRAS, 327, 459
* Madore & Steer (2008) Madore, B. F., & Steer, I. P. 2008, Master List of Galaxy Distances, http://nedwww.ipac.caltech.edu/level5/NED1D/intro.html
* Makovoz & Khan (2005) Makovoz, D., & Khan, I. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 81
* Makovoz et al. (2006) Makovoz, D., Roby, T., Khan, I., & Booth, H. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6274, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
* Márquez et al. (2004) Márquez, I., et al. 2004, A&A, 416, 475
* Marshall et al. (2007) Marshall, J. A., Herter, T. L., Armus, L., Charmandaris, V., Spoon, H. W. W., Bernard-Salas, J., & Houck, J. R. 2007, ApJ, 670, 129
* Mazzarella & Balzano (1986) Mazzarella, J. M., & Balzano, V. A. 1986, ApJS, 62, 751
* Miller & Goodrich (1990) Miller, J. S., & Goodrich, R. W. 1990, ApJ, 355, 456
* Moran et al. (2000) Moran, E. C., Barth, A. J., Kay, L. E., & Filippenko, A. V. 2000, ApJ, 540, L73
* Moustakas & Kennicutt (2006) Moustakas, J., & Kennicutt, Jr., R. C. 2006, ApJS, 164, 81
* Mulchaey et al. (1996) Mulchaey, J. S., Wilson, A. S., & Tsvetanov, Z. 1996, ApJS, 102, 309
* Nagar et al. (1999) Nagar, N. M., Wilson, A. S., Mulchaey, J. S., & Gallimore, J. F. 1999, ApJS, 120, 209
* Nenkova et al. (2002) Nenkova, M., Ivezić, Ž., & Elitzur, M. 2002, ApJ, 570, L9
* Nenkova et al. (2008a) Nenkova, M., Sirocky, M. M., Ivezić, Ž., & Elitzur, M. 2008a, ApJ, 685, 147
* Nenkova et al. (2008b) Nenkova, M., Sirocky, M. M., Nikutta, R., Ivezić, Ž., & Elitzur, M. 2008b, ApJ, 685, 160
* Ossenkopf et al. (1992) Ossenkopf, V., Henning, T., & Mathis, J. S. 1992, A&A, 261, 567
* Osterbrock (1977) Osterbrock, D. E. 1977, ApJ, 215, 733
* Osterbrock & Koski (1976) Osterbrock, D. E., & Koski, A. T. 1976, MNRAS, 176, 61P
* Osterbrock & Pogge (1985) Osterbrock, D. E., & Pogge, R. W. 1985, ApJ, 297, 166
* Peterson et al. (2000) Peterson, B. M., et al. 2000, ApJ, 542, 161
* Phillips et al. (1984) Phillips, M. M., Pagel, B. E. J., Edmunds, M. G., & Diaz, A. 1984, MNRAS, 210, 701
* Pier & Krolik (1992) Pier, E. A., & Krolik, J. H. 1992, ApJ, 401, 99
* Rafanelli et al. (1990) Rafanelli, P., Osterbrock, D. E., & Pogge, R. W. 1990, AJ, 99, 53
* Reach et al. (2005) Reach, W. T., et al. 2005, PASP, 117, 978
* Regan & Gruendl (1995) Regan, M. W., & Gruendl, R. A. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes, 335
* Reunanen et al. (2002) Reunanen, J., Kotilainen, J. K., & Prieto, M. A. 2002, MNRAS, 331, 154
* Rieke et al. (2004) Rieke, G. H., et al. 2004, ApJS, 154, 25
* Risaliti et al. (2007) Risaliti, G., Elvis, M., Fabbiano, G., Baldi, A., Zezas, A., & Salvati, M. 2007, ApJ, 659, L111
* Risaliti et al. (2002) Risaliti, G., Elvis, M., & Nicastro, F. 2002, ApJ, 571, 234
* Rush et al. (1996) Rush, B., Malkan, M. A., & Edelson, R. A. 1996, ApJ, 473, 130
* Rush et al. (1993) Rush, B., Malkan, M. A., & Spinoglio, L. 1993, ApJS, 89, 1
* Savitzky & Golay (1964) Savitzky, A., & Golay, M. J. E. 1964, Analytical Chemistry, 36, 1627
* Shields & Filippenko (1990) Shields, J. C., & Filippenko, A. V. 1990, AJ, 100, 1034
* Siebenmorgen & Krügel (2007) Siebenmorgen, R., & Krügel, E. 2007, A&A, 461, 445
* Sirocky et al. (2008) Sirocky, M. M., Levenson, N. A., Elitzur, M., Spoon, H. W. W., & Armus, L. 2008, ApJ, 678, 729
* Smith et al. (2007a) Smith, J. D. T., et al. 2007a, PASP, 119, 1133
* Smith et al. (2007b) —. 2007b, ApJ, 656, 770
* Smith et al. (2004) Smith, J. E., Robinson, A., Alexander, D. M., Young, S., Axon, D. J., & Corbett, E. A. 2004, MNRAS, 350, 140
* Spinoglio et al. (2002) Spinoglio, L., Andreani, P., & Malkan, M. A. 2002, ApJ, 572, 105
* Spoon et al. (2007) Spoon, H. W. W., Marshall, J. A., Houck, J. R., Elitzur, M., Hao, L., Armus, L., Brandl, B. R., & Charmandaris, V. 2007, ApJ, 654, L49
* Storchi-Bergmann et al. (1995) Storchi-Bergmann, T., Kinney, A. L., & Challis, P. 1995, ApJS, 98, 103
* Sturm et al. (2000) Sturm, E., Lutz, D., Tran, D., Feuchtgruber, H., Genzel, R., Kunze, D., Moorwood, A. F. M., & Thornley, M. D. 2000, A&A, 358, 481
* Sturm et al. (2002) Sturm, E., Lutz, D., Verma, A., Netzer, H., Sternberg, A., Moorwood, A. F. M., Oliva, E., & Genzel, R. 2002, A&A, 393, 821
* Sturm et al. (2006) Sturm, E., et al. 2006, ApJ, 653, L13
* Taylor et al. (1989) Taylor, D., Dyson, J. E., Axon, D. J., & Pedlar, A. 1989, MNRAS, 240, 487
* Thean et al. (2000) Thean, A., Pedlar, A., Kukula, M. J., Baum, S. A., & O’Dea, C. P. 2000, MNRAS, 314, 573
* Thean et al. (2001) —. 2001, MNRAS, 325, 737
* Tran (1995) Tran, H. D. 1995, ApJ, 440, 565
* Tran (2001) —. 2001, ApJ, 554, L19
* Tran (2003) —. 2003, ApJ, 583, 632
* Tristram et al. (2007) Tristram, K. R. W., et al. 2007, A&A, 474, 837
* Tully & Fisher (1988) Tully, R. B., & Fisher, J. R. 1988, Catalog of Nearby Galaxies (Cambridge: Cambridge University Press)
* Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803
* Vaceli et al. (1997) Vaceli, M. S., Viegas, S. M., Gruenwald, R., & de Souza, R. E. 1997, AJ, 114, 1345
* van Dokkum (2001) van Dokkum, P. G. 2001, PASP, 113, 1420
* Veilleux (1988) Veilleux, S. 1988, AJ, 95, 1695
* Veilleux et al. (1995) Veilleux, S., Kim, D.-C., Sanders, D. B., Mazzarella, J. M., & Soifer, B. T. 1995, ApJS, 98, 171
* Véron-Cetty & Véron (2006) Véron-Cetty, M.-P., & Véron, P. 2006, A&A, 455, 773
* Véron-Cetty et al. (2001) Véron-Cetty, M.-P., Véron, P., & Gonçalves, A. C. 2001, A&A, 372, 730
* Whittle et al. (1988) Whittle, M., Pedlar, A., Meurs, E. J. A., Unger, S. W., Axon, D. J., & Ward, M. J. 1988, ApJ, 326, 125
* Winkler (1988) Winkler, H. 1988, MNRAS, 234, 703
* Winkler (1992) —. 1992, MNRAS, 257, 677
* Wu et al. (2009) Wu, Y., Charmandaris, V., Huang, J., Spinoglio, L., & Tommasin, S. 2009, ApJ, 701, 658
* Young et al. (1996) Young, S., Hough, J. H., Efstathiou, A., Wills, B. J., Bailey, J. A., Ward, M. J., & Axon, D. J. 1996, MNRAS, 281, 1206
Table 1: AGNs from the extended 12 $\mu$m sample observed by the Spitzer Space
Telescope
Source ID | Class | Ref | RA (J2000) | Dec (J2000) | cz | D | PID
---|---|---|---|---|---|---|---
| | | (h:m:s) | (°:′:″) | (km s-1) | (Mpc) |
MRK335 | S1n | 1 | 00:06:19.53 | 20:12:10.5 | 7730 | 110.4 | 3269
MRK938 | HII | 2 | 00:11:06.56 | $-$12:06:27.3 | 5881 | 84.0 | 3269, 3672
E12-G21 | S1n | 3 | 00:40:45.93 | $-$79:14:24.2 | 9000 | 128.6 | 3269
MRK348 | S1h | 4 | 00:48:47.16 | 31:57:25.2 | 4507 | 64.4 | 3269
NGC424 | S1h | 5 | 01:11:27.66 | $-$38:05:00.0 | 3527 | 50.4 | 3269
NGC526A | S1.9 | 6 | 01:23:54.39 | $-$35:03:55.4 | 5725 | 81.8 | 86, 3269
NGC513 | S1h | 7 | 01:24:26.78 | 33:47:58.4 | 5859 | 83.7 | 3269
F01475-0740 | S1h | 8 | 01:50:02.69 | $-$07:25:48.4 | 5296 | 75.7 | 3269
NGC931 | S1.0-1.5 | 9 | 02:28:14.49 | 31:18:41.7 | 4992 | 71.3 | 3269
NGC1056 | HII | 10 | 02:42:48.29 | 28:34:26.1 | 1545 | 22.1 | 3269
NGC1097 | LINER | 11, 12 | 02:46:18.91 | $-$30:16:28.8 | 1271 | *17.5 | 159, 3269
NGC1125 | S2 | 13 | 02:51:40.44 | $-$16:39:02.4 | 3277 | 46.8 | 3269
NGC1143-4 | S2 | 8, 10 | 02:55:11.66 | $-$00:11:03.4 | 8648 | 123.5 | 21, 3269
M-2-8-39 | S1h | 8, 14 | 03:00:30.62 | $-$11:24:57.2 | 8962 | 128.0 | 3269
NGC1194 | S1.8-1.9 | 15, 16 | 03:03:49.12 | $-$01:06:13.2 | 4076 | 58.2 | 3269
NGC1241 | S2 | 8, 17 | 03:11:14.63 | $-$08:55:18.1 | 4052 | 57.9 | 3269
NGC1320 | S2 | 8, 18 | 03:24:48.69 | $-$03:02:32.0 | 2663 | *37.7 | 3269
NGC1365 | S1.8 | 19 | 03:33:36.39 | $-$36:08:25.8 | 1636 | *17.7 | 3269, 3672
NGC1386 | S1i | 20 | 03:36:46.20 | $-$35:59:57.0 | 868 | *16.2 | 3269
F03450+0055 | S1n | 1, 21 | 03:47:40.22 | 01:05:13.7 | 9294 | 132.8 | 3269
NGC1566 | S1.5 | 22, 23, 24 | 04:20:00.41 | $-$54:56:16.7 | 1504 | *11.8 | 159, 3269
F04385-0828 | S1h | 8 | 04:40:54.96 | $-$08:22:21.9 | 4527 | 64.7 | 3269
NGC1667 | S2 | 8 | 04:48:37.15 | $-$06:19:11.9 | 4547 | 65.0 | 3269
E33-G2 | S2 | 25 | 04:55:58.88 | $-$75:32:28.4 | 5426 | 77.5 | 3269
M-5-13-17 | S1.5 | 23 | 05:19:35.84 | $-$32:39:28.1 | 3790 | 54.1 | 3269
MRK6 | S1.5 | 26, 27 | 06:52:12.35 | 74:25:37.2 | 5640 | 80.6 | 3269
MRK79 | S1.2 | 28 | 07:42:32.84 | 49:48:34.5 | 6652 | 95.0 | 3269
NGC2639 | LINER | 29 | 08:43:38.06 | 50:12:20.4 | 3336 | 47.7 | 3269
MRK704 | S1.5 | 30 | 09:18:25.98 | 16:18:20.0 | 8764 | 125.2 | 3269
NGC2992 | S1i | 31 | 09:45:41.93 | $-$14:19:34.6 | 2311 | *30.5 | 96, 3269
MRK1239 | S1n | 32 | 09:52:19.09 | $-$01:36:43.5 | 5974 | 85.3 | 3269
NGC3079 | LINER | 33 | 10:01:57.85 | 55:40:46.9 | 1116 | *19.7 | 59, 3269
NGC3227 | S1.5 | 34 | 10:23:30.55 | 19:51:54.6 | 1157 | *20.9 | 96, 3269
NGC3511 | HII | 25 | 11:03:23.81 | $-$23:05:12.3 | 1109 | *14.6 | 3269
NGC3516 | S1.2 | 34 | 11:06:47.49 | 72:34:07.6 | 2649 | *38.9 | 3269
M+0-29-23 | HII | 10 | 11:21:12.27 | $-$02:59:02.5 | 7464 | 106.6 | 3269
NGC3660 | S1.8 | 8, 35 | 11:23:32.27 | $-$08:39:30.4 | 3679 | 52.6 | 3269
NGC3982 | S1.9 | 34 | 11:56:28.12 | 55:07:31.3 | 1109 | *21.8 | 3269
NGC4051 | S1n | 36 | 12:03:09.61 | 44:31:53.0 | 700 | *17.0 | 3269
UGC7064 | S1.9 | 37 | 12:04:43.32 | 31:10:38.1 | 7494 | 107.1 | 3269
NGC4151 | S1.5 | 34 | 12:10:32.57 | 39:24:21.0 | 995 | *20.3 | 3269
MRK766 | S1n | 32 | 12:18:26.51 | 29:48:46.9 | 3876 | 55.4 | 3269
NGC4388 | S1h | 38 | 12:25:46.81 | 12:39:43.3 | 2524 | *18.1 | 3269, 20695
NGC4501 | S2 | 8 | 12:31:59.18 | 14:25:13.3 | 2281 | *20.7 | 3269
NGC4579 | LINER | 34, 39 | 12:37:43.52 | 11:49:05.4 | 1519 | *16.8 | 159, 3269
NGC4593 | S1 | 40 | 12:39:39.44 | $-$05:20:39.0 | 2698 | *44.0 | 3269
NGC4594 | LINER | 34 | 12:39:59.44 | $-$11:37:22.9 | 1024 | *10.9 | 159, 3269
NGC4602 | HII | 40 | 12:40:36.98 | $-$05:07:58.5 | 2539 | *34.4 | 3269
TOL1238-364 | S1h | 8 | 12:40:52.86 | $-$36:45:21.2 | 3275 | 46.8 | 3269
M-2-33-34 | S1n | 37 | 12:52:12.49 | $-$13:24:53.0 | 4386 | 62.7 | 3269
NGC4941 | S2 | 5 | 13:04:13.13 | $-$05:33:05.8 | 1108 | *13.8 | 86, 3269
NGC4968 | S2 | 34 | 13:07:05.96 | $-$23:40:36.4 | 2957 | 42.2 | 3269
NGC5005 | LINER | 34 | 13:10:56.29 | 37:03:32.9 | 946 | *17.5 | 3269
NGC5033 | S1.8 | 34 | 13:13:27.49 | 36:35:37.6 | 875 | *20.6 | 159, 3269
NGC5135 | S2 | 41 | 13:25:44.04 | $-$29:50:00.2 | 4105 | 58.6 | 3269
M-6-30-15 | S1.5 | 23 | 13:35:53.78 | $-$34:17:44.2 | 2323 | 33.2 | 3269
NGC5256 | S2 | 42 | 13:38:17.25 | 48:16:32.4 | 8211 | 117.3 | 3269
IC4329A | S1.5 | 23 | 13:49:19.24 | $-$30:18:34.4 | 4813 | 68.8 | 3269, 30318
NGC5347 | S2 | 8, 43 | 13:53:17.80 | 33:29:27.3 | 2335 | *36.7 | 3269
NGC5506 | S1i | 31 | 14:13:14.87 | $-$03:12:27.6 | 1853 | *28.7 | 3269
NGC5548 | S1.5 | 34 | 14:17:59.52 | 25:08:12.6 | 5149 | 73.6 | 69, 86, 3269
MRK817 | S1.5 | 44 | 14:36:22.08 | 58:47:39.6 | 9430 | 134.7 | 3269
NGC5929 | S2 | 45, 47 | 15:26:06.20 | 41:40:14.5 | 2492 | *38.5 | 3269
NGC5953 | S2 | 47 | 15:34:32.39 | 15:11:37.2 | 1965 | *33.0 | 59, 3269
M-2-40-4 | S1.9 | 48, 49, 50 | 15:48:24.96 | $-$13:45:26.9 | 7553 | 107.9 | 3269
F15480-0344 | S1h | 8, 38 | 15:50:41.48 | $-$03:53:18.1 | 9084 | 129.8 | 3269
NGC6810 | HII | 25 | 19:43:34.42 | $-$58:39:20.3 | 2031 | *29.0 | 3269
NGC6860 | S1.5 | 23 | 20:08:46.90 | $-$61:05:59.6 | 4462 | 63.7 | 3269
NGC6890 | S2 | 5, 51 | 20:18:18.02 | $-$44:48:24.7 | 2419 | *31.8 | 3269
IC5063 | S1h | 52 | 20:52:02.29 | $-$57:04:07.5 | 3402 | 48.6 | 86, 3269
UGC11680 | S2 | 49 | 21:07:41.35 | 03:52:17.9 | 7791 | 111.3 | 3269
NGC7130 | S2 | 53, 54, 55, 56 | 21:48:19.52 | $-$34:57:04.8 | 4842 | 69.2 | 3269, 3672
NGC7172 | S2 | 41, 57 | 22:02:01.90 | $-$31:52:10.4 | 2603 | *33.9 | 86, 3269
NGC7213 | S1 | 58 | 22:09:16.21 | $-$47:09:59.7 | 1750 | *22.0 | 3269
NGC7314 | S1i | 59 | 22:35:46.21 | $-$26:03:01.5 | 1428 | *19.0 | 86, 3269
M-3-58-7 | S1h | 8 | 22:49:37.17 | $-$19:16:26.2 | 9432 | 134.7 | 3269
NGC7469 | S1.5 | 44 | 23:03:15.61 | 08:52:26.3 | 4892 | 69.9 | 32, 3269
NGC7496 | S2 | 38, 49, 60 | 23:09:47.29 | $-$43:25:40.2 | 1649 | *20.1 | 3269
NGC7582 | S1i | 61 | 23:18:23.63 | $-$42:22:13.1 | 1575 | *18.8 | 3269
NGC7590 | S2 | 60 | 23:18:54.81 | $-$42:14:20.0 | 1575 | *23.7 | 3269
NGC7603 | S1.5 | 44 | 23:18:56.67 | 00:14:38.1 | 8851 | 126.4 | 3269
NGC7674 | S1h | 4, 9 | 23:27:56.72 | 08:46:44.4 | 8671 | 123.9 | 3269, 3672
CGCG381-051 | HII | 8 | 23:48:41.74 | 02:14:23.5 | 9194 | 131.3 | 3269
Note. — The source ID is based on the primary source identification of Rush et
al. (1993). AGN classifications are from (1) Véron-Cetty et al. (2001), (2)
Mulchaey et al. (1996), (3) Clavel & Joly (1984), (4) Miller & Goodrich
(1990), (5) Moran et al. (2000), (6) Nagar et al. (1999), (7) Tran (1995), (8)
Tran (2003), (9) Veilleux (1988), (10) Veilleux et al. (1995), (11) Keel
(1983), (12) Phillips et al. (1984), (13) Winkler (1988), (14) Heisler et al.
(1989), (15) de Grijp et al. (1992), (16) Hao et al. (2005b), (17) Vaceli et
al. (1997), (18) De Robertis & Osterbrock (1986), (19) Alloin et al. (1981),
(20) Reunanen et al. (2002), (21) Giannuzzo & Stirpe (1996), (22) Alloin et
al. (1985), (23) Winkler (1992), (24) Agüero et al. (2004), (25) Kirhakos &
Steiner (1990), (26) Osterbrock & Koski (1976), (27) Feldmeier et al. (1999),
(28) Osterbrock (1977), (29) Alonso-Herrero et al. (2000), (30) Mazzarella &
Balzano (1986), (31) Goodrich et al. (1994), (32) Osterbrock & Pogge (1985),
(33) Heckman (1980), (34) Ho et al. (1997), (35) Kollatschny et al. (1983),
(36) Peterson et al. (2000), (37) Goodrich (1989), (38) Young et al. (1996),
(39) Filippenko & Sargent (1985), (40) Kollatschny & Fricke (1985), (41)
Heisler et al. (1997), (42) Kay (1994), (43) González Delgado & Perez (1996),
(44) Dahari & De Robertis (1988), (45) Whittle et al. (1988), (46) Taylor et
al. (1989), (47) Rafanelli et al. (1990), (48) Tran (2001), (49) Kewley et al.
(2001), (50) Lumsden et al. (2001), (51) Márquez et al. (2004), (52) Inglis et
al. (1993), (53) Shields & Filippenko (1990), (54) Kim et al. (1995), (55)
González Delgado et al. (1998), (56) Moustakas & Kennicutt (2006), (57) Coziol
et al. (1998), (58) Filippenko & Halpern (1984), (59) Dewangan & Griffiths
(2005), (60) Storchi-Bergmann et al. (1995), and (61) Aretxaga et al. (1999).
The classification naming convention broadly follows Véron-Cetty & Véron
(2006): S1.n refers to Seyfert types 1.n (1.2, 1.5, 1.8, or 1.9); S1h = hidden
Seyfert 1 revealed by spectropolarimetry; S1i = hidden Seyfert 1 revealed by
broad Paschen or Brackett lines or via another, non-polarimetric technique;
S1n = narrow-line Seyfert 1; S2 = Seyfert 2; HII = star-forming or starburst
galaxy. For NGC 1194 and NGC 5033, the classifications were based on
inspection of the spectra presented in the cited references and differ from
from the classification assigned in those references. M-2-40-4 presents a S1.9
spectrum, although spectropolarimetry reveals broader Balmer lines; we retain
the S1.9 classification. To the best of our knowledge, HBLRs have been
searched in all but three of the 20 S2s in this list: NGC 1125, E33-G2, and
NGC 4968 (Tran, 2003). Source coordinates are based on the astrometry of the
IRAC 8.0 $\mu$m image. Recessional velocities are from NED. The adopted
distances either were calculated using the Hubble Law assuming $H_{0}=70$ km
s-1 Mpc-1 or, where data were available in NED, taken from an average of
redshift-independent distance indicators. Redshift-independent distances,
compiled by Madore & Steer (2008), which, for this sample, commonly derive
solely from Tully & Fisher (1988), are marked by (*). PID lists the Spitzer
program codes that were used to construct the present data set. Spitzer PID
3269 indicates data from our original Cycle 1 proposal. The SINGS key project
is PID 159.
Table 2: IRAC Photometry of the 12 $\mu$m AGN sample
Object | [3.6] | unc | [4.5] | unc | [5.8] | unc | [8.0] | unc
---|---|---|---|---|---|---|---|---
MRK335 | 8.99 | 0.07 | 8.31 | 0.07 | 7.67 | 0.06 | 6.69 | 0.05
MRK938 | 9.55 | 0.05 | 9.24 | 0.04 | 7.74 | 0.05 | 5.09 | 0.02
E12-G21 | 9.63 | 0.09 | 9.14 | 0.10 | 8.31 | 0.08 | 6.22 | 0.07
MRK348 | 9.27 | 0.08 | 8.46 | 0.08 | 7.67 | 0.06 | 6.38 | 0.04
NGC424 | 7.83 | 0.05 | 6.94 | 0.04 | 6.11 | 0.03 | 4.95 | 0.02
NGC526A | 9.00 | 0.07 | 8.33 | 0.07 | 7.66 | 0.07 | 6.67 | 0.06
NGC5135 | 8.95 | 0.10 | 8.60 | 0.09 | 7.35 | 0.08 | 4.82 | 0.07
F01475-0740 | 10.83 | 0.16 | 9.89 | 0.14 | 8.59 | 0.10 | 7.16 | 0.06
NGC931 | 8.38 | 0.05 | 7.75 | 0.05 | 6.87 | 0.04 | 5.78 | 0.03
NGC1056 | 9.26 | 0.11 | 9.32 | 0.12 | 8.23 | 0.12 | 5.50 | 0.10
NGC1097 | 7.73 | 0.10 | 7.75 | 0.11 | 6.54 | 0.10 | 3.98 | 0.10
NGC1125 | 10.05 | 0.11 | 9.67 | 0.11 | 8.45 | 0.10 | 6.51 | 0.05
NGC1143-4 | 9.68 | 0.09 | 9.44 | 0.08 | 8.37 | 0.09 | 5.66 | 0.09
M-2-8-39 | 10.86 | 0.14 | 10.25 | 0.15 | 9.27 | 0.13 | 7.35 | 0.07
NGC1194 | 9.29 | 0.08 | 8.38 | 0.07 | 7.31 | 0.05 | 6.10 | 0.04
NGC1241 | 9.87 | 0.11 | 9.86 | 0.12 | 9.56 | 0.16 | 7.11 | 0.09
NGC1320 | 9.10 | 0.08 | 8.47 | 0.07 | 7.48 | 0.06 | 5.96 | 0.04
NGC1365 | 7.23 | 0.07 | 6.85 | 0.06 | 5.70 | 0.08 | 3.25 | 0.09
NGC1386 | 8.64 | 0.08 | 8.15 | 0.07 | 7.16 | 0.05 | 5.56 | 0.03
F03450+0055 | 9.11 | 0.08 | 8.46 | 0.07 | 7.74 | 0.06 | 6.61 | 0.04
NGC1566 | 8.67 | 0.09 | 8.59 | 0.08 | 7.94 | 0.07 | 6.14 | 0.05
F04385-0828 | 9.09 | 0.08 | 8.04 | 0.06 | 7.04 | 0.05 | 5.64 | 0.03
NGC1667 | 9.60 | 0.11 | 9.65 | 0.12 | 8.41 | 0.14 | 5.96 | 0.11
E33-G2 | 9.40 | 0.09 | 8.69 | 0.08 | 7.91 | 0.07 | 6.65 | 0.05
M-5-13-17 | 9.74 | 0.10 | 9.38 | 0.10 | 8.26 | 0.09 | 6.65 | 0.07
MRK6 | 8.71 | 0.06 | 8.12 | 0.06 | 7.60 | 0.05 | 6.63 | 0.05
MRK79 | 8.73 | 0.06 | 8.05 | 0.06 | 7.24 | 0.05 | 6.05 | 0.04
NGC2639 | 9.17 | 0.11 | 9.25 | 0.12 | 8.67 | 0.15 | 7.27 | 0.12
MRK704 | 8.84 | 0.07 | 8.06 | 0.06 | 7.35 | 0.05 | 6.08 | 0.04
NGC2992 | 8.46 | 0.06 | 8.07 | 0.05 | 7.25 | 0.05 | 5.35 | 0.05
MRK1239 | 7.70 | 0.04 | 7.03 | 0.04 | 6.31 | 0.03 | 5.27 | 0.03
NGC3079 | 8.12 | 0.09 | 8.09 | 0.09 | 6.61 | 0.09 | 3.78 | 0.08
NGC3227 | 8.48 | 0.06 | 8.16 | 0.06 | 7.18 | 0.06 | 5.36 | 0.04
NGC3511 | 10.03 | 0.12 | 10.13 | 0.13 | 9.41 | 0.21 | 6.20 | 0.11
NGC3516 | 8.42 | 0.06 | 7.96 | 0.06 | 7.27 | 0.06 | 6.16 | 0.04
M+0-29-23 | 9.93 | 0.11 | 9.65 | 0.11 | 8.30 | 0.09 | 5.59 | 0.06
NGC3660 | 10.68 | 0.13 | 10.66 | 0.16 | 10.35 | 0.34 | 7.65 | 0.12
NGC3982 | 9.91 | 0.11 | 9.95 | 0.12 | 8.81 | 0.14 | 6.36 | 0.10
NGC4051 | 8.53 | 0.06 | 7.94 | 0.06 | 7.10 | 0.04 | 5.56 | 0.03
UGC7064 | 9.98 | 0.10 | 9.67 | 0.11 | 8.88 | 0.09 | 6.62 | 0.12
NGC4151 | 7.42 | 0.05 | 6.67 | 0.03 | 5.86 | 0.03 | 4.52 | 0.02
MRK766 | 9.06 | 0.07 | 8.32 | 0.07 | 7.60 | 0.06 | 5.99 | 0.04
NGC4388 | 8.79 | 0.07 | 8.35 | 0.06 | 7.11 | 0.05 | 5.49 | 0.08
NGC4501 | 8.53 | 0.10 | 8.61 | 0.11 | 8.42 | 0.14 | 6.98 | 0.11
NGC4579 | 8.25 | 0.09 | 8.23 | 0.09 | 7.87 | 0.08 | 7.12 | 0.06
NGC4593 | 8.38 | 0.07 | 7.82 | 0.06 | 7.12 | 0.05 | 5.86 | 0.03
NGC4594 | 7.04 | 0.10 | 7.16 | 0.11 | 6.93 | 0.10 | 6.95 | 0.09
NGC4602 | 10.23 | 0.12 | 10.26 | 0.13 | 9.38 | 0.19 | 7.37 | 0.10
TOL1238-364 | 9.36 | 0.09 | 8.88 | 0.08 | 7.77 | 0.07 | 5.47 | 0.10
M-2-33-34 | 10.12 | 0.11 | 9.87 | 0.12 | 9.06 | 0.13 | 7.39 | 0.08
NGC4941 | 9.65 | 0.10 | 9.58 | 0.11 | 9.02 | 0.28 | 8.01 | 0.13
NGC4968 | 9.66 | 0.09 | 9.04 | 0.09 | 8.02 | 0.07 | 6.20 | 0.04
NGC5005 | 7.81 | 0.10 | 7.85 | 0.10 | 7.35 | 0.10 | 5.63 | 0.09
NGC5033 | 8.26 | 0.10 | 8.30 | 0.10 | 7.59 | 0.10 | 5.45 | 0.10
NGC513 | 9.68 | 0.10 | 9.45 | 0.10 | 8.72 | 0.13 | 6.32 | 0.09
M-6-30-15 | 8.55 | 0.06 | 7.87 | 0.06 | 7.16 | 0.05 | 6.07 | 0.04
NGC5256 | 10.19 | 0.12 | 10.01 | 0.12 | 8.56 | 0.11 | 5.67 | 0.06
IC4329A | 7.38 | 0.03 | 6.68 | 0.04 | 5.94 | 0.03 | 4.93 | 0.02
NGC5347 | 9.87 | 0.10 | 9.21 | 0.10 | 8.18 | 0.07 | 6.56 | 0.05
NGC5506 | 7.02 | 0.03 | 6.25 | 0.03 | 5.49 | 0.02 | 4.41 | 0.02
NGC5548 | 9.49 | 0.05 | 8.85 | 0.04 | 7.96 | 0.03 | 6.48 | 0.02
MRK817 | 9.10 | 0.07 | 8.45 | 0.07 | 7.61 | 0.06 | 6.33 | 0.04
NGC5929 | 10.33 | 0.12 | 10.36 | 0.14 | 9.59 | 0.20 | 8.52 | 0.14
NGC5953 | 9.13 | 0.10 | 9.24 | 0.10 | 7.70 | 0.11 | 4.85 | 0.10
M-2-40-4 | 8.51 | 0.06 | 7.86 | 0.06 | 7.12 | 0.04 | 5.58 | 0.04
F15480-0344 | 10.38 | 0.12 | 9.66 | 0.12 | 8.75 | 0.10 | 7.03 | 0.07
NGC6810 | 8.14 | 0.09 | 8.17 | 0.10 | 7.22 | 0.09 | 4.49 | 0.08
NGC6860 | 9.06 | 0.07 | 8.44 | 0.07 | 7.57 | 0.06 | 6.36 | 0.04
NGC6890 | 9.50 | 0.10 | 9.21 | 0.10 | 8.35 | 0.09 | 6.35 | 0.06
IC5063 | 8.79 | 0.07 | 7.90 | 0.06 | 6.80 | 0.05 | 5.14 | 0.03
UGC11680 | 10.02 | 0.10 | 9.64 | 0.11 | 9.09 | 0.11 | 7.28 | 0.10
NGC7130 | 9.43 | 0.08 | 9.16 | 0.07 | 7.92 | 0.07 | 5.15 | 0.06
NGC7172 | 8.36 | 0.07 | 7.88 | 0.06 | 7.00 | 0.07 | 5.31 | 0.06
NGC7213 | 8.02 | 0.08 | 7.79 | 0.08 | 7.31 | 0.07 | 6.43 | 0.06
NGC7314 | 9.94 | 0.10 | 9.53 | 0.11 | 8.87 | 0.19 | 7.37 | 0.08
M-3-58-7 | 8.94 | 0.07 | 8.25 | 0.07 | 7.49 | 0.05 | 6.04 | 0.04
NGC7469 | 8.18 | 0.05 | 7.69 | 0.04 | 6.57 | 0.05 | 4.33 | 0.04
NGC7496 | 9.99 | 0.11 | 9.83 | 0.11 | 8.46 | 0.10 | 5.93 | 0.06
NGC7582 | 7.57 | 0.05 | 7.00 | 0.04 | 5.87 | 0.05 | 3.73 | 0.05
NGC7590 | 9.44 | 0.11 | 9.55 | 0.12 | 8.85 | 0.16 | 6.37 | 0.11
NGC7603 | 8.23 | 0.05 | 7.65 | 0.05 | 7.02 | 0.04 | 5.83 | 0.03
NGC7674 | 8.99 | 0.04 | 8.20 | 0.03 | 7.23 | 0.03 | 5.40 | 0.02
CGCG381-051 | 11.14 | 0.15 | 10.83 | 0.17 | 9.67 | 0.20 | 6.91 | 0.07
Note. — IRAC photometry is presented in the Spitzer IRAC magnitude system.
Zero point flux densities are 280.9, 179.7, 115.0, and 64.13 Jy for the 3.6,
4.5, 5.8, and 8.0 $\mu$m channels, respectively. Color corrections and
corrections for extended emission have been applied. Uncertainties include
contributions from detector noise and photon counting statistics, noise in the
IRS spectra and numerical integration errors as they contribute to the color
correction, and the (10% of extended flux) uncertainty of the extended
emission correction.
Table 3: Comparison of Line Measurements using Optimal and Non-optimal
Weighting
Line | $\lambda$($\mu$m) | Non-opt. Flux | Opt. Flux | Non-opt. EQW | Opt. EQW
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
NGC 3079
H2S(1) | 17.05 | 51 | (5) | 51 | (4) | 137 | (14) | 135 | (10)
$[$NeII$]$ | 12.81 | 183 | (11) | 190 | (12) | 345 | (22) | 352 | (22)
$[$NeIII$]$ | 15.56 | 22 | (6) | 23 | (6) | 41 | (12) | 44 | (11)
$[$OIV$]$ | 25.91 | 30 | (7) | 31 | (6) | 38 | (8) | 39 | (8)
$[$SIII$]$ | 18.71 | 18 | (3) | 16 | (3) | 51 | (7) | 44 | (7)
$[$SIII$]$ | 33.50 | 63 | (14) | 61 | (15) | 43 | (10) | 42 | (11)
$[$SiII$]$ | 34.82 | 260 | (27) | 262 | (26) | 165 | (17) | 166 | (16)
PAH | 6.22 | 2520 | (110) | 2570 | (120) | 3890 | (170) | 3950 | (180)
PAH | 11.33 | 1380 | (50) | 1370 | (51) | 5200 | (190) | 5170 | (190)
F01475-0740
$[$NeII$]$ | 12.81 | 16 | (1) | 16 | (1) | 44 | (4) | 42 | (2)
$[$NeIII$]$ | 15.56 | 11 | (1) | 10 | (1) | 28 | (4) | 25 | (2)
$[$NeV$]$ | 14.32 | $<4$ | $\cdots$ | 2 | (1) | $<10$ | $\cdots$ | 6 | (2)
$[$SIII$]$ | 18.71 | $<4$ | $\cdots$ | 5 | (1) | $<9$ | $\cdots$ | 11 | (3)
PAH | 6.22 | $<56$ | $\cdots$ | 40 | (9) | $<160$ | $\cdots$ | 113 | (26)
PAH | 11.33 | 70 | (7) | 69 | (5) | 165 | (17) | 161 | (11)
Note. — Integrated fluxes are given in units of $10^{-17}$ W m-2, and
equivalent widths are given in units of nm. Uncertainties ($1\sigma$) are
listed in parentheses. Upper limits are $3\sigma$. Only lines with a detection
in the optimal extraction are presented; all lines that are detected in the
unweighted spectrum are also detected in the optimally extracted spectrum.
Measurements from the unweighted spectrum are given in columns (3) and (5);
measurements from the optimally weighted spectrum are given in the neighboring
columns (4) and (6).
Table 4: PAH Fluxes
Source ID | 6.2 $\mu$m | 7.4 $\mu$m | 7.6 $\mu$m | 7.8 $\mu$m | 8.3 $\mu$m | 8.6 $\mu$m | 11.2 $\mu$m | 11.3 $\mu$m | 12.0 $\mu$m | 12.6 $\mu$m | 17.0 $\mu$m
---|---|---|---|---|---|---|---|---|---|---|---
MRK335 | $<29$ | $\cdots$ | $<105$ | $\cdots$ | $<29$ | $\cdots$ | $58$ | (10) | $37$ | (11) | $28$ | (8) | $<7$ | $\cdots$ | $22$ | (6) | $27$ | (6) | $31$ | (4) | $<18$ | $\cdots$
MRK938 | $735$ | (34) | $1470$ | (150) | $789$ | (45) | $943$ | (40) | $415$ | (29) | $451$ | (21) | $152$ | (4) | $378$ | (12) | $223$ | (15) | $487$ | (20) | $225$ | (26)
E12-G21 | $208$ | (21) | $<201$ | $\cdots$ | $258$ | (18) | $263$ | (17) | $57$ | (11) | $148$ | (9) | $33$ | (4) | $190$ | (10) | $67$ | (8) | $142$ | (15) | $96$ | (8)
MRK348 | $<37$ | $\cdots$ | $244$ | (31) | $<44$ | $\cdots$ | $113$ | (16) | $<39$ | $\cdots$ | $58$ | (9) | $14$ | (3) | $51$ | (6) | $<28$ | $\cdots$ | $24$ | (7) | $<46$ | $\cdots$
NGC424 | $41$ | (12) | $509$ | (76) | $137$ | (33) | $189$ | (38) | $145$ | (24) | $88$ | (19) | $<21$ | $\cdots$ | $55$ | (14) | $57$ | (14) | $<56$ | $\cdots$ | $<60$ | $\cdots$
NGC526A | $22$ | (7) | $207$ | (29) | $35$ | (8) | $37$ | (11) | $43$ | (9) | $26$ | (6) | $<7$ | $\cdots$ | $30$ | (6) | $45$ | (5) | $26$ | (5) | $<31$ | $\cdots$
NGC513 | $186$ | (20) | $<312$ | $\cdots$ | $355$ | (31) | $236$ | (31) | $<84$ | $\cdots$ | $153$ | (19) | $99$ | (3) | $150$ | (10) | $76$ | (12) | $105$ | (12) | $109$ | (11)
F01475-0740 | $40$ | (9) | $90$ | (29) | $103$ | (10) | $<33$ | $\cdots$ | $<23$ | $\cdots$ | $30$ | (6) | $13$ | (2) | $69$ | (5) | $60$ | (6) | $66$ | (6) | $<27$ | $\cdots$
NGC931 | $78$ | (10) | $<176$ | $\cdots$ | $73$ | (22) | $118$ | (26) | $<50$ | $\cdots$ | $39$ | (12) | $16$ | (4) | $112$ | (7) | $35$ | (9) | $36$ | (9) | $36$ | (12)
NGC1056 | $648$ | (33) | $665$ | (88) | $898$ | (42) | $716$ | (48) | $225$ | (34) | $468$ | (12) | $202$ | (11) | $395$ | (30) | $157$ | (11) | $320$ | (16) | $205$ | (18)
NGC1097 | $2219$ | (96) | $2710$ | (240) | $2850$ | (150) | $3190$ | (180) | $1050$ | (130) | $1625$ | (36) | $642$ | (30) | $1703$ | (78) | $868$ | (41) | $1810$ | (140) | $548$ | (42)
NGC1125 | $167$ | (15) | $435$ | (82) | $206$ | (26) | $192$ | (21) | $93$ | (8) | $79$ | (6) | $51$ | (3) | $130$ | (7) | $52$ | (7) | $117$ | (8) | $56$ | (8)
NGC1143-4 | $364$ | (24) | $402$ | (48) | $484$ | (14) | $474$ | (12) | $125$ | (7) | $227$ | (6) | $83$ | (4) | $252$ | (11) | $110$ | (4) | $204$ | (10) | $142$ | (9)
M-2-8-39 | $<27$ | $\cdots$ | $<188$ | $\cdots$ | $<53$ | $\cdots$ | $<50$ | $\cdots$ | $<31$ | $\cdots$ | $23$ | (7) | $<5$ | $\cdots$ | $<15$ | $\cdots$ | $<20$ | $\cdots$ | $13$ | (4) | $84$ | (14)
NGC1194 | $32$ | (8) | $<138$ | $\cdots$ | $113$ | (17) | $96$ | (18) | $<46$ | $\cdots$ | $<34$ | $\cdots$ | $<8$ | $\cdots$ | $50$ | (7) | $48$ | (12) | $35$ | (7) | $64$ | (10)
NGC1241 | $112$ | (16) | $338$ | (93) | $<104$ | $\cdots$ | $190$ | (38) | $63$ | (18) | $62$ | (12) | $31$ | (3) | $79$ | (8) | $32$ | (10) | $45$ | (5) | $81$ | (9)
NGC1320 | $118$ | (10) | $<255$ | $\cdots$ | $213$ | (29) | $195$ | (32) | $61$ | (17) | $95$ | (13) | $39$ | (4) | $98$ | (8) | $25$ | (7) | $76$ | (10) | $<37$ | $\cdots$
NGC1365 | $3730$ | (180) | $4330$ | (580) | $4890$ | (260) | $5610$ | (310) | $1720$ | (190) | $2436$ | (84) | $839$ | (42) | $2510$ | (110) | $1243$ | (64) | $2590$ | (180) | $1220$ | (82)
NGC1386 | $85$ | (15) | $<264$ | $\cdots$ | $270$ | (28) | $274$ | (30) | $82$ | (16) | $93$ | (10) | $42$ | (4) | $220$ | (9) | $108$ | (14) | $148$ | (14) | $146$ | (21)
F03450+0055 | $32$ | (6) | $249$ | (36) | $<24$ | $\cdots$ | $69$ | (7) | $76$ | (6) | $28$ | (5) | $<6$ | $\cdots$ | $<16$ | $\cdots$ | $33$ | (6) | $36$ | (6) | $<30$ | $\cdots$
NGC1566 | $160$ | (14) | $286$ | (53) | $218$ | (17) | $303$ | (20) | $106$ | (14) | $122$ | (9) | $79$ | (3) | $227$ | (9) | $85$ | (6) | $142$ | (8) | $159$ | (7)
F04385-0828 | $54$ | (11) | $<134$ | $\cdots$ | $103$ | (18) | $<51$ | $\cdots$ | $<44$ | $\cdots$ | $<33$ | $\cdots$ | $<10$ | $\cdots$ | $104$ | (9) | $91$ | (12) | $119$ | (11) | $<45$ | $\cdots$
NGC1667 | $387$ | (29) | $563$ | (85) | $436$ | (35) | $518$ | (40) | $121$ | (29) | $290$ | (20) | $106$ | (5) | $279$ | (14) | $102$ | (12) | $230$ | (11) | $159$ | (17)
E33-G2 | $<40$ | $\cdots$ | $335$ | (54) | $<50$ | $\cdots$ | $<47$ | $\cdots$ | $55$ | (8) | $39$ | (7) | $<8$ | $\cdots$ | $33$ | (8) | $<33$ | $\cdots$ | $<18$ | $\cdots$ | $<21$ | $\cdots$
M-5-13-17 | $114$ | (9) | $<227$ | $\cdots$ | $96$ | (23) | $143$ | (24) | $44$ | (12) | $102$ | (9) | $56$ | (3) | $103$ | (7) | $35$ | (6) | $47$ | (6) | $61$ | (9)
MRK6 | $30$ | (7) | $<132$ | $\cdots$ | $51$ | (13) | $<24$ | $\cdots$ | $<30$ | $\cdots$ | $<28$ | $\cdots$ | $12$ | (2) | $<16$ | $\cdots$ | $<15$ | $\cdots$ | $<22$ | $\cdots$ | $73$ | (14)
MRK79 | $29$ | (6) | $335$ | (74) | $<62$ | $\cdots$ | $63$ | (14) | $57$ | (9) | $<21$ | $\cdots$ | $<9$ | $\cdots$ | $51$ | (7) | $40$ | (7) | $50$ | (10) | $<47$ | $\cdots$
NGC2639 | $68$ | (17) | $<249$ | $\cdots$ | $119$ | (28) | $110$ | (27) | $<66$ | $\cdots$ | $<51$ | $\cdots$ | $31$ | (3) | $80$ | (9) | $<36$ | $\cdots$ | $39$ | (8) | $61$ | (10)
MRK704 | $<29$ | $\cdots$ | $<92$ | $\cdots$ | $<39$ | $\cdots$ | $62$ | (16) | $<36$ | $\cdots$ | $<24$ | $\cdots$ | $<6$ | $\cdots$ | $<18$ | $\cdots$ | $<22$ | $\cdots$ | $<23$ | $\cdots$ | $<30$ | $\cdots$
NGC2992 | $352$ | (26) | $706$ | (96) | $434$ | (29) | $521$ | (34) | $193$ | (18) | $258$ | (13) | $119$ | (5) | $330$ | (14) | $142$ | (12) | $187$ | (27) | $274$ | (25)
MRK1239 | $45$ | (13) | $246$ | (58) | $133$ | (28) | $134$ | (17) | $90$ | (15) | $74$ | (13) | $24$ | (6) | $54$ | (13) | $59$ | (12) | $78$ | (11) | $<42$ | $\cdots$
NGC3079 | $2570$ | (120) | $4140$ | (340) | $3350$ | (150) | $3960$ | (190) | $1330$ | (140) | $1233$ | (73) | $361$ | (18) | $1372$ | (51) | $859$ | (45) | $1804$ | (95) | $949$ | (45)
NGC3227 | $329$ | (16) | $572$ | (62) | $370$ | (30) | $506$ | (22) | $164$ | (14) | $239$ | (12) | $120$ | (9) | $391$ | (24) | $171$ | (13) | $216$ | (18) | $328$ | (27)
NGC3511 | $321$ | (22) | $315$ | (69) | $420$ | (25) | $407$ | (29) | $143$ | (25) | $230$ | (15) | $103$ | (4) | $213$ | (11) | $78$ | (8) | $170$ | (9) | $144$ | (11)
NGC3516 | $<38$ | $\cdots$ | $<256$ | $\cdots$ | $75$ | (24) | $<67$ | $\cdots$ | $58$ | (14) | $<28$ | $\cdots$ | $11$ | (3) | $31$ | (6) | $<28$ | $\cdots$ | $<27$ | $\cdots$ | $<37$ | $\cdots$
M+0-29-23 | $409$ | (23) | $787$ | (69) | $487$ | (22) | $538$ | (25) | $202$ | (14) | $300$ | (12) | $112$ | (4) | $305$ | (8) | $138$ | (6) | $281$ | (14) | $193$ | (13)
NGC3660 | $57$ | (15) | $201$ | (47) | $<64$ | $\cdots$ | $175$ | (23) | $45$ | (10) | $36$ | (8) | $23$ | (2) | $27$ | (5) | $21$ | (6) | $43$ | (6) | $<32$ | $\cdots$
NGC3982 | $258$ | (32) | $480$ | (110) | $443$ | (46) | $352$ | (52) | $122$ | (33) | $218$ | (12) | $102$ | (4) | $187$ | (10) | $67$ | (10) | $130$ | (13) | $153$ | (11)
NGC4051 | $139$ | (10) | $<176$ | $\cdots$ | $260$ | (20) | $214$ | (22) | $77$ | (14) | $96$ | (10) | $66$ | (5) | $171$ | (12) | $75$ | (10) | $143$ | (9) | $133$ | (16)
UGC7064 | $122$ | (16) | $<254$ | $\cdots$ | $229$ | (26) | $134$ | (21) | $44$ | (11) | $132$ | (9) | $35$ | (4) | $110$ | (12) | $46$ | (10) | $70$ | (6) | $66$ | (10)
NGC4151 | $55$ | (14) | $<465$ | $\cdots$ | $287$ | (55) | $472$ | (64) | $276$ | (47) | $235$ | (41) | $45$ | (14) | $<78$ | $\cdots$ | $<77$ | $\cdots$ | $168$ | (40) | $<208$ | $\cdots$
MRK766 | $102$ | (13) | $594$ | (48) | $56$ | (17) | $192$ | (19) | $99$ | (14) | $118$ | (10) | $35$ | (4) | $103$ | (8) | $56$ | (9) | $<40$ | $\cdots$ | $150$ | (22)
NGC4388 | $233$ | (9) | $440$ | (140) | $393$ | (53) | $451$ | (53) | $134$ | (16) | $186$ | (14) | $82$ | (7) | $350$ | (19) | $180$ | (24) | $270$ | (29) | $244$ | (24)
NGC4501 | $<75$ | $\cdots$ | $<197$ | $\cdots$ | $79$ | (25) | $102$ | (22) | $<52$ | $\cdots$ | $<37$ | $\cdots$ | $36$ | (4) | $124$ | (9) | $58$ | (12) | $83$ | (10) | $90$ | (7)
NGC4579 | $63$ | (11) | $213$ | (41) | $<43$ | $\cdots$ | $55$ | (15) | $<31$ | $\cdots$ | $23$ | (6) | $24$ | (2) | $95$ | (4) | $53$ | (7) | $55$ | (10) | $63$ | (5)
NGC4593 | $90$ | (10) | $<171$ | $\cdots$ | $145$ | (20) | $170$ | (20) | $76$ | (15) | $72$ | (12) | $30$ | (4) | $94$ | (9) | $27$ | (9) | $58$ | (11) | $<47$ | $\cdots$
NGC4594 | $34$ | (10) | $<172$ | $\cdots$ | $<51$ | $\cdots$ | $84$ | (20) | $<33$ | $\cdots$ | $<16$ | $\cdots$ | $7$ | (2) | $16$ | (3) | $<10$ | $\cdots$ | $<17$ | $\cdots$ | $<7$ | $\cdots$
TOL1238-364 | $335$ | (19) | $550$ | (89) | $399$ | (35) | $394$ | (40) | $94$ | (15) | $221$ | (10) | $97$ | (6) | $324$ | (16) | $168$ | (13) | $253$ | (18) | $<58$ | $\cdots$
NGC4602 | $107$ | (11) | $353$ | (64) | $<74$ | $\cdots$ | $145$ | (21) | $<52$ | $\cdots$ | $104$ | (14) | $39$ | (3) | $82$ | (6) | $47$ | (8) | $65$ | (8) | $59$ | (6)
M-2-33-34 | $40$ | (7) | $<229$ | $\cdots$ | $76$ | (22) | $56$ | (16) | $<33$ | $\cdots$ | $<33$ | $\cdots$ | $24$ | (2) | $67$ | (4) | $25$ | (4) | $55$ | (5) | $57$ | (8)
NGC4941 | $36$ | (5) | $173$ | (22) | $35$ | (8) | $<24$ | $\cdots$ | $28$ | (4) | $20$ | (2) | $<9$ | $\cdots$ | $40$ | (6) | $30$ | (6) | $26$ | (7) | $<29$ | $\cdots$
NGC4968 | $133$ | (9) | $364$ | (41) | $181$ | (17) | $156$ | (11) | $108$ | (16) | $92$ | (12) | $41$ | (4) | $145$ | (9) | $77$ | (9) | $83$ | (9) | $59$ | (16)
NGC5005 | $326$ | (22) | $722$ | (76) | $350$ | (29) | $534$ | (34) | $194$ | (29) | $246$ | (17) | $195$ | (10) | $525$ | (27) | $226$ | (13) | $386$ | (28) | $358$ | (13)
NGC5033 | $483$ | (28) | $579$ | (94) | $650$ | (39) | $751$ | (44) | $243$ | (27) | $357$ | (12) | $179$ | (6) | $472$ | (16) | $187$ | (12) | $383$ | (18) | $319$ | (15)
NGC5135 | $854$ | (26) | $1150$ | (130) | $1085$ | (38) | $1205$ | (43) | $387$ | (29) | $645$ | (20) | $276$ | (12) | $691$ | (31) | $349$ | (16) | $638$ | (51) | $368$ | (33)
M-6-30-15 | $30$ | (9) | $223$ | (60) | $<57$ | $\cdots$ | $84$ | (15) | $43$ | (9) | $29$ | (6) | $<10$ | $\cdots$ | $61$ | (8) | $44$ | (6) | $25$ | (6) | $<29$ | $\cdots$
NGC5256 | $364$ | (23) | $661$ | (62) | $570$ | (18) | $471$ | (22) | $139$ | (11) | $241$ | (8) | $76$ | (4) | $250$ | (12) | $137$ | (13) | $248$ | (24) | $160$ | (16)
IC4329A | $53$ | (14) | $572$ | (70) | $<92$ | $\cdots$ | $141$ | (32) | $168$ | (22) | $98$ | (17) | $<17$ | $\cdots$ | $65$ | (14) | $114$ | (15) | $<66$ | $\cdots$ | $<106$ | $\cdots$
NGC5347 | $51$ | (10) | $<180$ | $\cdots$ | $<52$ | $\cdots$ | $70$ | (15) | $48$ | (9) | $27$ | (7) | $12$ | (3) | $73$ | (6) | $62$ | (7) | $85$ | (11) | $<34$ | $\cdots$
NGC5506 | $161$ | (21) | $<443$ | $\cdots$ | $429$ | (50) | $580$ | (41) | $<92$ | $\cdots$ | $82$ | (21) | $<39$ | $\cdots$ | $352$ | (38) | $295$ | (49) | $329$ | (42) | $132$ | (34)
NGC5548 | $59$ | (8) | $149$ | (39) | $<41$ | $\cdots$ | $65$ | (15) | $<40$ | $\cdots$ | $48$ | (12) | $15$ | (3) | $92$ | (6) | $<26$ | $\cdots$ | $68$ | (6) | $<31$ | $\cdots$
MRK817 | $71$ | (13) | $234$ | (60) | $97$ | (16) | $147$ | (11) | $<38$ | $\cdots$ | $60$ | (10) | $12$ | (3) | $54$ | (7) | $47$ | (8) | $46$ | (8) | $<37$ | $\cdots$
NGC5929 | $41$ | (9) | $<213$ | $\cdots$ | $<77$ | $\cdots$ | $<69$ | $\cdots$ | $<33$ | $\cdots$ | $<24$ | $\cdots$ | $20$ | (3) | $53$ | (7) | $<17$ | $\cdots$ | $25$ | (7) | $<22$ | $\cdots$
NGC5953 | $1030$ | (56) | $1080$ | (200) | $1437$ | (82) | $1424$ | (96) | $439$ | (56) | $749$ | (22) | $322$ | (17) | $748$ | (44) | $301$ | (21) | $604$ | (48) | $431$ | (36)
M-2-40-4 | $133$ | (12) | $<291$ | $\cdots$ | $282$ | (32) | $193$ | (26) | $37$ | (11) | $121$ | (9) | $50$ | (5) | $186$ | (9) | $69$ | (8) | $145$ | (11) | $57$ | (14)
F15480-0344 | $<37$ | $\cdots$ | $<202$ | $\cdots$ | $<67$ | $\cdots$ | $75$ | (14) | $<21$ | $\cdots$ | $<17$ | $\cdots$ | $22$ | (3) | $47$ | (6) | $38$ | (5) | $30$ | (6) | $<39$ | $\cdots$
NGC6810 | $1238$ | (58) | $1600$ | (170) | $1600$ | (66) | $1699$ | (80) | $484$ | (37) | $819$ | (25) | $385$ | (19) | $895$ | (49) | $408$ | (20) | $837$ | (55) | $420$ | (64)
NGC6860 | $87$ | (11) | $256$ | (62) | $<63$ | $\cdots$ | $134$ | (10) | $56$ | (14) | $63$ | (12) | $16$ | (3) | $100$ | (6) | $36$ | (11) | $62$ | (10) | $57$ | (10)
NGC6890 | $134$ | (13) | $<74$ | $\cdots$ | $254$ | (27) | $296$ | (30) | $64$ | (12) | $85$ | (10) | $55$ | (4) | $158$ | (11) | $76$ | (6) | $101$ | (8) | $118$ | (12)
IC5063 | $<31$ | $\cdots$ | $397$ | (73) | $<114$ | $\cdots$ | $129$ | (30) | $<55$ | $\cdots$ | $67$ | (15) | $<17$ | $\cdots$ | $106$ | (16) | $118$ | (21) | $85$ | (26) | $<78$ | $\cdots$
UGC11680 | $59$ | (14) | $<252$ | $\cdots$ | $<85$ | $\cdots$ | $125$ | (23) | $66$ | (12) | $<28$ | $\cdots$ | $16$ | (3) | $49$ | (7) | $28$ | (6) | $19$ | (5) | $33$ | (8)
NGC7130 | $630$ | (20) | $1070$ | (120) | $752$ | (39) | $907$ | (30) | $319$ | (16) | $486$ | (12) | $188$ | (6) | $500$ | (16) | $248$ | (13) | $445$ | (24) | $258$ | (20)
NGC7172 | $262$ | (23) | $898$ | (81) | $210$ | (55) | $452$ | (62) | $<153$ | $\cdots$ | $<47$ | $\cdots$ | $27$ | (4) | $175$ | (12) | $127$ | (16) | $211$ | (11) | $217$ | (14)
NGC7213 | $76$ | (11) | $299$ | (72) | $<76$ | $\cdots$ | $90$ | (24) | $<43$ | $\cdots$ | $78$ | (11) | $30$ | (4) | $159$ | (10) | $103$ | (7) | $72$ | (11) | $<37$ | $\cdots$
NGC7314 | $25$ | (6) | $<120$ | $\cdots$ | $<72$ | $\cdots$ | $53$ | (10) | $<24$ | $\cdots$ | $<19$ | $\cdots$ | $<8$ | $\cdots$ | $31$ | (5) | $<14$ | $\cdots$ | $<19$ | $\cdots$ | $57$ | (6)
M-3-58-7 | $87$ | (9) | $283$ | (32) | $122$ | (17) | $197$ | (11) | $124$ | (12) | $53$ | (10) | $28$ | (3) | $99$ | (6) | $42$ | (6) | $66$ | (7) | $56$ | (14)
NGC7469 | $1209$ | (39) | $1780$ | (130) | $1447$ | (44) | $1565$ | (32) | $471$ | (20) | $806$ | (17) | $316$ | (12) | $858$ | (31) | $382$ | (19) | $675$ | (37) | $383$ | (56)
NGC7496 | $329$ | (19) | $335$ | (59) | $434$ | (18) | $361$ | (17) | $136$ | (17) | $182$ | (15) | $81$ | (4) | $230$ | (11) | $89$ | (28) | $195$ | (26) | $146$ | (11)
NGC7582 | $2042$ | (88) | $2900$ | (270) | $2860$ | (110) | $3300$ | (130) | $1162$ | (84) | $1390$ | (46) | $517$ | (27) | $1625$ | (74) | $935$ | (48) | $1780$ | (110) | $730$ | (58)
NGC7590 | $252$ | (22) | $380$ | (100) | $343$ | (32) | $293$ | (30) | $91$ | (21) | $191$ | (13) | $85$ | (6) | $191$ | (14) | $61$ | (15) | $125$ | (8) | $129$ | (10)
NGC7603 | $123$ | (12) | $<186$ | $\cdots$ | $100$ | (20) | $177$ | (11) | $64$ | (11) | $59$ | (8) | $44$ | (3) | $138$ | (7) | $84$ | (8) | $108$ | (8) | $72$ | (9)
NGC7674 | $230$ | (13) | $248$ | (69) | $398$ | (22) | $375$ | (20) | $113$ | (20) | $167$ | (14) | $63$ | (5) | $195$ | (11) | $85$ | (10) | $152$ | (13) | $126$ | (25)
CGCG381-051 | $111$ | (15) | $215$ | (57) | $114$ | (15) | $112$ | (10) | $<32$ | $\cdots$ | $110$ | (8) | $29$ | (3) | $66$ | (7) | $34$ | (7) | $64$ | (8) | $<22$ | $\cdots$
Note. — Integrated fluxes are given in units of $10^{-17}$ W m-2.
Uncertainties ($1\sigma$) are listed in parentheses. Upper limits are
$3\sigma$.
Table 5: PAH Feature EQWs
Source ID | 6.2 $\mu$m | 7.4 $\mu$m | 7.6 $\mu$m | 7.8 $\mu$m | 8.3 $\mu$m | 8.6 $\mu$m | 11.2 $\mu$m | 11.3 $\mu$m | 12.0 $\mu$m | 12.6 $\mu$m | 17.0 $\mu$m
---|---|---|---|---|---|---|---|---|---|---|---
MRK335 | $<32$ | $\cdots$ | $<167$ | $\cdots$ | $<48$ | $\cdots$ | $102$ | (18) | $71$ | (21) | $56$ | (16) | $<14$ | $\cdots$ | $50$ | (13) | $71$ | (15) | $91$ | (11) | $<65$ | $\cdots$
MRK938 | $2700$ | (130) | $5880$ | (590) | $3150$ | (180) | $3730$ | (160) | $1620$ | (110) | $1797$ | (82) | $596$ | (17) | $1429$ | (44) | $622$ | (41) | $1133$ | (48) | $388$ | (45)
E12-G21 | $500$ | (51) | $<554$ | $\cdots$ | $723$ | (50) | $757$ | (50) | $174$ | (35) | $465$ | (27) | $139$ | (15) | $803$ | (42) | $303$ | (36) | $682$ | (74) | $640$ | (54)
MRK348 | $<44$ | $\cdots$ | $362$ | (46) | $<67$ | $\cdots$ | $181$ | (25) | $<66$ | $\cdots$ | $104$ | (16) | $25$ | (5) | $96$ | (12) | $<52$ | $\cdots$ | $44$ | (13) | $<85$ | $\cdots$
NGC424 | $12$ | (3) | $191$ | (29) | $53$ | (13) | $76$ | (15) | $63$ | (10) | $38$ | (8) | $<10$ | $\cdots$ | $26$ | (6) | $30$ | (7) | $<31$ | $\cdots$ | $<41$ | $\cdots$
NGC526A | $33$ | (9) | $381$ | (53) | $66$ | (14) | $72$ | (22) | $87$ | (19) | $51$ | (13) | $<13$ | $\cdots$ | $57$ | (11) | $96$ | (11) | $58$ | (11) | $<87$ | $\cdots$
NGC513 | $613$ | (67) | $<1145$ | $\cdots$ | $1330$ | (120) | $900$ | (120) | $<337$ | $\cdots$ | $632$ | (79) | $541$ | (18) | $832$ | (55) | $445$ | (72) | $648$ | (71) | $804$ | (80)
F01475-0740 | $113$ | (26) | $251$ | (81) | $287$ | (27) | $<92$ | $\cdots$ | $<65$ | $\cdots$ | $84$ | (16) | $30$ | (5) | $161$ | (11) | $153$ | (15) | $173$ | (16) | $<62$ | $\cdots$
NGC931 | $57$ | (7) | $<146$ | $\cdots$ | $62$ | (19) | $102$ | (23) | $<44$ | $\cdots$ | $35$ | (11) | $17$ | (4) | $124$ | (8) | $41$ | (10) | $44$ | (10) | $56$ | (18)
NGC1056 | $2340$ | (120) | $2630$ | (350) | $3580$ | (170) | $2880$ | (190) | $920$ | (140) | $1943$ | (50) | $998$ | (56) | $1960$ | (150) | $821$ | (56) | $1745$ | (86) | $1230$ | (110)
NGC1097 | $1734$ | (75) | $2370$ | (210) | $2530$ | (130) | $2880$ | (160) | $970$ | (120) | $1490$ | (33) | $668$ | (31) | $1794$ | (82) | $1010$ | (48) | $2280$ | (180) | $650$ | (50)
NGC1125 | $767$ | (71) | $2620$ | (490) | $1270$ | (160) | $1220$ | (130) | $615$ | (55) | $541$ | (41) | $296$ | (16) | $733$ | (40) | $231$ | (32) | $448$ | (30) | $167$ | (25)
NGC1143-4 | $1342$ | (88) | $1600$ | (190) | $1940$ | (57) | $1924$ | (49) | $530$ | (28) | $1009$ | (27) | $563$ | (28) | $1688$ | (73) | $642$ | (25) | $1121$ | (58) | $812$ | (49)
M-2-8-39 | $<87$ | $\cdots$ | $<599$ | $\cdots$ | $<167$ | $\cdots$ | $<158$ | $\cdots$ | $<98$ | $\cdots$ | $72$ | (21) | $<14$ | $\cdots$ | $<41$ | $\cdots$ | $<55$ | $\cdots$ | $36$ | (11) | $256$ | (42)
NGC1194 | $30$ | (7) | $<155$ | $\cdots$ | $131$ | (20) | $115$ | (21) | $<61$ | $\cdots$ | $<52$ | $\cdots$ | $<19$ | $\cdots$ | $121$ | (16) | $98$ | (26) | $64$ | (13) | $164$ | (25)
NGC1241 | $750$ | (100) | $2910$ | (810) | $<919$ | $\cdots$ | $1730$ | (340) | $590$ | (170) | $600$ | (120) | $285$ | (26) | $725$ | (72) | $275$ | (82) | $372$ | (38) | $661$ | (70)
NGC1320 | $125$ | (11) | $<290$ | $\cdots$ | $246$ | (33) | $229$ | (37) | $74$ | (20) | $116$ | (16) | $48$ | (4) | $121$ | (10) | $32$ | (9) | $101$ | (13) | $<53$ | $\cdots$
NGC1365 | $1235$ | (58) | $1420$ | (190) | $1600$ | (84) | $1830$ | (100) | $566$ | (63) | $817$ | (28) | $341$ | (17) | $1016$ | (46) | $482$ | (25) | $985$ | (68) | $446$ | (30)
NGC1386 | $68$ | (12) | $<216$ | $\cdots$ | $223$ | (23) | $230$ | (25) | $73$ | (14) | $91$ | (9) | $50$ | (5) | $264$ | (10) | $115$ | (15) | $149$ | (14) | $185$ | (27)
F03450+0055 | $42$ | (8) | $431$ | (63) | $<42$ | $\cdots$ | $127$ | (13) | $144$ | (12) | $51$ | (10) | $<10$ | $\cdots$ | $<27$ | $\cdots$ | $64$ | (12) | $75$ | (12) | $<71$ | $\cdots$
NGC1566 | $303$ | (26) | $640$ | (120) | $500$ | (40) | $720$ | (47) | $268$ | (35) | $321$ | (24) | $270$ | (12) | $788$ | (31) | $307$ | (21) | $534$ | (30) | $719$ | (33)
F04385-0828 | $33$ | (6) | $<84$ | $\cdots$ | $66$ | (12) | $<33$ | $\cdots$ | $<32$ | $\cdots$ | $<26$ | $\cdots$ | $<13$ | $\cdots$ | $137$ | (12) | $105$ | (14) | $128$ | (12) | $<55$ | $\cdots$
NGC1667 | $2450$ | (180) | $3890$ | (590) | $3000$ | (240) | $3550$ | (280) | $820$ | (200) | $1950$ | (130) | $745$ | (37) | $1975$ | (97) | $731$ | (83) | $1693$ | (81) | $1300$ | (140)
E33-G2 | $<55$ | $\cdots$ | $550$ | (88) | $<84$ | $\cdots$ | $<82$ | $\cdots$ | $102$ | (15) | $72$ | (13) | $<16$ | $\cdots$ | $68$ | (15) | $<71$ | $\cdots$ | $<42$ | $\cdots$ | $<59$ | $\cdots$
M-5-13-17 | $327$ | (25) | $<718$ | $\cdots$ | $309$ | (75) | $466$ | (77) | $148$ | (41) | $352$ | (32) | $182$ | (9) | $337$ | (23) | $120$ | (22) | $163$ | (20) | $199$ | (28)
MRK6 | $34$ | (8) | $<209$ | $\cdots$ | $85$ | (22) | $<41$ | $\cdots$ | $<58$ | $\cdots$ | $<56$ | $\cdots$ | $23$ | (4) | $<32$ | $\cdots$ | $<32$ | $\cdots$ | $<48$ | $\cdots$ | $146$ | (27)
MRK79 | $24$ | (4) | $363$ | (80) | $<69$ | $\cdots$ | $74$ | (17) | $71$ | (12) | $<26$ | $\cdots$ | $<12$ | $\cdots$ | $72$ | (10) | $60$ | (11) | $81$ | (16) | $<89$ | $\cdots$
NGC2639 | $470$ | (120) | $<2493$ | $\cdots$ | $1220$ | (280) | $1150$ | (280) | $<715$ | $\cdots$ | $<559$ | $\cdots$ | $378$ | (39) | $990$ | (110) | $<465$ | $\cdots$ | $540$ | (120) | $1220$ | (200)
MRK704 | $<22$ | $\cdots$ | $<86$ | $\cdots$ | $<37$ | $\cdots$ | $63$ | (16) | $<38$ | $\cdots$ | $<26$ | $\cdots$ | $<8$ | $\cdots$ | $<22$ | $\cdots$ | $<31$ | $\cdots$ | $<34$ | $\cdots$ | $<61$ | $\cdots$
NGC2992 | $382$ | (28) | $940$ | (130) | $588$ | (39) | $723$ | (47) | $275$ | (26) | $368$ | (19) | $152$ | (7) | $420$ | (18) | $175$ | (15) | $223$ | (32) | $345$ | (32)
MRK1239 | $15$ | (4) | $112$ | (26) | $63$ | (13) | $66$ | (8) | $47$ | (7) | $39$ | (6) | $15$ | (3) | $34$ | (8) | $42$ | (8) | $64$ | (9) | $<47$ | $\cdots$
NGC3079 | $3950$ | (180) | $9210$ | (750) | $7650$ | (340) | $9310$ | (450) | $3330$ | (350) | $3400$ | (200) | $1417$ | (72) | $5170$ | (190) | $2180$ | (110) | $3600$ | (190) | $2540$ | (120)
NGC3227 | $294$ | (14) | $562$ | (60) | $370$ | (30) | $517$ | (22) | $174$ | (15) | $259$ | (12) | $137$ | (10) | $447$ | (27) | $195$ | (15) | $245$ | (20) | $340$ | (28)
NGC3511 | $2980$ | (210) | $3040$ | (670) | $4020$ | (240) | $3850$ | (270) | $1320$ | (230) | $2100$ | (130) | $1006$ | (41) | $2070$ | (110) | $769$ | (83) | $1742$ | (94) | $1940$ | (140)
NGC3516 | $<29$ | $\cdots$ | $<243$ | $\cdots$ | $73$ | (23) | $<68$ | $\cdots$ | $64$ | (15) | $<32$ | $\cdots$ | $15$ | (4) | $43$ | (7) | $<43$ | $\cdots$ | $<42$ | $\cdots$ | $<63$ | $\cdots$
M+0-29-23 | $1314$ | (74) | $2530$ | (220) | $1569$ | (71) | $1740$ | (81) | $666$ | (48) | $1023$ | (40) | $465$ | (15) | $1247$ | (33) | $487$ | (22) | $920$ | (44) | $634$ | (43)
NGC3660 | $710$ | (190) | $2800$ | (650) | $<893$ | $\cdots$ | $2460$ | (320) | $630$ | (140) | $480$ | (110) | $311$ | (26) | $361$ | (73) | $309$ | (90) | $688$ | (96) | $<481$ | $\cdots$
NGC3982 | $1850$ | (230) | $3360$ | (770) | $3020$ | (320) | $2350$ | (350) | $780$ | (210) | $1382$ | (77) | $636$ | (25) | $1166$ | (65) | $412$ | (62) | $796$ | (81) | $924$ | (67)
NGC4051 | $99$ | (7) | $<134$ | $\cdots$ | $201$ | (15) | $168$ | (17) | $63$ | (12) | $79$ | (8) | $58$ | (4) | $153$ | (11) | $71$ | (9) | $144$ | (8) | $155$ | (19)
UGC7064 | $521$ | (68) | $<1222$ | $\cdots$ | $1120$ | (120) | $670$ | (100) | $227$ | (56) | $685$ | (48) | $172$ | (21) | $544$ | (58) | $235$ | (52) | $365$ | (34) | $328$ | (48)
NGC4151 | $13$ | (3) | $<123$ | $\cdots$ | $77$ | (15) | $129$ | (17) | $76$ | (13) | $65$ | (11) | $10$ | (3) | $<18$ | $\cdots$ | $<19$ | $\cdots$ | $43$ | (10) | $<57$ | $\cdots$
MRK766 | $114$ | (15) | $855$ | (69) | $83$ | (25) | $294$ | (29) | $157$ | (22) | $187$ | (16) | $46$ | (4) | $137$ | (10) | $75$ | (12) | $<52$ | $\cdots$ | $178$ | (26)
NGC4388 | $184$ | (7) | $360$ | (110) | $330$ | (44) | $388$ | (46) | $125$ | (14) | $188$ | (15) | $109$ | (8) | $458$ | (24) | $201$ | (27) | $272$ | (29) | $217$ | (22)
NGC4501 | $<225$ | $\cdots$ | $<856$ | $\cdots$ | $360$ | (110) | $490$ | (110) | $<273$ | $\cdots$ | $<205$ | $\cdots$ | $335$ | (34) | $1168$ | (87) | $560$ | (120) | $850$ | (110) | $1580$ | (130)
NGC4579 | $124$ | (22) | $630$ | (120) | $<131$ | $\cdots$ | $179$ | (48) | $<111$ | $\cdots$ | $83$ | (21) | $104$ | (7) | $429$ | (20) | $293$ | (41) | $338$ | (64) | $455$ | (33)
NGC4593 | $69$ | (7) | $<152$ | $\cdots$ | $132$ | (19) | $157$ | (18) | $72$ | (14) | $69$ | (11) | $32$ | (4) | $101$ | (10) | $33$ | (10) | $75$ | (14) | $<82$ | $\cdots$
NGC4594 | $26$ | (8) | $<240$ | $\cdots$ | $<76$ | $\cdots$ | $139$ | (33) | $<64$ | $\cdots$ | $<33$ | $\cdots$ | $26$ | (6) | $65$ | (13) | $<48$ | $\cdots$ | $<100$ | $\cdots$ | $<71$ | $\cdots$
TOL1238-364 | $453$ | (26) | $750$ | (120) | $548$ | (49) | $545$ | (55) | $133$ | (21) | $316$ | (14) | $115$ | (7) | $384$ | (18) | $191$ | (15) | $276$ | (19) | $<50$ | $\cdots$
NGC4602 | $1230$ | (130) | $4570$ | (830) | $<955$ | $\cdots$ | $1870$ | (280) | $<659$ | $\cdots$ | $1310$ | (170) | $506$ | (32) | $1052$ | (71) | $620$ | (110) | $880$ | (110) | $826$ | (80)
M-2-33-34 | $177$ | (31) | $<1134$ | $\cdots$ | $380$ | (110) | $291$ | (84) | $<180$ | $\cdots$ | $<183$ | $\cdots$ | $151$ | (11) | $422$ | (26) | $156$ | (25) | $345$ | (32) | $334$ | (48)
NGC4941 | $158$ | (20) | $930$ | (120) | $190$ | (41) | $<134$ | $\cdots$ | $168$ | (26) | $120$ | (12) | $<55$ | $\cdots$ | $256$ | (36) | $185$ | (34) | $160$ | (40) | $<153$ | $\cdots$
NGC4968 | $220$ | (14) | $648$ | (73) | $325$ | (31) | $283$ | (21) | $199$ | (29) | $170$ | (22) | $64$ | (5) | $226$ | (14) | $117$ | (14) | $123$ | (13) | $81$ | (22)
NGC5005 | $373$ | (25) | $1120$ | (120) | $564$ | (46) | $908$ | (57) | $368$ | (55) | $504$ | (34) | $717$ | (36) | $1928$ | (98) | $809$ | (47) | $1410$ | (100) | $1860$ | (65)
NGC5033 | $844$ | (49) | $1220$ | (200) | $1402$ | (85) | $1663$ | (98) | $568$ | (62) | $870$ | (30) | $638$ | (21) | $1693$ | (58) | $680$ | (45) | $1443$ | (69) | $1684$ | (80)
NGC5135 | $1194$ | (37) | $1690$ | (180) | $1604$ | (56) | $1802$ | (64) | $593$ | (45) | $1010$ | (31) | $475$ | (20) | $1177$ | (52) | $549$ | (26) | $946$ | (76) | $437$ | (39)
M-6-30-15 | $25$ | (7) | $237$ | (64) | $<62$ | $\cdots$ | $95$ | (17) | $51$ | (11) | $35$ | (7) | $<12$ | $\cdots$ | $78$ | (9) | $61$ | (8) | $37$ | (8) | $<49$ | $\cdots$
NGC5256 | $2160$ | (140) | $4000$ | (380) | $3440$ | (110) | $2830$ | (130) | $841$ | (65) | $1478$ | (49) | $493$ | (28) | $1587$ | (73) | $730$ | (68) | $1180$ | (110) | $573$ | (59)
IC4329A | $14$ | (4) | $198$ | (24) | $<32$ | $\cdots$ | $52$ | (12) | $65$ | (8) | $38$ | (6) | $<6$ | $\cdots$ | $26$ | (6) | $50$ | (6) | $<30$ | $\cdots$ | $<56$ | $\cdots$
NGC5347 | $97$ | (19) | $<324$ | $\cdots$ | $<93$ | $\cdots$ | $124$ | (27) | $84$ | (16) | $47$ | (12) | $19$ | (4) | $116$ | (10) | $98$ | (11) | $137$ | (18) | $<57$ | $\cdots$
NGC5506 | $30$ | (3) | $<104$ | $\cdots$ | $104$ | (12) | $147$ | (10) | $<26$ | $\cdots$ | $26$ | (6) | $<20$ | $\cdots$ | $181$ | (20) | $132$ | (22) | $138$ | (18) | $70$ | (18)
NGC5548 | $92$ | (12) | $267$ | (70) | $<74$ | $\cdots$ | $121$ | (27) | $<77$ | $\cdots$ | $92$ | (23) | $25$ | (5) | $160$ | (11) | $<47$ | $\cdots$ | $133$ | (12) | $<64$ | $\cdots$
MRK817 | $96$ | (18) | $400$ | (100) | $167$ | (28) | $259$ | (19) | $<67$ | $\cdots$ | $103$ | (18) | $17$ | (4) | $82$ | (11) | $77$ | (13) | $78$ | (13) | $<59$ | $\cdots$
NGC5929 | $480$ | (100) | $<2941$ | $\cdots$ | $<1085$ | $\cdots$ | $<995$ | $\cdots$ | $<500$ | $\cdots$ | $<371$ | $\cdots$ | $417$ | (58) | $1100$ | (150) | $<367$ | $\cdots$ | $570$ | (150) | $<483$ | $\cdots$
NGC5953 | $3170$ | (170) | $3310$ | (610) | $4370$ | (250) | $4270$ | (290) | $1290$ | (160) | $2184$ | (65) | $1000$ | (51) | $2330$ | (140) | $942$ | (67) | $1920$ | (150) | $1420$ | (120)
M-2-40-4 | $89$ | (8) | $<210$ | $\cdots$ | $208$ | (23) | $146$ | (20) | $30$ | (9) | $103$ | (7) | $56$ | (5) | $210$ | (11) | $80$ | (9) | $181$ | (14) | $94$ | (22)
F15480-0344 | $<90$ | $\cdots$ | $<521$ | $\cdots$ | $<175$ | $\cdots$ | $200$ | (36) | $<57$ | $\cdots$ | $<48$ | $\cdots$ | $50$ | (5) | $110$ | (14) | $95$ | (12) | $75$ | (14) | $<89$ | $\cdots$
NGC6810 | $1485$ | (70) | $1960$ | (210) | $1955$ | (81) | $2063$ | (97) | $580$ | (44) | $974$ | (30) | $408$ | (20) | $944$ | (52) | $425$ | (21) | $861$ | (56) | $310$ | (48)
NGC6860 | $98$ | (13) | $374$ | (90) | $<95$ | $\cdots$ | $213$ | (16) | $96$ | (24) | $112$ | (21) | $34$ | (5) | $214$ | (12) | $84$ | (27) | $154$ | (24) | $169$ | (31)
NGC6890 | $280$ | (27) | $<164$ | $\cdots$ | $588$ | (64) | $701$ | (72) | $159$ | (31) | $215$ | (24) | $164$ | (13) | $474$ | (32) | $231$ | (18) | $312$ | (24) | $381$ | (39)
IC5063 | $<13$ | $\cdots$ | $160$ | (30) | $<46$ | $\cdots$ | $53$ | (12) | $<23$ | $\cdots$ | $29$ | (6) | $<8$ | $\cdots$ | $49$ | (7) | $54$ | (9) | $38$ | (12) | $<35$ | $\cdots$
UGC11680 | $219$ | (54) | $<1114$ | $\cdots$ | $<386$ | $\cdots$ | $590$ | (110) | $326$ | (59) | $<139$ | $\cdots$ | $83$ | (15) | $258$ | (35) | $157$ | (35) | $113$ | (31) | $207$ | (49)
NGC7130 | $1614$ | (52) | $2740$ | (310) | $1914$ | (99) | $2285$ | (75) | $784$ | (39) | $1176$ | (30) | $363$ | (12) | $957$ | (30) | $442$ | (22) | $747$ | (40) | $332$ | (26)
NGC7172 | $278$ | (24) | $1210$ | (110) | $290$ | (76) | $642$ | (88) | $<238$ | $\cdots$ | $<84$ | $\cdots$ | $126$ | (17) | $780$ | (52) | $350$ | (43) | $470$ | (25) | $853$ | (56)
NGC7213 | $87$ | (13) | $510$ | (120) | $<134$ | $\cdots$ | $170$ | (45) | $<87$ | $\cdots$ | $156$ | (23) | $44$ | (5) | $242$ | (15) | $193$ | (12) | $151$ | (24) | $<77$ | $\cdots$
NGC7314 | $116$ | (25) | $<638$ | $\cdots$ | $<387$ | $\cdots$ | $286$ | (57) | $<132$ | $\cdots$ | $<105$ | $\cdots$ | $<40$ | $\cdots$ | $159$ | (28) | $<68$ | $\cdots$ | $<85$ | $\cdots$ | $285$ | (28)
M-3-58-7 | $92$ | (9) | $387$ | (44) | $171$ | (24) | $284$ | (16) | $184$ | (18) | $78$ | (14) | $43$ | (4) | $153$ | (8) | $68$ | (9) | $112$ | (12) | $99$ | (24)
NGC7469 | $733$ | (24) | $1142$ | (83) | $934$ | (29) | $1019$ | (21) | $309$ | (13) | $526$ | (11) | $178$ | (6) | $483$ | (18) | $214$ | (11) | $370$ | (20) | $160$ | (24)
NGC7496 | $1533$ | (87) | $1270$ | (220) | $1600$ | (68) | $1282$ | (59) | $455$ | (58) | $597$ | (49) | $241$ | (13) | $679$ | (34) | $249$ | (77) | $516$ | (69) | $281$ | (21)
NGC7582 | $757$ | (33) | $1240$ | (120) | $1248$ | (47) | $1484$ | (57) | $563$ | (41) | $731$ | (24) | $369$ | (20) | $1137$ | (52) | $549$ | (28) | $943$ | (56) | $345$ | (28)
NGC7590 | $1590$ | (140) | $2910$ | (780) | $2690$ | (250) | $2320$ | (230) | $730$ | (170) | $1530$ | (100) | $693$ | (47) | $1580$ | (120) | $530$ | (130) | $1156$ | (69) | $1490$ | (110)
NGC7603 | $83$ | (8) | $<166$ | $\cdots$ | $93$ | (18) | $173$ | (11) | $69$ | (12) | $65$ | (9) | $69$ | (5) | $220$ | (11) | $159$ | (14) | $229$ | (17) | $242$ | (30)
NGC7674 | $184$ | (10) | $225$ | (63) | $367$ | (20) | $356$ | (19) | $113$ | (20) | $173$ | (15) | $61$ | (4) | $191$ | (11) | $83$ | (9) | $151$ | (13) | $124$ | (25)
CGCG381-051 | $750$ | (100) | $1330$ | (360) | $698$ | (89) | $674$ | (62) | $<178$ | $\cdots$ | $571$ | (43) | $120$ | (11) | $279$ | (29) | $163$ | (32) | $327$ | (42) | $<85$ | $\cdots$
Note. — PAH equivalent widths are given in units of nm. Uncertainties
($1\sigma$) are listed in parentheses. Upper limits are $3\sigma$.
Table 6: H2 Line Fluxes
Source ID | S(7) | S(6) | S(5) | S(4) | S(3) | S(2) | S(1) | S(0)
---|---|---|---|---|---|---|---|---
MRK335 | $<72$ | $\cdots$ | $<88$ | $\cdots$ | $<64$ | $\cdots$ | $<55$ | $\cdots$ | $53$ | (12) | $21$ | (6) | $17$ | (5) | $<18$ | $\cdots$
MRK938 | $160$ | (32) | $<205$ | $\cdots$ | $151$ | (22) | $249$ | (57) | $128$ | (13) | $143$ | (22) | $195$ | (22) | $<153$ | $\cdots$
E12-G21 | $55$ | (15) | $<132$ | $\cdots$ | $<67$ | $\cdots$ | $<83$ | $\cdots$ | $40$ | (11) | $<34$ | $\cdots$ | $<21$ | $\cdots$ | $<22$ | $\cdots$
MRK348 | $<79$ | $\cdots$ | $<85$ | $\cdots$ | $62$ | (17) | $<74$ | $\cdots$ | $<26$ | $\cdots$ | $<28$ | $\cdots$ | $<39$ | $\cdots$ | $23$ | (6)
NGC424 | $<96$ | $\cdots$ | $<76$ | $\cdots$ | $<68$ | $\cdots$ | $<177$ | $\cdots$ | $<55$ | $\cdots$ | $<54$ | $\cdots$ | $<35$ | $\cdots$ | $<56$ | $\cdots$
NGC526A | $76$ | (19) | $40$ | (11) | $73$ | (10) | $55$ | (11) | $29$ | (9) | $22$ | (4) | $33$ | (9) | $<28$ | $\cdots$
NGC513 | $<94$ | $\cdots$ | $<135$ | $\cdots$ | $<75$ | $\cdots$ | $<137$ | $\cdots$ | $81$ | (18) | $<47$ | $\cdots$ | $41$ | (9) | $<26$ | $\cdots$
F01475-0740 | $<111$ | $\cdots$ | $<54$ | $\cdots$ | $47$ | (11) | $<49$ | $\cdots$ | $<24$ | $\cdots$ | $29$ | (8) | $<23$ | $\cdots$ | $<43$ | $\cdots$
NGC931 | $<130$ | $\cdots$ | $<78$ | $\cdots$ | $65$ | (20) | $<112$ | $\cdots$ | $<47$ | $\cdots$ | $<25$ | $\cdots$ | $<29$ | $\cdots$ | $<42$ | $\cdots$
NGC1056 | $<144$ | $\cdots$ | $<196$ | $\cdots$ | $<97$ | $\cdots$ | $<197$ | $\cdots$ | $85$ | (17) | $<51$ | $\cdots$ | $<47$ | $\cdots$ | $<32$ | $\cdots$
NGC1097 | $<235$ | $\cdots$ | $<595$ | $\cdots$ | $186$ | (43) | $<772$ | $\cdots$ | $319$ | (31) | $<503$ | $\cdots$ | $374$ | (36) | $<82$ | $\cdots$
NGC1125 | $<81$ | $\cdots$ | $<79$ | $\cdots$ | $<52$ | $\cdots$ | $<44$ | $\cdots$ | $<22$ | $\cdots$ | $<30$ | $\cdots$ | $<21$ | $\cdots$ | $<52$ | $\cdots$
NGC1143-4 | $<47$ | $\cdots$ | $<160$ | $\cdots$ | $<41$ | $\cdots$ | $<55$ | $\cdots$ | $62$ | (7) | $51$ | (13) | $91$ | (7) | $<45$ | $\cdots$
M-2-8-39 | $<150$ | $\cdots$ | $65$ | (20) | $<81$ | $\cdots$ | $82$ | (26) | $<39$ | $\cdots$ | $<22$ | $\cdots$ | $<35$ | $\cdots$ | $20$ | (6)
NGC1194 | $<57$ | $\cdots$ | $<54$ | $\cdots$ | $<50$ | $\cdots$ | $<67$ | $\cdots$ | $<23$ | $\cdots$ | $<40$ | $\cdots$ | $<25$ | $\cdots$ | $<31$ | $\cdots$
NGC1241 | $<171$ | $\cdots$ | $<92$ | $\cdots$ | $<78$ | $\cdots$ | $<144$ | $\cdots$ | $<34$ | $\cdots$ | $42$ | (9) | $47$ | (8) | $<18$ | $\cdots$
NGC1320 | $<96$ | $\cdots$ | $<68$ | $\cdots$ | $<51$ | $\cdots$ | $<83$ | $\cdots$ | $<41$ | $\cdots$ | $<30$ | $\cdots$ | $<27$ | $\cdots$ | $<56$ | $\cdots$
NGC1365 | $<259$ | $\cdots$ | $<1070$ | $\cdots$ | $<351$ | $\cdots$ | $<1203$ | $\cdots$ | $382$ | (49) | $<465$ | $\cdots$ | $<377$ | $\cdots$ | $<342$ | $\cdots$
NGC1386 | $<181$ | $\cdots$ | $<89$ | $\cdots$ | $82$ | (24) | $<75$ | $\cdots$ | $82$ | (21) | $96$ | (19) | $<54$ | $\cdots$ | $<88$ | $\cdots$
F03450+0055 | $60$ | (15) | $<51$ | $\cdots$ | $<62$ | $\cdots$ | $<37$ | $\cdots$ | $<25$ | $\cdots$ | $23$ | (7) | $<26$ | $\cdots$ | $<25$ | $\cdots$
NGC1566 | $<80$ | $\cdots$ | $<85$ | $\cdots$ | $<35$ | $\cdots$ | $<78$ | $\cdots$ | $121$ | (11) | $77$ | (10) | $149$ | (10) | $<22$ | $\cdots$
F04385-0828 | $<90$ | $\cdots$ | $66$ | (19) | $<76$ | $\cdots$ | $<82$ | $\cdots$ | $<34$ | $\cdots$ | $55$ | (15) | $<38$ | $\cdots$ | $<58$ | $\cdots$
NGC1667 | $<148$ | $\cdots$ | $<188$ | $\cdots$ | $<82$ | $\cdots$ | $<192$ | $\cdots$ | $<60$ | $\cdots$ | $73$ | (15) | $79$ | (15) | $<20$ | $\cdots$
E33-G2 | $<103$ | $\cdots$ | $<96$ | $\cdots$ | $<60$ | $\cdots$ | $<70$ | $\cdots$ | $<40$ | $\cdots$ | $46$ | (9) | $20$ | (6) | $<20$ | $\cdots$
M-5-13-17 | $<90$ | $\cdots$ | $<60$ | $\cdots$ | $<51$ | $\cdots$ | $<52$ | $\cdots$ | $37$ | (9) | $<26$ | $\cdots$ | $<22$ | $\cdots$ | $<29$ | $\cdots$
MRK6 | $<77$ | $\cdots$ | $<43$ | $\cdots$ | $71$ | (16) | $<40$ | $\cdots$ | $59$ | (10) | $27$ | (7) | $<35$ | $\cdots$ | $<52$ | $\cdots$
MRK79 | $<105$ | $\cdots$ | $<58$ | $\cdots$ | $<69$ | $\cdots$ | $<62$ | $\cdots$ | $61$ | (17) | $<25$ | $\cdots$ | $54$ | (13) | $<28$ | $\cdots$
NGC2639 | $<103$ | $\cdots$ | $<100$ | $\cdots$ | $<100$ | $\cdots$ | $<121$ | $\cdots$ | $<74$ | $\cdots$ | $<41$ | $\cdots$ | $<25$ | $\cdots$ | $17$ | (5)
MRK704 | $<94$ | $\cdots$ | $<54$ | $\cdots$ | $<48$ | $\cdots$ | $<68$ | $\cdots$ | $<39$ | $\cdots$ | $<27$ | $\cdots$ | $<23$ | $\cdots$ | $<26$ | $\cdots$
NGC2992 | $<176$ | $\cdots$ | $<154$ | $\cdots$ | $100$ | (24) | $<129$ | $\cdots$ | $108$ | (13) | $<106$ | $\cdots$ | $187$ | (26) | $<76$ | $\cdots$
MRK1239 | $<86$ | $\cdots$ | $<102$ | $\cdots$ | $<90$ | $\cdots$ | $<39$ | $\cdots$ | $<53$ | $\cdots$ | $<31$ | $\cdots$ | $<35$ | $\cdots$ | $<51$ | $\cdots$
NGC3079 | $<254$ | $\cdots$ | $<612$ | $\cdots$ | $666$ | (45) | $<770$ | $\cdots$ | $365$ | (21) | $450$ | (110) | $506$ | (39) | $<179$ | $\cdots$
NGC3227 | $<93$ | $\cdots$ | $<97$ | $\cdots$ | $240$ | (38) | $134$ | (42) | $197$ | (24) | $112$ | (24) | $277$ | (31) | $<87$ | $\cdots$
NGC3511 | $<166$ | $\cdots$ | $<140$ | $\cdots$ | $<86$ | $\cdots$ | $<126$ | $\cdots$ | $<64$ | $\cdots$ | $<34$ | $\cdots$ | $29$ | (9) | $26$ | (7)
NGC3516 | $<100$ | $\cdots$ | $<94$ | $\cdots$ | $78$ | (22) | $105$ | (29) | $<49$ | $\cdots$ | $42$ | (11) | $<31$ | $\cdots$ | $<49$ | $\cdots$
M+0-29-23 | $<71$ | $\cdots$ | $<142$ | $\cdots$ | $98$ | (24) | $<114$ | $\cdots$ | $34$ | (10) | $56$ | (11) | $70$ | (13) | $<52$ | $\cdots$
NGC3660 | $108$ | (28) | $<97$ | $\cdots$ | $<88$ | $\cdots$ | $<106$ | $\cdots$ | $<44$ | $\cdots$ | $<24$ | $\cdots$ | $<27$ | $\cdots$ | $19$ | (4)
NGC3982 | $<152$ | $\cdots$ | $<191$ | $\cdots$ | $<92$ | $\cdots$ | $<201$ | $\cdots$ | $<58$ | $\cdots$ | $<46$ | $\cdots$ | $<28$ | $\cdots$ | $<25$ | $\cdots$
NGC4051 | $<64$ | $\cdots$ | $<66$ | $\cdots$ | $<45$ | $\cdots$ | $<89$ | $\cdots$ | $123$ | (18) | $<28$ | $\cdots$ | $<41$ | $\cdots$ | $<77$ | $\cdots$
UGC7064 | $<77$ | $\cdots$ | $<96$ | $\cdots$ | $<100$ | $\cdots$ | $<89$ | $\cdots$ | $<32$ | $\cdots$ | $<30$ | $\cdots$ | $<25$ | $\cdots$ | $<29$ | $\cdots$
NGC4151 | $<228$ | $\cdots$ | $<65$ | $\cdots$ | $<59$ | $\cdots$ | $<233$ | $\cdots$ | $<129$ | $\cdots$ | $<150$ | $\cdots$ | $<178$ | $\cdots$ | $200$ | (60)
MRK766 | $<108$ | $\cdots$ | $<89$ | $\cdots$ | $40$ | (9) | $<71$ | $\cdots$ | $<28$ | $\cdots$ | $<49$ | $\cdots$ | $<56$ | $\cdots$ | $<71$ | $\cdots$
NGC4388 | $<199$ | $\cdots$ | $57$ | (17) | $121$ | (25) | $<210$ | $\cdots$ | $143$ | (14) | $172$ | (38) | $135$ | (30) | $<127$ | $\cdots$
NGC4501 | $<121$ | $\cdots$ | $<146$ | $\cdots$ | $83$ | (23) | $177$ | (32) | $129$ | (20) | $76$ | (15) | $192$ | (9) | $16$ | (5)
NGC4579 | $<70$ | $\cdots$ | $87$ | (23) | $191$ | (16) | $115$ | (21) | $290$ | (14) | $115$ | (12) | $201$ | (6) | $<12$ | $\cdots$
NGC4593 | $<76$ | $\cdots$ | $<61$ | $\cdots$ | $64$ | (15) | $<81$ | $\cdots$ | $55$ | (16) | $<22$ | $\cdots$ | $59$ | (17) | $<52$ | $\cdots$
NGC4594 | $<70$ | $\cdots$ | $<53$ | $\cdots$ | $<54$ | $\cdots$ | $<91$ | $\cdots$ | $<26$ | $\cdots$ | $<21$ | $\cdots$ | $5$ | (1) | $<8$ | $\cdots$
TOL1238-364 | $<76$ | $\cdots$ | $<108$ | $\cdots$ | $<48$ | $\cdots$ | $<152$ | $\cdots$ | $55$ | (16) | $<64$ | $\cdots$ | $<46$ | $\cdots$ | $<99$ | $\cdots$
NGC4602 | $<173$ | $\cdots$ | $<66$ | $\cdots$ | $74$ | (24) | $<94$ | $\cdots$ | $<60$ | $\cdots$ | $<25$ | $\cdots$ | $33$ | (7) | $23$ | (6)
M-2-33-34 | $<88$ | $\cdots$ | $<49$ | $\cdots$ | $<61$ | $\cdots$ | $<44$ | $\cdots$ | $<22$ | $\cdots$ | $<17$ | $\cdots$ | $<21$ | $\cdots$ | $<44$ | $\cdots$
NGC4941 | $108$ | (11) | $72$ | (7) | $77$ | (8) | $51$ | (8) | $45$ | (10) | $22$ | (6) | $<25$ | $\cdots$ | $<32$ | $\cdots$
NGC4968 | $<135$ | $\cdots$ | $<53$ | $\cdots$ | $<55$ | $\cdots$ | $<80$ | $\cdots$ | $<41$ | $\cdots$ | $54$ | (11) | $<40$ | $\cdots$ | $<45$ | $\cdots$
NGC5005 | $187$ | (52) | $<127$ | $\cdots$ | $431$ | (28) | $<167$ | $\cdots$ | $376$ | (21) | $187$ | (36) | $393$ | (15) | $61$ | (19)
NGC5033 | $<87$ | $\cdots$ | $<167$ | $\cdots$ | $95$ | (24) | $<162$ | $\cdots$ | $118$ | (11) | $85$ | (18) | $137$ | (13) | $30$ | (10)
NGC5135 | $<138$ | $\cdots$ | $<172$ | $\cdots$ | $<143$ | $\cdots$ | $<186$ | $\cdots$ | $116$ | (19) | $<167$ | $\cdots$ | $262$ | (29) | $<120$ | $\cdots$
M-6-30-15 | $<88$ | $\cdots$ | $<60$ | $\cdots$ | $<46$ | $\cdots$ | $<61$ | $\cdots$ | $<23$ | $\cdots$ | $<19$ | $\cdots$ | $<25$ | $\cdots$ | $<29$ | $\cdots$
NGC5256 | $<121$ | $\cdots$ | $<139$ | $\cdots$ | $<75$ | $\cdots$ | $<94$ | $\cdots$ | $96$ | (12) | $90$ | (25) | $87$ | (14) | $<81$ | $\cdots$
IC4329A | $<181$ | $\cdots$ | $<106$ | $\cdots$ | $<47$ | $\cdots$ | $<159$ | $\cdots$ | $<64$ | $\cdots$ | $<44$ | $\cdots$ | $<90$ | $\cdots$ | $<91$ | $\cdots$
NGC5347 | $<65$ | $\cdots$ | $<51$ | $\cdots$ | $59$ | (16) | $64$ | (16) | $<38$ | $\cdots$ | $<31$ | $\cdots$ | $<29$ | $\cdots$ | $<37$ | $\cdots$
NGC5506 | $<332$ | $\cdots$ | $<127$ | $\cdots$ | $<58$ | $\cdots$ | $<157$ | $\cdots$ | $<77$ | $\cdots$ | $166$ | (45) | $<55$ | $\cdots$ | $<167$ | $\cdots$
NGC5548 | $<51$ | $\cdots$ | $67$ | (17) | $44$ | (14) | $<70$ | $\cdots$ | $71$ | (17) | $44$ | (10) | $44$ | (14) | $<37$ | $\cdots$
MRK817 | $<155$ | $\cdots$ | $<97$ | $\cdots$ | $<64$ | $\cdots$ | $<56$ | $\cdots$ | $<63$ | $\cdots$ | $<33$ | $\cdots$ | $<32$ | $\cdots$ | $<52$ | $\cdots$
NGC5929 | $<88$ | $\cdots$ | $61$ | (16) | $57$ | (16) | $<85$ | $\cdots$ | $74$ | (11) | $34$ | (7) | $61$ | (6) | $<12$ | $\cdots$
NGC5953 | $<131$ | $\cdots$ | $<329$ | $\cdots$ | $<90$ | $\cdots$ | $<361$ | $\cdots$ | $112$ | (25) | $<125$ | $\cdots$ | $141$ | (31) | $<51$ | $\cdots$
M-2-40-4 | $<95$ | $\cdots$ | $<64$ | $\cdots$ | $<60$ | $\cdots$ | $101$ | (26) | $<32$ | $\cdots$ | $62$ | (13) | $<34$ | $\cdots$ | $<43$ | $\cdots$
F15480-0344 | $<66$ | $\cdots$ | $<74$ | $\cdots$ | $52$ | (7) | $<57$ | $\cdots$ | $<31$ | $\cdots$ | $<16$ | $\cdots$ | $46$ | (11) | $29$ | (9)
NGC6810 | $<177$ | $\cdots$ | $<352$ | $\cdots$ | $<146$ | $\cdots$ | $<290$ | $\cdots$ | $168$ | (18) | $<143$ | $\cdots$ | $359$ | (55) | $<178$ | $\cdots$
NGC6860 | $<126$ | $\cdots$ | $<56$ | $\cdots$ | $69$ | (20) | $<44$ | $\cdots$ | $<33$ | $\cdots$ | $<40$ | $\cdots$ | $41$ | (12) | $<21$ | $\cdots$
NGC6890 | $<95$ | $\cdots$ | $<77$ | $\cdots$ | $<62$ | $\cdots$ | $<116$ | $\cdots$ | $<31$ | $\cdots$ | $<32$ | $\cdots$ | $46$ | (13) | $<39$ | $\cdots$
IC5063 | $<95$ | $\cdots$ | $<61$ | $\cdots$ | $88$ | (16) | $<138$ | $\cdots$ | $<67$ | $\cdots$ | $<58$ | $\cdots$ | $<67$ | $\cdots$ | $<129$ | $\cdots$
UGC11680 | $<126$ | $\cdots$ | $<85$ | $\cdots$ | $<90$ | $\cdots$ | $<106$ | $\cdots$ | $<57$ | $\cdots$ | $<18$ | $\cdots$ | $23$ | (7) | $<19$ | $\cdots$
NGC7130 | $<71$ | $\cdots$ | $<111$ | $\cdots$ | $<114$ | $\cdots$ | $134$ | (37) | $117$ | (19) | $94$ | (24) | $122$ | (24) | $<137$ | $\cdots$
NGC7172 | $<80$ | $\cdots$ | $<135$ | $\cdots$ | $72$ | (15) | $<273$ | $\cdots$ | $47$ | (4) | $56$ | (17) | $51$ | (12) | $<36$ | $\cdots$
NGC7213 | $<122$ | $\cdots$ | $<85$ | $\cdots$ | $<71$ | $\cdots$ | $<73$ | $\cdots$ | $88$ | (20) | $<27$ | $\cdots$ | $98$ | (13) | $<30$ | $\cdots$
NGC7314 | $77$ | (12) | $44$ | (6) | $62$ | (7) | $42$ | (12) | $26$ | (4) | $18$ | (3) | $15$ | (4) | $<27$ | $\cdots$
M-3-58-7 | $56$ | (17) | $<61$ | $\cdots$ | $<71$ | $\cdots$ | $74$ | (22) | $<32$ | $\cdots$ | $<27$ | $\cdots$ | $<35$ | $\cdots$ | $<53$ | $\cdots$
NGC7469 | $<149$ | $\cdots$ | $<248$ | $\cdots$ | $<143$ | $\cdots$ | $<119$ | $\cdots$ | $252$ | (31) | $184$ | (36) | $189$ | (60) | $<342$ | $\cdots$
NGC7496 | $<78$ | $\cdots$ | $<109$ | $\cdots$ | $99$ | (26) | $99$ | (25) | $<160$ | $\cdots$ | $238$ | (35) | $62$ | (17) | $<75$ | $\cdots$
NGC7582 | $<228$ | $\cdots$ | $<548$ | $\cdots$ | $<145$ | $\cdots$ | $<495$ | $\cdots$ | $242$ | (31) | $350$ | (100) | $310$ | (60) | $<341$ | $\cdots$
NGC7590 | $<179$ | $\cdots$ | $<128$ | $\cdots$ | $<99$ | $\cdots$ | $<124$ | $\cdots$ | $<59$ | $\cdots$ | $<40$ | $\cdots$ | $38$ | (8) | $27$ | (6)
NGC7603 | $71$ | (11) | $<66$ | $\cdots$ | $<55$ | $\cdots$ | $<62$ | $\cdots$ | $57$ | (14) | $31$ | (9) | $40$ | (12) | $22$ | (6)
NGC7674 | $<87$ | $\cdots$ | $<78$ | $\cdots$ | $74$ | (22) | $<107$ | $\cdots$ | $93$ | (16) | $<49$ | $\cdots$ | $<64$ | $\cdots$ | $<101$ | $\cdots$
CGCG381-051 | $<75$ | $\cdots$ | $<98$ | $\cdots$ | $76$ | (24) | $64$ | (20) | $22$ | (6) | $<27$ | $\cdots$ | $33$ | (11) | $<31$ | $\cdots$
Note. — Integrated fluxes are given in units of $10^{-18}$ W m-2.
Uncertainties ($1\sigma$) are listed in parentheses. Upper limits are
$3\sigma$. The line wavelengths for the S(7) – S(0) series are 5.512, 6.159,
6.859, 8.004, 9.656, 12.329, 17.063, and 28.171 $\mu$m, respectively.
Table 7: H2 EQWs
Source ID | S(7) | S(6) | S(5) | S(4) | S(3) | S(2) | S(1) | S(0)
---|---|---|---|---|---|---|---|---
MRK335 | $<7$ | $\cdots$ | $<10$ | $\cdots$ | $<9$ | $\cdots$ | $<10$ | $\cdots$ | $9$ | (2) | $6$ | (2) | $6$ | (2) | $<15$ | $\cdots$
MRK938 | $50$ | (10) | $<74$ | $\cdots$ | $60$ | (9) | $98$ | (22) | $77$ | (8) | $37$ | (6) | $34$ | (4) | $<11$ | $\cdots$
E12-G21 | $12$ | (3) | $<31$ | $\cdots$ | $<18$ | $\cdots$ | $<24$ | $\cdots$ | $14$ | (4) | $<16$ | $\cdots$ | $<14$ | $\cdots$ | $<19$ | $\cdots$
MRK348 | $<8$ | $\cdots$ | $<10$ | $\cdots$ | $8$ | (2) | $<12$ | $\cdots$ | $<5$ | $\cdots$ | $<5$ | $\cdots$ | $<7$ | $\cdots$ | $10$ | (3)
NGC424 | $<3$ | $\cdots$ | $<2$ | $\cdots$ | $<2$ | $\cdots$ | $<7$ | $\cdots$ | $<2$ | $\cdots$ | $<3$ | $\cdots$ | $<2$ | $\cdots$ | $<11$ | $\cdots$
NGC526A | $9$ | (2) | $6$ | (2) | $12$ | (2) | $11$ | (2) | $5$ | (2) | $5$ | (1) | $9$ | (3) | $<27$ | $\cdots$
NGC513 | $<28$ | $\cdots$ | $<44$ | $\cdots$ | $<26$ | $\cdots$ | $<54$ | $\cdots$ | $38$ | (9) | $<28$ | $\cdots$ | $30$ | (7) | $<23$ | $\cdots$
F01475-0740 | $<35$ | $\cdots$ | $<16$ | $\cdots$ | $13$ | (3) | $<14$ | $\cdots$ | $<6$ | $\cdots$ | $8$ | (2) | $<5$ | $\cdots$ | $<18$ | $\cdots$
NGC931 | $<8$ | $\cdots$ | $<6$ | $\cdots$ | $5$ | (2) | $<10$ | $\cdots$ | $<5$ | $\cdots$ | $<3$ | $\cdots$ | $<5$ | $\cdots$ | $<12$ | $\cdots$
NGC1056 | $<45$ | $\cdots$ | $<70$ | $\cdots$ | $<37$ | $\cdots$ | $<80$ | $\cdots$ | $37$ | (8) | $<27$ | $\cdots$ | $<28$ | $\cdots$ | $<15$ | $\cdots$
NGC1097 | $<16$ | $\cdots$ | $<46$ | $\cdots$ | $16$ | (4) | $<70$ | $\cdots$ | $28$ | (3) | $<61$ | $\cdots$ | $44$ | (4) | $<7$ | $\cdots$
NGC1125 | $<30$ | $\cdots$ | $<35$ | $\cdots$ | $<28$ | $\cdots$ | $<28$ | $\cdots$ | $<19$ | $\cdots$ | $<12$ | $\cdots$ | $<6$ | $\cdots$ | $<13$ | $\cdots$
NGC1143-4 | $<15$ | $\cdots$ | $<58$ | $\cdots$ | $<16$ | $\cdots$ | $<23$ | $\cdots$ | $45$ | (5) | $29$ | (7) | $52$ | (4) | $<20$ | $\cdots$
M-2-8-39 | $<53$ | $\cdots$ | $22$ | (7) | $<26$ | $\cdots$ | $26$ | (8) | $<12$ | $\cdots$ | $<6$ | $\cdots$ | $<11$ | $\cdots$ | $17$ | (5)
NGC1194 | $<5$ | $\cdots$ | $<5$ | $\cdots$ | $<5$ | $\cdots$ | $<8$ | $\cdots$ | $<7$ | $\cdots$ | $<8$ | $\cdots$ | $<6$ | $\cdots$ | $<14$ | $\cdots$
NGC1241 | $<90$ | $\cdots$ | $<59$ | $\cdots$ | $<61$ | $\cdots$ | $<133$ | $\cdots$ | $<36$ | $\cdots$ | $35$ | (8) | $39$ | (6) | $<16$ | $\cdots$
NGC1320 | $<10$ | $\cdots$ | $<7$ | $\cdots$ | $<6$ | $\cdots$ | $<10$ | $\cdots$ | $<5$ | $\cdots$ | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $<13$ | $\cdots$
NGC1365 | $<8$ | $\cdots$ | $<35$ | $\cdots$ | $<12$ | $\cdots$ | $<39$ | $\cdots$ | $16$ | (2) | $<18$ | $\cdots$ | $<14$ | $\cdots$ | $<9$ | $\cdots$
NGC1386 | $<14$ | $\cdots$ | $<7$ | $\cdots$ | $7$ | (2) | $<6$ | $\cdots$ | $12$ | (3) | $10$ | (2) | $<7$ | $\cdots$ | $<15$ | $\cdots$
F03450+0055 | $7$ | (2) | $<7$ | $\cdots$ | $<10$ | $\cdots$ | $<7$ | $\cdots$ | $<4$ | $\cdots$ | $5$ | (1) | $<6$ | $\cdots$ | $<13$ | $\cdots$
NGC1566 | $<13$ | $\cdots$ | $<16$ | $\cdots$ | $<7$ | $\cdots$ | $<19$ | $\cdots$ | $36$ | (3) | $28$ | (4) | $68$ | (4) | $<13$ | $\cdots$
F04385-0828 | $<6$ | $\cdots$ | $4$ | (1) | $<5$ | $\cdots$ | $<6$ | $\cdots$ | $<6$ | $\cdots$ | $6$ | (2) | $<5$ | $\cdots$ | $<9$ | $\cdots$
NGC1667 | $<75$ | $\cdots$ | $<117$ | $\cdots$ | $<56$ | $\cdots$ | $<131$ | $\cdots$ | $<41$ | $\cdots$ | $53$ | (11) | $65$ | (12) | $<14$ | $\cdots$
E33-G2 | $<13$ | $\cdots$ | $<13$ | $\cdots$ | $<9$ | $\cdots$ | $<13$ | $\cdots$ | $<7$ | $\cdots$ | $10$ | (2) | $6$ | (2) | $<14$ | $\cdots$
M-5-13-17 | $<24$ | $\cdots$ | $<17$ | $\cdots$ | $<16$ | $\cdots$ | $<17$ | $\cdots$ | $13$ | (3) | $<9$ | $\cdots$ | $<7$ | $\cdots$ | $<14$ | $\cdots$
MRK6 | $<7$ | $\cdots$ | $<5$ | $\cdots$ | $10$ | (2) | $<7$ | $\cdots$ | $12$ | (2) | $6$ | (2) | $<7$ | $\cdots$ | $<22$ | $\cdots$
MRK79 | $<8$ | $\cdots$ | $<5$ | $\cdots$ | $<7$ | $\cdots$ | $<7$ | $\cdots$ | $8$ | (2) | $<4$ | $\cdots$ | $10$ | (3) | $<10$ | $\cdots$
NGC2639 | $<46$ | $\cdots$ | $<66$ | $\cdots$ | $<90$ | $\cdots$ | $<129$ | $\cdots$ | $<84$ | $\cdots$ | $<54$ | $\cdots$ | $<50$ | $\cdots$ | $41$ | (12)
MRK704 | $<7$ | $\cdots$ | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $<7$ | $\cdots$ | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $<5$ | $\cdots$ | $<15$ | $\cdots$
NGC2992 | $<16$ | $\cdots$ | $<16$ | $\cdots$ | $12$ | (3) | $<18$ | $\cdots$ | $14$ | (2) | $<13$ | $\cdots$ | $24$ | (3) | $<14$ | $\cdots$
MRK1239 | $<3$ | $\cdots$ | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $<2$ | $\cdots$ | $<3$ | $\cdots$ | $<2$ | $\cdots$ | $<4$ | $\cdots$ | $<15$ | $\cdots$
NGC3079 | $<27$ | $\cdots$ | $<89$ | $\cdots$ | $133$ | (9) | $<184$ | $\cdots$ | $216$ | (12) | $100$ | (25) | $135$ | (10) | $<19$ | $\cdots$
NGC3227 | $<8$ | $\cdots$ | $<9$ | $\cdots$ | $23$ | (4) | $14$ | (4) | $22$ | (3) | $13$ | (3) | $29$ | (3) | $<13$ | $\cdots$
NGC3511 | $<125$ | $\cdots$ | $<127$ | $\cdots$ | $<85$ | $\cdots$ | $<118$ | $\cdots$ | $<60$ | $\cdots$ | $<34$ | $\cdots$ | $39$ | (12) | $34$ | (9)
NGC3516 | $<7$ | $\cdots$ | $<7$ | $\cdots$ | $7$ | (2) | $11$ | (3) | $<6$ | $\cdots$ | $6$ | (2) | $<5$ | $\cdots$ | $<15$ | $\cdots$
M+0-29-23 | $<22$ | $\cdots$ | $<46$ | $\cdots$ | $32$ | (8) | $<37$ | $\cdots$ | $16$ | (5) | $19$ | (4) | $23$ | (4) | $<13$ | $\cdots$
NGC3660 | $116$ | (30) | $<119$ | $\cdots$ | $<118$ | $\cdots$ | $<149$ | $\cdots$ | $<53$ | $\cdots$ | $<37$ | $\cdots$ | $<41$ | $\cdots$ | $28$ | (6)
NGC3982 | $<94$ | $\cdots$ | $<136$ | $\cdots$ | $<67$ | $\cdots$ | $<132$ | $\cdots$ | $<38$ | $\cdots$ | $<28$ | $\cdots$ | $<17$ | $\cdots$ | $<15$ | $\cdots$
NGC4051 | $<4$ | $\cdots$ | $<5$ | $\cdots$ | $<3$ | $\cdots$ | $<7$ | $\cdots$ | $10$ | (2) | $<3$ | $\cdots$ | $<5$ | $\cdots$ | $<16$ | $\cdots$
UGC7064 | $<30$ | $\cdots$ | $<41$ | $\cdots$ | $<46$ | $\cdots$ | $<45$ | $\cdots$ | $<15$ | $\cdots$ | $<16$ | $\cdots$ | $<12$ | $\cdots$ | $<20$ | $\cdots$
NGC4151 | $<5$ | $\cdots$ | $<2$ | $\cdots$ | $<2$ | $\cdots$ | $<6$ | $\cdots$ | $<3$ | $\cdots$ | $<4$ | $\cdots$ | $<5$ | $\cdots$ | $14$ | (4)
MRK766 | $<10$ | $\cdots$ | $<10$ | $\cdots$ | $5$ | (1) | $<11$ | $\cdots$ | $<4$ | $\cdots$ | $<7$ | $\cdots$ | $<7$ | $\cdots$ | $<12$ | $\cdots$
NGC4388 | $<16$ | $\cdots$ | $5$ | (1) | $10$ | (2) | $<18$ | $\cdots$ | $24$ | (2) | $18$ | (4) | $12$ | (3) | $<12$ | $\cdots$
NGC4501 | $<25$ | $\cdots$ | $<41$ | $\cdots$ | $32$ | (9) | $88$ | (16) | $102$ | (16) | $75$ | (15) | $338$ | (15) | $39$ | (12)
NGC4579 | $<10$ | $\cdots$ | $16$ | (4) | $48$ | (4) | $39$ | (7) | $91$ | (4) | $67$ | (7) | $146$ | (4) | $<12$ | $\cdots$
NGC4593 | $<5$ | $\cdots$ | $<5$ | $\cdots$ | $5$ | (1) | $<8$ | $\cdots$ | $5$ | (2) | $<3$ | $\cdots$ | $10$ | (3) | $<16$ | $\cdots$
NGC4594 | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $<6$ | $\cdots$ | $<16$ | $\cdots$ | $<7$ | $\cdots$ | $<11$ | $\cdots$ | $5$ | (2) | $<17$ | $\cdots$
TOL1238-364 | $<10$ | $\cdots$ | $<15$ | $\cdots$ | $<7$ | $\cdots$ | $<21$ | $\cdots$ | $8$ | (2) | $<7$ | $\cdots$ | $<4$ | $\cdots$ | $<11$ | $\cdots$
NGC4602 | $<156$ | $\cdots$ | $<74$ | $\cdots$ | $95$ | (30) | $<120$ | $\cdots$ | $<75$ | $\cdots$ | $<33$ | $\cdots$ | $46$ | (10) | $30$ | (8)
M-2-33-34 | $<36$ | $\cdots$ | $<22$ | $\cdots$ | $<29$ | $\cdots$ | $<23$ | $\cdots$ | $<13$ | $\cdots$ | $<11$ | $\cdots$ | $<12$ | $\cdots$ | $<34$ | $\cdots$
NGC4941 | $41$ | (4) | $31$ | (3) | $38$ | (4) | $30$ | (5) | $29$ | (7) | $13$ | (3) | $<13$ | $\cdots$ | $<23$ | $\cdots$
NGC4968 | $<22$ | $\cdots$ | $<9$ | $\cdots$ | $<10$ | $\cdots$ | $<15$ | $\cdots$ | $<7$ | $\cdots$ | $8$ | (2) | $<6$ | $\cdots$ | $<12$ | $\cdots$
NGC5005 | $17$ | (5) | $<14$ | $\cdots$ | $59$ | (4) | $<30$ | $\cdots$ | $121$ | (7) | $68$ | (13) | $205$ | (8) | $20$ | (6)
NGC5033 | $<12$ | $\cdots$ | $<29$ | $\cdots$ | $19$ | (5) | $<36$ | $\cdots$ | $37$ | (3) | $31$ | (7) | $73$ | (7) | $16$ | (5)
NGC5135 | $<18$ | $\cdots$ | $<24$ | $\cdots$ | $<21$ | $\cdots$ | $<28$ | $\cdots$ | $21$ | (3) | $<26$ | $\cdots$ | $31$ | (3) | $<11$ | $\cdots$
M-6-30-15 | $<6$ | $\cdots$ | $<5$ | $\cdots$ | $<4$ | $\cdots$ | $<7$ | $\cdots$ | $<3$ | $\cdots$ | $<3$ | $\cdots$ | $<4$ | $\cdots$ | $<10$ | $\cdots$
NGC5256 | $<66$ | $\cdots$ | $<82$ | $\cdots$ | $<46$ | $\cdots$ | $<56$ | $\cdots$ | $76$ | (9) | $45$ | (12) | $31$ | (5) | $<18$ | $\cdots$
IC4329A | $<4$ | $\cdots$ | $<3$ | $\cdots$ | $<2$ | $\cdots$ | $<6$ | $\cdots$ | $<2$ | $\cdots$ | $<2$ | $\cdots$ | $<5$ | $\cdots$ | $<12$ | $\cdots$
NGC5347 | $<13$ | $\cdots$ | $<10$ | $\cdots$ | $11$ | (3) | $11$ | (3) | $<7$ | $\cdots$ | $<5$ | $\cdots$ | $<5$ | $\cdots$ | $<11$ | $\cdots$
NGC5506 | $<5$ | $\cdots$ | $<2$ | $\cdots$ | $<1$ | $\cdots$ | $<4$ | $\cdots$ | $<5$ | $\cdots$ | $7$ | (2) | $<3$ | $\cdots$ | $<12$ | $\cdots$
NGC5548 | $<7$ | $\cdots$ | $11$ | (3) | $7$ | (2) | $<13$ | $\cdots$ | $12$ | (3) | $8$ | (2) | $9$ | (3) | $<16$ | $\cdots$
MRK817 | $<18$ | $\cdots$ | $<13$ | $\cdots$ | $<10$ | $\cdots$ | $<10$ | $\cdots$ | $<9$ | $\cdots$ | $<6$ | $\cdots$ | $<5$ | $\cdots$ | $<12$ | $\cdots$
NGC5929 | $<83$ | $\cdots$ | $70$ | (19) | $74$ | (21) | $<126$ | $\cdots$ | $133$ | (20) | $75$ | (16) | $135$ | (14) | $<23$ | $\cdots$
NGC5953 | $<35$ | $\cdots$ | $<100$ | $\cdots$ | $<28$ | $\cdots$ | $<107$ | $\cdots$ | $34$ | (8) | $<39$ | $\cdots$ | $47$ | (10) | $<12$ | $\cdots$
M-2-40-4 | $<6$ | $\cdots$ | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $8$ | (2) | $<4$ | $\cdots$ | $7$ | (2) | $<6$ | $\cdots$ | $<11$ | $\cdots$
F15480-0344 | $<16$ | $\cdots$ | $<18$ | $\cdots$ | $13$ | (2) | $<15$ | $\cdots$ | $<8$ | $\cdots$ | $<4$ | $\cdots$ | $10$ | (3) | $11$ | (3)
NGC6810 | $<19$ | $\cdots$ | $<42$ | $\cdots$ | $<18$ | $\cdots$ | $<35$ | $\cdots$ | $19$ | (2) | $<15$ | $\cdots$ | $27$ | (4) | $<14$ | $\cdots$
NGC6860 | $<13$ | $\cdots$ | $<6$ | $\cdots$ | $9$ | (3) | $<7$ | $\cdots$ | $<6$ | $\cdots$ | $<10$ | $\cdots$ | $12$ | (4) | $<16$ | $\cdots$
NGC6890 | $<19$ | $\cdots$ | $<16$ | $\cdots$ | $<14$ | $\cdots$ | $<28$ | $\cdots$ | $<9$ | $\cdots$ | $<10$ | $\cdots$ | $15$ | (4) | $<17$ | $\cdots$
IC5063 | $<4$ | $\cdots$ | $<3$ | $\cdots$ | $4$ | (1) | $<6$ | $\cdots$ | $<4$ | $\cdots$ | $<3$ | $\cdots$ | $<3$ | $\cdots$ | $<9$ | $\cdots$
UGC11680 | $<42$ | $\cdots$ | $<31$ | $\cdots$ | $<37$ | $\cdots$ | $<51$ | $\cdots$ | $<27$ | $\cdots$ | $<10$ | $\cdots$ | $15$ | (5) | $<21$ | $\cdots$
NGC7130 | $<17$ | $\cdots$ | $<28$ | $\cdots$ | $<29$ | $\cdots$ | $34$ | (9) | $27$ | (4) | $16$ | (4) | $16$ | (3) | $<14$ | $\cdots$
NGC7172 | $<6$ | $\cdots$ | $<14$ | $\cdots$ | $9$ | (2) | $<40$ | $\cdots$ | $36$ | (3) | $14$ | (4) | $20$ | (5) | $<13$ | $\cdots$
NGC7213 | $<10$ | $\cdots$ | $<9$ | $\cdots$ | $<10$ | $\cdots$ | $<14$ | $\cdots$ | $12$ | (3) | $<5$ | $\cdots$ | $21$ | (3) | $<17$ | $\cdots$
NGC7314 | $30$ | (5) | $20$ | (3) | $32$ | (3) | $23$ | (6) | $17$ | (3) | $8$ | (2) | $8$ | (2) | $<20$ | $\cdots$
M-3-58-7 | $5$ | (2) | $<6$ | $\cdots$ | $<9$ | $\cdots$ | $11$ | (3) | $<4$ | $\cdots$ | $<5$ | $\cdots$ | $<6$ | $\cdots$ | $<14$ | $\cdots$
NGC7469 | $<9$ | $\cdots$ | $<15$ | $\cdots$ | $<9$ | $\cdots$ | $<8$ | $\cdots$ | $15$ | (2) | $10$ | (2) | $8$ | (3) | $<14$ | $\cdots$
NGC7496 | $<37$ | $\cdots$ | $<51$ | $\cdots$ | $41$ | (11) | $34$ | (9) | $<52$ | $\cdots$ | $65$ | (10) | $12$ | (3) | $<10$ | $\cdots$
NGC7582 | $<7$ | $\cdots$ | $<20$ | $\cdots$ | $<6$ | $\cdots$ | $<23$ | $\cdots$ | $23$ | (3) | $20$ | (6) | $15$ | (3) | $<11$ | $\cdots$
NGC7590 | $<83$ | $\cdots$ | $<78$ | $\cdots$ | $<73$ | $\cdots$ | $<99$ | $\cdots$ | $<46$ | $\cdots$ | $<36$ | $\cdots$ | $45$ | (9) | $30$ | (7)
NGC7603 | $4$ | (1) | $<4$ | $\cdots$ | $<4$ | $\cdots$ | $<6$ | $\cdots$ | $7$ | (2) | $6$ | (2) | $13$ | (4) | $18$ | (5)
NGC7674 | $<7$ | $\cdots$ | $<6$ | $\cdots$ | $6$ | (2) | $<10$ | $\cdots$ | $10$ | (2) | $<5$ | $\cdots$ | $<6$ | $\cdots$ | $<15$ | $\cdots$
CGCG381-051 | $<55$ | $\cdots$ | $<67$ | $\cdots$ | $48$ | (15) | $38$ | (12) | $9$ | (2) | $<13$ | $\cdots$ | $13$ | (4) | $<14$ | $\cdots$
Note. — H2 equivalent widths are given in units of nm. Uncertainties
($1\sigma$) are listed in parentheses. Upper limits are $3\sigma$. The line
wavelengths for the S(7) – S(0) series are 5.512, 6.159, 6.859, 8.004, 9.656,
12.329, 17.063, and 28.171 $\mu$m, respectively.
Table 8: Fine Structure Line Fluxes
Source ID | [OIV] | [SiII] | [NeII] | [NeIII] | [NeV] | [SIII] | [SIII] | [SIV] | [ArII]
---|---|---|---|---|---|---|---|---|---
| 25.91 $\mu$m | 34.815 $\mu$m | 12.813 $\mu$m | 15.555 $\mu$m | 14.322 $\mu$m | 18.713 $\mu$m | 33.48 $\mu$m | 10.511 $\mu$m | 6.985 $\mu$m
MRK335 | $32$ | (7) | $<80$ | $\cdots$ | $24$ | (5) | $44$ | (9) | $<19$ | $\cdots$ | $<32$ | $\cdots$ | $<87$ | $\cdots$ | $27$ | (8) | $<70$ | $\cdots$
MRK938 | $<126$ | $\cdots$ | $<796$ | $\cdots$ | $563$ | (25) | $<82$ | $\cdots$ | $<56$ | $\cdots$ | $52$ | (15) | $<895$ | $\cdots$ | $<55$ | $\cdots$ | $209$ | (30)
E12-G21 | $202$ | (13) | $169$ | (52) | $140$ | (19) | $61$ | (8) | $52$ | (9) | $90$ | (9) | $<174$ | $\cdots$ | $47$ | (12) | $<69$ | $\cdots$
MRK348 | $179$ | (14) | $131$ | (35) | $131$ | (10) | $178$ | (16) | $54$ | (11) | $42$ | (12) | $<110$ | $\cdots$ | $62$ | (9) | $69$ | (14)
NGC424 | $145$ | (29) | $<244$ | $\cdots$ | $73$ | (22) | $136$ | (30) | $<58$ | $\cdots$ | $<53$ | $\cdots$ | $<244$ | $\cdots$ | $<54$ | $\cdots$ | $<68$ | $\cdots$
NGC526A | $203$ | (10) | $<155$ | $\cdots$ | $51$ | (11) | $135$ | (13) | $<25$ | $\cdots$ | $56$ | (10) | $66$ | (19) | $61$ | (12) | $57$ | (10)
NGC513 | $110$ | (8) | $308$ | (36) | $237$ | (15) | $93$ | (9) | $24$ | (8) | $100$ | (8) | $217$ | (38) | $<48$ | $\cdots$ | $<94$ | $\cdots$
F01475-0740 | $<65$ | $\cdots$ | $<182$ | $\cdots$ | $157$ | (9) | $96$ | (10) | $21$ | (6) | $47$ | (11) | $<185$ | $\cdots$ | $<30$ | $\cdots$ | $62$ | (19)
NGC931 | $365$ | (23) | $<124$ | $\cdots$ | $67$ | (12) | $141$ | (17) | $78$ | (15) | $<37$ | $\cdots$ | $167$ | (48) | $156$ | (20) | $<56$ | $\cdots$
NGC1056 | $52$ | (10) | $715$ | (67) | $503$ | (20) | $127$ | (22) | $<30$ | $\cdots$ | $218$ | (14) | $490$ | (32) | $<83$ | $\cdots$ | $94$ | (30)
NGC1097 | $<97$ | $\cdots$ | $2900$ | (240) | $3200$ | (170) | $90$ | (25) | $<183$ | $\cdots$ | $593$ | (31) | $1030$ | (250) | $<170$ | $\cdots$ | $1281$ | (57)
NGC1125 | $252$ | (21) | $223$ | (55) | $311$ | (9) | $243$ | (11) | $77$ | (10) | $136$ | (10) | $220$ | (53) | $107$ | (8) | $95$ | (17)
NGC1143-4 | $74$ | (11) | $614$ | (78) | $499$ | (14) | $110$ | (9) | $26$ | (7) | $267$ | (8) | $454$ | (75) | $30$ | (9) | $216$ | (16)
M-2-8-39 | $138$ | (13) | $64$ | (19) | $66$ | (6) | $134$ | (17) | $62$ | (11) | $<36$ | $\cdots$ | $<48$ | $\cdots$ | $44$ | (8) | $<90$ | $\cdots$
NGC1194 | $115$ | (13) | $<63$ | $\cdots$ | $31$ | (9) | $94$ | (15) | $41$ | (11) | $<41$ | $\cdots$ | $<90$ | $\cdots$ | $62$ | (8) | $101$ | (17)
NGC1241 | $69$ | (7) | $208$ | (24) | $139$ | (7) | $107$ | (10) | $<19$ | $\cdots$ | $43$ | (6) | $138$ | (30) | $65$ | (17) | $<80$ | $\cdots$
NGC1320 | $347$ | (23) | $<127$ | $\cdots$ | $87$ | (11) | $71$ | (16) | $69$ | (16) | $<46$ | $\cdots$ | $<134$ | $\cdots$ | $96$ | (12) | $<49$ | $\cdots$
NGC1365 | $1000$ | (160) | $6520$ | (790) | $5000$ | (220) | $440$ | (110) | $255$ | (74) | $1669$ | (85) | $2800$ | (500) | $446$ | (48) | $2130$ | (140)
NGC1386 | $891$ | (52) | $450$ | (140) | $192$ | (17) | $419$ | (23) | $277$ | (26) | $196$ | (25) | $<199$ | $\cdots$ | $285$ | (14) | $156$ | (22)
F03450+0055 | $<25$ | $\cdots$ | $<78$ | $\cdots$ | $<21$ | $\cdots$ | $37$ | (11) | $<27$ | $\cdots$ | $<26$ | $\cdots$ | $<79$ | $\cdots$ | $<27$ | $\cdots$ | $<66$ | $\cdots$
NGC1566 | $78$ | (8) | $194$ | (18) | $168$ | (9) | $86$ | (7) | $<15$ | $\cdots$ | $64$ | (5) | $100$ | (16) | $30$ | (6) | $61$ | (14)
F04385-0828 | $<60$ | $\cdots$ | $<301$ | $\cdots$ | $120$ | (13) | $<50$ | $\cdots$ | $<38$ | $\cdots$ | $<38$ | $\cdots$ | $<316$ | $\cdots$ | $<22$ | $\cdots$ | $<81$ | $\cdots$
NGC1667 | $87$ | (8) | $79$ | (21) | $364$ | (14) | $66$ | (15) | $<24$ | $\cdots$ | $159$ | (12) | $<68$ | $\cdots$ | $<54$ | $\cdots$ | $191$ | (31)
E33-G2 | $159$ | (11) | $<77$ | $\cdots$ | $30$ | (8) | $57$ | (11) | $25$ | (7) | $<34$ | $\cdots$ | $<83$ | $\cdots$ | $113$ | (15) | $<54$ | $\cdots$
M-5-13-17 | $130$ | (12) | $131$ | (24) | $119$ | (7) | $71$ | (9) | $<25$ | $\cdots$ | $<26$ | $\cdots$ | $85$ | (25) | $34$ | (10) | $<59$ | $\cdots$
MRK6 | $302$ | (17) | $334$ | (41) | $240$ | (10) | $463$ | (23) | $75$ | (10) | $140$ | (12) | $165$ | (44) | $172$ | (11) | $113$ | (16)
MRK79 | $447$ | (18) | $161$ | (43) | $96$ | (14) | $204$ | (14) | $82$ | (13) | $53$ | (13) | $166$ | (47) | $114$ | (12) | $<77$ | $\cdots$
NGC2639 | $23$ | (6) | $111$ | (20) | $135$ | (11) | $42$ | (11) | $<20$ | $\cdots$ | $48$ | (9) | $54$ | (13) | $<57$ | $\cdots$ | $<96$ | $\cdots$
MRK704 | $102$ | (11) | $53$ | (14) | $<32$ | $\cdots$ | $75$ | (12) | $<43$ | $\cdots$ | $<34$ | $\cdots$ | $<46$ | $\cdots$ | $<29$ | $\cdots$ | $<47$ | $\cdots$
NGC2992 | $1258$ | (40) | $1090$ | (100) | $657$ | (32) | $756$ | (37) | $249$ | (25) | $419$ | (20) | $710$ | (110) | $344$ | (14) | $276$ | (25)
MRK1239 | $83$ | (17) | $<157$ | $\cdots$ | $66$ | (18) | $118$ | (19) | $<37$ | $\cdots$ | $<50$ | $\cdots$ | $<171$ | $\cdots$ | $<42$ | $\cdots$ | $<102$ | $\cdots$
NGC3079 | $311$ | (64) | $2620$ | (260) | $1860$ | (120) | $232$ | (57) | $<115$ | $\cdots$ | $155$ | (25) | $610$ | (150) | $112$ | (28) | $589$ | (48)
NGC3227 | $573$ | (50) | $740$ | (150) | $752$ | (24) | $619$ | (30) | $135$ | (22) | $253$ | (24) | $<391$ | $\cdots$ | $268$ | (31) | $294$ | (34)
NGC3511 | $<23$ | $\cdots$ | $402$ | (29) | $248$ | (13) | $<34$ | $\cdots$ | $<17$ | $\cdots$ | $95$ | (8) | $151$ | (28) | $<50$ | $\cdots$ | $<78$ | $\cdots$
NGC3516 | $591$ | (23) | $147$ | (29) | $65$ | (10) | $189$ | (18) | $72$ | (12) | $81$ | (13) | $160$ | (29) | $159$ | (12) | $<72$ | $\cdots$
M+0-29-23 | $<39$ | $\cdots$ | $446$ | (56) | $407$ | (18) | $71$ | (13) | $<28$ | $\cdots$ | $109$ | (12) | $<186$ | $\cdots$ | $<43$ | $\cdots$ | $150$ | (22)
NGC3660 | $41$ | (9) | $74$ | (17) | $51$ | (7) | $38$ | (9) | $<16$ | $\cdots$ | $<27$ | $\cdots$ | $<62$ | $\cdots$ | $45$ | (11) | $<87$ | $\cdots$
NGC3982 | $39$ | (8) | $531$ | (34) | $301$ | (16) | $67$ | (15) | $<30$ | $\cdots$ | $133$ | (10) | $238$ | (33) | $74$ | (16) | $130$ | (34)
NGC4051 | $297$ | (32) | $295$ | (94) | $184$ | (16) | $107$ | (18) | $<28$ | $\cdots$ | $85$ | (20) | $<152$ | $\cdots$ | $64$ | (16) | $88$ | (20)
UGC7064 | $117$ | (13) | $<214$ | $\cdots$ | $117$ | (8) | $57$ | (9) | $47$ | (10) | $75$ | (11) | $<220$ | $\cdots$ | $<36$ | $\cdots$ | $<90$ | $\cdots$
NGC4151 | $1860$ | (110) | $1230$ | (300) | $1459$ | (62) | $1528$ | (82) | $353$ | (58) | $761$ | (68) | $<623$ | $\cdots$ | $892$ | (58) | $441$ | (42)
MRK766 | $387$ | (29) | $<181$ | $\cdots$ | $210$ | (15) | $231$ | (37) | $121$ | (24) | $98$ | (24) | $265$ | (66) | $92$ | (14) | $48$ | (8)
NGC4388 | $3401$ | (74) | $1200$ | (210) | $845$ | (35) | $1201$ | (83) | $482$ | (32) | $526$ | (26) | $730$ | (150) | $695$ | (35) | $270$ | (23)
NGC4501 | $30$ | (5) | $185$ | (16) | $104$ | (13) | $87$ | (8) | $<20$ | $\cdots$ | $28$ | (7) | $38$ | (12) | $<55$ | $\cdots$ | $<68$ | $\cdots$
NGC4579 | $68$ | (5) | $371$ | (24) | $196$ | (12) | $99$ | (4) | $<8$ | $\cdots$ | $44$ | (4) | $58$ | (11) | $26$ | (5) | $173$ | (15)
NGC4593 | $80$ | (17) | $<117$ | $\cdots$ | $75$ | (11) | $<42$ | $\cdots$ | $<37$ | $\cdots$ | $<32$ | $\cdots$ | $<123$ | $\cdots$ | $<30$ | $\cdots$ | $<68$ | $\cdots$
NGC4594 | $35$ | (2) | $271$ | (21) | $129$ | (7) | $144$ | (6) | $<6$ | $\cdots$ | $56$ | (4) | $94$ | (23) | $<19$ | $\cdots$ | $91$ | (22)
TOL1238-364 | $<131$ | $\cdots$ | $<438$ | $\cdots$ | $553$ | (21) | $268$ | (31) | $<41$ | $\cdots$ | $170$ | (23) | $<413$ | $\cdots$ | $93$ | (12) | $167$ | (25)
NGC4602 | $<16$ | $\cdots$ | $184$ | (21) | $127$ | (9) | $<13$ | $\cdots$ | $<13$ | $\cdots$ | $72$ | (8) | $123$ | (22) | $28$ | (8) | $82$ | (26)
M-2-33-34 | $692$ | (32) | $352$ | (24) | $138$ | (7) | $320$ | (12) | $131$ | (9) | $137$ | (9) | $233$ | (27) | $218$ | (8) | $75$ | (19)
NGC4941 | $203$ | (12) | $180$ | (25) | $160$ | (9) | $202$ | (8) | $43$ | (8) | $93$ | (9) | $<72$ | $\cdots$ | $135$ | (12) | $71$ | (8)
NGC4968 | $293$ | (24) | $188$ | (42) | $247$ | (12) | $252$ | (17) | $95$ | (15) | $64$ | (15) | $161$ | (38) | $107$ | (13) | $61$ | (19)
NGC5005 | $127$ | (22) | $920$ | (100) | $615$ | (36) | $183$ | (14) | $<39$ | $\cdots$ | $59$ | (11) | $<163$ | $\cdots$ | $<67$ | $\cdots$ | $169$ | (29)
NGC5033 | $179$ | (8) | $949$ | (48) | $534$ | (23) | $140$ | (13) | $33$ | (7) | $148$ | (7) | $255$ | (54) | $<45$ | $\cdots$ | $192$ | (24)
NGC5135 | $680$ | (50) | $1450$ | (170) | $1421$ | (64) | $620$ | (33) | $130$ | (26) | $391$ | (21) | $960$ | (190) | $260$ | (21) | $596$ | (53)
M-6-30-15 | $123$ | (15) | $<79$ | $\cdots$ | $60$ | (10) | $<34$ | $\cdots$ | $<26$ | $\cdots$ | $<37$ | $\cdots$ | $<87$ | $\cdots$ | $66$ | (14) | $45$ | (10)
NGC5256 | $636$ | (32) | $720$ | (86) | $616$ | (29) | $398$ | (17) | $63$ | (18) | $265$ | (14) | $452$ | (79) | $156$ | (12) | $218$ | (27)
IC4329A | $1233$ | (71) | $302$ | (93) | $219$ | (33) | $512$ | (51) | $365$ | (46) | $<78$ | $\cdots$ | $380$ | (97) | $308$ | (43) | $59$ | (15)
NGC5347 | $<50$ | $\cdots$ | $<113$ | $\cdots$ | $71$ | (13) | $<35$ | $\cdots$ | $<23$ | $\cdots$ | $<34$ | $\cdots$ | $<128$ | $\cdots$ | $31$ | (9) | $100$ | (17)
NGC5506 | $2324$ | (99) | $1170$ | (170) | $826$ | (53) | $1192$ | (81) | $343$ | (41) | $501$ | (35) | $700$ | (180) | $772$ | (40) | $331$ | (37)
NGC5548 | $80$ | (11) | $<129$ | $\cdots$ | $99$ | (8) | $97$ | (12) | $<30$ | $\cdots$ | $48$ | (11) | $<137$ | $\cdots$ | $<42$ | $\cdots$ | $<53$ | $\cdots$
MRK817 | $<45$ | $\cdots$ | $<114$ | $\cdots$ | $76$ | (10) | $70$ | (13) | $<40$ | $\cdots$ | $<39$ | $\cdots$ | $<128$ | $\cdots$ | $<42$ | $\cdots$ | $<60$ | $\cdots$
NGC5929 | $53$ | (6) | $205$ | (23) | $99$ | (9) | $93$ | (9) | $<17$ | $\cdots$ | $64$ | (7) | $82$ | (20) | $<37$ | $\cdots$ | $68$ | (17)
NGC5953 | $233$ | (18) | $1690$ | (120) | $1303$ | (60) | $227$ | (35) | $52$ | (14) | $449$ | (17) | $700$ | (120) | $<104$ | $\cdots$ | $347$ | (33)
M-2-40-4 | $158$ | (20) | $222$ | (57) | $204$ | (13) | $150$ | (18) | $60$ | (15) | $52$ | (13) | $<184$ | $\cdots$ | $<45$ | $\cdots$ | $<63$ | $\cdots$
F15480-0344 | $426$ | (17) | $<115$ | $\cdots$ | $80$ | (8) | $161$ | (17) | $133$ | (15) | $<37$ | $\cdots$ | $<120$ | $\cdots$ | $96$ | (12) | $<48$ | $\cdots$
NGC6810 | $<148$ | $\cdots$ | $1620$ | (200) | $1479$ | (68) | $<119$ | $\cdots$ | $<59$ | $\cdots$ | $567$ | (32) | $810$ | (190) | $<79$ | $\cdots$ | $522$ | (48)
NGC6860 | $110$ | (9) | $118$ | (33) | $77$ | (9) | $81$ | (9) | $35$ | (10) | $33$ | (8) | $<100$ | $\cdots$ | $<28$ | $\cdots$ | $<72$ | $\cdots$
NGC6890 | $88$ | (14) | $207$ | (30) | $185$ | (11) | $61$ | (13) | $33$ | (8) | $92$ | (10) | $145$ | (23) | $<56$ | $\cdots$ | $63$ | (17)
IC5063 | $904$ | (90) | $<713$ | $\cdots$ | $213$ | (34) | $620$ | (46) | $<106$ | $\cdots$ | $155$ | (43) | $<721$ | $\cdots$ | $529$ | (34) | $78$ | (16)
UGC11680 | $23$ | (6) | $<88$ | $\cdots$ | $55$ | (7) | $55$ | (9) | $<20$ | $\cdots$ | $<18$ | $\cdots$ | $<49$ | $\cdots$ | $<48$ | $\cdots$ | $<95$ | $\cdots$
NGC7130 | $124$ | (37) | $890$ | (140) | $1010$ | (28) | $308$ | (27) | $111$ | (18) | $232$ | (19) | $680$ | (160) | $107$ | (15) | $415$ | (50)
NGC7172 | $549$ | (17) | $612$ | (46) | $211$ | (14) | $184$ | (15) | $72$ | (14) | $89$ | (10) | $308$ | (38) | $89$ | (10) | $141$ | (34)
NGC7213 | $<33$ | $\cdots$ | $207$ | (31) | $227$ | (13) | $129$ | (10) | $<23$ | $\cdots$ | $36$ | (11) | $<99$ | $\cdots$ | $<51$ | $\cdots$ | $116$ | (17)
NGC7314 | $521$ | (20) | $114$ | (26) | $74$ | (8) | $246$ | (16) | $108$ | (8) | $89$ | (8) | $94$ | (20) | $188$ | (10) | $68$ | (7)
M-3-58-7 | $<58$ | $\cdots$ | $<169$ | $\cdots$ | $94$ | (9) | $73$ | (13) | $57$ | (13) | $<35$ | $\cdots$ | $<188$ | $\cdots$ | $39$ | (10) | $74$ | (23)
NGC7469 | $<187$ | $\cdots$ | $1660$ | (340) | $2005$ | (46) | $242$ | (75) | $200$ | (44) | $638$ | (51) | $1590$ | (360) | $172$ | (26) | $824$ | (61)
NGC7496 | $<73$ | $\cdots$ | $<486$ | $\cdots$ | $424$ | (32) | $34$ | (11) | $<26$ | $\cdots$ | $232$ | (15) | $377$ | (96) | $48$ | (15) | $215$ | (22)
NGC7582 | $2210$ | (150) | $2790$ | (650) | $2840$ | (130) | $1153$ | (70) | $274$ | (67) | $904$ | (55) | $1700$ | (540) | $458$ | (36) | $1057$ | (62)
NGC7590 | $39$ | (6) | $492$ | (20) | $174$ | (10) | $59$ | (12) | $<25$ | $\cdots$ | $107$ | (9) | $242$ | (23) | $<59$ | $\cdots$ | $<112$ | $\cdots$
NGC7603 | $21$ | (6) | $130$ | (32) | $127$ | (9) | $70$ | (11) | $<23$ | $\cdots$ | $<23$ | $\cdots$ | $64$ | (17) | $<30$ | $\cdots$ | $<61$ | $\cdots$
NGC7674 | $492$ | (40) | $<298$ | $\cdots$ | $287$ | (15) | $380$ | (24) | $181$ | (22) | $175$ | (21) | $<313$ | $\cdots$ | $156$ | (15) | $179$ | (26)
CGCG381-051 | $<41$ | $\cdots$ | $127$ | (36) | $184$ | (11) | $<28$ | $\cdots$ | $18$ | (5) | $102$ | (11) | $<115$ | $\cdots$ | $<23$ | $\cdots$ | $86$ | (27)
Note. — Integrated fluxes are given in units of $10^{-18}$ W m-2.
Uncertainties ($1\sigma$) are listed in parentheses. Upper limits are
$3\sigma$.
Table 9: Fine Structure Line EQWs
Source ID | [OIV] | [SiII] | [NeII] | [NeIII] | [NeV] | [SIII] | [SIII] | [SIV] | [ArII]
---|---|---|---|---|---|---|---|---|---
| 25.91 $\mu$m | 34.815 $\mu$m | 12.813 $\mu$m | 15.555 $\mu$m | 14.322 $\mu$m | 18.713 $\mu$m | 33.48 $\mu$m | 10.511 $\mu$m | 6.985 $\mu$m
MRK335 | $22$ | (5) | $<101$ | $\cdots$ | $7$ | (2) | $15$ | (3) | $<6$ | $\cdots$ | $<12$ | $\cdots$ | $<102$ | $\cdots$ | $5$ | (2) | $<9$ | $\cdots$
MRK938 | $<9$ | $\cdots$ | $<46$ | $\cdots$ | $125$ | (6) | $<13$ | $\cdots$ | $<10$ | $\cdots$ | $8$ | (2) | $<53$ | $\cdots$ | $<30$ | $\cdots$ | $83$ | (12)
E12-G21 | $170$ | (11) | $158$ | (49) | $68$ | (9) | $36$ | (5) | $28$ | (5) | $64$ | (6) | $<163$ | $\cdots$ | $18$ | (5) | $<18$ | $\cdots$
MRK348 | $65$ | (5) | $83$ | (22) | $24$ | (2) | $33$ | (3) | $10$ | (2) | $8$ | (3) | $<64$ | $\cdots$ | $12$ | (2) | $9$ | (2)
NGC424 | $23$ | (5) | $<70$ | $\cdots$ | $4$ | (1) | $8$ | (2) | $<3$ | $\cdots$ | $<4$ | $\cdots$ | $<65$ | $\cdots$ | $<2$ | $\cdots$ | $<2$ | $\cdots$
NGC526A | $148$ | (7) | $<290$ | $\cdots$ | $11$ | (3) | $35$ | (3) | $<6$ | $\cdots$ | $18$ | (3) | $108$ | (31) | $10$ | (2) | $9$ | (2)
NGC513 | $94$ | (7) | $256$ | (30) | $149$ | (9) | $66$ | (7) | $16$ | (5) | $75$ | (6) | $184$ | (32) | $<24$ | $\cdots$ | $<33$ | $\cdots$
F01475-0740 | $<23$ | $\cdots$ | $<100$ | $\cdots$ | $41$ | (2) | $24$ | (3) | $5$ | (2) | $11$ | (3) | $<96$ | $\cdots$ | $<6$ | $\cdots$ | $17$ | (5)
NGC931 | $89$ | (6) | $<48$ | $\cdots$ | $8$ | (2) | $20$ | (2) | $10$ | (2) | $<6$ | $\cdots$ | $61$ | (18) | $16$ | (2) | $<4$ | $\cdots$
NGC1056 | $25$ | (5) | $272$ | (26) | $278$ | (11) | $77$ | (13) | $<17$ | $\cdots$ | $127$ | (8) | $192$ | (12) | $<38$ | $\cdots$ | $36$ | (12)
NGC1097 | $<8$ | $\cdots$ | $196$ | (16) | $412$ | (22) | $11$ | (3) | $<24$ | $\cdots$ | $65$ | (3) | $72$ | (17) | $<16$ | $\cdots$ | $108$ | (5)
NGC1125 | $62$ | (5) | $57$ | (14) | $115$ | (4) | $72$ | (3) | $24$ | (3) | $39$ | (3) | $56$ | (14) | $79$ | (6) | $53$ | (10)
NGC1143-4 | $33$ | (5) | $228$ | (29) | $270$ | (7) | $58$ | (5) | $13$ | (4) | $147$ | (5) | $176$ | (29) | $22$ | (7) | $84$ | (6)
M-2-8-39 | $96$ | (9) | $91$ | (27) | $18$ | (2) | $39$ | (5) | $17$ | (3) | $<12$ | $\cdots$ | $<61$ | $\cdots$ | $12$ | (2) | $<28$ | $\cdots$
NGC1194 | $45$ | (5) | $<36$ | $\cdots$ | $5$ | (2) | $20$ | (3) | $7$ | (2) | $<11$ | $\cdots$ | $<50$ | $\cdots$ | $16$ | (2) | $10$ | (2)
NGC1241 | $60$ | (6) | $180$ | (21) | $112$ | (6) | $83$ | (8) | $<15$ | $\cdots$ | $36$ | (5) | $120$ | (26) | $66$ | (17) | $<64$ | $\cdots$
NGC1320 | $75$ | (5) | $<39$ | $\cdots$ | $11$ | (2) | $10$ | (2) | $9$ | (2) | $<7$ | $\cdots$ | $<39$ | $\cdots$ | $12$ | (1) | $<5$ | $\cdots$
NGC1365 | $26$ | (4) | $144$ | (18) | $189$ | (8) | $16$ | (4) | $9$ | (3) | $58$ | (3) | $64$ | (11) | $18$ | (2) | $70$ | (5)
NGC1386 | $142$ | (8) | $83$ | (25) | $19$ | (2) | $47$ | (3) | $28$ | (3) | $27$ | (4) | $<35$ | $\cdots$ | $39$ | (2) | $12$ | (2)
F03450+0055 | $<11$ | $\cdots$ | $<66$ | $\cdots$ | $<4$ | $\cdots$ | $8$ | (3) | $<6$ | $\cdots$ | $<6$ | $\cdots$ | $<61$ | $\cdots$ | $<4$ | $\cdots$ | $<10$ | $\cdots$
NGC1566 | $45$ | (5) | $115$ | (11) | $63$ | (4) | $36$ | (3) | $<5$ | $\cdots$ | $30$ | (2) | $59$ | (9) | $9$ | (2) | $12$ | (3)
F04385-0828 | $<8$ | $\cdots$ | $<54$ | $\cdots$ | $12$ | (1) | $<5$ | $\cdots$ | $<4$ | $\cdots$ | $<5$ | $\cdots$ | $<55$ | $\cdots$ | $<3$ | $\cdots$ | $<5$ | $\cdots$
NGC1667 | $63$ | (6) | $43$ | (11) | $270$ | (11) | $53$ | (12) | $<19$ | $\cdots$ | $129$ | (9) | $<42$ | $\cdots$ | $<37$ | $\cdots$ | $131$ | (21)
E33-G2 | $92$ | (7) | $<76$ | $\cdots$ | $7$ | (2) | $14$ | (3) | $6$ | (2) | $<10$ | $\cdots$ | $<77$ | $\cdots$ | $21$ | (3) | $<8$ | $\cdots$
M-5-13-17 | $55$ | (5) | $77$ | (14) | $41$ | (3) | $23$ | (3) | $<8$ | $\cdots$ | $<8$ | $\cdots$ | $48$ | (14) | $11$ | (3) | $<18$ | $\cdots$
MRK6 | $110$ | (6) | $192$ | (24) | $53$ | (2) | $100$ | (5) | $16$ | (2) | $29$ | (3) | $90$ | (24) | $32$ | (2) | $16$ | (2)
MRK79 | $138$ | (5) | $72$ | (19) | $15$ | (2) | $36$ | (3) | $14$ | (2) | $11$ | (3) | $71$ | (20) | $14$ | (2) | $<7$ | $\cdots$
NGC2639 | $54$ | (13) | $239$ | (43) | $189$ | (15) | $76$ | (19) | $<32$ | $\cdots$ | $104$ | (20) | $119$ | (28) | $<66$ | $\cdots$ | $<87$ | $\cdots$
MRK704 | $49$ | (5) | $52$ | (14) | $<4$ | $\cdots$ | $13$ | (2) | $<7$ | $\cdots$ | $<8$ | $\cdots$ | $<41$ | $\cdots$ | $<3$ | $\cdots$ | $<4$ | $\cdots$
NGC2992 | $230$ | (7) | $200$ | (19) | $77$ | (4) | $88$ | (4) | $28$ | (3) | $57$ | (3) | $132$ | (20) | $44$ | (2) | $34$ | (3)
MRK1239 | $20$ | (4) | $<64$ | $\cdots$ | $5$ | (2) | $12$ | (2) | $<3$ | $\cdots$ | $<6$ | $\cdots$ | $<65$ | $\cdots$ | $<2$ | $\cdots$ | $<4$ | $\cdots$
NGC3079 | $39$ | (8) | $166$ | (16) | $352$ | (22) | $44$ | (11) | $<19$ | $\cdots$ | $44$ | (7) | $42$ | (11) | $61$ | (15) | $120$ | (10)
NGC3227 | $76$ | (7) | $112$ | (23) | $85$ | (3) | $65$ | (3) | $14$ | (2) | $26$ | (3) | $<59$ | $\cdots$ | $30$ | (4) | $27$ | (3)
NGC3511 | $<31$ | $\cdots$ | $489$ | (35) | $257$ | (13) | $<42$ | $\cdots$ | $<19$ | $\cdots$ | $132$ | (11) | $188$ | (35) | $<48$ | $\cdots$ | $<76$ | $\cdots$
NGC3516 | $165$ | (6) | $58$ | (11) | $10$ | (2) | $32$ | (3) | $12$ | (2) | $14$ | (2) | $61$ | (11) | $21$ | (2) | $<6$ | $\cdots$
M+0-29-23 | $<10$ | $\cdots$ | $105$ | (13) | $130$ | (6) | $21$ | (4) | $<8$ | $\cdots$ | $35$ | (4) | $<44$ | $\cdots$ | $<20$ | $\cdots$ | $48$ | (7)
NGC3660 | $60$ | (13) | $105$ | (24) | $83$ | (11) | $62$ | (14) | $<27$ | $\cdots$ | $<40$ | $\cdots$ | $<88$ | $\cdots$ | $56$ | (13) | $<118$ | $\cdots$
NGC3982 | $23$ | (5) | $303$ | (19) | $184$ | (10) | $40$ | (9) | $<18$ | $\cdots$ | $78$ | (6) | $139$ | (19) | $47$ | (10) | $94$ | (25)
NGC4051 | $56$ | (6) | $79$ | (25) | $18$ | (2) | $12$ | (2) | $<3$ | $\cdots$ | $10$ | (3) | $<39$ | $\cdots$ | $5$ | (1) | $6$ | (2)
UGC7064 | $76$ | (9) | $<170$ | $\cdots$ | $61$ | (4) | $28$ | (5) | $24$ | (5) | $38$ | (6) | $<170$ | $\cdots$ | $<17$ | $\cdots$ | $<41$ | $\cdots$
NGC4151 | $108$ | (6) | $119$ | (29) | $38$ | (2) | $42$ | (2) | $9$ | (2) | $23$ | (2) | $<56$ | $\cdots$ | $20$ | (1) | $11$ | (1)
MRK766 | $62$ | (5) | $<35$ | $\cdots$ | $27$ | (2) | $27$ | (4) | $15$ | (3) | $12$ | (3) | $51$ | (13) | $12$ | (2) | $6$ | (1)
NGC4388 | $305$ | (7) | $138$ | (24) | $82$ | (4) | $104$ | (7) | $42$ | (3) | $45$ | (2) | $81$ | (17) | $106$ | (5) | $21$ | (2)
NGC4501 | $68$ | (12) | $387$ | (34) | $109$ | (14) | $127$ | (12) | $<24$ | $\cdots$ | $55$ | (15) | $84$ | (27) | $<49$ | $\cdots$ | $<26$ | $\cdots$
NGC4579 | $69$ | (5) | $350$ | (23) | $123$ | (8) | $72$ | (3) | $<5$ | $\cdots$ | $34$ | (3) | $56$ | (11) | $9$ | (2) | $44$ | (4)
NGC4593 | $22$ | (5) | $<44$ | $\cdots$ | $10$ | (2) | $<6$ | $\cdots$ | $<5$ | $\cdots$ | $<6$ | $\cdots$ | $<45$ | $\cdots$ | $<2$ | $\cdots$ | $<5$ | $\cdots$
NGC4594 | $70$ | (5) | $471$ | (36) | $81$ | (4) | $144$ | (6) | $<5$ | $\cdots$ | $72$ | (5) | $170$ | (41) | $<5$ | $\cdots$ | $10$ | (3)
TOL1238-364 | $<13$ | $\cdots$ | $<60$ | $\cdots$ | $59$ | (2) | $24$ | (3) | $<4$ | $\cdots$ | $14$ | (2) | $<55$ | $\cdots$ | $11$ | (2) | $22$ | (3)
NGC4602 | $<21$ | $\cdots$ | $249$ | (29) | $173$ | (12) | $<17$ | $\cdots$ | $<18$ | $\cdots$ | $98$ | (11) | $165$ | (29) | $35$ | (10) | $105$ | (34)
M-2-33-34 | $508$ | (23) | $297$ | (20) | $86$ | (5) | $190$ | (7) | $79$ | (5) | $82$ | (5) | $196$ | (23) | $135$ | (5) | $35$ | (9)
NGC4941 | $133$ | (8) | $153$ | (21) | $95$ | (6) | $109$ | (4) | $24$ | (5) | $49$ | (5) | $<59$ | $\cdots$ | $87$ | (8) | $35$ | (4)
NGC4968 | $67$ | (6) | $62$ | (14) | $36$ | (2) | $35$ | (2) | $13$ | (2) | $9$ | (2) | $51$ | (12) | $17$ | (2) | $10$ | (3)
NGC5005 | $47$ | (8) | $204$ | (23) | $226$ | (13) | $83$ | (7) | $<15$ | $\cdots$ | $31$ | (6) | $<38$ | $\cdots$ | $<24$ | $\cdots$ | $23$ | (4)
NGC5033 | $99$ | (5) | $438$ | (22) | $204$ | (9) | $66$ | (6) | $14$ | (3) | $82$ | (4) | $122$ | (26) | $<15$ | $\cdots$ | $38$ | (5)
NGC5135 | $61$ | (5) | $115$ | (14) | $207$ | (9) | $76$ | (4) | $17$ | (4) | $43$ | (2) | $78$ | (16) | $48$ | (4) | $85$ | (8)
M-6-30-15 | $38$ | (5) | $<40$ | $\cdots$ | $9$ | (1) | $<5$ | $\cdots$ | $<4$ | $\cdots$ | $<6$ | $\cdots$ | $<41$ | $\cdots$ | $7$ | (2) | $4$ | (1)
NGC5256 | $147$ | (8) | $133$ | (16) | $283$ | (14) | $144$ | (6) | $24$ | (7) | $87$ | (5) | $86$ | (15) | $119$ | (9) | $133$ | (16)
IC4329A | $140$ | (8) | $63$ | (20) | $10$ | (2) | $26$ | (3) | $17$ | (2) | $<4$ | $\cdots$ | $74$ | (19) | $12$ | (2) | $1$ | (1)
NGC5347 | $<13$ | $\cdots$ | $<52$ | $\cdots$ | $11$ | (2) | $<5$ | $\cdots$ | $<3$ | $\cdots$ | $<6$ | $\cdots$ | $<54$ | $\cdots$ | $5$ | (1) | $18$ | (3)
NGC5506 | $151$ | (6) | $96$ | (14) | $34$ | (2) | $56$ | (4) | $15$ | (2) | $28$ | (2) | $56$ | (14) | $45$ | (2) | $7$ | (1)
NGC5548 | $29$ | (4) | $<77$ | $\cdots$ | $19$ | (2) | $20$ | (3) | $<6$ | $\cdots$ | $10$ | (3) | $<78$ | $\cdots$ | $<7$ | $\cdots$ | $<9$ | $\cdots$
MRK817 | $<9$ | $\cdots$ | $<31$ | $\cdots$ | $13$ | (2) | $11$ | (2) | $<7$ | $\cdots$ | $<6$ | $\cdots$ | $<33$ | $\cdots$ | $<6$ | $\cdots$ | $<9$ | $\cdots$
NGC5929 | $102$ | (12) | $338$ | (38) | $224$ | (21) | $212$ | (21) | $<40$ | $\cdots$ | $134$ | (14) | $139$ | (35) | $<72$ | $\cdots$ | $89$ | (23)
NGC5953 | $56$ | (4) | $344$ | (25) | $417$ | (19) | $76$ | (12) | $17$ | (5) | $139$ | (5) | $149$ | (26) | $<32$ | $\cdots$ | $109$ | (10)
M-2-40-4 | $36$ | (5) | $61$ | (16) | $25$ | (2) | $23$ | (3) | $8$ | (2) | $9$ | (2) | $<50$ | $\cdots$ | $<5$ | $\cdots$ | $<4$ | $\cdots$
F15480-0344 | $146$ | (6) | $<56$ | $\cdots$ | $20$ | (2) | $38$ | (4) | $32$ | (4) | $<8$ | $\cdots$ | $<56$ | $\cdots$ | $22$ | (3) | $<12$ | $\cdots$
NGC6810 | $<11$ | $\cdots$ | $130$ | (16) | $151$ | (7) | $<10$ | $\cdots$ | $<5$ | $\cdots$ | $39$ | (2) | $65$ | (15) | $<8$ | $\cdots$ | $64$ | (6)
NGC6860 | $69$ | (6) | $122$ | (34) | $19$ | (2) | $22$ | (3) | $9$ | (3) | $11$ | (3) | $<98$ | $\cdots$ | $<5$ | $\cdots$ | $<9$ | $\cdots$
NGC6890 | $36$ | (6) | $95$ | (14) | $57$ | (3) | $19$ | (4) | $10$ | (3) | $30$ | (3) | $67$ | (11) | $<16$ | $\cdots$ | $13$ | (4)
IC5063 | $55$ | (6) | $<62$ | $\cdots$ | $9$ | (2) | $27$ | (2) | $<4$ | $\cdots$ | $7$ | (2) | $<59$ | $\cdots$ | $26$ | (2) | $3$ | (1)
UGC11680 | $22$ | (6) | $<125$ | $\cdots$ | $32$ | (4) | $33$ | (6) | $<12$ | $\cdots$ | $<12$ | $\cdots$ | $<67$ | $\cdots$ | $<23$ | $\cdots$ | $<39$ | $\cdots$
NGC7130 | $12$ | (4) | $79$ | (13) | $166$ | (5) | $42$ | (4) | $16$ | (3) | $28$ | (2) | $62$ | (14) | $22$ | (3) | $107$ | (13)
NGC7172 | $198$ | (6) | $191$ | (14) | $45$ | (3) | $50$ | (4) | $16$ | (3) | $38$ | (4) | $99$ | (12) | $66$ | (7) | $18$ | (4)
NGC7213 | $<16$ | $\cdots$ | $173$ | (26) | $49$ | (3) | $29$ | (2) | $<5$ | $\cdots$ | $8$ | (3) | $<77$ | $\cdots$ | $<6$ | $\cdots$ | $17$ | (3)
NGC7314 | $357$ | (14) | $90$ | (20) | $33$ | (4) | $110$ | (7) | $47$ | (4) | $48$ | (5) | $75$ | (15) | $108$ | (6) | $35$ | (3)
M-3-58-7 | $<13$ | $\cdots$ | $<51$ | $\cdots$ | $16$ | (2) | $12$ | (2) | $10$ | (2) | $<6$ | $\cdots$ | $<55$ | $\cdots$ | $5$ | (1) | $9$ | (3)
NGC7469 | $<7$ | $\cdots$ | $66$ | (14) | $109$ | (3) | $11$ | (3) | $9$ | (2) | $25$ | (2) | $64$ | (15) | $9$ | (2) | $51$ | (4)
NGC7496 | $<10$ | $\cdots$ | $<62$ | $\cdots$ | $110$ | (8) | $7$ | (2) | $<6$ | $\cdots$ | $40$ | (3) | $49$ | (12) | $15$ | (5) | $88$ | (9)
NGC7582 | $72$ | (5) | $77$ | (18) | $146$ | (7) | $52$ | (3) | $12$ | (3) | $41$ | (3) | $48$ | (15) | $39$ | (3) | $43$ | (3)
NGC7590 | $45$ | (7) | $504$ | (21) | $164$ | (9) | $68$ | (14) | $<26$ | $\cdots$ | $127$ | (11) | $253$ | (24) | $<46$ | $\cdots$ | $<83$ | $\cdots$
NGC7603 | $14$ | (4) | $133$ | (33) | $27$ | (2) | $21$ | (3) | $<6$ | $\cdots$ | $<8$ | $\cdots$ | $64$ | (16) | $<3$ | $\cdots$ | $<5$ | $\cdots$
NGC7674 | $67$ | (6) | $<53$ | $\cdots$ | $28$ | (2) | $37$ | (2) | $17$ | (2) | $18$ | (2) | $<54$ | $\cdots$ | $15$ | (2) | $15$ | (2)
CGCG381-051 | $<16$ | $\cdots$ | $71$ | (20) | $96$ | (6) | $<12$ | $\cdots$ | $8$ | (2) | $37$ | (4) | $<60$ | $\cdots$ | $<9$ | $\cdots$ | $54$ | (17)
Note. — Equivalent widths are given in units of nm. Uncertainties ($1\sigma$)
are listed in parentheses. Upper limits are $3\sigma$. Uncertainties
($1\sigma$) are listed in parentheses. Upper limits are $3\sigma$.
Table 10: Dust Opacities and MIPS Aperture Corrections determined by PAHFIT
Source ID | Model | $\tau_{10}$ | Ap. Corr.
---|---|---|---
(1) | (2) | (3) | (4)
MRK335 | Screen | 0.10 | (0.07) | 1.00 | $\cdots$
MRK938 | Screen | 0.91 | (0.03) | 1.27 | (0.02)
E12-G21 | Mixed | 1.21 | (0.26) | 1.00 | $\cdots$
MRK348 | Screen | 0.22 | (0.08) | 1.66 | (0.06)
NGC424 | $\cdots$ | 0.00 | $\cdots$ | 1.14 | (0.07)
NGC526A | $\cdots$ | 0.00 | $\cdots$ | 1.81 | $\cdots$
NGC513 | Screen | 0.24 | (0.02) | 1.23 | (0.02)
F01475-0740 | Screen | 0.09 | (0.10) | 1.00 | $\cdots$
NGC931 | $\cdots$ | 0.00 | $\cdots$ | 1.24 | (0.03)
NGC1056 | $\cdots$ | 0.00 | $\cdots$ | 1.38 | (0.02)
NGC1097 | Screen | 0.04 | (0.08) | 1.81 | $\cdots$
NGC1125 | Mixed | 1.79 | (0.53) | 1.40 | (0.05)
NGC1143-4 | Screen | 0.70 | (0.02) | 1.00 | $\cdots$
M-2-8-39 | Mixed | 4.01 | (0.52) | 1.03 | (0.04)
NGC1194 | Screen | 2.35 | (0.06) | 1.14 | (0.08)
NGC1241 | Mixed | 0.31 | (0.09) | 1.81 | $\cdots$
NGC1320 | Screen | 0.69 | (0.13) | 1.24 | (0.05)
NGC1365 | Mixed | 0.83 | (0.07) | 1.30 | (0.02)
NGC1386 | Mixed | 9.41 | (1.44) | 1.23 | (0.06)
F03450+0055 | Screen | 0.58 | (0.11) | $\cdots$ | $\cdots$
NGC1566 | $\cdots$ | 0.00 | $\cdots$ | 1.42 | (0.07)
F04385-0828 | Screen | 1.67 | (0.03) | 1.14 | (0.03)
NGC1667 | Mixed | 0.09 | (0.08) | 1.81 | $\cdots$
E33-G2 | Screen | 0.34 | (0.09) | 1.46 | (0.07)
M-5-13-17 | Screen | 0.01 | (0.04) | 1.22 | (0.05)
MRK6 | $\cdots$ | 0.00 | (0.01) | 1.00 | $\cdots$
MRK79 | Screen | 0.41 | (0.11) | 1.14 | (0.04)
NGC2639 | $\cdots$ | 0.00 | $\cdots$ | 1.81 | $\cdots$
MRK704 | Screen | 1.07 | (0.08) | 1.07 | (0.04)
NGC2992 | Mixed | 0.52 | (0.11) | 1.62 | (0.05)
MRK1239 | Screen | 0.64 | (0.17) | 1.29 | (0.05)
NGC3079 | Mixed | 6.19 | (0.19) | $\cdots$ | $\cdots$
NGC3227 | $\cdots$ | 0.00 | (0.05) | 1.12 | (0.02)
NGC3511 | Mixed | 0.20 | (0.09) | 1.81 | $\cdots$
NGC3516 | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
M+0-29-23 | Mixed | 0.26 | (0.17) | 1.21 | (0.09)
NGC3660 | $\cdots$ | 0.00 | $\cdots$ | 1.39 | (0.11)
NGC3982 | Mixed | 0.19 | (0.07) | 1.81 | $\cdots$
NGC4051 | Mixed | 1.33 | (0.28) | 1.33 | (0.06)
UGC7064 | $\cdots$ | 0.00 | $\cdots$ | 1.63 | (0.12)
NGC4151 | $\cdots$ | 0.00 | $\cdots$ | 1.01 | (0.02)
MRK766 | $\cdots$ | 0.00 | $\cdots$ | 1.26 | (0.04)
NGC4388 | Mixed | 3.32 | (0.13) | 1.55 | (0.02)
NGC4501 | Mixed | 0.62 | (0.10) | 1.81 | $\cdots$
NGC4579 | $\cdots$ | 0.00 | $\cdots$ | 1.24 | (0.02)
NGC4593 | Screen | 0.35 | (0.21) | 1.65 | (0.09)
NGC4594 | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
NGC4602 | $\cdots$ | 0.00 | (0.09) | 1.81 | $\cdots$
TOL1238-364 | Mixed | 1.34 | (0.14) | 1.14 | (0.02)
M-2-33-34 | Mixed | 1.30 | (0.07) | 1.11 | (0.03)
NGC4941 | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
NGC4968 | Screen | 0.17 | (0.11) | 1.06 | (0.05)
NGC5005 | Mixed | 0.96 | (0.06) | 1.70 | (0.04)
NGC5033 | Mixed | 0.54 | (0.04) | 1.00 | $\cdots$
NGC5135 | $\cdots$ | 0.00 | $\cdots$ | 1.38 | (0.08)
M-6-30-15 | Mixed | 1.50 | (0.32) | 1.00 | $\cdots$
NGC5256 | Mixed | 1.17 | (0.08) | 1.25 | (0.03)
IC4329A | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
NGC5347 | Screen | 1.11 | (0.19) | 1.45 | (0.08)
NGC5506 | Screen | 2.16 | (0.04) | 1.20 | (0.03)
NGC5548 | Screen | 0.56 | (0.11) | 1.00 | $\cdots$
MRK817 | Screen | 0.05 | (0.07) | 1.25 | (0.05)
NGC5929 | Mixed | 0.08 | (0.14) | 1.81 | $\cdots$
NGC5953 | Screen | 0.09 | (0.02) | $\cdots$ | $\cdots$
M-2-40-4 | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
F15480-0344 | Screen | 1.10 | (0.09) | 1.00 | $\cdots$
NGC6810 | Screen | 0.02 | (0.05) | 1.31 | (0.02)
NGC6860 | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
NGC6890 | $\cdots$ | 0.00 | $\cdots$ | 1.34 | (0.03)
IC5063 | Screen | 0.74 | (0.09) | 1.00 | $\cdots$
UGC11680 | Mixed | 0.30 | (0.22) | 1.05 | (0.06)
NGC7130 | Mixed | 0.08 | (0.04) | 1.34 | (0.02)
NGC7172 | Screen | 2.65 | (0.04) | 1.23 | (0.04)
NGC7213 | $\cdots$ | 0.00 | $\cdots$ | 1.03 | (0.06)
NGC7314 | Screen | 1.17 | (0.15) | 1.81 | $\cdots$
M-3-58-7 | $\cdots$ | 0.00 | $\cdots$ | 1.00 | $\cdots$
NGC7469 | $\cdots$ | 0.00 | $\cdots$ | 1.26 | (0.02)
NGC7496 | Mixed | 0.24 | (0.04) | 1.35 | (0.03)
NGC7582 | Screen | 1.39 | (0.06) | 1.22 | (0.03)
NGC7590 | $\cdots$ | 0.00 | $\cdots$ | 1.81 | $\cdots$
NGC7603 | Screen | 0.75 | (0.23) | 1.00 | $\cdots$
NGC7674 | Screen | 0.97 | (0.12) | 1.21 | (0.04)
CGCG381-051 | $\cdots$ | 0.00 | $\cdots$ | 1.44 | (0.05)
Note. — (1) Source ID. (2) Dust extinction model: Screen models place all of
the obscuring dust between the observer and the infrared continuum sources,
and Mixed models assume that the obscuring dust is uniformly mixed with the
infrared continuum sources. (3) Opacity of the obscuring dust component at 10
$\mu$m; this opacity affects all of the continuum model components except for
the optically-thin hot dust component. (4) The aperture correction for the
MIPS-SED data; the MIPS-SED data need to be divided by this correction factor
to reconcile the model fit with the smaller IRS and IRAC apertures. The
correction factor was restricted to the range 1 – 1.81 during the fitting
procedure.
Table 11: Silicate Strengths
Source ID | $S_{10}$ | $S_{18}$
---|---|---
MRK335 | $0.295$ | (0.028) | $0.089$ | (0.009)
MRK938 | $-0.536$ | (0.046) | $-0.268$ | (0.009)
E12-G21 | $0.036$ | (0.050) | $-0.001$ | (0.019)
MRK348 | $-0.120$ | (0.022) | $0.079$ | (0.011)
NGC424 | $0.111$ | (0.018) | $0.085$ | (0.011)
NGC526A | $0.252$ | (0.031) | $0.098$ | (0.011)
NGC513 | $0.030$ | (0.067) | $0.089$ | (0.095)
F01475-0740 | $0.143$ | (0.017) | $0.163$ | (0.007)
NGC931 | $0.013$ | (0.010) | $0.013$ | (0.007)
NGC1056 | $0.061$ | (0.030) | $-0.050$ | (0.006)
NGC1097 | $0.207$ | (0.054) | $0.015$ | (0.014)
NGC1125 | $-0.458$ | (0.070) | $-0.114$ | (0.016)
NGC1143-4 | $-0.465$ | (0.042) | $-0.144$ | (0.009)
M-2-8-39 | $0.014$ | (0.021) | $0.080$ | (0.012)
NGC1194 | $-0.797$ | (0.010) | $-0.145$ | (0.012)
NGC1241 | $-0.104$ | (0.066) | $-0.053$ | (0.018)
NGC1320 | $-0.020$ | (0.028) | $0.036$ | (0.010)
NGC1365 | $-0.169$ | (0.019) | $-0.066$ | (0.012)
NGC1386 | $-0.534$ | (0.023) | $-0.079$ | (0.013)
F03450+0055 | $0.328$ | (0.024) | $0.068$ | (0.011)
NGC1566 | $-0.045$ | (0.020) | $-0.015$ | (0.006)
F04385-0828 | $-0.769$ | (0.017) | $-0.102$ | (0.011)
NGC1667 | $0.049$ | (0.058) | $-0.091$ | (0.009)
E33-G2 | $0.109$ | (0.028) | $0.013$ | (0.008)
M-5-13-17 | $0.007$ | (0.033) | $-0.013$ | (0.013)
MRK6 | $0.137$ | (0.015) | $0.197$ | (0.009)
MRK79 | $0.175$ | (0.034) | $0.052$ | (0.014)
NGC2639 | $0.065$ | (0.088) | $-0.002$ | (0.033)
MRK704 | $0.082$ | (0.020) | $-0.016$ | (0.012)
NGC2992 | $0.042$ | (0.023) | $-0.022$ | (0.010)
MRK1239 | $0.238$ | (0.021) | $0.135$ | (0.014)
NGC3079 | $-0.923$ | (0.058) | $-0.561$ | (0.012)
NGC3227 | $-0.032$ | (0.027) | $0.052$ | (0.015)
NGC3511 | $0.015$ | (0.076) | $-0.109$ | (0.012)
NGC3516 | $-0.007$ | (0.012) | $0.052$ | (0.006)
M+0-29-23 | $-0.366$ | (0.029) | $-0.187$ | (0.012)
NGC3660 | $0.315$ | (0.075) | $0.194$ | (0.042)
NGC3982 | $-0.065$ | (0.056) | $0.007$ | (0.014)
NGC4051 | $0.030$ | (0.018) | $0.049$ | (0.011)
UGC7064 | $0.044$ | (0.040) | $0.086$ | (0.018)
NGC4151 | $0.117$ | (0.009) | $0.145$ | (0.016)
MRK766 | $0.167$ | (0.024) | $0.038$ | (0.010)
NGC4388 | $-0.603$ | (0.015) | $-0.089$ | (0.010)
NGC4501 | $-0.202$ | (0.065) | $-0.122$ | (0.016)
NGC4579 | $0.400$ | (0.036) | $0.105$ | (0.010)
NGC4593 | $0.229$ | (0.074) | $0.080$ | (0.036)
NGC4594 | $0.347$ | (0.016) | $0.196$ | (0.013)
TOL1238-364 | $-0.127$ | (0.016) | $0.033$ | (0.011)
NGC4602 | $0.095$ | (0.088) | $-0.049$ | (0.011)
M-2-33-34 | $-0.039$ | (0.018) | $-0.020$ | (0.006)
NGC4941 | $-0.026$ | (0.043) | $-0.016$ | (0.015)
NGC4968 | $-0.071$ | (0.024) | $0.056$ | (0.008)
NGC5005 | $-0.320$ | (0.037) | $-0.108$ | (0.011)
NGC5033 | $-0.127$ | (0.026) | $-0.108$ | (0.008)
NGC5135 | $-0.183$ | (0.016) | $-0.056$ | (0.009)
M-6-30-15 | $0.143$ | (0.021) | $0.049$ | (0.006)
NGC5256 | $-0.352$ | (0.070) | $-0.137$ | (0.011)
IC4329A | $0.114$ | (0.017) | $0.042$ | (0.014)
NGC5347 | $-0.069$ | (0.021) | $0.019$ | (0.007)
NGC5506 | $-0.703$ | (0.034) | $-0.046$ | (0.070)
NGC5548 | $0.136$ | (0.021) | $0.081$ | (0.007)
MRK817 | $0.271$ | (0.033) | $0.071$ | (0.011)
NGC5929 | $-0.051$ | (0.146) | $0.013$ | (0.032)
NGC5953 | $0.015$ | (0.027) | $-0.079$ | (0.010)
M-2-40-4 | $-0.221$ | (0.014) | $0.032$ | (0.007)
F15480-0344 | $-0.009$ | (0.025) | $0.095$ | (0.007)
NGC6810 | $0.034$ | (0.025) | $0.105$ | (0.023)
NGC6860 | $0.171$ | (0.019) | $0.054$ | (0.007)
NGC6890 | $-0.037$ | (0.015) | $0.018$ | (0.009)
IC5063 | $-0.237$ | (0.014) | $-0.014$ | (0.010)
UGC11680 | $0.067$ | (0.038) | $0.087$ | (0.018)
NGC7130 | $-0.036$ | (0.028) | $-0.001$ | (0.010)
NGC7172 | $-1.679$ | (0.008) | $-0.387$ | (0.012)
NGC7213 | $0.587$ | (0.023) | $0.236$ | (0.007)
NGC7314 | $-0.205$ | (0.056) | $-0.118$ | (0.018)
M-3-58-7 | $0.189$ | (0.027) | $0.021$ | (0.010)
NGC7469 | $0.068$ | (0.018) | $0.048$ | (0.010)
NGC7496 | $-0.091$ | (0.037) | $-0.024$ | (0.008)
NGC7582 | $-0.657$ | (0.026) | $-0.156$ | (0.010)
NGC7590 | $0.213$ | (0.085) | $0.086$ | (0.068)
NGC7603 | $0.205$ | (0.020) | $0.033$ | (0.015)
NGC7674 | $-0.098$ | (0.019) | $0.025$ | (0.012)
CGCG381-051 | $0.381$ | (0.018) | $0.126$ | (0.006)
Note. — Sil 10 $\mu$m and 18 $\mu$m strengths. Sil absorption is indicated by
negative strengths, emission, positive. Formal statistical uncertainties are
given in parentheses.
Table 12: Infrared Continuum Spectral Indices
Source ID | $\alpha$(20–30 $\mu$m) | $\alpha$(55-90 $\mu$m)
---|---|---
MRK335 | $-$1.75 | (0.03) | $-$1.49 | (0.29)
MRK938 | 1.58 | (0.02) | $-$2.03 | (0.03)
E12-G21 | $-$0.62 | (0.02) | $-$0.59 | (0.10)
MRK348 | $-$1.87 | (0.01) | $-$2.62 | (0.23)
NGC424 | $-$2.23 | (0.02) | $-$2.60 | (0.11)
NGC526A | $-$2.91 | (0.03) | $-$1.50 | (0.44)
NGC513 | $-$0.40 | (0.03) | $-$0.72 | (0.06)
F01475-0740 | $-$1.36 | (0.03) | $-$2.23 | (0.19)
NGC931 | $-$1.45 | (0.02) | $-$1.61 | (0.07)
NGC1056 | 0.62 | (0.03) | $-$1.08 | (0.06)
NGC1097 | 0.69 | (0.01) | $-$1.05 | (0.02)
NGC1125 | 0.21 | (0.03) | $-$2.08 | (0.06)
NGC1143-4 | 0.62 | (0.01) | $-$0.60 | (0.05)
M-2-8-39 | $-$2.36 | (0.02) | $-$2.80 | (0.43)
NGC1194 | $-$1.13 | (0.02) | $-$2.82 | (0.20)
NGC1241 | 0.00 | (0.03) | $-$0.47 | (0.06)
NGC1320 | $-$1.06 | (0.02) | $-$1.79 | (0.06)
NGC1365 | 0.59 | (0.01) | $-$1.05 | (0.03)
NGC1386 | $-$0.45 | (0.02) | $-$1.52 | (0.03)
F03450+0055 | $-$1.89 | (0.03) | $\cdots$ | $\cdots$
NGC1566 | $-$0.50 | (0.02) | $-$0.29 | (0.07)
F04385-0828 | $-$0.50 | (0.03) | $-$2.85 | (0.07)
NGC1667 | 0.42 | (0.03) | $-$0.37 | (0.06)
E33-G2 | $-$1.96 | (0.02) | $-$1.39 | (0.21)
M-5-13-17 | $-$0.98 | (0.03) | $-$1.30 | (0.12)
MRK6 | $-$1.72 | (0.02) | $-$1.95 | (0.14)
MRK79 | $-$1.32 | (0.02) | $-$1.54 | (0.12)
NGC2639 | 0.04 | (0.06) | 0.36 | (0.07)
MRK704 | $-$2.29 | (0.02) | $-$3.87 | (0.38)
NGC2992 | $-$0.73 | (0.03) | $-$1.19 | (0.04)
MRK1239 | $-$1.82 | (0.02) | $-$2.45 | (0.11)
NGC3079 | 2.45 | (0.04) | $\cdots$ | $\cdots$
NGC3227 | $-$0.83 | (0.02) | $-$1.11 | (0.04)
NGC3511 | 0.24 | (0.05) | $-$0.12 | (0.06)
NGC3516 | $-$1.35 | (0.02) | $-$1.97 | (0.11)
M+0-29-23 | 0.41 | (0.01) | $-$0.98 | (0.04)
NGC3660 | 0.11 | (0.04) | $-$0.79 | (0.15)
NGC3982 | $-$0.02 | (0.04) | $-$0.59 | (0.07)
NGC4051 | $-$1.26 | (0.03) | $-$1.16 | (0.08)
UGC7064 | $-$0.75 | (0.04) | $-$1.05 | (0.12)
NGC4151 | $-$2.14 | (0.02) | $-$2.50 | (0.05)
MRK766 | $-$0.81 | (0.02) | $-$2.03 | (0.04)
NGC4388 | $-$0.53 | (0.03) | $-$1.29 | (0.09)
NGC4501 | $-$0.28 | (0.05) | 0.34 | (0.08)
NGC4579 | $-$0.67 | (0.02) | $-$0.36 | (0.08)
NGC4593 | $-$1.11 | (0.02) | $-$1.33 | (0.09)
NGC4594 | $-$0.74 | (0.06) | $-$0.12 | (0.46)
NGC4602 | 0.04 | (0.05) | $-$0.19 | (0.13)
TOL1238-364 | $-$0.90 | (0.03) | $-$1.51 | (0.06)
M-2-33-34 | $-$0.72 | (0.04) | $-$1.30 | (0.12)
NGC4941 | $-$0.90 | (0.03) | $-$1.33 | (0.14)
NGC4968 | $-$1.47 | (0.02) | $-$1.58 | (0.10)
NGC5005 | 1.45 | (0.03) | $-$0.45 | (0.04)
NGC5033 | 0.10 | (0.02) | $-$0.19 | (0.18)
NGC5135 | 0.44 | (0.02) | $-$0.72 | (0.06)
M-6-30-15 | $-$1.74 | (0.02) | $-$1.67 | (0.16)
NGC5256 | 0.85 | (0.02) | $-$1.39 | (0.04)
IC4329A | $-$2.14 | (0.02) | $-$2.70 | (0.10)
NGC5347 | $-$1.51 | (0.04) | $-$1.67 | (0.12)
NGC5506 | $-$0.67 | (0.02) | $-$2.11 | (0.05)
NGC5548 | $-$1.65 | (0.03) | $-$1.88 | (0.13)
MRK817 | $-$0.80 | (0.02) | $-$2.51 | (0.08)
NGC5929 | 0.45 | (0.07) | $-$1.26 | (0.09)
NGC5953 | 0.51 | (0.03) | $\cdots$ | $\cdots$
M-2-40-4 | $-$0.77 | (0.01) | $-$1.28 | (0.05)
F15480-0344 | $-$1.21 | (0.02) | $-$1.82 | (0.10)
NGC6810 | $-$0.42 | (0.02) | $-$1.07 | (0.03)
NGC6860 | $-$2.06 | (0.02) | $-$0.61 | (0.13)
NGC6890 | $-$0.66 | (0.02) | $-$0.66 | (0.05)
IC5063 | $-$1.01 | (0.02) | $-$2.70 | (0.06)
UGC11680 | $-$1.39 | (0.03) | $-$0.59 | (0.27)
NGC7130 | 0.44 | (0.02) | $-$1.24 | (0.03)
NGC7172 | 0.40 | (0.02) | $-$0.65 | (0.04)
NGC7213 | $-$2.30 | (0.03) | $-$0.76 | (0.10)
NGC7314 | $-$0.77 | (0.02) | $-$0.82 | (0.13)
M-3-58-7 | $-$0.87 | (0.02) | $-$1.69 | (0.05)
NGC7469 | $-$0.11 | (0.01) | $-$1.55 | (0.03)
NGC7496 | 0.47 | (0.02) | $-$1.57 | (0.04)
NGC7582 | 0.71 | (0.02) | $-$1.38 | (0.03)
NGC7590 | 0.10 | (0.03) | $-$0.34 | (0.06)
NGC7603 | $-$1.75 | (0.02) | $-$1.01 | (0.16)
NGC7674 | $-$0.94 | (0.02) | $-$1.66 | (0.04)
CGCG381-051 | $-$0.77 | (0.02) | $-$1.69 | (0.08)
Note. — Continuum spectral indices over the wavelength ranges 20–30 $\mu$m and
55-90 $\mu$m. Formal statistical uncertainties are provided in parentheses.The
convention is $F_{\lambda}\propto\lambda^{\alpha}$.
Figure 1: A comparison of AGN classifications from Rush et al. (1993) and our
revised classifications collected from the literature, Table 1. The top panel
illustrates the re-classification of Rush et al. Type 1 AGNs, and the bottom
panel similarly shows the re-classification of Type 2 AGNs. Seyfert 2s are
split into two groups: (1) HBLR S2s, which are known to harbor an HBLR, and
(2) non-HBLR S2s, in which no HBLR has yet been detected.
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2
Figure 2: Spitzer spectrophotometry of the 12 $\mu$m AGN sample. The SEDs are
plotted as $\lambda F_{\lambda}$ vs. rest $\lambda$ and in order of increasing
RA. IRAC data are plotted as diamonds with horizontal errorbars representing
the camera bandwidth and vertical errorbars indicating a combination of
statistical uncertainties, uncertainty of the extended flux correction, and
uncertainty of the color corrections. IRS data are traced by the solid line;
each camera and order is plotted independently, hence there are gaps appearing
around 15 $\mu$m, which separates the SL and LL spectrographs. Scaled,
staring-mode IRS data are indicated by dash-dot lines. MIPS SED data are show
as vertical error bars, which represent statistical uncertainties of the MIPS-
SED photometry, but do not include the systematic 10-15% uncertainty
associated with the flux calibration. The IRAC and IRS data were extracted
from 20″ diameter synthetic apertures, and the MIPS-SED data were extracted
from 30″ aperture along a 20″ wide slit.
Figure 3: IRAC images of NGC 4151, shown to illustrate the results of our
post-BCD artifact removal technique. Each row corresponds to a single IRAC
detector, stacked in order of increasing wavelength. The left column shows the
S14.0 pipeline image, and the right column shows the matching image after
artifact removal. The images are displayed with a log stretch from 0 to 1 MJy
sr-1 for the 3.6 $\mu$m and 4.5 $\mu$m images and 0 to 20 MJy sr-1 for the 5.8
and 8.0 $\mu$m images, where zero surface brightness has been adjusted roughly
to the image background.
Figure 4: An illustration of IRS optimal spectral extraction for NGC 3079. The
wavelength axis is in the observed frame. The fractional differences between
the optimally-weighted and non-weighted spectra, calculated as a fraction of
the optimally-weighted spectrum, are plotted as open circles. The optimally-
extracted spectrum, $F_{\nu}$, has been scaled to fit the plotting range and
is shown as a thick line. Even though this is an extended source and further
shows extended, strong PAH features, optimal weighting has preserved the shape
and strength of the PAH 6.2 $\mu$m feature. The relative differences typically
fall in the range 2–4%, comparable to the statistical uncertainties of the
non-weighted extraction. In this high signal-to-noise case, optimum extraction
reduced the statistical uncertainties by $\sim 60\%$. Figure 5: An
illustration of IRS optimal spectral extraction. The IRS short-low spectrum of
F01475-0740 is shown. The wavelength axis is in the observed frame. The
optimally-weighted spectrum is plotted as a thick line, and non-weighted
spectrum as a thin line. Notice that the continuum and PAH features retain
their shape with optimal weighting; however, the formal statistical
uncertainties are reduced by a factor of 2–3 over the SL spectral range.
Figure 6: An illustration of spectral fringe reduction. Shown is the extracted
spectrum of MRK 1239, both with (filled circles connected by solid-lines) and
without (dashed-line) fringe reduction. Wavelengths are in the observed frame.
The data shown here are not yet flux-calibrated, as fringe mitigation is
performed prior to flux calibration in our processing. Note that the fringes
were fit to wavelength ranges avoiding spectral lines; the apparent [O IV]
$\lambda$26 $\mu$m line in the uncorrected data is spurious.
Figure 7: Frequency distribution of the flux density ratio $F_{\nu}({\rm
IRS})/F_{\nu}({\rm IRAC})$ at the overlap wavelengths 5.8 $\mu$m (IRAC channel
3 = 5.731 $\mu$m) and 8.0 $\mu$m (IRAC channel 4 = 7.872 $\mu$m). Figure 8:
IRS / IRAC 8 $\mu$m flux ratio vs. IRAC point source fraction. The point
source fraction is the ratio of the point source flux determined by wavelet
convolution to the total flux in the 20″ synthetic aperture. Here, the
subscript “4” refers to IRAC channel 4 = 8 $\mu$m.
Figure 9: Example spectrum decomposition for NGC 4151 (left) and NGC 7213
(right). The vertical axis is in $\nu F_{\nu}$ units. The measurements are
represented by square symbols; the uncertainties are plotted as vertical
error-bars and are usually smaller than the symbol size. Continuous lines
represent the best fit model (green), underlying dust continuum (gray), and
individual PAH features (blue) and emission lines (purple). The broken lines
mark the following components: stars (pink dashed), dust continuum (red dash-
dot-dot-dot), and optically-thin, warm dust emission (black dotted). The
baseline used for Sil index measurements (stars + dust continuum + warm, thin
dust continuum) is shown as the black dash-dot-dash line. Figure 10:
Comparison of the PAHFIT spectrum decomposition analysis ($y$-axis on these
plots) and the local continuum analysis of Wu et al. (2009) ($x$-axis on these
plots). Only data with nearly matching apertures are included in these plots.
The individual comparisons are top left: PAH 6.2 $\mu$m flux, top right: PAH
6.2 $\mu$m eqw, bottom left: flux of the PAH 11.3 $\mu$m which includes the
11.22 $\mu$m and 11.33 $\mu$m PAH features, and bottom right: PAH 11.3 $\mu$m
complex eqw. Data that include upper limits are indicated by open circles, and
detections are marked by filled circles. The solid line indicates loci of
equivalent measurements. On the flux-flux diagrams, shaded regions bisected by
a dashed line mark the average ratio and $\pm 1\,\sigma$ range reported by
Smith et al. (2007b) for galaxies in the SINGS sample. On the eqw diagrams,
the dashed line illustrates a factor of 3 (6.2 $\mu$m) or 2.5 (11.3 $\mu$m)
enhancement for PAHFIT compared to the spline technique; this line is purely
for illustration and is not based on a fit to the data. Notice that the
PAHFIT-derived fluxes and eqws are, on average, systematically higher, because
PAHFIT removes the contamination of neighboring, weak PAH features. Figure 11:
Comparison of Sil 10 $\mu$m feature strengths based on the PAHFIT spectrum
decomposition analysis ($y$-axis) and the spline continuum approximation
($y$-axis) used by Wu et al. (2009). The solid line indicates loci of
equivalent measurements, and the dashed line illustrates a 0.15 dex
enhancement of Sil strengths measured by PAHFIT compared to the spline
technique; this line is purely for illustration and is not based on a fit to
the data. Figure 12: SEDs averaged by optical classification. All SEDs were
normalized to $F$(5–35$\mu$m) prior to averaging. The gray filled regions
indicate the median absolute deviation among objects within that
classification bin. PAH, H2, and fine-structure lines included in the PAHFIT
spectral decomposition are annotated. Figure 13: PAH, H2, and fine-structure
line-subtracted SEDs averaged by optical classification. All SEDs were
normalized to $F$(5–35$\mu$m) prior to averaging. The gray filled regions
indicate the median absolute deviation among objects within that
classification bin. The dashed lines trace a Rayleigh-Jeans continuum
spectrum, representing an approximate spectrum for stellar photospheres,
anchored to 3.6 $\mu$m. Sil features appear in emission for the S1-1.5 & S1n
class but in absorption in the averaged SEDs both the HBLR S2s (S1h and S1i)
and of non-HBLR S2s (S2s where a HBLR has not yet been detected).
|
arxiv-papers
| 2010-01-27T15:33:24 |
2024-09-04T02:49:08.032188
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. F. Gallimore, A. Yzaguirre, J. Jakoboski, M. J. Stevenosky, D. J.\n Axon, S. A. Baum, C. L. Buchanan, M. Elitzur, M. Elvis, C. P. O'Dea, and A.\n Robinson",
"submitter": "Jack F. Gallimore",
"url": "https://arxiv.org/abs/1001.4974"
}
|
1001.4987
|
2010323-334Nancy, France 323 Martin Dyer Leslie Ann Goldberg Markus Jalsenius
David Richerby
# The Complexity of Approximating
Bounded-Degree Boolean #CSP
M. Dyer School of Computing, University of Leeds, Leeds, LS2 9JT, U.K.
M.E.Dyer,D.M.Richerby@leeds.ac.uk , L. A. Goldberg Department of Computer
Science, University of Liverpool, Liverpool, L69 3BX, U.K.
L.A.Goldberg@liverpool.ac.uk , M. Jalsenius Current address: Department of
Computer Science, University of Bristol, Merchant Venturers Building, Woodland
Road, Bristol, BS8 1UB, U.K. M.Jalsenius@bristol.ac.uk and D. M. Richerby
###### Abstract.
The degree of a CSP instance is the maximum number of times that a variable
may appear in the scope of constraints. We consider the approximate counting
problem for Boolean CSPs with bounded-degree instances, for constraint
languages containing the two unary constant relations $\\{0\\}$ and $\\{1\\}$.
When the maximum degree is at least $25$ we obtain a complete classification
of the complexity of this problem. It is exactly solvable in polynomial-time
if every relation in the constraint language is affine. It is equivalent to
the problem of approximately counting independent sets in bipartite graphs if
every relation can be expressed as conjunctions of $\\{0\\}$, $\\{1\\}$ and
binary implication. Otherwise, there is no FPRAS unless
$\mathbf{NP}=\mathbf{RP}$. For lower degree bounds, additional cases arise in
which the complexity is related to the complexity of approximately counting
independent sets in hypergraphs.
###### Key words and phrases:
Boolean constraint satisfaction problem, generalized satisfiability, counting,
approximation algorithms.
###### 1991 Mathematics Subject Classification:
F.2.2, G.2.1
Funded in part by the EPSRC grant “The Complexity of Counting in Constraint
Satisfaction Problems”.
## 1\. Introduction
In the constraint satisfaction problem (CSP), we seek to assign values from
some domain to a set of variables, while satisfying given constraints on the
combinations of values that certain subsets of the variables may take.
Constraint satisfaction problems are ubiquitous in computer science, with
close connections to graph theory, database query evaluation, type inference,
satisfiability, scheduling and artificial intelligence [20, 22, 25]. CSP can
also be reformulated in terms of homomorphisms between relational structures
[14] and conjunctive query containment in database theory [20]. Weighted
versions of CSP appear in statistical physics, where they correspond to
partition functions of spin systems [31].
We give formal definitions in Section 2 but, for now, consider an undirected
graph $G$ and the CSP where the domain is
$\\{\mathrm{red},\mathrm{green},\mathrm{blue}\\}$, the variables are the
vertices of $G$ and the constraints specify that, for every edge $xy\in G$,
$x$ and $y$ must be assigned different values. Thus, in a satisfying
assignment, no two adjacent vertices are given the same colour: the CSP is
satisfiable if, and only if, the graph is 3-colourable. As a second example,
given a formula in 3-CNF, we can write a system of constraints over the
variables, with domain $\\{\mathrm{true},\mathrm{false}\\}$, that requires the
assignment to each clause to satisfy at least one literal. Clearly, the
resulting CSP is directly equivalent to the original satisfiability problem.
### 1.1. Decision CSP
In the _uniform constraint satisfaction problem_ , we are given the set of
constraints explicitly, as lists of allowable combinations for given subsets
of the variables; these lists can be considered as relations over the domain.
Since it includes problems such as 3-sat and 3-colourability, uniform CSP is
$\mathbf{NP}$-complete. However, uniform CSP also includes problems in
$\mathbf{P}$, such as 2-sat and 2-colourability, raising the natural question
of what restrictions lead to tractable problems. There are two natural ways to
restrict CSP: we can restrict the form of the instances and we can restrict
the form of the constraints.
The most common restriction to CSP is to allow only certain fixed relations in
the constraints. The list of allowed relations is known as the _constraint
language_ and we write $\mathrm{CSP}(\Gamma)$ for the so-called _non-uniform_
CSP in which each constraint states that the values assigned to some tuple of
variables must be a tuple in a specified relation in $\Gamma$.
The classic example of this is Schaefer’s dichotomy for Boolean constraint
languages $\Gamma$ (i.e., those with domain $\\{0,1\\}$; often called
“generalized satisfiability”) [26]. He showed that $\mathrm{CSP}(\Gamma)$ is
in $\mathbf{P}$ if $\Gamma$ is included in one of six classes and is
$\mathbf{NP}$-complete, otherwise. More recently, Bulatov has produced a
corresponding dichotomy for the three-element domain [2]. These two results
restrict the size of the domain but allow relations of arbitrary arity in the
constraint language. The converse restriction — relations of restricted arity,
especially binary relations, over arbitrary finite domains — has also been
studied in depth [16, 17].
For all $\Gamma$ studied so far, $\mathrm{CSP}(\Gamma)$ has been either in
$\mathbf{P}$ or $\mathbf{NP}$-complete and Feder and Vardi have conjectured
that this holds for every constraint language [14]. Ladner has shown that it
is not the case that every problem in $\mathbf{NP}$ is either in $\mathbf{P}$
or $\mathbf{NP}$-complete since, if $\mathbf{P}{}\neq\mathbf{NP}{}$, there is
an infinite, strict hierarchy between the two [23]. However, there are
problems in $\mathbf{NP}$, such as graph Hamiltonicity and even connectedness,
that cannot be expressed as $\mathrm{CSP}(\Gamma)$ for any finite
$\Gamma\,$111This follows from results on the expressive power of existential
monadic second-order logic [12]. and Ladner’s diagonalization does not seem to
be expressible in CSP [14], so a dichotomy for CSP appears possible.
Restricting the tree-width of instances has also been a fruitful direction of
research [15, 21]. In contrast, little is known about restrictions on the
degree of instances, i.e., the maximum number of times that any variable may
appear. Dalmau and Ford have shown that, for any fixed Boolean constraint
language $\Gamma$ containing the constant unary relations
$R_{\mathrm{zero}}=\\{0\\}$ and $R_{\mathrm{one}}=\\{1\\}$, the complexity of
$\mathrm{CSP}(\Gamma)$ for instances of degree at most three is exactly the
same as the complexity of $\mathrm{CSP}(\Gamma)$ with no degree restriction
[6]. The case where variables may appear at most twice has not yet been
completely classified; it is known that degree-2 $\mathrm{CSP}(\Gamma)$ is as
hard as general $\mathrm{CSP}(\Gamma)$ whenever $\Gamma$ contains
$R_{\mathrm{zero}}$ and $R_{\mathrm{one}}$ and some relation that is not a
$\Delta$-matroid [13]; the known polynomial-time cases come from restrictions
on the kinds of $\Delta$-matroids that appear in $\Gamma$ [6].
### 1.2. Counting CSP
A generalization of classical CSP is to ask how many satisfying solutions
there are. This is referred to as counting CSP, $\\#\mathrm{CSP}$. Clearly,
the decision problem is reducible to counting: if we can efficiently count the
solutions, we can efficiently determine whether there is at least one. The
converse does not hold: for example, we can determine in polynomial time
whether a graph admits a perfect matching but it is $\\#\mathbf{P}$-complete
to count the perfect matchings, even in a bipartite graph [29].
$\\#\mathbf{P}$ is the class of functions $f$ for which there is a
nondeterministic, polynomial-time Turing machine that has exactly $f(x)$
accepting paths for input $x$ [28]. It is easily seen that the counting
version of any $\mathbf{NP}$ decision problem is in $\\#\mathbf{P}$ and
$\\#\mathbf{P}$ can be considered the counting “analogue” of $\mathbf{NP}$.
Note, though that problems that are $\\#\mathbf{P}$-complete under appropriate
reductions are, under standard complexity-theoretic assumptions, considerably
harder than $\mathbf{NP}$-complete problems: $\mathbf{P}^{\\#\mathbf{P}}$
includes the whole of the polynomial hierarchy [27], whereas
$\mathbf{P}^{\mathbf{NP}}$ is generally thought not to.
Although no dichotomy is known for CSP, Bulatov has recently shown that, for
all $\Gamma\\!$, $\\#\mathrm{CSP}(\Gamma)$ is either computable in polynomial
time or $\\#\mathbf{P}$-complete [3]. However, Bulatov’s dichotomy sheds
little light on which constraint languages yield polynomial-time counting CSPs
and which do not. The criterion of the dichotomy is based on “defects” in a
certain infinite algebra built up from the polymorphisms of $\Gamma$ and it is
open whether the characterization is even decidable. It also seems not to
apply to bounded-degree $\\#\mathrm{CSP}$.
So, although there is a full dichotomy for $\\#\mathrm{CSP}(\Gamma)$, results
for restricted forms of constraint language are still of interest. Creignou
and Hermann have shown that only one of Schaefer’s polynomial-time cases for
Boolean languages survives the transition to counting:
$\\#\mathrm{CSP}(\Gamma)\in\mathbf{FP}$ (i.e., has a polynomial time
algorithm) if $\Gamma$ is affine (i.e., each relation is the solution set of a
system of linear equations over $\mathrm{GF}_{2}$) and is
$\\#\mathbf{P}$-complete, otherwise [5]. This result has been extended to
rational and even complex-weighted instances [10, 4] and, in the latter case,
the dichotomy is shown to hold for the restriction of the problem in which
instances have degree $3$. This implies that the degree-3 problem
$\\#\mathrm{CSP}_{3}(\Gamma)$ ($\\#\mathrm{CSP}(\Gamma)$ restricted to
instances of degree 3) is in $\mathbf{FP}$ if $\Gamma$ is affine and is
$\\#\mathbf{P}$-complete, otherwise.
### 1.3. Approximate counting
Since $\\#\mathrm{CSP}(\Gamma)$ is very often $\\#\mathbf{P}$-complete,
approximation algorithms play an important role. The key concept is that of a
_fully polynomial randomized approximation scheme_ (FPRAS). This is a
randomized algorithm for computing some function $f(x)$, taking as its input
$x$ and a constant $\epsilon>0$, and computing a value $Y$ such that
$e^{-\epsilon}\leqslant Y/f(x)\leqslant e^{\epsilon}$ with probability at
least $\tfrac{3}{4}$, in time polynomial in both $|x|$ and ${\epsilon}^{-1}$.
(See Section 2.4.)
Dyer, Goldberg and Jerrum have classified the complexity of approximately
computing $\\#\mathrm{CSP}(\Gamma)$ for Boolean constraint languages [9]. When
all relations in $\Gamma$ are affine, $\\#\mathrm{CSP}(\Gamma)$ can be
computed exactly in polynomial time by the result of Creignou and Hermann
discussed above [5]. Otherwise, if every relation in $\Gamma$ can be defined
by a conjunction of pins (i.e., assertions $v=0$ or $v=1$) and Boolean
implications, then $\\#\mathrm{CSP}(\Gamma)$ is as hard to approximate as the
problem $\\#\mathrm{BIS}$ of counting independent sets in a bipartite graph;
otherwise, $\\#\mathrm{CSP}(\Gamma)$ is as hard to approximate as the problem
$\\#\mathrm{SAT}$ of counting the satisfying truth assignments of a Boolean
formula. Dyer, Goldberg, Greenhill and Jerrum have shown that the latter
problem is complete for $\\#\mathbf{P}$ under appropriate approximation-
preserving reductions (see Section 2.4) and has no FPRAS unless
$\mathbf{NP}=\mathbf{RP}$ [8], which is thought to be unlikely. The complexity
of $\\#\mathrm{BIS}$ is currently open: there is no known FPRAS but it is not
known to be $\\#\mathbf{P}$-complete, either. $\\#\mathrm{BIS}$ is known to be
complete for a logically-defined subclass of $\\#\mathbf{P}$ with respect to
approximation-preserving reductions [8].
### 1.4. Our result
We consider the complexity of approximately solving Boolean $\\#\mathrm{CSP}$
problems when instances have bounded degree. Following Dalmau and Ford [6] and
Feder [13] we consider the case in which $R_{\mathrm{zero}}=\\{0\\}$ and
$R_{\mathrm{one}}=\\{1\\}$ are available. We proceed by showing that any
Boolean relation that is not definable as a conjunction of ORs or NANDs can be
used in low-degree instances to assert equalities between variables. Thus, we
can side-step degree restrictions by replacing high-degree variables with
distinct variables asserted to be equal.
Our main result, Corollary 6.7, is a trichotomy for the case in which
instances have maximum degree $d$ for some $d\geqslant 25$. If every relation
in $\Gamma$ is affine, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\\{R_{\mathrm{zero}},R_{\mathrm{one}}\\})$ is
solvable in polynomial time. Otherwise, if every relation in $\Gamma$ can be
defined as a conjunction of $R_{\mathrm{zero}}$, $R_{\mathrm{one}}$ and binary
implications, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\\{R_{\mathrm{zero}},R_{\mathrm{one}}\\})$ is
equivalent in approximation complexity to $\\#\mathrm{BIS}{}$. Otherwise, it
has no FPRAS unless $\mathbf{NP}=\mathbf{RP}$. Theorem 6.5 gives a partial
classification of the complexity when $d<25$. In the new cases that arise
here, the complexity is given in terms of the complexity of counting
independent sets in hypergraphs with bounded degree and bounded hyper-edge
size. The complexity of this problem is not fully understood and we explain
what is known about it in Section 6.
## 2\. Preliminaries
### 2.1. Basic notation
We write $\overline{a}$ for the tuple
$\left\langle\,{a_{1},\dots,a_{r}}\,\right\rangle$, which we often shorten to
$\overline{a}=a_{1}\dots a_{r}$. We write $a^{r}$ for the $r$-tuple $a\dots a$
and $\overline{a}\overline{b}$ for the tuple formed from the elements of
$\overline{a}$ followed by those of $\overline{b}$. The _bit-wise complement_
of a relation $R\subseteq\\{0,1\\}^{r}$ is the relation
${\widetilde{R}}=\\{\left\langle\,{a_{1}\oplus 1,\dots,a_{r}\oplus
1}\,\right\rangle\mid\overline{a}\in R\\}$, where $\oplus$ denotes addition
modulo 2.
We say that a relation $R$ is _ppp-definable_ 222This should not be confused
with the concept of primitive positive definability (pp-definability) which
appears in algebraic treatments of CSP and $\\#\mathrm{CSP}$, for example in
the work of Bulatov [3]. in a relation $R^{\prime}$ and write
$R\leqslant_{\mathrm{ppp}}R^{\prime}$ if $R$ can be obtained from $R^{\prime}$
by some sequence of the following operations:
* •
permutation of columns (for notational convenience only);
* •
pinning (taking sub-relations of the form $R_{i\mapsto c}=\\{\overline{a}\in
R\mid a_{i}=c\\}$ for some $i$ and some $c\in\\{0,1\\}$); and
* •
projection (“deleting the $i$th column” to give the relation $\\{a_{1}\dots
a_{i-1}a_{i+1}\dots a_{r}\mid a_{1}\dots a_{r}\in R\\}$).
It is easy to see that $\leqslant_{\mathrm{ppp}}$ is reflexive and transitive
and that, if $R\leqslant_{\mathrm{ppp}}R^{\prime}\\!$, then $R$ can be
obtained from $R^{\prime}$ by first permuting the columns, then making some
pins and then projecting.
We write $R_{=}=\\{00,11\\}$, $R_{\neq}=\\{01,10\\}$,
$R_{\mathrm{OR}}=\\{01,10,11\\}$, $R_{\mathrm{NAND}}=\\{00,01,10\\}$,
$R_{\rightarrow}=\\{00,01,11\\}$ and $R_{\leftarrow}=\\{00,10,11\\}$. For
$k\geqslant 2$, we write $R_{{=},{k}}=\\{0^{k}\\!,1^{k}\\}$,
$R_{\mathrm{OR},{k}}=\\{0,1\\}^{k}\setminus\\{0^{k}\\}$ and
$R_{\mathrm{NAND},{k}}=\\{0,1\\}^{k}\setminus\\{1^{k}\\}$ (i.e., $k$-ary
equality, $\mathrm{OR}$ and $\mathrm{NAND}$).
### 2.2. Boolean constraint satisfaction problems
A _constraint language_ is a set $\Gamma=\\{R_{1},\dots,R_{m}\\}$ of named
Boolean relations. Given a set $V$ of variables, the set of _constraints_ over
$\Gamma$ is the set $\mathrm{Cons}(V,\Gamma)$ which contains $R(\overline{v})$
for every relation $R\in\Gamma$ with arity $r$ and every $\overline{v}\in
V^{r}\\!$. Note that $v=v^{\prime}$ and $v\neq v^{\prime}$ are not constraints
unless the appropriate relations are included in $\Gamma\\!$. The _scope_ of a
constraint $R(\overline{v})$ is the tuple $\overline{v}$, which need not
consist of distinct variables.
An _instance_ of the constraint satisfaction problem (CSP) over $\Gamma$ is a
set $V$ of variables and a set $C\subseteq\mathrm{Cons}(V,\Gamma)$ of
constraints. An _assignment_ to a set $V$ of variables is a function
$\sigma\colon V\to\\{0,1\\}$. An assignment to $V$ _satisfies_ an instance
$(V,C)$ if
$\left\langle\,{\sigma(v_{1}),\dots,\sigma(v_{r})}\,\right\rangle\in R$ for
every constraint $R(v_{1},\dots,v_{r})$. We write $Z(I)$ for the number of
satisfying assignments to a CSP instance $I$. We study the counting CSP
problem $\\#\mathrm{CSP}(\Gamma)$, parameterized by $\Gamma\\!$, in which we
must compute $Z(I)$ for an instance $I=(V,C)$ of CSP over $\Gamma$.
The _degree_ of an instance is the greatest number of times any variable
appears among its constraints. Note that the variable $v$ appears twice in the
constraint $R(v,v)$. Our specific interest in this paper is in classifying the
complexity of bounded-degree counting CSPs. For a constraint language $\Gamma$
and a positive integer $d$, define $\\#\mathrm{CSP}_{d}(\Gamma)$ to be the
restriction of $\\#\mathrm{CSP}(\Gamma)$ to instances of degree at most $d$.
Instances of degree 1 are trivial.
###### Theorem 2.1.
For any $\Gamma\\!$, $\\#\mathrm{CSP}_{1}(\Gamma)\in\mathbf{FP}$. ∎
When considering $\\#\mathrm{CSP}_{d}$ for $d\geqslant 2$, we follow
established practice by allowing _pinning_ in the constraint language [6, 13].
We write $R_{\mathrm{zero}}=\\{0\\}$ and $R_{\mathrm{one}}=\\{1\\}$ for the
two singleton unary relations. We refer to constraints in $R_{\mathrm{zero}}$
and $R_{\mathrm{one}}$ as _pins_. To make notation easier, we will sometimes
write constraints using constants instead of explicit pins. That is, we will
allow the constants 0 and 1 to appear in the place of variables in the scopes
of constraints. Such constraints can obviously be rewritten as a set of
“proper” constraints, without increasing degree. We let
$\Gamma_{\mathrm{\\!pin}}$ denote the constraint language
$\\{R_{\mathrm{zero}},R_{\mathrm{one}}\\}$.
### 2.3. Hypergraphs
A _hypergraph_ $H=(V,E)$ is a set $V=V(H)$ of vertices and a set
$E=E(H)\subseteq{\mathcal{P}({V})}$ of non-empty _hyper-edges_. The _degree_
of a vertex $v\in V(H)$ is the number $d(v)=|\\{e\in E(H)\mid v\in e\\}|$ and
the degree of a hypergraph is the maximum degree of its vertices. If
$w=\max\\{|e|\mid e\in E(H)\\}$, we say that $H$ has _width_ $w$. An
_independent set_ in a hypergraph $H$ is a set $S\subseteq V(H)$ such that
$e\nsubseteq S$ for every $e\in E(H)$. Note that an independent set may
contain more than one vertex from any hyper-edge of size at least three.
We write $\\#w\mathrm{\text{-}HIS}$ for the problem of counting the
independent sets in a width-$w$ hypergraph $H$, and
$\\#w\mathrm{\text{-}HIS}_{d}$ for the restriction of
$\\#w\mathrm{\text{-}HIS}$ to inputs of degree at most $d$.
### 2.4. Approximation complexity
A _randomized approximation scheme_ (RAS) for a function
$f\colon\Sigma^{*}\rightarrow\mathbb{N}$ is a probabilistic Turing machine
that takes as input a pair $(x,\epsilon)\in\Sigma^{*}\times(0,1)$, and
produces, on an output tape, an integer random variable $Y$ with
$\Pr(e^{-\epsilon}\leqslant Y/f(x)\leqslant
e^{\epsilon})\geqslant\frac{3}{4}$.333The choice of the value $\frac{3}{4}$ is
inconsequential: the same class of problems has an FPRAS if we choose any
probability $p$ with $\frac{1}{2}<p<1$ [18]. A _fully polynomial randomized
approximation scheme (FPRAS)_ is a RAS that runs in time
$\mathrm{poly}(|x|,\epsilon^{-1})$.
To compare the complexity of approximate counting problems, we use the AP-
reductions of [8]. Suppose $f$ and $g$ are two functions from some input
domain $\Sigma^{*}$ to the natural numbers and we wish to compare the
complexity of approximately computing $f$ to that of approximately computing
$g$. An _approximation-preserving_ reduction from $f$ to $g$ is a
probabilistic oracle Turing machine $M$ that takes as input a pair
$(x,\epsilon)\in\Sigma^{*}\times(0,1)$, and satisfies the following three
conditions: (i) every oracle call made by $M$ is of the form $(w,\delta)$
where $w\in\Sigma^{*}$ is an instance of $g$, and $0<\delta<1$ is an error
bound satisfying $\delta^{-1}\leqslant\mathrm{poly}(|x|,\epsilon^{-1})$; (ii)
$M$ is a randomized approximation scheme for $f$ whenever the oracle is a
randomized approximation scheme for $g$; and (iii) the run-time of $M$ is
polynomial in $|x|$ and $\epsilon^{-1}$.
If there is an approximation-preserving reduction from $f$ to $g$, we write
$f\leqslant_{\mathrm{AP}}g$ and say that $f$ is _AP-reducible_ to $g$. If $g$
has an FPRAS, then so does $f$. If $f\leqslant_{\mathrm{AP}}g$ and
$g\leqslant_{\mathrm{AP}}f$, then we say that $f$ and $g$ are _AP-
interreducible_ and write $f\equiv_{\mathrm{AP}}g$.
## 3\. Classes of relations
A relation $R\subseteq\\{0,1\\}^{r}$ is _affine_ if it is the set of solutions
to some system of linear equations over $\mathrm{GF}_{2}$. That is, there is a
set $\Sigma$ of equations in variables $x_{1},\dots,x_{r}$, each of the form
$x_{i_{1}}\oplus\dots\oplus x_{i_{n}}=c$, where $\oplus$ denotes addition
modulo 2 and $c\in\\{0,1\\}$, such that $\overline{a}\in R$ if, and only if,
the assignment $x_{1}\mapsto a_{1},\dots,x_{r}\mapsto a_{r}$ satisfies every
equation in $\Sigma$. Note that the empty and complete relations are affine.
We define IM-conj to be the class of relations defined by a conjunction of
pins and (binary) implications. This class is called $\text{IM}_{2}$ in [9].
###### Lemma 3.1.
If $R\in\text{IM-conj}$ is not affine, then
$R_{\rightarrow}\leqslant_{\mathrm{ppp}}R$.∎
Let $\mathrm{OR}{}\text{-conj}$ be the set of Boolean relations that are
defined by a conjunction of pins and $\mathrm{OR}$s of any arity and
$\mathrm{NAND}{}\text{-conj}$ the set of Boolean relations definable by
conjunctions of pins and $\mathrm{NAND}$s (i.e., negated conjunctions) of any
arity. We say that one of the defining formulae of these relations is
_normalized_ if no pinned variable appears in any $\mathrm{OR}$ or
$\mathrm{NAND}$, the arguments of each individual $\mathrm{OR}$ and
$\mathrm{NAND}$ are distinct, every $\mathrm{OR}$ or $\mathrm{NAND}$ has at
least two arguments and no $\mathrm{OR}$ or $\mathrm{NAND}$’s arguments are a
subset of any other’s.
###### Lemma 3.2.
Every $\mathrm{OR}{}\text{-conj}$ (respectively,
$\mathrm{NAND}{}\text{-conj}$) relation is defined by a unique normalized
formula.∎
Given the uniqueness of defining normalized formulae, we define the _width_ of
an $\mathrm{OR}{}\text{-conj}$ or $\mathrm{NAND}{}\text{-conj}$ relation $R$
to be $0pt{R}$, the greatest number of arguments to any of the $\mathrm{OR}$s
or $\mathrm{NAND}$s in the normalized formula that defines it. Note that, from
the definition of normalized formulae, there are no relations of width 1.
###### Lemma 3.3.
If $R\in\mathrm{OR}{}\text{-conj}$ has width $w$, then
$R_{\mathrm{OR},{2}},\dots,R_{\mathrm{OR},{w}}\leqslant_{\mathrm{ppp}}R$.
Similarly, if $R\in\mathrm{NAND}{}\text{-conj}$ has width $w$, then
$R_{\mathrm{NAND},{2}},\dots,R_{\mathrm{NAND},{w}}\leqslant_{\mathrm{ppp}}R$.∎
Given tuples $\overline{a},\overline{b}\in\\{0,1\\}^{r}\\!$, we write
$\overline{a}\leqslant\overline{b}$ if $a_{i}\leqslant b_{i}$ for all
$i\in[1,r]$. If $\overline{a}\leqslant\overline{b}$ and
$\overline{a}\neq\overline{b}$, we write $\overline{a}<\overline{b}$. We say
that a relation $R\subseteq\\{0,1\\}^{r}$ is _monotone_ if, whenever
$\overline{a}\in R$ and $\overline{a}\leqslant\overline{b}$, then
$\overline{b}\in R$. We say that $R$ is _antitone_ if, whenever
$\overline{a}\in R$ and $\overline{b}\leqslant\overline{a}$, then
$\overline{b}\in R$. Clearly, $R$ is monotone if, and only if,
${\widetilde{R}}$ is antitone. Call a relation _pseudo-monotone_
(respectively, _pseudo-antitone_) if its restriction to non-constant columns
is monotone (respectively, antitone). The following is a consequence of
results in [19, Chapter 7.1.1].
###### Proposition 3.4.
A relation $R\subseteq\\{0,1\\}^{r}$ is in $\mathrm{OR}{}\text{-conj}$
(respectively, $\mathrm{NAND}{}\text{-conj}$) if, and only if, it is pseudo-
monotone (respectively, pseudo-antitone).∎
## 4\. Simulating equality
An important ingredient in bounded-degree dichotomy theorems [4] is expressing
equality using constraints from a language that does not necessarily include
the equality relation.
A constraint language $\Gamma$ is said to _simulate_ the $k$-ary equality
relation $R_{{=},{k}}$ if, for some $\ell\geqslant k$, there is a
$(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$-CSP instance $I$ with variables
$x_{1},\dots,x_{\ell}$ that has exactly $m\geqslant 1$ satisfying assignments
$\sigma$ with $\sigma(x_{1})=\dots=\sigma(x_{k})=0$, exactly $m$ with
$\sigma(x_{1})=\dots=\sigma(x_{k})=1$ and no other satisfying assignments. If,
further, the degree of $I$ is $d$ and the degree of each variable
$x_{1},\dots,x_{k}$ is at most $d-1$, we say that $\Gamma$ _$d$ -simulates_
$R_{{=},{k}}$. We say that $\Gamma$ _$d$ -simulates equality_ if it
$d$-simulates $R_{{=},{k}}$ for all $k\geqslant 2$.
The point is that, if $\Gamma$ $d$-simulates equality, we can express the
constraint $y_{1}=\dots=y_{r}$ in $\Gamma\cup\Gamma_{\mathrm{\\!pin}}$ and
then use each $y_{i}$ in one further constraint, while still having an
instance of degree $d$. The variables $x_{k+1},\dots,x_{\ell}$ in the
definition function as auxiliary variables and are not used in any other
constraint. Simulating equality makes degree bounds moot.
###### Proposition 4.1.
If $\Gamma$ $d$-simulates equality, then
$\\#\mathrm{CSP}(\Gamma)\leqslant_{\mathrm{AP}}\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$.∎
We now investigate which relations simulate equality.
###### Lemma 4.2.
$R\in\\{0,1\\}^{r}$ 3-simulates equality if $R_{=}\leqslant_{\mathrm{ppp}}R$,
$R_{\neq}\leqslant_{\mathrm{ppp}}R$ or
$R_{\rightarrow}\leqslant_{\mathrm{ppp}}R$.
###### Proof 4.3.
For each $k\geqslant 2$, we show how to 3-simulate $R_{{=},{k}}$. We may
assume without loss of generality that the ppp-definition of $R_{=}$,
$R_{\neq}$ or $R_{\rightarrow}$ from $R$ involves applying the identity
permutation to the columns, pinning columns 3 to $3+p-1$ inclusive to zero,
pinning columns $3+p$ to $3+p+q-1$ inclusive to one (that is, pinning
$p\geqslant 0$ columns to zero and $q\geqslant 0$ to one) and then projecting
away all but the first two columns.
Suppose first that $R_{=}\leqslant_{\mathrm{ppp}}R$ or
$R_{\rightarrow}\leqslant_{\mathrm{ppp}}R$. $R$ must contain $\alpha\geqslant
1$ tuples that begin $000^{p}1^{q}$, $\beta\geqslant 0$ that begin
$010^{p}1^{q}$ and $\gamma\geqslant 1$ that begin $110^{p}1^{q}$, with
$\beta=0$ unless we are ppp-defining $R_{\rightarrow}$. We consider, first,
the case where $\alpha=\gamma$, and show that we can 3-simulate $R_{{=},{k}}$,
expressing the constraint $R_{{=},{k}}(x_{1},\dots,x_{k})$ with the
constraints
$R(x_{1}x_{2}0^{p}1^{q}*),\ R(x_{2}x_{3}0^{p}1^{q}*),\dots,\
R(x_{k-1}x_{k}0^{p}1^{q}*),\ R(x_{k}x_{1}0^{p}1^{q}*)\,,$
where $*$ denotes a fresh $(r-2-p-q)$-tuple of variables in each constraint.
These constraints are equivalent to $x_{1}=\dots=x_{k}=x_{1}$ or to
$x_{1}\rightarrow\dots\rightarrow x_{k}\rightarrow x_{1}$ so constrain the
variables $x_{1},\dots,x_{k}$ to have the same value, as required. Every
variable appears at most twice and there are $\alpha^{k}$ solutions to these
constraints that put $x_{1}=\dots=x_{k}=0$, $\gamma^{k}=\alpha^{k}$ solutions
with $x_{1}=\dots=x_{k}=1$ and no other solutions. Hence, $R$ 3-simulates
$R_{{=},{k}}$, as required.
We now show, by induction on $r$, that we can 3-simulate $R_{{=},{k}}$ even in
the case that $\alpha\neq\gamma$. For the base case, $r=2$, we have
$\alpha=\gamma=1$ and we are done. For the inductive step, let $r>2$ and
assume, w.l.o.g. that $\alpha>\gamma$ ($\alpha<\gamma$ is symmetric). In
particular, we have $\alpha\geqslant 2$, so there are distinct tuples
$000^{p}1^{q}\overline{a}$, and $000^{p}1^{q}\overline{b}$ and
$110^{p}1^{q}\overline{c}$ in $R$. Choose $j$ such that $a_{j}\neq b_{j}$.
Pinning the $(2+p+q+j)$th column of $R$ to $c_{j}$ and projecting out the
resulting constant column gives a relation $R^{\prime}$ of arity $r-1$
containing at least one tuple beginning $000^{p}1^{q}$ and at least one
beginning $110^{p}1^{q}$: by the inductive hypothesis, $R^{\prime}$
3-simulates $R_{{=},{k}}$.
Finally, we consider the case that $R_{\neq}\leqslant_{\mathrm{ppp}}R$. $R$
contains $\alpha\geqslant 1$ tuples beginning $010^{p}1^{q}$ and
$\beta\geqslant 1$ beginning $100^{p}1^{q}$. We express the constraint
$R_{{=},{k}}(x_{1},\dots,x_{k})$ by introducing fresh variables
$y_{1},\dots,y_{k}$ and using the constraints
$\begin{array}[]{ccccc}R(x_{1}y_{1}0^{p}1^{q}*),&R(x_{2}y_{2}0^{p}1^{q}*),&\ldots,&R(x_{k-1}y_{k-1}0^{p}1^{q}*),&R(x_{k}y_{k}0^{p}1^{q}*),\\\
R(y_{1}x_{2}0^{p}1^{q}*),&R(y_{2}x_{3}0^{p}1^{q}*),&\ldots,&R(y_{k-1}x_{k}0^{p}1^{q}*),&R(y_{k}x_{1}0^{p}1^{q}*)\,.\end{array}$
There are $\alpha^{k}\beta^{k}$ solutions when $x_{1}=\dots=x_{k}=0$ (and
$y_{1}=\dots=y_{k}=1$) and $\beta^{k}\alpha^{k}$ solutions when the $x$s are 1
and the $y$s are 0. There are no other solutions and no variable is used more
than twice.
For $c\in\\{0,1\\}$, an $r$-ary relation is _$c$ -valid_ if it contains the
tuple $c^{r}\\!$.
###### Lemma 4.4.
Let $r\geqslant 2$ and let $R\subseteq\\{0,1\\}^{r}$ be 0- and 1-valid but not
complete. Then $R$ 3-simulates equality.∎
In the following lemma, we do not require $R$ and $R^{\prime}$ to be distinct.
The technique is to assert $x_{1}=\dots=x_{k}$ by simulating the formula
$\mathrm{OR}(x_{1},y_{1})\wedge\mathrm{NAND}(y_{1},x_{2})\wedge\mathrm{OR}(x_{2},y_{2})\wedge\mathrm{NAND}(y_{2},x_{3})\wedge\cdots\wedge\mathrm{OR}(x_{k},y_{k})\wedge\mathrm{NAND}(y_{k},x_{1})$.
###### Lemma 4.5.
If $R_{\mathrm{OR}}\leqslant_{\mathrm{ppp}}R$ and
$R_{\mathrm{NAND}}\leqslant_{\mathrm{ppp}}R^{\prime}\\!$, then
$\\{R,R^{\prime}\\}$ 3-simulates equality.∎
## 5\. Classifying relations
We are now ready to prove that every Boolean relation $R$ is in
$\mathrm{OR}{}\text{-conj}$, in $\mathrm{NAND}{}\text{-conj}$ or 3-simulates
equality. If $R_{0}$ and $R_{1}$ are $r$-ary, let
$R_{0}+R_{1}=\\{0\overline{a}\mid\overline{a}\in
R_{0}\\}\cup\\{1\overline{a}\mid\overline{a}\in R_{1}\\}$.
###### Lemma 5.1.
Let $R_{0},R_{1}\in\mathrm{OR}{}\text{-conj}$ and let $R=R_{0}+R_{1}$. Then
$R\in\mathrm{OR}{}\text{-conj}$, $R\in\mathrm{NAND}{}\text{-conj}$ or $R$
3-simulates equality.
###### Proof 5.2.
Let $R_{0}$ and $R_{1}$ have arity $r$. We may assume that $R$ has no constant
columns. If it does, let $R^{\prime}$ be the relation that results from
projecting them away. $R^{\prime}=R^{\prime}_{0}+R^{\prime}_{1}$, where both
$R^{\prime}_{0}$ and $R^{\prime}_{1}$ are $\mathrm{OR}{}\text{-conj}$
relations. By the remainder of the proof,
$R^{\prime}\in\mathrm{OR}{}\text{-conj}$,
$R^{\prime}\in\mathrm{NAND}{}\text{-conj}$ or $R^{\prime}$ 3-simulates
equality. Re-instating the constant columns does not alter this. For $R$
without constant columns, there are two cases.
_Case 1. $R_{0}\subseteq R_{1}$._ Suppose $R_{i}$ is defined by the normalized
$\mathrm{OR}{}\text{-conj}$ formula $\phi_{i}$ in variables
$x_{2},\dots,x_{r+1}$. Then $R$ is defined by the formula
$\phi_{0}\vee(x_{1}=1\wedge\phi_{1})\equiv(\phi_{0}\vee
x_{1}=1)\wedge(\phi_{0}\vee\phi_{1})\equiv(\phi_{0}\vee
x_{1}=1)\wedge\phi_{1}\,,$ (1)
where the second equivalence is because $\phi_{0}$ implies $\phi_{1}$, because
$R_{0}\subseteq R_{1}$. $R_{1}$ has no constant column, since such a column
would have to be constant with the same value in $R_{0}$, contradicting our
assumption that $R$ has no constant columns. There are two cases.
_Case 1.1. $R_{0}$ has no constant columns._ $x_{1}=1$ is equivalent to
$\mathrm{OR}(x_{1})$ and $\phi_{0}$ contains no pins, so we can rewrite
$\phi_{0}\vee x_{1}=1$ in CNF. Therefore, (1) is $\mathrm{OR}{}\text{-conj}$.
_Case 1.2. $R_{0}$ has a constant column._ Suppose first that the $k$th column
of $R_{0}$ is constant-zero. $R_{1}$ has no constant columns, so the
projection of $R$ onto its first and $(k+1)$st columns gives the relation
$R_{\leftarrow}$, and $R$ 3-simulates equality by Lemma 4.2. Otherwise, all
constant columns of $R_{0}$ contain ones. Then $\phi_{0}$ is in CNF, since
every pin $x_{i}=1$ in $\phi_{0}$ can be written $\mathrm{OR}(x_{i})$. Thus,
we can write $\phi_{0}\vee x_{1}=1$ in CNF, so (1) defines an
$\mathrm{OR}{}\text{-conj}$ relation.
_Case 2. $R_{0}\nsubseteq R_{1}$._ We will show that $R$ 3-simulates equality
or is in $\mathrm{NAND}{}\text{-conj}$. We consider two cases (recall that no
relation has width 1).
_Case 2.1. At least one of $R_{0}$ and $R_{1}$ has positive width._ There are
two sub-cases.
_Case 2.1.1. $R_{1}$ has a constant column._ Suppose the $k$th column of
$R_{1}$ is constant. If the $k$th column of $R_{0}$ is also constant, then the
projection of $R$ to its first and $(k+1)$st columns is either equality or
disequality (since the corresponding column of $R$ is not constant) so $R$
3-simulates equality by Lemma 4.2. Otherwise, if the projection of $R$ to the
first and $(k+1)$st columns is $R_{\rightarrow}$, then $R$ 3-simulates
equality by Lemma 4.2. Otherwise, that projection must be $R_{\mathrm{NAND}}$.
By Lemma 3.3 and the assumption of Case 2.1, $R_{\mathrm{OR}}$ is ppp-
definable in at least one of $R_{0}$ and $R_{1}$ so $R$ 3-simulates equality
by Lemma 4.5.
_Case 2.1.2. $R_{1}$ has no constant columns._ By Proposition 3.4, $R_{1}$ is
monotone. Let $\overline{a}\in R_{0}\setminus R_{1}$: by applying the same
permutation to the columns of $R_{0}$ and $R_{1}$, we may assume that
$\overline{a}=0^{\ell}1^{r-\ell}$. We must have $\ell\geqslant 1$ as every
non-empty $r$-ary monotone relation contains the tuple $1^{r}\\!$. Let
$\overline{b}\in R_{1}$ be a tuple such that $a_{i}=b_{i}$ for a maximal
initial segment of $[1,r]$. By monotonicity of $R_{1}$, we may assume that
$\overline{b}=0^{k}1^{r-k}$. Further, we must have $k<\ell$, since, otherwise,
we would have $\overline{b}<\overline{a}$, contradicting our choice of
$\overline{a}\notin R_{1}$.
Now, consider the relation $R^{\prime}=\\{a_{0}a_{1}\dots a_{\ell-k}\mid
a_{0}0^{k}a_{1}\dots a_{\ell-k}1^{r-\ell}\in R\\}$, which is the result of
pinning columns 2 to $(k+1)$ of $R$ to zero and columns $(r-\ell+1)$ to
$(r+1)$ to one and discarding the resulting constant columns. $R^{\prime}$
contains $0^{\ell-k+1}$ and $1^{\ell-k+1}$ but is not complete, since
$10^{\ell-k}\notin R^{\prime}\\!$. By Lemma 4.4, $R^{\prime}$ and, hence, $R$
3-simulates equality.
_Case 2.2. Both $R_{0}$ and $R_{1}$ have width zero,_ i.e., are complete
relations, possibly padded with constant columns. For $i\in[1,r]$, let
$R^{\prime}_{i}$ be the relation obtained from $R$ by projecting onto its
first and $(i+1)$st columns. Since $R$ has no constant columns,
$R^{\prime}_{i}$ is either complete, $R_{=}$, $R_{\neq}$, $R_{\mathrm{OR}}$,
$R_{\mathrm{NAND}}$, $R_{\rightarrow}$ or $R_{\leftarrow}$. If there is a $k$
such that $R^{\prime}_{k}$ is $R_{=}$, $R_{\neq}$, $R_{\rightarrow}$ or
$R_{\leftarrow}$, then $R_{=}$, $R_{\neq}$ or $R_{\rightarrow}$ is ppp-
definable in $R$ and hence $R$ 3-simulates equality by Lemma 4.2. If there are
$k_{1}$ and $k_{2}$ such that $R^{\prime}_{k_{1}}=R_{\mathrm{OR}}$ and
$R^{\prime}_{k_{2}}=R_{\mathrm{NAND}}$, then $R$ 3-simulates equality by Lemma
4.5. It remains to consider the following two cases.
_Case 2.2.1. Each $R^{\prime}_{i}$ is either $R_{\mathrm{OR}}$ or complete._
$R_{1}$ must be complete, which contradicts the assumption that
$R_{0}\not\subseteq R_{1}$.
_Case 2.2.1. Each $R^{\prime}_{i}$ is either $R_{\mathrm{NAND}}$ or complete._
$R_{0}$ must be complete. Let $I=\\{i\mid
R^{\prime}_{i}=R_{\mathrm{NAND}}\\}$. Then $R=\bigwedge_{i\in
I}\mathrm{NAND}(x_{1},x_{i+1})$, so $R\in\mathrm{NAND}{}\text{-conj}$.
Using the duality between $\mathrm{OR}{}\text{-conj}$ and
$\mathrm{NAND}{}\text{-conj}$ relations, we can prove the corresponding result
for $R_{0},R_{1}\in\mathrm{NAND}{}\text{-conj}$. The proof of the
classification is completed by a simple induction on the arity of $R$.
Decomposing $R$ as $R_{0}+R_{1}$ and assuming inductively that $R_{0}$ and
$R_{1}$ are of one of the stated types, we use the previous results in this
section and Lemma 4.5 to show that $R$ is.
###### Theorem 5.3.
Every Boolean relation is $\mathrm{OR}{}\text{-conj}$ or
$\mathrm{NAND}{}\text{-conj}$ or 3-simulates equality.∎
## 6\. Complexity
The complexity of approximating $\\#\mathrm{CSP}(\Gamma)$ where the degree of
instances is unbounded is given by Dyer, Goldberg and Jerrum [9, Theorem 3].
###### Theorem 6.1.
Let $\Gamma$ be a Boolean constraint language.
* •
If every $R\in\Gamma$ is affine, then $\\#\mathrm{CSP}(\Gamma)\in\mathbf{FP}$.
* •
Otherwise, if $\Gamma\subseteq\text{IM-conj}$, then
$\\#\mathrm{CSP}(\Gamma)\equiv_{\mathrm{AP}}\\#\mathrm{BIS}$.
* •
Otherwise, $\\#\mathrm{CSP}(\Gamma)\equiv_{\mathrm{AP}}\\#\mathrm{SAT}$.
Working towards our classification of the approximation complexity of
$\\#\mathrm{CSP}(\Gamma)$, we first deal with subcases. The IM-conj case and
$\mathrm{OR}{}\text{-conj}$/$\mathrm{NAND}{}\text{-conj}$ cases are based on
links between those classes of relations and the problems of counting
independent sets in bipartite and general graphs, respectively[9, 8], the
latter extended to hypergraphs.
###### Proposition 6.2.
If $\Gamma\subseteq\text{IM-conj}$ contains at least one non-affine relation,
then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{BIS}$
for all $d\geqslant 3$. ∎
###### Proposition 6.3.
Let $R$ be an $\mathrm{OR}{}\text{-conj}$ or $\mathrm{NAND}{}\text{-conj}$
relation of width $w$. Then, for $d\geqslant 2$,
$\\#w\mathrm{\text{-}HIS}_{d}{}\leqslant_{\mathrm{AP}}\\#\mathrm{CSP}_{d}(\\{R\\}\cup\Gamma_{\mathrm{\\!pin}})$.∎
###### Proposition 6.4.
Let $R$ be an $\mathrm{OR}{}\text{-conj}$ or $\mathrm{NAND}{}\text{-conj}$
relation of width $w$. Then, for $d\geqslant 2$,
$\\#\mathrm{CSP}_{d}(\\{R\\}\cup\Gamma_{\mathrm{\\!pin}})\leqslant_{\mathrm{AP}}\\#w\mathrm{\text{-}HIS}_{kd}$,
where $k$ is the greatest number of times that any variable appears in the
normalized formula defining $R$. ∎
We now give the complexity of approximating
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ for $d\geqslant 3$.
###### Theorem 6.5.
Let $\Gamma$ be a Boolean constraint language and let $d\geqslant 3$.
* •
If every $R\in\Gamma$ is affine, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\in\mathbf{FP}$.
* •
Otherwise, if $\Gamma\subseteq\text{IM-conj}$, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{BIS}$.
* •
Otherwise, if $\Gamma\subseteq\mathrm{OR}{}\text{-conj}$ or
$\Gamma\subseteq\mathrm{NAND}{}\text{-conj}$, then let $w$ be the greatest
width of any relation in $\Gamma$ and let $k$ be the greatest number of times
that any variable appears in the normalized formulae defining the relations of
$\Gamma$. Then
$\\#w\mathrm{\text{-}HIS}_{d}\leqslant_{\mathrm{AP}}\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\leqslant_{\mathrm{AP}}\\#w\mathrm{\text{-}HIS}_{kd}$.
* •
Otherwise,
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{SAT}$.
###### Proof 6.6.
The affine case is immediate from Theorem 6.1.
($\Gamma\cup\Gamma_{\mathrm{\\!pin}}$ is affine if, and only if, $\Gamma$ is.)
Otherwise, if $\Gamma\subseteq\text{IM-conj}$ and some $R\in\Gamma$ is not
affine, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{BIS}$
by Proposition 6.2. Otherwise, if $\Gamma\subseteq\mathrm{OR}{}\text{-conj}$
or $\Gamma\subseteq\mathrm{NAND}{}\text{-conj}$, then
$\\#w\mathrm{\text{-}HIS}_{d}\leqslant_{\mathrm{AP}}\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\leqslant_{\mathrm{AP}}\\#w\mathrm{\text{-}HIS}_{kd}$
by Propositions 6.3 and 6.4.
Finally, suppose that $\Gamma$ is not affine, $\Gamma\nsubseteq\text{IM-
conj}$, $\Gamma\nsubseteq\mathrm{OR}{}\text{-conj}$ and
$\Gamma\nsubseteq\mathrm{NAND}{}\text{-conj}$. Since
$(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ is neither affine or a subset of IM-
conj, we have
$\\#\mathrm{CSP}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{SAT}$
by Theorem 6.1 so, if we can show that $\Gamma$ $d$-simulates equality, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{CSP}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$
by Proposition 4.1 and we are done. If $\Gamma$ contains a $R$ relation that
is neither $\mathrm{OR}{}\text{-conj}$ nor $\mathrm{NAND}{}\text{-conj}$, then
$R$ 3-simulates equality by Theorem 5.3. Otherwise, $\Gamma$ must contain
distinct relations $R_{1}\in\mathrm{OR}{}\text{-conj}$ and
$R_{2}\in\mathrm{NAND}{}\text{-conj}$ that are non-affine so have width at
least two. So $\Gamma$ 3-simulates equality by Lemma 4.5.
Unless $\mathbf{NP}=\mathbf{RP}$, there is no FPRAS for counting independent
sets in graphs of maximum degree at least 25 [7], and, therefore, no FPRAS for
$\\#w\mathrm{\text{-}HIS}_{d}$ with $r\geqslant 2$ and $d\geqslant 25$.
Further, since $\\#\mathrm{SAT}$ is complete for $\\#\mathbf{P}$ under AP-
reductions [8], $\\#\mathrm{SAT}$ cannot have an FPRAS unless
$\mathbf{NP}=\mathbf{RP}$. From Theorem 6.5 above we have the following
corollary.
###### Corollary 6.7.
Let $\Gamma$ be a Boolean constraint language and let $d\geqslant 25$.
* •
If every $R\in\Gamma$ is affine, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\in\mathbf{FP}$.
* •
Otherwise, if $\Gamma\subseteq\text{IM-conj}$, then
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})\equiv_{\mathrm{AP}}\\#\mathrm{BIS}$.
* •
Otherwise there is no FPRAS for
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$, unless
$\mathbf{NP}=\mathbf{RP}$.
$\Gamma\cup\Gamma_{\mathrm{\\!pin}}$ is affine (respectively, in
$\mathrm{OR}{}\text{-conj}$ or in $\mathrm{NAND}{}\text{-conj}$) if, and only
if $\Gamma$ is, so the case for large-degree instances ($d\geqslant 25$)
corresponds exactly in complexity to the unbounded case [9]. The case for
lower degree bounds is more complex. To put Theorem 6.5 in context, we
summarize the known approximability of $\\#w\mathrm{\text{-}HIS}_{d}$,
parameterized by $d$ and $w$.
The case $d=1$ is clearly in $\mathbf{FP}$ (Theorem 2.1) and so is the case
$d=w=2$, which corresponds to counting independent sets in graphs of maximum
degree two. For $d=2$ and width $w\geqslant 3$, Dyer and Greenhill have shown
that there is an FPRAS for $\\#w\mathrm{\text{-}HIS}_{d}$ [11]. For $d=3$,
they have shown that there is an FPRAS if the the width $w$ is at most 3. For
larger width, the approximability of $\\#w\mathrm{\text{-}HIS}_{3}$ is still
not known. With the width restricted to $w=2$ (normal graphs), Weitz has shown
that, for degree $d\in\\{3,4,5\\}$, there is a deterministic approximation
scheme that runs in polynomial time (a PTAS) [30]. This extends a result of
Luby and Vigoda, who gave an FPRAS for $d\leqslant 4$ [24]. For $d>5$,
approximating $\\#w\mathrm{\text{-}HIS}_{d}$ becomes considerably harder. More
precisely, Dyer, Frieze and Jerrum have shown that for $d=6$ the Monte Carlo
Markov chain technique is likely to fail, in the sense that “cautious” Markov
chains are provably slowly mixing [7]. They also showed that, for $d=25$,
there can be no polynomial-time algorithm for approximate counting, unless
$\mathbf{NP}=\mathbf{RP}$. These results imply that for $d\in\\{6,\dots,24\\}$
and $w\geqslant 2$ the Monte Carlo Markov chain technique is likely to fail
and for $d\geqslant 25$ and $w\geqslant 2$, there can be no FPRAS unless
$\mathbf{NP}=\mathbf{RP}$. Table 1 summarizes the results.
Degree $d$ | Width $w$ | Approximability of $\\#w\mathrm{\text{-}HIS}_{d}$
---|---|---
$1$ | $\geqslant 2$ | $\mathbf{FP}$
$2$ | $2$ | $\mathbf{FP}$
$2$ | $\geqslant 3$ | FPRAS [11]
$3$ | $2,3$ | FPRAS [11]
$3,4,5$ | $2$ | PTAS [30]
$6,\dots,24$ | $\geqslant 2$ | The MCMC method is likely to fail [7]
$\geqslant 25$ | $\geqslant 2$ | No FPRAS unless $\mathbf{NP}=\mathbf{RP}$ [7]
Table 1. Approximability of $\\#w\mathrm{\text{-}HIS}_{d}$ (still open for all
other values of $d$ and $w$).
Returning to bounded-degree $\\#\mathrm{CSP}$, the case $d=2$ seems to be
rather different to degree bounds three and higher. This is also the case for
decision CSP — recall that degree-$d$
$\mathrm{CSP}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ has the same complexity as
unbounded-degree $\mathrm{CSP}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ for all
$d\geqslant 3$ [6], while degree-2
$\mathrm{CSP}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ is often easier than the
unbounded-degree case [6, 13] but the complexity of degree-2
$\mathrm{CSP}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ is still open for some
$\Gamma\\!$.
Our key techniques for determining the complexity of
$\\#\mathrm{CSP}_{d}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ for $d\geqslant 3$
were the 3-simulation of equality and Theorem 5.3, which says that every
Boolean relation is in $\mathrm{OR}{}\text{-conj}$, in
$\mathrm{NAND}{}\text{-conj}$ or 3-simulates equality. However, it seems that
not all relations that 3-simulate equality also 2-simulate equality so the
corresponding classification of relations does not appear to hold. It seems
that different techniques will be required for the degree-2 case. For example,
it is possible that there is no FPRAS for
$\\#\mathrm{CSP}_{3}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ except when $\Gamma$
is affine. However, Bubley and Dyer have shown that there is an FPRAS for
degree-2 $\\#\mathrm{SAT}$, even though the exact counting problem is
$\\#\mathbf{P}$-complete [1]. This shows that there is a class $\mathcal{C}$
of constraint languages for which
$\\#\mathrm{CSP}_{2}(\Gamma\cup\Gamma_{\mathrm{\\!pin}})$ has an FPRAS for
every $\Gamma\in\mathcal{C}$ but for which no exact polynomial-time algorithm
is known.
We leave the complexity of degree-2 $\\#\mathrm{CSP}$ and of $\\#\mathrm{BIS}$
and the the various parameterized versions of the counting hypergraph
independent sets problem as open questions.
## References
* [1] R. Bubley and M. Dyer. Graph orientations with no sink and an approximation for a hard case of #SAT. In 8th ACM–SIAM Symp. on Discrete Algorithms (SODA 1997), pages 248–257, 1997.
* [2] A. A. Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element domain. J. ACM, 53(1):66–120, 2006.
* [3] A. A. Bulatov. The complexity of the counting constraint satisfaction problem. In 35th Intl Colloq. on Automata, Languages and Programming (ICALP 2008) Part I, volume 5125 of LNCS, pages 646–661. Springer, 2008\.
* [4] J.-Y. Cai, P. Lu, and M. Xia. The complexity of complex weighted Boolean #CSP. Upcoming journal submission, 2009.
* [5] N. Creignou and M. Hermann. Complexity of generalized satisfiablility counting problems. Inform. and Comput., 125(1):1–12, 1996.
* [6] V. Dalmau and D. K. Ford. Generalized satisfiability with limited occurrences per variable: A study through Delta-matroid parity. In Math. Founds of Comput. Sci. (MFCS 2003), volume 2747 of LNCS, pages 358–367. Springer, 2003.
* [7] M. Dyer, A. Frieze, and M. Jerrum. On counting independent sets in sparse graphs. SIAM J. Computing, 31(5):1527–1541, 2002.
* [8] M. Dyer, L. A. Goldberg, C. S. Greenhill, and M. Jerrum. The relative complexity of approximate counting problems. Algorithmica, 38(3):471–500, 2003.
* [9] M. Dyer, L. A. Goldberg, and M. Jerrum. An approximation trichotomy for Boolean #CSP. To appear in _J. Comput. Sys. Sci._ http://arxiv.org/abs/0710.4272, 2007.
* [10] M. Dyer, L. A. Goldberg, and M. Jerrum. The complexity of weighted Boolean CSP. SIAM J. Comput., 38(5):1970–1986, 2009.
* [11] M. Dyer and C. S. Greenhill. On Markov chains for independent sets. J. Algorithms, 35(1):17–49, 2000.
* [12] R. Fagin, L. J. Stockmeyer, and M. Y. Vardi. On monadic NP vs monadic co-NP. Inform. and Comput., 120(1):78–92, 1995.
* [13] T. Feder. Fanout limitations on constraint systems. Theor. Comput. Sci., 255(1–2):281–293, 2001.
* [14] T. Feder and M. Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM J. Comput., 28(1):57–104, 1998.
* [15] E. C. Freuder. Complexity of $k$-tree structured constraint satisfaction problems. In 8th Conf. of American Assoc. for Art. Intelligence, pages 4–9. AAAI Press/MIT Press, 1990.
* [16] P. Hell and J. Nešetřil. On the complexity of $h$-coloring. J. Combin. Theory B, 48(1):92–110, 1990.
* [17] P. Hell and J. Nešetřil. Graphs and Homomorphisms. Oxford University Press, 2004.
* [18] M. Jerrum, L. G. Valiant, and V. V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci., 43:169–188, 1986.
* [19] D. E. Knuth. The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms. In preparation.
* [20] Ph. G. Kolaitis and M. Y. Vardi. Conjunctive query containment and constraint satisfaction. J. Comput. Sys. Sci., 61(2):302–332, 2000.
* [21] Ph. G. Kolaitis and M. Y. Vardi. A game-theoretic approach to constraint satisfaction. In 17th Conf. of American Assoc. for Artif. Intelligence, pages 175–181. AAAI Press/MIT Press, 2000.
* [22] V. Kumar. Algorithms for constraint satisfaction problems: A survey. AI Magazine, 13(1):33–42, 1992.
* [23] R. E. Ladner. On the structure of polynomial time reducibility. J. ACM, 22(1):155–171, 1975.
* [24] M. Luby and E. Vigoda. Fast convergence of the Glauber dynamics for sampling independent sets. Random Structures and Algorithms, 15(3–4):229–241, 1999.
* [25] U. Montanari. Networks of constraints: Fundamental properties and applications to picture processing. Inform. Sci., 7:95–135, 1974.
* [26] T. J. Schaefer. The complexity of satisfiability problems. In 10th ACM Symp. on Theory of Computing, pages 216–226, 1978.
* [27] S. Toda. On the computational power of PP and $\bigoplus$P. In 30th Ann. Symp. on Founds of Comput. Sci. (FOCS 1989), pages 514–519. IEEE Computer Society, 1989.
* [28] L. G. Valiant. The complexity of computing the permanent. Theor. Comput. Sci., 8:189–201, 1979.
* [29] L. G. Valiant. The complexity of enumeration and reliability problems. SIAM J. Comput., 8(3):410–421, 1979.
* [30] D. Weitz. Counting independent sets up to the tree threshold. In 38th ACM Symp. on Theory of Computing, pages 140–149, 2006.
* [31] D. Welsh. Complexity: Knots, Colourings and Counting. Cambridge University Press, 1993.
|
arxiv-papers
| 2010-01-27T17:07:40 |
2024-09-04T02:49:08.074077
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Martin E. Dyer, Leslie Ann Goldberg, Markus Jalsenius and David\n Richerby",
"submitter": "David Richerby",
"url": "https://arxiv.org/abs/1001.4987"
}
|
1001.5157
|
# QCD motivated approach to soft interactions at high energy: inclusive
production
E. Gotsman, E. Levin, U. Maor
Department of Particle Physics, School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Science
Tel Aviv University, Tel Aviv, 69978, Israel
###### Abstract
We extend our two component Pomeron model (GLMM) for soft high energy
scattering to single inclusive cross sections. To this end we present a
suitable formulation which also includes the semi enhanced Pomeron-particle
vertex corrections. The available data on single inclusive density
$(1/\sigma_{in})d\sigma/dy$ in the c.m. energy range of 200-1800 $GeV$ are
well reproduced by our model. The just published ALICE collaboration point at
900 $GeV$ and the CMS collaboration measurements at 900 and 2360 $GeV$ are in
excellent agreement with the calculations of our model . We also present
predictions covering the complete LHC energy range which can be readily tested
in the early low luminosity LHC runs. The results presented in this
communication provide additional support to our Pomeron model approach.
###### pacs:
13.85.-t, 13.85.Hd, 11.55.-m, 11.55.Bq
††preprint: TAUP -2916-10 [hep-ph]
In this paper we expand our approach to soft hadron interactions, developed in
Ref.GLMM , to obtain estimates for single inclusive cross sections. We have
two main objectives: First, we wish to reproduce the existing data and predict
the single inclusive experimental distributions which will be measured in the
preliminary low luminosity LHC runs. The data for $\sqrt{s}$ = 200-2360 GeV
are well reproduced by our model. The first published LHC experimental
outputAlice , provides data on p-p single inclusive cross section at
$\sqrt{s}$ = 900 GeV which is in accord with the results of our model. The
recently published data by the CMS collaboration CMS at 900 GeV and the
higher energy of $\sqrt{s}$ = 2.36 TeV are also in agreement with our
predictions. Second, the successful data analysis presented in this paper
follows directly from our GLMM model and its fitted parametersGLMM . As such,
it provides additional support to the validity of our hypothesis.
In our approach to soft Pomeron interactions we combine two elements: A two
channel Good-Walker mechanismGW with a super critical Regge-like Pomeron with
an intercept $\Delta_{I\\!\\!P}\,>\,0$, to which we add the enhanced (multi)
Pomeron interactions. Our formulation is based on two main assumptions:
1) We assume that the slope of the Pomeron trajectory
$\alpha^{\prime}_{I\\!\\!P}=0$. This assumption is strongly supported by the
global data analysis we have presented in Ref.GLMM , in which the fitted value
of $\alpha^{\prime}_{I\\!\\!P}$ is exceedingly small. A consequence of the
small value of $\alpha^{\prime}_{I\\!\\!P}$ is a relatively large
$\Delta_{I\\!\\!P}\simeq 0.35$.
2) In our calculations of enhanced (and semi enhanced) Pomeron interactions we
only take into account the triple Pomeron vertex.
These assumptions are compatible with the main features of the Pomeron in N=4
super symmetric Yang-Mills theory, in which the Pomeron has an intercept of
$\Delta_{I\\!\\!P}=1-2/\sqrt{\lambda}$ at large values of $\lambda$, and
$\alpha^{\prime}_{I\\!\\!P}\,=\,0$. Note that the fitted value
$\Delta_{I\\!\\!P}\simeq 0.35$ obtained in our model corresponds to a large
value of $\lambda\simeq 10$. In this approach, the main contributions to the
total cross section are the elastic and diffractive cross sections. This is a
consequence of the Good-Walker mechanismGW coupled to the vanishing of the
cross sections initiated by multi Pomeron interactions (for details see
Ref.GLMM ). The strength of the Pomeron interaction is proportional to
$2/\sqrt{\lambda}$, which can be taken into account by introducing a triple
Pomeron vertex.
Our Pomeron model assumptions provide a natural matching between soft Pomeron
dynamics and high density QCD (hdQCD), see Refs.BFKL ; LI ; GLR ; MUQI ; MV ;
B ; K ; JIMWLK . The only hdQCD dimensional scale which is responsible for
high energy interactions is $Q_{s}$, the saturation scale. This scale
increases with energy leading to $\alpha^{\prime}_{I\\!\\!P}\propto
1/Q^{2}_{s}(x)\to 0$ at high enough energies where $x\rightarrow 0$. Recall
that the triple BFKL Pomeron vertex plays a decisive role in small $x$ pQCD
GLR ; B ; K ; BRN ; BART ; MUCD ; LALE ; LELU ; LMP . The consequent emerging
compatibility of soft and hard Pomeron dynamics and their similar formulations
are the main results of our Pomeron studies.
In the framework of Pomeron calculusGRIBRT (see also Refs. COL ; SOFT ; LEREG
), single inclusive cross sections can be calculated using the Mueller
diagrams MUDI shown in Fig. 1-a. They lead to
$\displaystyle\frac{1}{\sigma_{in}}\frac{d\sigma}{dy}=\frac{1}{\sigma_{in}(Y)}\left\\{a_{PP}(\alpha^{2}g_{1}+\beta^{2}g_{2})^{2}G_{enh}\left(T\left(Y/2-y\right)\right)\times
G_{enh}\left(T\left(Y/2+y\right)\right)\right.$ (1)
$\displaystyle-\left.a_{RP}(\alpha^{2}g^{R}_{1}+\beta^{2}g^{R}_{2})(\alpha^{2}g_{1}+\beta^{2}g_{2})\right.$
$\displaystyle\left.\left[e^{(\Delta_{R}(Y/2-y)}\times
G_{enh}\left(T\left(Y/2+y\right)\right)+e^{(\Delta_{R}(Y/2-y)}\times
G_{enh}\left(T\left(Y/2+y\right)\right)\right]\right\\},$
where the Pomeron Green’s function is
$G_{enh}\left(Y\right)\,=\,1-\\\
exp\left(\frac{1}{T\left(Y\right)}\right)\,\frac{1}{T\left(Y\right)}\,\Gamma\left(0,\frac{1}{T\left(Y\right)}\right).$
(2)
Following GribovGRIBRT , we take into account in Eq. (1), the sum of the
Pomeron enhanced diagrams, considering them as a first approximation for the
exact Green function of the Pomeron (Fig. 1-b). Eq. (2) gives the explicit
form of this Green function for $\alpha^{\prime}_{I\\!\\!P}=0$. Also included
in Eq. (1) are the contributions of the secondary Reggeons.
Figure 1: Mueller diagramsMUDI for a single inclusive cross section. A bold
waving line presents the exact Pomeron Green function of (2), which is the sum
of the enhanced diagrams of Fig. 1-b. A zig-zag line corresponds to the
exchange of a Reggeon.
In Eq. (1) we have introduced two new phenomenological parameters,
$a_{{I\\!\\!P}{I\\!\\!P}}$ and
$a_{{I\\!\\!P}{I\\!\\!R}}$=$a_{{I\\!\\!R}{I\\!\\!P}}$, for the description of
hadron emission from the Pomeron and Reggeon. There is an additional
dimensional parameter, denoted by $Q$, which represents the average transverse
momentum of the produced minijets. $Q_{0}Q$ is the effective mass squared of
these minijets, with $Q_{0}$ = 2 GeV (see Ref.KL for details). $Q$ and
$Q_{0}$ are needed to calculate the pseudorapidity $\eta$ which replaces the
rapidity $y$. The relation between $y$ and $\eta$ is well known (see, for
example, Ref.KL ),
$y\left(\eta,Q\right)\,\,=\,\,\frac{1}{2}\ln\left\\{\frac{\sqrt{\frac{Q_{0}Q+Q^{2}}{Q^{2}}\,+\,\sinh^{2}\eta}\,\,+\,\,\sinh\eta}{\sqrt{\frac{Q_{0}Q+Q^{2}}{Q^{2}}\,+\,\sinh^{2}\eta}\,\,-\,\,\sinh\eta}\right\\},$
(3)
with the Jacobian
$h\left(\eta,Q\right)\,=\,\frac{\cosh\eta}{\sqrt{\frac{Q_{0}+Q}{Q}\,+\,\sinh^{2}\eta}}.$
(4)
In the parametrization of Ref.GLMM , the value of the Pomeron-particle
vertices are large. To compensate, we also sum the semi-enhanced diagrams
which contribute to the exact vertex of the Pomeron-particle interaction (see
Fig. 1-c). This vertex is equal SCH ; BORY to
$G_{enh}\left(y\right)g_{i}\left(b\right)\,\to\,g_{i}\left(b,y\right)\,=\,g_{i}\,G_{enh}(y)\,S_{i}(b)/(1+g_{i}G_{enh}(y)\,S_{i}(b)),$
(5)
whereGLMM ,
$S_{i}(b)=\frac{m^{2}_{i}}{4\pi}\,b\,m_{i}\,K_{1}(m_{i}\,b).$ (6)
Using Eq. (5), we obtain
$\displaystyle\frac{1}{\sigma_{in}}\,\frac{d\sigma}{dy}\,\,=\,\,\frac{1}{\sigma_{in}(Y)}\,\left\\{a_{PP}\left(\int
d^{2}b(\alpha^{2}\,g_{1}(b,Y/2-y)+\beta^{2}g_{2}(b,Y/2-y))\right.\right.$
$\displaystyle\left.\left.\times\,\int
d^{2}b(\alpha^{2}\,g_{1}(b,Y/2+y)+\beta^{2}g_{2}(b,Y/2+y)\right)\right.$
$\displaystyle\left.\,\,\,-\,\,a_{RP}\,\,(\alpha^{2}\,g^{R}_{1}+\beta^{2}g^{R}_{2})\,(\alpha^{2}\,\int
d^{2}b(\alpha^{2}\,g_{1}(b,Y/2-y)+\beta^{2}g_{2}(b,Y/2-y))\,e^{\Delta_{R}\,(Y/2+y)}\right.$
$\displaystyle\left.+\,\int
d^{2}b(\alpha^{2}\,g_{1}(b,Y/2+y)+\beta^{2}g_{2}(b,Y/2+y))\,e^{\Delta_{R}\,(Y/2-y)})\right\\}.$
Figure 2: Single inclusive density versus energy. The dotted data were taken
from Ref.PDG . The square
data points correspond to the experimental data from LHC by Alice
CollaborationAlice at W = 900 GeV and the CMS collaborationCMS at W = 900
and 2360 GeV.
Introducing a new notation,
$V(y)\,=\,\int d^{2}b\tilde{V}(b,y)\,\,=\,\,\int
d^{2}b(\alpha^{2}\,g_{1}(b,Y/2-y)+\beta^{2}g_{2}(b,Y/2-y)),$ (8)
we obtain a more compact expression for Eq. (7)
$\displaystyle\frac{1}{\sigma_{in}}\,\frac{d\sigma}{dy}\,\,=\,\,\frac{1}{\sigma_{in}(Y)}\,\left\\{a_{PP}V(y/2-y)V(Y/2+y)\,\right.$
$\displaystyle\left.-\,a_{RP}\,\,(\alpha^{2}\,g^{R}_{1}+\beta^{2}g^{R}_{2})\,(V(Y/2-y)\,e^{(\Delta_{R}\,(Y/2+y)}\,+\,V(Y/2+y)\,e^{(\Delta_{R}\,(Y/2-y)})\right\\}.$
(9)
Eq. (9) enables us to calculate the single inclusive density as a function of
the pseudo rapidity $\eta$.
As noted, this calculation entails three additional parameters. The
determination of these parameters from existing data PDG is not trivial.
Comparing the numbers corresponding to the data shown in Fig. 2, it is evident
that a conventional overall $\chi^{2}$ analysis is impractical, owing to the
quoted error bars of the 546 GeV data points, which are considerably smaller
than the error bars quoted for the other energies. The full lines in Fig. 2
are the results derived from a $\chi^{2}$ fit to the 200-1800 GeV data,
excluding the 546 GeV points. This fit yields a seemingly poor
$\chi^{2}/d.o.f=3.2$. Despite this, we consider this fit to be acceptable, as
the data points ”oscillate” about a uniform line with error bars which are
much smaller than their deviation from a smooth average. The results of this
fit are $a_{{I\\!\\!P}{I\\!\\!P}}$ = 75.7, $a_{{I\\!\\!P}{I\\!\\!R}}$ = 0.12
and Q = 3.8 GeV. In our procedure, the line for 546 GeV in Fig. 2 is
calculated with the model parameters and is visually compatible with the
experimental data points. Note that both the axes of Fig. 2 are linear, and
that our calculation coincides with the LHC experimental results Alice and
CMS . We have also made predictions for the higher energies at which the LHC
is expected to run, see Fig. 2. The contributions of the secondary Regge
trajectories are minimal. The experimental values for
$\sigma_{in}=\sigma_{tot}-\sigma_{el}-\sigma_{diff}$ were taken from Refs.PDG
; Alice ; CMS . For our predictions we have used the values of $\sigma_{in}$
calculated in our GLMM model. Our output over-estimates the few data points
with $\eta>4$ data at 546 and 900 GeV by up to 20$\%$. This is to be expected,
as we have not taken into account the parton correlations due to energy
conservation, which are important in the fragmentation region, but difficult
to include in the framework of Pomeron calculus.
To summarize, we have presented a theoretical formulation for single inclusive
hadron-hadron interactions based on our GLMM model. We have reproduced the p-p
dataPDG ; Alice ; CMS on single inclusive density as a function of the pseudo
rapidity. Our results provide additional support for our proposed Pomeron
approach. We have also presented predictions for the LHC energy range. These
predictions may soon be tested during the preliminary low luminosity LHC runs.
Acknowledgement: This research was supported in part by BSF grant #20004019.
## References
* (1) E. Gotsman, E. Levin, U. Maor and J.S. Miller, Eur. Phys. J. C57 (2008) 689 [arXiv:0805.2799 [hep-ph]]; [arXiv:0903.0247 [hep-ph]].
* (2) Alice collaboration, Eur. Phys. J. C. 65 (2010) 111 [arXiv:0911.5430 [hep-ex]].
* (3) CMS Collaboration, JHEP 02 (2010) 041 [arXiv:1002.0621 [hep-ex]].
* (4) M. L. Good and W. D. Walker, Phys. Rev. 120 (1960) 1857.
* (5) E.A. Kuraev, L.N. Lipatov, and F.S. Fadin, Sov. Phys. JETP 45, (1977) 199; Ya.Ya. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28, (1978) 22.
* (6) L.N. Lipatov, Phys. Rep. 286 (1997) 131; Sov. Phys. JETP 63 (1986) 904, and references therein.
* (7) L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep. 100 (1983) 1.
* (8) A. H. Mueller and J. Qiu, Nucl. Phys. B268 (1986) 427.
* (9) L. McLerran and R. Venugopalan, Phys. Rev. D49 (1994) 2233, 3352; D50 (1994) 2225; D53 (1996) 458; D59 (1999) 09400.
* (10) I. Balitsky, [arXiv:hep-ph/9509348]; Phys. Rev. D60 (1999) 014020 [arXiv:hep-ph/9812311].
* (11) Y.V. Kovchegov, Phys. Rev. D60, (1999) 034008 [arXiv:hep-ph/9901281].
* (12) J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, Phys. Rev. D59 (1999) 014014 [arXiv:hep-ph/9706377]; Nucl. Phys. B504 (1997) 415 [arXiv:hep-ph/9701284]; J. Jalilian-Marian, A. Kovner and H. Weigert, Phys. Rev. D59 (1999) 014015 [arXiv:hep-ph/9709432]; A. Kovner, J.G. Milhano and H. Weigert, Phys. Rev. D62 (2000) 114005 [arXiv:hep-ph/0004014]; E. Iancu, A. Leonidov and L.D. McLerran, Phys. Lett. B510 (2001) 133 (2001) [arXiv:hep-ph/0102009]; Nucl. Phys. A692 (2001) 583 [arXiv:hep-ph/0011241]; E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nucl. Phys. A703 (2002) 489 [arXiv:hep-ph/0109115]; H. Weigert, Nucl. Phys. A703, (2002) [arXiv:hep-ph/0004044].
* (13) M. A. Braun, Phys. Lett. B632 (2006) 297 [arXiv:hep-ph/0512057]; Eur. Phys. J. C16 (2000) 337 [arXiv:hep-ph/0001268]; Phys. Lett. B483 (2000) 115 [arXiv:hep-ph/0003004]; Eur. Phys. J. C33 (2004) 113 [arXiv:hep-ph/0309293]; C6, 321 (1999) [arXiv:hep-ph/9706373]. M. A. Braun and G. P. Vacca, Eur. Phys. J. C6 (1999) 147 [arXiv:hep-ph/9711486].
* (14) J. Bartels, M. Braun and G.P. Vacca, Eur. Phys. J. C40 (2005) 419 [arXiv:hep-ph/0412218]; J. Bartels and C. Ewerz, JHEP 9909 026 (1999) [arXiv:hep-ph/9908454]; J. Bartels and M. Wusthoff, Z. Phys. C66, (1995) 157; A. H. Mueller and B. Patel, Nucl. Phys. B425 (1994) 471 [arXiv:hep-ph/9403256]; J. Bartels, Z. Phys. C60 (1993) 471.
* (15) A. H. Mueller, Nucl. Phys. B415 (1994) 373; B437 (1995) 107.
* (16) E. Laenen and E. Levin, Nucl. Phys. B451 (1995) 207.
* (17) E. Levin and M. Lublinsky, Nucl. Phys. A763 (2005) 172 [arXiv:hep-ph/0501173]; Phys. Lett. B607 (2005) 131 [arXiv:hep-ph/0411121]; Nucl. Phys. A730 (2004) 191 [arXiv:hep-ph/0308279].
* (18) E. Levin, J. Miller and A. Prygarin, Nucl. Phys. A806 (2008) 245 [arXiv:0706.2944 [hep-ph]].
* (19) V.N. Gribov, Sov. Phys. JETP 26 (1968) 414 [Zh. Eksp. Teor. Fiz. 53 (1967) 654].
* (20) P.D.B. Collins, ”An introduction to Regge theory and high energy physics”, Cambridge University Press 1977.
* (21) Luca Caneschi (editor), ”Regge Theory of Low -$p_{T}$ Hadronic Interaction”, North-Holland 1989.
* (22) E. Levin, [arXiv:hep-ph/9808486]; [arXiv:hep-ph/9710546].
* (23) A.H. Mueller, Phys. Rev. D2 (1970) 2963.
* (24) D. Kharzeev and E. Levin, Phys. Lett. B523 (2001) 79 [arXiv:nucl-th/0108006].
* (25) A. Schwimmer, Nucl. Phys. B94 (1975) 445.
* (26) K.G. Boreskov, A.B. Kaidalov, V.A. Khoze, A.D. Martin and M.G. Ryskin, Eur. Phys. J. C44 (2005) 523 [arXiv:hep-ph/0506211].
* (27) C. Amsler et al. (Particle Data Group), Physics Letters, B667 (2008) 1.
|
arxiv-papers
| 2010-01-28T11:40:53 |
2024-09-04T02:49:08.082130
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Gotsman, E. Levin and U. Maor",
"submitter": "Errol Gotsman",
"url": "https://arxiv.org/abs/1001.5157"
}
|
1001.5249
|
# Absolute Monte Carlo estimation of integrals and partition functions
Artur B. Adib adiba@mail.nih.gov Laboratory of Chemical Physics, NIDDK,
National Institutes of Health, Bethesda, Maryland 20892-0520, USA
###### Abstract
Owing to their favorable scaling with dimensionality, Monte Carlo (MC) methods
have become the tool of choice for numerical integration across the
quantitative sciences. Almost invariably, efficient MC integration schemes are
strictly designed to compute ratios of integrals, their efficiency being
intimately tied to the degree of overlap between the given integrands.
Consequently, substantial user insight is required prior to the use of such
methods, either to mitigate the oft-encountered lack of overlap in ratio
computations, or to find closely related integrands of known quadrature in
absolute integral estimation. Here a simple physical idea—measuring the volume
of a container by filling it up with an ideal gas—is exploited to design a new
class of MC integration schemes that can yield efficient, absolute integral
estimates for a broad class of integrands with simple transition matrices as
input. The methods are particularly useful in cases where existing (importance
sampling) strategies are most demanding, namely when the integrands are
concentrated in relatively small and unknown regions of configuration space
(e.g. physical systems in ordered/low-temperature phases). Examples ranging
from a volume with infinite support to the partition function of the 2D Ising
model are provided to illustrate the application and scope of the methods.
## I Introduction
To introduce and place the ideas of the present contribution in context,
consider the paradigmatic problem of estimating the volume $V$ of a given
region as shown in Figure 1. In its most rudimentary form, MC estimates of $V$
proceed using the “hit-or-miss” idea, whereby the user designs a reference
region of known volume $V_{0}$ that fully overlaps with $V$, and draws random
points uniformly distributed in $V_{0}$ (cf. Fig. 1, top panel). An estimate
of $V$ can then be obtained from $V/V_{0}=f$, where $f$ is the fraction of
such points that fall inside $V$. As is immediately apparent, the efficiency
of this simple method hinges upon the amount of overlap between the volumes
$V_{0}$ and $V$: the tighter $V_{0}$ bounds $V$, the more the hits and hence
the better the quality of the estimate. Conversely, bounding volumes that have
little overlap with $V$ give rise to more misses than hits and hence to large
errors in the estimate. This poses a major obstacle to the implementation of
such methods, especially in cases where it is difficult to guess the precise
location of the volume and its boundaries, often leading to the design of an
unnecessarily conservative (large) reference region, and hence to very
inefficient estimates of $V$.
Despite its simplicity, the above example captures the central issues
pertinent to integral estimation problems in general, for which more
sophisticated methods exist liu-book ; frenkel02 ; gregory-book . Notably, in
most efficient Monte Carlo techniques for normalizing constant or free energy
estimation, a suitable family of intermediate integrands is required to
interpolate between the two desired integrands (or between the available
reference system and the integrand of interest, in absolute integral
estimation), the extent of overlap between the integrands dictating the
efficiency of the methods much like in the example above bennett76 ; gelman98
; jarzynski02-lncse . Such requirements become particularly daunting when the
integrands in question are highly concentrated in unknown and disparate
regions of configuration space, as is typically the case in most interesting
physical problems, thereby demanding substantial user insight prior to the
applicability of such methods.
In the present contribution, a new family of MC integration strategies that
can greatly alleviate the demand for such types of insights will be
introduced. As expressed by the central results underpinning these methods,
Eqs. (4) and (7), the integrals of interest ($Z$, Eq. (1)) are computed
individually, thus fundamentally departing from the aforementioned
alternatives that rely on ratios of similar integrals. At the core of these
new strategies is the use of integration spaces of enlarged dimensionality
(“replicas”), a concept already widely invoked to speed up Markov chain
simulations (parallel tempering deem05 , evolutionary Monte Carlo liu-book ,
etc.), combined with the replacement of reference integrals by normalized
transition matrices frenkel02 (also known as transition functions liu-book
or kernels gelman98 in the Markov chain literature). Two main variants will
be presented: the first, more physically intuitive, requires a fluctuating
number of replicas (Fig. 1, bottom panel); the second, more abstract but also
more easily adaptable to existing replica simulations, uses a fixed number of
replicas along with “virtual” insertions/deletions of replicas. These two
variants are complementary to each other much like grand-canonical Monte Carlo
(GCMC) and Widom’s test particle insertion method in simulations involving
chemical potentials frenkel02 .
## II Theory
To see how in principle the simultaneous use of multiple configurations
(replicas) allows for the absolute computation of integrals, let us go back to
the volume estimation problem of Fig. 1. As illustrated in the bottom panel of
that figure, the volume $V$ of interest can be estimated by equilibrating it
with an infinite reservoir of ideal gas particles (replicas) of density
$\rho_{0}$, and measuring the average number $\langle N\rangle$ of particles
inside $V$, to obtain $V=\langle N\rangle/\rho_{0}$. This physical idea can be
implemented computationally by means of a generalized form of the usual grand-
canonical Monte Carlo method frenkel02 , where the particle reservoir is at
density $\rho_{0}$ (or, equivalently, at the corresponding chemical
potential), and attempted particle insertions/deletions take place in the
neighborhood of an existing particle rather than inside a fixed volume that
bounds $V$, that neighborhood being defined by a transition matrix
$T(x^{\prime}|x)$. These are the central ideas that motivated the development
of the two versions described below.
In general, we would like to estimate integrals of the type
$Z=\int_{\Omega}dx\,\pi(x),$ (1)
where $\Omega$ is the support of the integral, and $\pi(x)$ is a positive-
definite function (see below, however) of the $d$-dimensional vector $x$; e.g.
$\pi(x)=e^{-E(x)}$ for most physical problems with energy function $E(x)$.
Partition functions of discrete systems, i.e. $Z=\sum_{x\in\Omega}\pi(x)$, can
be dealt with in an analogous fashion. We are given a (normalized) transition
matrix $T(x^{\prime}|x)$, such as those routinely used in the trial part of
the Metropolis algorithm frenkel02 ; for example, $T(x^{\prime}|x)$ could be a
uniform probability distribution up to some distance $|x^{\prime}-x|$ from $x$
(see dashed circle in Fig. 1), or a Gaussian function centered about $x$. As
with Metropolis and other Markov chain simulations, the width of these
distributions can be chosen during a preliminary run of the simulation (see
below).
For non-positive definite integrands, one can invoke the identity
$Z=Z_{|\pi|}\,\langle\text{sgn}(\pi(x))\rangle_{|\pi|},$ (2)
where $Z$ is defined by Eq. (1), $Z_{|\pi|}=\int_{\Omega}dx\,|\pi(x)|$, and
the average of the sign function of $\pi(x)$ is with respect to points $x$
sampled from $|\pi(x)|$. Provided they exist, both quantities on the right
hand side of this identity are immediately available from the methods below.
### II.1 Varying number of replicas
For the first version of the method, we would like to simulate a system of
replicas in contact with a reservoir of ideal (non-interacting) replicas of
specified density $\rho_{0}$, such that each replica $x_{i}$ in $\Omega$
independently samples the distribution $\pi(x_{i})$. The corresponding grand-
canonical partition function is thus
$\displaystyle Q$
$\displaystyle=\sum_{N=1}^{\infty}\frac{\rho_{0}^{N}}{N!}\,\int_{\Omega}dx_{1}\cdots\int_{\Omega}dx_{N}\,\pi(x_{1})\cdots\pi(x_{N})$
$\displaystyle=\sum_{N=1}^{\infty}\frac{(\rho_{0}Z)^{N}}{N!}=e^{\rho_{0}Z}-1.$
(3)
Note that, unlike the traditional case where $0\leq N\leq\infty$, here the sum
starts at $N=1$, as in the present method (see below) replica
insertions/deletions take place in the neighborhood of at least one existing
replica. Provided we can simulate according to this partition function, the
desired integral $Z$ can be found from the equality
$\frac{\langle N^{2}\rangle}{\langle N\rangle}=1+\rho_{0}Z,$ (4)
which follows straightforwardly from Eq. (3) by computing each moment of $N$
separately. (Alternatively, one can use $\langle
N\rangle=\rho_{0}Z/(1-e^{-\rho_{0}Z})$ or any other relationship between
moments of $N$ and $Z$, and numerically solve the transcendental equation for
$Z$; the question of which $Z$ estimator is more efficient will be left for
future studies).
An algorithm that samples according to the above grand-canonical partition
function goes as follows. At the beginning of the algorithm we are given at
least one point $x_{1}$ that belongs to $\Omega$; let us assume the general
case where we already have $N$ replicas in $\Omega$, and let us denote the
vector of coordinates of the replicas by $x^{N}\equiv(x_{1},\ldots,x_{N})$. We
then decide whether to insert or remove a replica, typically—but not
necessarily (see Methods section)—with equal probability. If the decision was
an insertion, we sample a new replica coordinate $x_{N+1}$ from
$T(x_{N+1}|x_{i})$, where $x_{i}$ is a randomly chosen coordinate from the
existing $N$, and accept its insertion with probability
$p_{\text{acc}}(x^{N+1}|x^{N})=\min\left\\{1,\frac{N}{N+1}\,\frac{\rho_{0}\,\pi(x_{N+1})}{\sum_{i=1}^{N}T(x_{N+1}|x_{i})}\right\\},$
(5)
except when $x_{N+1}$ lies outside $\Omega$, in which case an immediate
rejection takes place. Similarly, if the decision was to try a deletion, we
randomly pick a replica $x_{j}$ from the existing $N$, and accept its deletion
with probability
$p_{\text{acc}}(x^{N-1}|x^{N})=\min\left\\{1,\frac{N}{N-1}\,\frac{\sum_{i\neq
j}^{N}T(x_{j}|x_{i})}{\rho_{0}\,\pi(x_{j})}\right\\},$ (6)
where $x^{N-1}$ is $x^{N}$ excluding $x_{j}$, and the sum over $i$ excludes
$i=j$. An exception is the case where only one replica remains, which is
always rejected. A proof that this algorithm samples according to Eq. (3)
follows by detailed balance (see Methods section). When the replicas
correspond to particles inserted uniformly in a fixed volume $V$, i.e.
$T(x^{\prime}|x)=1/V$, the above acceptance probabilities reduce to those of
Ref. frenkel02 (with the due mappings between $\rho_{0}$ and chemical
potential, and between $\pi(x)$ and the Boltzmann factor). Of course, it is
also possible to perform ordinary $\pi$-preserving Monte Carlo moves on each
replica before attempted insertion/deletions frenkel02 (Fig. 3 uses this
idea).
By repeating the above procedure a number of times, the simulation will
eventually equilibrate, and the number of replicas will fluctuate about its
mean value $\langle N\rangle$. If the equilibration is too slow, i.e. too few
replica insertions/deletions are accepted, the width of the distribution
$T(x^{\prime}|x)$ about $x$ can be adjusted accordingly during a preliminary
run, in a fashion analogous to what is done in Metropolis Monte Carlo to keep
the rate of accepted trial moves within a reasonable range frenkel02 .
Likewise, if the number of replicas starts to grow beyond one’s computational
capabilities, or diminish until it hardly departs from unity, $\rho_{0}$ can
be adjusted so that $\langle N\rangle$ is a reasonable number consistent with
one’s computing power. In practice, for many-particle systems with extensive
free energies (i.e. $Z\sim e^{N}$), it is best to write
$\rho_{0}=e^{\mu_{0}N}$ and adjust $\mu_{0}<0$ instead.
Note that the present algorithm generalizes standard grand-canonical
simulation frenkel02 in two crucial ways. First, it inserts and removes
entire replicas of the system of interest as opposed to individual particles
of a many-body system. This conceptual difference is essentially what allows
one to relate moments of $N$ to the partition function of the system of
interest (Eq. (4)). Second, the replicas are introduced in the neighborhood of
an existing replica instead of inside a fixed region, as prescribed by the
transition matrix. This allows for efficient simulation when the integrand of
interest is sharply peaked about unknown regions of configuration space (cf.
Fig. 5). Although the use of arbitrary transition matrices in grand-canonical
simulations is known in the mathematical literature moller-book , to our
knowledge the use of such ideas for integral/partition function estimation is
new.
### II.2 Fixed number of replicas
The second version of the replica gas method introduces two important
advantages. First, the number of replicas is constant as opposed to
fluctuating, making it more convenient for parallel computing architectures,
and second the replicas can be simulated at different temperatures. These
features also make the method easily implemented in existing parallel
tempering (replica exchange) simulations, thereby benefiting from the greatly
enhanced equilibration rates of these simulations deem05 . The integrals of
interest can then be estimated by computing two separate averages involving
both the integrand $\pi(x)$ and the transition matrix $T(x^{\prime}|x)$, as
described below.
To introduce the method in its simplest form, let us first assume that only
two replicas exist ($N=2$), each of them independently sampling the
distributions $\pi$ and $\tilde{\pi}$, via e.g. Metropolis. The integral of
interest is $Z$, as in Eq. (1), and
$\tilde{Z}=\int_{\Omega}dx\,\tilde{\pi}(x)$ is an auxiliary integral; the
auxiliary distribution $\tilde{\pi}$ is arbitrary (for example, it could be
$\pi$ itself), but in typical applications it corresponds to $\pi$ at a higher
temperature, i.e. $\tilde{\pi}(x)=\pi^{\beta}(x)$, where $0<\beta<1$. Then the
following identity holds:
$\displaystyle Z$
$\displaystyle=\frac{\int_{\Omega}dx[\tilde{\pi}(x)/\tilde{Z}]\int_{\Omega}dx^{\prime}\,T(x^{\prime}|x)\cdot\pi(x^{\prime})}{\int_{\Omega}dx[\tilde{\pi}(x)/\tilde{Z}]\int_{\Omega}dx^{\prime}[\pi(x^{\prime})/Z]\cdot
T(x^{\prime}|x)}$
$\displaystyle\equiv\frac{\left\langle\pi(x^{\prime})\right\rangle_{\tilde{\pi},T}}{\left\langle
T(x^{\prime}|x)\right\rangle_{\tilde{\pi},\pi}},$ (7)
where the average $\langle\mathcal{O}(x,x^{\prime})\rangle_{f,g}$ of an
observable $\mathcal{O}(x,x^{\prime})$ means that configurations $x$
($x^{\prime}$) are sampled from the distribution $f$ ($g$). Thus, the
numerator in the above result requires $x$ to be sampled from $\tilde{\pi}(x)$
while $x^{\prime}$ is sampled from $T(x^{\prime}|x)$ (“virtual replica
insertion,” in analogy with Widom’s method frenkel02 ), and for each such pair
of configurations one computes the value of $\pi(x^{\prime})$. Likewise, for
the average in the denominator, $x$ is sampled from $\tilde{\pi}(x)$ while
$x^{\prime}$ is sampled from $\pi(x^{\prime})$, and for each pair one
evaluates $T(x^{\prime}|x)$ (“virtual replica deletion”). In the limit of
infinite samples, the ratio of these two averages converges to $Z$ as
expressed by Eq. (7).
For simulations with multiple replicas at different temperatures
($\beta_{1},\ldots,\beta_{N}$), one can simply combine (i.e. sum) the above
result for each pair of replicas. Thus, for the Ising model results in Fig. 4,
the equation
$Z(\beta_{i})=\frac{\sum_{j\neq i}\left\langle
e^{-\beta_{i}E(x^{\prime})}\right\rangle_{\beta_{j},T}}{\sum_{j\neq
i}\left\langle T(x^{\prime}|x)\right\rangle_{\beta_{j},\beta_{i}}}$ (8)
was used. The sums run over each replica $j$ at temperature $\beta_{j}$,
except $j=i$. The energy function $E(x)$ for a spin configuration $x$ is the
usual Ising model function $E(x)=-\sum_{\langle k,l\rangle}x_{k}x_{l}$ with
periodic boundary conditions krauth-book . The transition matrix adopted
$T(x^{\prime}|x)$ generates a new spin configuration $x^{\prime}$ by flipping
each spin of $x$ with probability $p_{\text{flip}}$. Thus,
$T(x^{\prime}|x)=(p_{\text{flip}})^{\parallel
x^{\prime}-x\parallel}(1-p_{\text{flip}})^{N-\parallel
x^{\prime}-x\parallel}$, where $\parallel x^{\prime}-x\parallel$ is the
Hamming distance (number of spin mismatches) between the configurations $x$
and $x^{\prime}$, and $N$ is the total number of spins.
## III Results and Discussion
For illustrative purposes, the results of a volume estimation problem in two
spatial dimensions using the version with varying number of replicas are
reported in Figure 2. This example was chosen due to its infinite support, a
property that would render the use of importance sampling methods difficult,
as they would require the design of a non-trivial reference volume $V_{0}$
with similar support and known quadrature (recall that in principle we do not
know where the integrand is concentrated, or where its boundaries are). The
present method performs well in such circumstances without any prior
information concerning the support of the integrand, by using a simple uniform
transition matrix $T(x^{\prime}|x)$ (Fig. 2, dashed square).
As an application to integrals more general than simple volumes, in Fig. 3 a
representative estimate of $Z=\int_{-\infty}^{\infty}dx\sin(x)/x$ is shown.
Note that this integrand is non-positive definite, so Eq. (2) was used. The
quantities on the right hand side of that equation were estimated using the
varying number of replicas version of the method, with the positive-definite
integrand $|\pi(x)|=|\sin(x)/x|$.
In Figure 4 the partition function of the two-dimensional Ising model is
computed to demonstrate the version with fixed number of replicas (similar
results are obtained with the non-fixed version). At low temperatures, i.e.
ordered states, the replicas are densely localized about the spin-up and spin-
down states, and hence a local transition matrix $T(x^{\prime}|x)$ is
sufficient to ensure efficient convergence of the averages. Conversely, for
higher temperatures close to the disordered state and above, the relevant
configuration space—and hence the spread of the replicas—grows beyond the
reach of the local transition matrix adopted, thus causing the averages to
converge more slowly (cf. right side of Fig. 5).
To understand such convergence issues in more detail, consider for simplicity
Eq. (7) when $\tilde{\pi}=\pi$ (see below for the version with varying number
of replicas). In order for the averages in Eq. (7) to converge efficiently,
the transition matrix $T(x^{\prime}|x)$ has to be such that:
* (a)
Most configurations $x^{\prime}$ sampled from $T(x^{\prime}|x)$ fall in
typical regions of $\pi$ for any typical configuration $x$ sampled from $\pi$
(so that the numerator is not dominated by those rare configurations with high
values of $\pi(x^{\prime})$);
* (b)
Most configurations $x$,$x^{\prime}$ sampled from $\pi$ fall in typical
regions of $T(x^{\prime}|x)$ (so that the denominator is not dominated by
those rare events that cause $T(x^{\prime}|x)$ to be of appreciable value).
In the Ising model example of Fig. 4, where $T(x^{\prime}|x)$ typically flips
only a few spins of $x$, the low temperature estimates converge faster as
typical spin configurations only differ by a few spins, thereby satisfying
both requirements, while at higher temperatures close to $T_{c}$ and above,
any two typical spin configurations differ by a substantial number of spins,
and hence the requirement (b) is violated. Adding more replicas, increasing
the value of $p_{\text{flip}}$ for higher temperatures, or using non-local
transition matrices (such as those of cluster algorithms krauth-book ) can
alleviate the problem, but such issues will be left for future studies.
Note that analogous convergence issues arise in the version with varying
number of replicas. Indeed, as can be seen by inspection of Eqs. (5) and (6),
the acceptance probabilities for insertion and deletion are affected by the
choice of $T(x^{\prime}|x)$ much like the averages in Eq. (7) are affected by
criteria (a) and (b) above. (A separate issue is how well the replicas explore
the energy landscape. In the fixed number of replicas version, different
temperatures are used to overcome energy barriers. Although replica exchange
operations can be combined with the varying number of replicas method, as a
proof of concept for the convergence issues above, it suffices to start a
population of replicas that populate spin-up and spin-down states equally).
As illustrated by the above example, the most attractive use of the present
methods lies in problems where the integrand is sufficiently localized, so
that a general-purpose, local transition matrix can be used to yield efficient
results with moderate numbers of replicas. These are precisely the problems
for which existing importance sampling-based methods are most demanding, and
thus these methods can be seen as complementary to each other (see Fig. 5). It
should be noted that the so-called “flat histogram” Monte Carlo methods
wang01a ; shell07 are also able to bypass some of the difficulties with
importance sampling strategies in some cases, especially for discrete systems.
However, the required human input and scope of such integration methods are
rather different: they require the existence and knowledge of suitable order
parameters and their ranges (this being particularly difficult for entropic
problems, such as that of Fig. 2), knowledge of ground state degeneracies, and
for continuum systems suffer from systematic errors due to discretization
schemes, although attempts to alleviate some of these problems have been put
forward troster05 .
In summary, the present contribution has introduced two variants of a novel
Monte Carlo strategy for estimating integrals and partition functions. Both
versions can be seen as complementary to existing importance sampling or free
energy methods liu-book ; gelman98 ; jarzynski02-lncse , in that their utility
is generally best when the integrands are concentrated in relatively small and
unknown regions of configuration space. Both continuum and discrete systems
are equally amenable to their use. By shifting focus from importance sampling
functions to transition matrices, it is expected that these methods will
encourage a change of paradigm in Monte Carlo integration.
## IV Methods
In this section it will be shown that Eqs. (5) and (6) satisfy the detailed
balance condition
$p(x^{N})\cdot p_{\text{tr}}(x^{N+1}|x^{N})\cdot
p_{\text{acc}}(x^{N+1}|x^{N})=p(x^{N+1})\cdot
p_{\text{tr}}(x^{N}|x^{N+1})\cdot p_{\text{acc}}(x^{N}|x^{N+1}).$ (9)
According to the grand-canonical partition function Eq. (3), the probability
of observing the microstate $x^{N}$ is given by
$p(x^{N})\propto\frac{\rho_{0}^{N}\pi(x_{1})\cdots\pi(x_{N})}{N!},$ (10)
where the proportionality constant does not depend on the replica coordinates
or $N$. Note carefully the difference between the distribution of the labeled
microstate $x^{N}$, corresponding to replica with label “1” being at $x_{1}$,
“2” at $x_{2}$, etc, and that of the unordered set of coordinates
$\\{x^{N}\\}=\\{x_{1},\ldots,x_{N}\\}$, corresponding any replica being at
$x_{1}$, another arbitrary replica at $x_{2}$, etc. This probability is given
by
$p(\\{x^{N}\\})=\sum_{P}p(x^{N})\propto\rho_{0}^{N}\pi(x_{1})\cdots\pi(x_{N}),$
(11)
where $\sum_{P}$ is the sum over all possible permutations of
$x_{1},\ldots,x_{N}$. Since the replicas are indistinguishable, we have
$p(\\{x^{N}\\})=N!\,p(x^{N})$, as in above. Of course, it is possible to use
either description (labeled or unlabeled), provided the correct probability
distributions are used (Eq. (10) or Eq. (11), respectively). In this section,
following filinov69 , we will only show the proof using labeled states, i.e.
Eq. (10). It is a simple exercise to modify the development below for the
unlabeled case; the acceptance probabilities, of course, are unchanged.
Our acceptance probabilities are of the Metropolis-Hastings type, which by
construction satisfy detailed balance. In the present notation, the formulas
are
$p_{\text{acc}}(x^{N+1}|x^{N})=\text{min}\left\\{1,\frac{p_{\text{tr}}(x^{N}|x^{N+1})}{p_{\text{tr}}(x^{N+1}|x^{N})}\frac{p(x^{N+1})}{p(x^{N})}\right\\},$
(12)
and analogously for $p_{\text{acc}}(x^{N}|x^{N+1})$. According to the
insertion/deletion algorithm described before Eq. (6), the trial probabilities
for going between the states $x^{N}=(x_{1},\ldots,x_{N})$ and
$x^{N+1}=(x_{1},\ldots,x_{N},\xi)$ are given by
$(x_{1},\ldots,x_{N})\xrightleftharpoons[(1-q)\cdot\frac{1}{N+1}]{q\cdot\frac{1}{N+1}\cdot\sum_{i=1}^{N}\frac{1}{N}T(\xi|x_{i})}(x_{1},\ldots,x_{N},\xi),$
(13)
where $p_{\text{tr}}(x^{N+1}|x^{N})$ is given by the expression above the
arrows, and $p_{\text{tr}}(x^{N}|x^{N+1})$ by the one below them. In the
insertion trial probability, $q=1/2$ is the probability to try an insertion as
opposed to a deletion, $1/(N+1)$ is the probability that the new coordinate
$\xi$ is inserted in a given slot of the vector $x^{N+1}$ (in the above case,
the last slot), $1/N$ is the probability to pick coordinate $x_{i}$ as
reference, and $T(\xi|x_{i})$ is the probability to sample the candidate
position $\xi$ given the chosen reference. In the deletion trial probability,
$(1-q)=1/2$ is the probability to try a deletion, and $1/(N+1)$ is the
probability that the replica at $\xi$ will be chosen for attempted removal
among the existing ones. Plugging these trial probabilities in Eq. (12), we
obtain Eq. (5) (an analogous procedure gives Eq. (6)). The case where $q\neq
1/2$ can be easily taken care of by modifying the acceptance probabilities
accordingly.
Alternatively, detailed balance can be directly proven by plugging the trial
probabilities in Eq. (13) and the acceptance probabilities given by Eqs. (5)
and (6) into Eq. (9).
###### Acknowledgements.
The author would like to thank Attila Szabo, Gerhard Hummer, and David Minh
for discussions and suggestions. This research was supported by the Intramural
Research Program of the NIH, NIDDK.
## References
* (1)
* (2) Liu, JS (2004) _Monte Carlo Strategies in Scientific Computing_ (Springer, New York).
* (3) Frenkel, D, Smit, B (2002) _Understanding Molecular Simulation: From Algorithms to Applications_ (Academic, San Diego), 2nd edn.
* (4) Gregory, PC (2005) _Bayesian Logical Data Analysis for the Physical Sciences_ (Cambridge, Cambrige, UK).
* (5) Bennett, CH (1976) Efficient estimation of free energy differences from Monte Carlo data. _J. Comp. Phys._ 22:245-268.
* (6) Gelman, A & Meng, XL (1998) Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. _Stat. Sci._ 13:163-185.
* (7) Jarzynski, C (2002) Equilibrium and nonequilibrium foundations of free energy computational methods. In Schlick, T. & Gan, H. H. (eds.) _Computational Methods for Macromolecules_ , of _Lecture Notes in Computational Science and Engineering_ (Springer, Berlin), 24:287-303.
* (8) Earl, DJ & Deem, MW (2005) Parallel tempering: Theory, applications, and new perspectives. _Phys. Chem. Chem. Phys._ 7:3910-3916.
* (9) Möller, J, Waagepetersen, RP (2004) _Statistical Inference and Simulation for Spatial Point Processes_ (Chapman Hall/CRC, Boca Raton)
* (10) Krauth, W (2006) _Statistical Mechanics: Algorithms and Computations_ (Oxford, New York).
* (11) Wang, F & Landau, DP (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. _Phys. Rev. Lett._ 86:2050-2053.
* (12) Shell, MS, Panagiotopoulos, A & Pohorille, A (2007) Methods based on probability distributions and histograms. In Chipot, C. & Pohorille, A. (eds.) _Free Energy Calculations: Theory and Applications in Chemistry and Biology_ , of _Springer Series in Chemical Physics_ (Springer), 86:77-118.
* (13) Tröster, A & Dellago, C (2005) Wang-landau sampling with self-adaptive range. _Phys. Rev. E_ 71:066705.
* (14) Norman, GE & Filinov, VS (1969) Investigations of phase transitions by a Monte Carlo method. _High. Temp. (USSR)_ 7:216.
Figure 1: Monte Carlo estimation of a volume $V$ (shaded region) by means of
sampling from a reference volume $V_{0}$ (top), and equilibration with a
hypothetical, infinite reservoir of ideal gas particles at density $\rho_{0}$
(bottom). In the former, one draws random points uniformly from $V_{0}$ and
counts the fraction $f$ that lands in $V$, to obtain $V=V_{0}f$. In the
latter, one performs a grand-canonical Monte Carlo simulation frenkel02 at
reservoir density $\rho_{0}$, and monitors the average number $\langle
N\rangle$ of ideal gas particles in $V$; upon equilibration, the density of
particles in $V$ equals that of the reservoir, and thus $V=\langle
N\rangle/\rho_{0}$. (In practice, due to the constraint $N\geq 1$, this
formula for $V$ needs to be modified slightly; see Eq. (4)). Each particle
corresponds to a point (“replica”) residing in $V$, and attempted replica
insertions/removals take place in the neighborhood (dashed circle) of an
existing replica $x$, defined by the transition matrix $T(x^{\prime}|x)$ of
the method. A version of the algorithm with fixed number of replicas—possibly
at different temperatures—is also described in the text. Figure 2:
Estimation of a two-dimensional volume $Z$ (yellow region) using the replica
gas method with varying number of replicas, Eq. (4). The volume $Z$ is defined
by the region $|x_{2}|\leq e^{-|x_{1}|+1}$ for $|x_{1}|>1$, and $|x_{2}|\leq
1-\ln|x_{1}|$ for $|x_{1}|\leq 1$. The points correspond to the replica
configurations at the end of one simulation, and the dashed square defines the
boundaries of the adopted transition matrix $T(x^{\prime}|x)$ (uniform
distribution, each side of length unity) for the particular configuration $x$
shown. Inset: Histogram of $20$ independent estimates of $Z$ using the replica
gas method with $10^{6}$ attempted insertions/deletions, and $\rho_{0}=5$. The
exact value of $Z$, obtained by analytic quadrature, is $Z=12$. Figure 3: An
illustrative non-positive definite integrand, $\pi(x)=\sin(x)/x$. The scale in
the center of the graph corresponds to the parameter $\Delta=5$ in the uniform
transition matrix $T(x^{\prime}|x)=1/\Delta$ for $|x^{\prime}-x|<\Delta/2$
(zero otherwise) adopted for the results shown in the inset. Inset: Running
estimate of $Z=\int_{-\infty}^{\infty}dx\,\pi(x)$ using Eq. (2), where
$Z_{|\pi|}$ is estimated via the replica gas method with varying number of
replicas, Eq. (4). The mean sign function of $\pi$ required by this last
equality, $\langle\text{sgn}(\pi(x))\rangle_{|\pi|}$, is also obtained from
this run, by averaging $\text{sgn}(\pi(x))$ over all replicas $x$ during the
simulation. The dashed red line is the exact result $Z=\pi=3.141592...$. For
this example, $\rho_{0}=1$, and $10$ ordinary Monte Carlo moves per replica
are performed between every attempted insertion/deletion (GCMC step). The
ensuing number of replicas fluctuated about $N=10$. Figure 4: Natural
logarithm of the partition function $Z$ of the two-dimensional Ising model
with $N=32\times 32$ spins, according to the replica gas method (circles), and
the exact Kaufman formula krauth-book (dashed curve). The replica gas results
were obtained using the fixed number of replicas version, Eq. (8), with $20$
replicas at the temperatures corresponding to the data points shown (similar
results are obtained with varying number of replicas, see text), with error
bars indicating the standard deviation of 8 independent runs. Importance
sampling results were obtained using the ideal (non-interacting) spin
partition function $Z_{\text{id}}=2^{N}$ and
$Z_{\text{Ising}}/Z_{\text{id}}=\langle e^{-\beta
E_{\text{Ising}}(x)}\rangle_{\text{id}}$, with $10^{5}$ independent
configurations $x$ sampled from the ideal reference system (increasing this
number to $10^{6}$ does not lead to appreciable changes in the results). For
disordered states (i.e. $k_{B}T$ close to or higher than the critical
temperature $k_{B}T_{c}=2.269$), the partition function is no longer dominated
by a small fraction of the configuration space, and the replica gas method
converges more slowly with the adopted (local) transition matrix (see also
Fig. 5). The replica exchange simulation took $10^{5}$ MC steps, with an
attempted exchange every $100$ steps, where each MC step is a simple spin
flip. For the transition matrix sampling, $p_{\text{flip}}=1/N$ (cf.
discussion after Eq. (8)). Figure 5: Comparison of the merits of importance
sampling (left) and replica gas (right) methods for sparse (top) and localized
(bottom) integrands. Integrands are represented by their densest regions
(shaded curvy shapes). For physical systems, sparse integrands correspond to
Boltzmann factors at high temperatures, while at low temperatures the
integrands tend to be localized in a small fraction of configuration space
(e.g. magnetized spin systems, crystals, proteins in their native state, etc).
In importance sampling, one typically has at their disposal a general-purpose
sparse reference system (polygon on the left) such as an ideal gas, which is
generally sufficient to ensure proper sampling at high temperatures, but not
at lower ones. Conversely, in replica gas methods one typically has at their
disposal a local transition matrix (small boxes on the right) that is
generally sufficient to sparingly “cover” the integrand of interest (cf.
efficiency criteria (a) and (b) discussed in the text) at low temperatures,
but not at higher temperatures.
|
arxiv-papers
| 2010-01-28T19:34:54 |
2024-09-04T02:49:08.087093
|
{
"license": "Public Domain",
"authors": "Artur B. Adib",
"submitter": "Artur Adib",
"url": "https://arxiv.org/abs/1001.5249"
}
|
1001.5290
|
# Classification of quantum relativistic orientable objects
D.M. Gitmana and A.L. Shelepinb
aInstituto de Física, Universidade de São Paulo,
Caixa Postal 66318-CEP, 05315-970 São Paulo, S.P., Brazil
bMoscow Institute of Radio Engineering, Electronics and Automation,
Prospect Vernadskogo, 78, 117454, Moscow, Russia E-mail:
gitman@dfn.if.usp.brE-mail: alex@shelepin.msk.ru
###### Abstract
Started from our work "Fields on the Poincaré Group and Quantum Description of
Orientable Objects" (EPJC,2009), we consider here a classification of
orientable relativistic quantum objects in $3+1$ dimensions. In such a
classification, one uses a maximal set of $10$ commuting operators (generators
of left and right transformations) in the space of functions on the Poincaré
group. In addition to usual $6$ quantum numbers related to external symmetries
(given by left generators), there appear additional quantum numbers related to
internal symmetries (given by right generators). We believe that the proposed
approach can be useful for description of elementary spinning particles
considering as orientable objects. In particular, their classification in the
framework of the approach under consideration reproduces the usual
classification but is more comprehensive. This allows one to give a group-
theoretical interpretation to some facts of the existing phenomenological
classification of known spinning particles.
## Introduction
In our previous work [1], we discussed an new approach for description
orientable quantum relativistic objects in $3+1$-dimensions. In such an
approach, the orientable object is associated with a scalar field $f(h)$ on
the Poincaré group $M(3,1)=T(4){\times\\!\\!\\!\\!\\!\\!\supset}{\rm
Spin}(3,1)$, $h\in M(3,1)$. The field depends on a $4$-vector $x^{\mu},$ which
gives a position of the object, and on a $6$-parameter matrix
$Z\in\mathrm{Spin}(3,1)$, which describes the object orientation.
The field $f(h)$ admits two kinds of transformations, corresponding to a
change of the laboratory, or space-fixed reference frame (s.r.f.), as well as
to a change of the local, or body-fixed reference frame (b.r.f.),
$T(g_{l},g_{r})f(h)=f(g_{l}^{-1}hg_{r}).$ (1)
Here left multiplication by $g_{l}^{-1}$ corresponds to a change of the s.r.f.
(Lorentz transformations), whereas right multiplication by $g_{r}$ corresponds
to a change of the b.r.f.. There are two sets of transformation generators –
right and left ones, and they are used to construct a maximal set of commuting
operators in the space of functions $f(h)$.
The set of all the transformations (1) form the direct product $M(3,1)\times
M(3,1).$ Possible external symmetries correspond to the left transformations,
whereas some of possible internal symmetries correspond to the right
transformations. Indeed, external symmetries are usually defined as symmetries
of the enclosing space, i.e., symmetries with respect to a change of the
s.r.f., while internal ones are defined as symmetries of the body itself, in
particular, symmetries with respect to a change of the b.r.f..
We believe that the proposed approach can be useful for description of known
elementary spinning particles. In particular, their classification within this
approach (considering spinning particles as orientable objects) could be more
complete and consistent.
A classification of orientable objects is natural to define with the help of a
maximal set of commuting operators, which are constructed from the generators
of transformations (1). This set contains $10$ commuting operators (according
to the number of the group parameters) and consists of $4$ operator functions
of the left generators, $4$ operator functions of the right generators and $2$
Casimir operators, which can be constructed from the left generators, as well
as from the right generators. Such a classification attributes $10$ quantum
numbers to an orientable quantum object.
On the other side, in relativistic quantum theory of point-like objects there
exists the Wigner’s classification [2], based on the left generators
(generators of external symmetry transformations). Two Casimir operators
determine the representation (mass and spin), while the remaining $4$
operators determine, for instance, helicity and momentum. Thus, the Wigner’s
classification attributes only $6$ quantum numbers to a relativistic quantum
point-like object.
One ought to mention that in the 1960s, attempts were made to unite internal
and external symmetries in the framework of one group. Soon, however, the so-
called no-go theorem [3] was proved (under some very general assumptions),
stating that the symmetry group of the $S$-matrix is locally isomorphic to a
direct product of the Poincaré group and the group of internal symmetries.
However, on this basis, one often makes too strong conclusion that a
nontrivial relation between internal and external symmetries is impossible.
As was already said, the transformations (1) of a field $f(h)$ form the direct
product of groups of internal and external symmetries, in agreement with
mentioned no-go theorem. Nevertheless, as will be demonstrated below, a
nontrivial relation between internal and external quantum numbers is possible.
Both transformation groups, corresponding to a change of the s.r.f. and
b.r.f., act in the same space of $10$-parameter functions $f(h)$ and have the
same Casimir operators which define the mass and the spin. By fixing
eigenvalues of the Casimir operators, and therefore fixing the representation,
we obviously impose some conditions on the spectra of both left and right
operators that enter the maximal set. Thus, in spite of the fact that the left
and the right operators commute, their spectra are not independent.
Following our work [1], studying relativistic orientable objects, we often
appeal to the intuitively clear example of a three-dimensional rotator,
described by a field on the group $SO(3)\sim SU(2)$. The left
$\hat{J}_{1},\hat{J}_{2},\hat{J}_{3}$ and right
$\hat{I}_{1},\hat{I}_{2},\hat{I}_{3}$ generators of the group $SU(2)$ (being
the operators of angular momentum in the s.r.f. and b.r.f.) commute with each
other and have the same spectrum at a fixed eigenvalue of the Casimir operator
$\hat{\mathbf{J}}^{2}=\hat{J}_{1}^{2}+\hat{J}_{2}^{2}+\hat{J}_{3}^{2}=\hat{I}_{1}^{2}+\hat{I}_{2}^{2}+\hat{I}_{3}^{2}$.
Therefore, as long as we know the spectrum of the operator $\hat{J}_{3}$, we
also know the spectrum of the operator $\hat{I}_{3}$, and, furthermore, these
spectra must coincide.
Note that if $SU(2)$-subgroups of the direct product $SU(2)\times SU(2)$ act
in different spaces, then their Casimir operators $\hat{\mathbf{J}}^{2}$ and
$\hat{\mathbf{I}}^{2}$ (and the spectra of $\hat{J}_{3}$ and $\hat{I}_{3}$)
would be obviously independent of each other. However, in our case both groups
act in the same space (the space of functions of three coordinates of an
orientable object in the s.r.f.), which implies the equality of
$\hat{\mathbf{J}}^{2}$ and $\hat{\mathbf{I}}^{2}$.
We also note that $Z\in SL(2,C)=\mathrm{Spin}(3,1)$ is, in some sense,
redundant for a description of spin. Orientation is given by $6$ parameters,
whereas a description of spin (spin and projection) requires only $2$
parameters. There remain another $4$ quantum numbers related to the
orientation; these numbers, corresponding to right generators, are internal
ones.
The article is organized in the following way:
In section 1, we present a brief summary concerning the field on the Poincaré
group (details can be found in [1]). In sections 2 and 3, we examine two sets
of commuting operators in the space of functions on the Poincaré group that
correspond to states with a fixed parity and chirality. We then consider
properties of right generators from these sets. In section 4, we consider
possible physical interpretation of given classification of orientable
objects. In sections 5 and 6, some sets of commuting operators are applied to
a classification of orientable objects with spin $1/2$ and $1$. In section 7,
we consider classification of the fields on the homogeneous spaces of the
Poincaré group.
We emphasize the fact that we examine only non-unitary finite-dimensional
representations of the group $\mathrm{Spin}(3,1)$, and, accordingly, those of
the group $M(3,1)=T(4){\times\\!\\!\\!\\!\\!\\!\supset}\mathrm{Spin}(3,1)$,
which corresponds to finite-component relativistic wave equations
(“relativistic quantum mechanics” or “one-particle sector”). Consideration of
unitary representations goes beyond the scope of the present article.
## 1 Orientable objects. Right and left transformations.
As was already mentioned, for a description of orientable objects it is
convenient to use two reference frames: the laboratory (or s.r.f., related to
the observer), with an orthobasis ${e}_{\mu}$, and the local (or b.r.f.,
related to the body), with the orthobasis ${\xi}_{\underline{n}}$,
${\xi}_{\underline{n}}=v^{\mu}_{\;\;{\underline{n}}}{e}_{\mu}$. For Euclidean
spaces, $({e}_{i},{e}_{j})=\delta_{ij}$, and thus the elements of the matrix
$V=\|v_{\;k}^{i}\|$ satisfy the condition
$\sum_{i}v_{\;k}^{i}v_{\;l}^{i}=\delta_{kl}$, that is, the matrix $V$ is
orthogonal, $V^{-1}=V^{T}$. For pseudo-Euclidean spaces (in particular, the
4-dimensional Minkowski space) the matrix $V$ is pseudo-orthogonal,
$V^{-1}=\eta V^{T}\eta$, $\eta=\mathop{{\rm diag}}(1,-1,\dots,-1)$.
In Minkowski space, by using the homomorphism $SL(2,C)\sim SO_{0}(3,1)$, one
can describe the orientation by the matrix
$Z=\left(\begin{array}[]{cc}z_{\;\,\underline{1}}^{1}&z_{\;\,\underline{2}}^{1}\\\
z_{\;\,\underline{1}}^{2}&z_{\;\,\underline{2}}^{2}\end{array}\right)\in
SL(2,C),$ (2)
$\Xi=Z^{\dagger}EZ$, where $E=\sigma^{\mu}{e}_{\mu}$ and
$\Xi=\sigma^{\underline{n}}{\xi}_{\underline{n}}$.
The quantities $v^{\nu}_{\;\;{\underline{m}}}\in SO_{0}(3,1)$ are expressed in
terms of $z$ [1],
$v^{\mu}_{\;\;{\underline{n}}}=\frac{1}{2}(\sigma^{\mu})_{\dot{\beta}\alpha}(\bar{\sigma}_{\underline{n}})^{{\underline{a}}\dot{{\underline{b}}}}z^{\alpha}_{\;\;{\underline{a}}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{\dot{\beta}}_{\;\;\dot{{\underline{b}}}}.$
(3)
We note that $v^{\mu}_{\;{\underline{n}}}$ are tetrads, i.e., objects
transformed as vectors (with respect to the 1st index, $\mu$) under the change
of the s.r.f., being, at the same time, objects transformed as vectors (with
respect to the 2nd index, ${\underline{n}}$) under change of the b.r.f.111We
underline “right” indices in order to avoid confusion, since we shall consider
quantities at fixed values of the indices (for instance, spinors
$z_{\;\;\underline{1}}^{\alpha}$ and $z_{\;\;\underline{2}}^{\alpha}$)..
The position of an orientable object in Minkowski space is therefore given by
a 4-vector $x$ (being coordinates of the origin of the b.r.f. in the s.r.f.)
and by the matrix of orientation $Z$. It is known that each 4-vector $x$ can
be associated with a hermitian $2\times 2$ matrix222We use two sets of
$2\times 2$ matrices $\sigma_{\mu}=(\sigma_{0},\sigma_{k})$ and
$\bar{\sigma}_{\mu}=(\sigma_{0},-\sigma_{k})$, where $\sigma_{0}$ is a unity
matrix and $\sigma_{k}$ are the Pauli matrices,
$\sigma_{0}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),\quad\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (4) $X$,
$X=x^{\mu}\sigma_{\mu}=\left(\begin{array}[]{cc}x^{0}+x^{3}&x^{1}-ix^{2}\\\
x^{1}+ix^{2}&x^{0}-x^{3}\end{array}\right),\quad\det X=x_{\mu}x^{\mu},\quad
x^{\mu}=\frac{1}{2}\mathop{\rm Tr}(X\bar{\sigma}^{\mu}).$ (5)
Thus, the pair $(X,Z)\in M(3,1)$ uniquely determines the position and
orientation of the b.r.f. with respect to the s.r.f.; in addition a change of
the s.r.f. corresponds to left multiplication by $(A,U)^{-1}$, whereas a
change of the b.r.f. corresponds to right multiplication by
$(\underline{A},\underline{U})$:
$(X^{\prime},Z^{\prime})=(A,U)^{-1}(X,Z)(\underline{A},\underline{U})=(U^{-1}(X-A)(U^{\dagger})^{-1}+Z\underline{A}Z^{\dagger},\,U^{-1}Z\underline{U}),$
(6)
where $A=\sigma_{\mu}a^{\mu}$ and
$\underline{A}=\sigma_{\underline{m}}\underline{a}^{\underline{m}}$ correspond
to translations, while $U,\underline{U}\in SL(2,C)$ correspond to rotations
and boosts.
Let us now consider functions of coordinates and orientation – functions on
the Poincaré group $f(q)$, $q\in M(3,1)$. The action of the group
$M(3,1)_{\mathrm{ext}}\times M(3,1)_{\mathrm{int}}$ (here we use the
subscripts $\mathrm{ext}$ and $\mathrm{int}$, according to the interpretation
of left transformations as external ones and that of right transformations as
internal ones) in the space of functions $f(q)$ is given by
$\displaystyle\mathbb{T}(g,h)f(q)=f^{\prime}(q)=f(g^{-1}qh),$ (7)
$\displaystyle q\leftrightarrow(X,Z),\quad g\leftrightarrow(A,U),\quad
h\leftrightarrow(\underline{A},\underline{U}).$ (8)
As a consequence of (7), we have
$f^{\prime}(q^{\prime})=f(q),\quad q^{\prime}=gqh^{-1}.$ (9)
The mapping $q\leftrightarrow(X,Z)$ gives rise to the correspondence
$\displaystyle q\leftrightarrow(x,z),\quad\hbox{where}\quad
x=(x^{\mu}),\;z=(z^{\alpha}_{\;\,{\underline{b}}}),$ (10)
$\displaystyle\mu=0,1,2,3,\quad\alpha,b=1,2,\;\quad
z_{\;\,\underline{1}}^{1}z_{\;\,\underline{2}}^{2}-z_{\;\,\underline{1}}^{2}z_{\;\,\underline{2}}^{1}=1,$
and relation (9) takes the form
$f^{\prime}(x^{\prime},z^{\prime})=f(x,z),\quad(x^{\prime},z^{\prime})\leftrightarrow
q^{\prime}=gqh^{-1}.$ (11)
Using such a parameterization, we find the following relations for left and
right transformations, corresponding to changes of s.r.f. and b.r.f.:
$\displaystyle T_{L}(g)f(x,z)=f(g^{-1}x,\;g^{-1}z),\;\;g^{-1}x\leftrightarrow
U^{-1}(X-A)(U^{-1})^{\dagger},\;\;g^{-1}z\leftrightarrow U^{-1}Z,$ (12)
$\displaystyle T_{R}(g)f(x,z)=f(xg,\;zg),\quad xg\leftrightarrow
X+ZAZ^{\dagger},\quad zg\leftrightarrow ZU.$ (13)
According to (12), $x$ carries the the vector representation of the Lorentz
group, while $z$ carries the spinor representation of this group. If one
restricts the consideration to functions independent of $z$, then (12) reduces
to transformations of the left quasiregular representation, corresponding to
the case of a usual scalar field $f^{\prime}(x^{\prime})=f(x)$. If one
restricts the consideration to functions independent of $x$, then (12) reduces
to transformations of the left generalized regular representation (GRR) of the
Lorentz group.
Generators that correspond to translations and rotations have the form
$\displaystyle\hat{p}_{\mu}=i\partial/\partial
x^{\mu},\quad\hat{J}_{\mu\nu}=\hat{L}_{\mu\nu}+\hat{S}_{\mu\nu},$ (14)
$\displaystyle\hat{p}_{{\underline{m}}}^{R}=-v^{\nu}_{\;\;{\underline{m}}}\hat{p}_{\nu},\quad\hat{J}_{{\underline{m}}{\underline{n}}}^{R}=\hat{S}_{{\underline{m}}{\underline{n}}}^{R}.$
(15)
where $\hat{L}_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$ are
the operators of orbital momentum projections and $\hat{S}_{\mu\nu}$ are the
operators of spin projections. The operators of right translations can also be
presented in the form ${\hat{P}}^{R}=-Z\hat{P}Z^{\dagger}$; the operators
$\hat{S}_{\mu\nu}$ and $\hat{S}^{R}_{{\underline{m}}{\underline{n}}}$ depend
only on $z$ and $\partial/\partial z$,
$\displaystyle\hat{S}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
i\left((\sigma_{\mu\nu})^{\;\;\beta}_{\alpha}z^{\alpha}_{\;\;{\underline{a}}}\partial_{\beta}^{\;\;{\underline{a}}}+(\bar{\sigma}_{\mu\nu})^{\dot{\alpha}}_{\;\;{\dot{\beta}}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}^{\;\;\dot{{\underline{a}}}}\partial^{\dot{\beta}}_{\;\;\dot{{\underline{a}}}}\right),$
(16) $\displaystyle\hat{S}_{{\underline{m}}{\underline{n}}}^{R}$
$\displaystyle=$ $\displaystyle
i\left((\sigma_{{\underline{m}}{\underline{n}}})_{\;\;{\underline{b}}}^{\underline{a}}z^{\alpha}_{\;\;{\underline{a}}}\partial_{\alpha}^{\;\;{\underline{b}}}+(\bar{\sigma}_{{\underline{m}}{\underline{n}}})_{\dot{\underline{a}}}^{\;\;\dot{\underline{b}}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}^{\;\;\dot{{\underline{a}}}}\partial^{\dot{\alpha}}_{\;\;\dot{{\underline{b}}}}\right),$
(17)
where
$(\sigma_{\mu\nu})_{\alpha}^{\;\;\beta}=\frac{1}{4}(\sigma_{\mu}\bar{\sigma}_{\nu}-\sigma_{\nu}\bar{\sigma}_{\mu})_{\alpha}^{\;\;\beta},\quad(\bar{\sigma}_{\mu\nu})^{\dot{\alpha}}_{\;\;{\dot{\beta}}}=\frac{1}{4}(\bar{\sigma}_{\mu}\sigma_{\nu}-\bar{\sigma}_{\nu}\sigma_{\mu})^{\dot{\alpha}}_{\;\;{\dot{\beta}}}.$
(18)
In addition, it is convenient to present the spin operators in terms of three-
dimensional vector notation,
$\hat{S}_{k}=\frac{1}{2}\epsilon_{ijk}\hat{S}^{ij}$,
$\hat{B}_{k}=\hat{S}_{0k}$, see formulae (Appendix. Generators and weight
diagrams of the Lorentz group)–(61) of the Appendix.
Below, we also consider phase transformations of $Z$ (being symmetry
transformations for a field on the Poincaré group [1]),
$Z^{\prime}=Ze^{i\phi},$ (19)
with the generator (chirality operator)
$\hat{\Gamma}^{5}=-i\partial/\partial\phi={\textstyle\frac{1}{2}}\left(z^{\alpha}_{\;\;{\underline{b}}}\partial_{\alpha}^{\;\;{\underline{b}}}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}}^{\;\;\dot{\underline{b}}}\partial^{{\dot{\alpha}}}_{\;\;\dot{\underline{b}}}\right).$
(20)
In fact, this means a transition to an analysis of the group
$M(3,1)_{\mathrm{ext}}\times M(3,1)_{\mathrm{int}}\times U(1)$.
## 2 Maximal sets of commuting operators
A maximal set of commuting operators in the space of the functions on a group
contains Casimir operators and an equal number operators that are some
functions of left and right generators [4].
Casimir operators that label irreps can be composed of both left and right
generators, so that the “left” and “right” mass and spin are the same. For the
Poincaré group $M(3,1),$ we have
$\displaystyle\hat{\rm
p}^{2}=\eta^{\mu\nu}\hat{p}_{\mu}\hat{p}_{\nu}=\eta^{{\underline{m}}{\underline{n}}}\hat{p}^{R}_{\underline{m}}\hat{p}^{R}_{\underline{n}},$
(21) $\displaystyle\hat{\rm
W}^{2}=\eta_{\mu\nu}\hat{W}^{\mu}\hat{W}^{\nu}=\eta_{{\underline{m}}{\underline{n}}}\hat{W}_{R}^{\underline{m}}\hat{W}_{R}^{\underline{n}},\quad$
(22)
$\displaystyle\hbox{where}\quad\hat{W}^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\hat{p}_{\nu}\hat{J}_{\rho\sigma}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\hat{p}_{\nu}\hat{S}_{\rho\sigma},\qquad\hat{W}_{R}^{\underline{m}}=\frac{1}{2}\epsilon^{{\underline{m}}{\underline{n}}\underline{r}\underline{s}}\hat{p}^{R}_{\underline{n}}\hat{S}^{R}_{\underline{r}\underline{s}}.$
(23)
As four operators, composed of left generators (generators of the group
$M(3,1)_{\mathrm{ext}}$), one can choose the Casimir operator
$\hat{\mathbf{p}}\hat{\mathbf{S}}=\hat{p_{k}}\hat{S^{k}}$ and the generators
$\hat{p}_{k}$ of the subgroup $M(3)_{\mathrm{ext}}$. The latter correspond to
additive quantum numbers (if the quantum number is additive, then this quantum
number of composite system is the sum of the corresponding quantum numbers of
subsystems). Eigenfunctions of these operators correspond to definite values
of the helicity and the momentum.
Functions of right generators can be chosen in different ways, for instance,
the set $\hat{\mathbf{p}}\mathstrut^{R}\hat{\mathbf{S}}\mathstrut^{R}$,
$\hat{p}^{R}_{\underline{m}}$ (analogous to the set
$\hat{\mathbf{p}}\hat{\mathbf{S}}$, $\hat{p}_{\mu}$ for
$M(3,1)_{\mathrm{ext}}$), corresponding to reduction
$M(3,1)_{\mathrm{int}}\supset M(3)_{\mathrm{int}}\supset T(3)_{\mathrm{int}}$.
However, as it follows from the explicit form of the operator
$\hat{p}^{R}_{\underline{m}}$, their eigenfunctions $f(x,z)$ contain
arbitrarily large powers of $z$ such that the corresponding representation of
the Lorentz group is infinite-dimensional.
We choose sets, corresponding to reduction $M(3,1)_{\mathrm{int}}\supset
SL(2,C)_{\mathrm{int}}$. Two Casimir operators of the “right” Lorentz group
$SL(2,C)_{\mathrm{int}}$ have the form
$\hat{\mathbf{S}}^{2}-\hat{\mathbf{B}}^{2}=\frac{1}{2}\hat{S}^{R}_{{\underline{m}}{\underline{n}}}\hat{S}_{R}^{{\underline{m}}{\underline{n}}}=\frac{1}{2}\hat{S}_{\mu\nu}\hat{S}^{\mu\nu},\quad\hat{\mathbf{S}}\hat{\mathbf{B}}=\frac{1}{16}\epsilon^{{\underline{m}}{\underline{n}}\underline{r}\underline{s}}\hat{S}^{R}_{{\underline{m}}{\underline{n}}}\hat{S}^{R}_{\underline{r}\underline{s}}=\frac{1}{16}\epsilon^{\mu\nu\rho\sigma}\hat{S}_{\mu\nu}\hat{S}_{\rho\sigma}.$
(24)
where $\hat{S}^{i}$ and $\hat{B}^{i}$ are operators of spin and boost
projections, see Appendix. In contrast to
$\hat{S}_{{\underline{m}}{\underline{n}}}^{R}$, being the generators of the
group $M(3,1)_{\mathrm{int}}$, the operators of spin projections
$\hat{S}_{\mu\nu}$ are not generators of the group $M(3,1)_{\mathrm{ext}}$,
see (14) and (15), and therefore operators (24) are functions of right (but
not left) generators of the Poincaré group.
For $SL(2,C)_{\mathrm{int}}$ we use two sets of commuting operators
corresponding to the reduction schemes
$\displaystyle SL(2,C)_{\mathrm{int}}\supset U(1)\times U(1),$ $\displaystyle
SL(2,C)_{\mathrm{int}}\supset SU(2)\supset U(1).$
In the first case two generators $\hat{S}_{3}^{R}$ and $\hat{B}_{3}^{R}$ of
the maximal commutative (Cartan) subgroup of $SL(2,C)_{\mathrm{int}}$
correspond to additive quantum numbers. In the second case the set includes
Casimir operator $\hat{\mathbf{S}}_{R}^{2}$ of $SU(2)_{\mathrm{int}}$ and
$\hat{S}_{3}^{R}$.
Therefore, we shall consider two sets of 10 commuting operators on the group
$M(3,1)$:
$\displaystyle\hat{\mathrm{W}}^{2},\;\hat{p}_{\mu},\;\hat{\mathbf{p}}\hat{\mathbf{S}}\;(\hat{S}_{3}\,\,\hbox{in
the rest
frame}),\;\hat{\mathbf{S}}^{2}-\hat{\mathbf{B}}^{2},\;\hat{\mathbf{S}}\hat{\mathbf{B}},\;\;\hat{S}_{3}^{R},\;\hat{B}_{3}^{R},$
(25)
$\displaystyle\hat{\mathrm{W}}^{2},\;\hat{p}_{\mu},\;\hat{\mathbf{p}}\hat{\mathbf{S}}\;(\hat{S}_{3}\,\,\hbox{in
the rest
frame}),\;\hat{\mathbf{S}}^{2}-\hat{\mathbf{B}}^{2},\;\hat{\mathbf{S}}\hat{\mathbf{B}},\;\;\hat{\mathbf{S}}_{R}^{2},\;\hat{S}_{3}^{R},$
(26)
including the Lubanski–Pauli operator $\hat{W}^{2}$, four left generators
$\hat{p}_{\mu}$ (the eigenvalue of the Casimir operator
$\hat{\mathrm{p}}^{2}$, is evidently expressed through their eigenvalues), and
helicity $\hat{\mathbf{p}}\hat{\mathbf{S}}$, expressed through the left
generators. The Casimir operators $\hat{\mathbf{S}}^{2}-\hat{\mathbf{B}}^{2}$
and $\hat{\mathbf{S}}\hat{\mathbf{B}}$ of the subgroup
$SL(2,C)_{\mathrm{int}}$ determine characteristics $j_{1},j_{2}$ of the irreps
$T_{[{j_{1}j_{2}}]}$ of the Lorentz group (see Appendix).
Eigenfunctions of the maximal sets of operators (25) are at the same time
eigenfunctions of the chirality operator $\hat{\Gamma}^{5}$ (20). Indeed, in
the irreps of the Lorentz group $T_{[j_{1}j_{2}]}$ an eigenvalue of the
chirality operator is $\Gamma^{5}=j_{1}-j_{2}$.
Besides the states of definite chirality, the states of definite internal
parity are obviously also of interest. The operator of space reflection
$\hat{P}$ anticommutes with the chirality operator $\hat{\Gamma}^{5}$, and
also with the operators $\hat{\mathbf{S}}\hat{\mathbf{B}}$ and
$\hat{B}_{3}^{R}$. Therefore, eigenfunctions of these the latter three
operators change their sign under the action of $\hat{P}$. The set (26) is
more convenient to describe states of definite internal parity, because only
one operator $\hat{\mathbf{S}}\hat{\mathbf{B}}$ from this set doesn’t commute
with $\hat{P}$. Eigenvalues of $\hat{\mathbf{S}}\hat{\mathbf{B}}$ are
proportional to $(j_{1}-j_{2})(j_{1}+j_{2}+1)$, see (63). Thus, one can use
quantum numbers, corresponding to the set (26), to characterize eigenstates of
$\hat{P}$, with only one change (replacement the sign of $j_{1}-j_{2}$ by
internal parity $\eta$).
## 3 Right generators and charges. Spectra
Despite the commutativity of the left and right generators, the corresponding
quantum numbers are not independent. Since the same Casimir operators are
constructed from left and right generators uniformly, within the framework of
a representation (fixing their eigenvalues), the spectra of the corresponding
left and right generators are the same.
The relation between left and right generators can be easily seen on the
example of the group of three-dimensional rotations $SU(2)$, describing a
three-dimensional non-relativistic rotator [5, 6, 7]. Until recently, it has
been the only example of a well-developed (by Wigner, Casimir and Eckart, back
in the 1930’s) theory, based on use of both left and right transformations.
The concept of two coordinate systems is always present in the problem of
rotation of a solid body, independently of the fact whether it is described
classically or quantum mechanically. One coordinate system (laboratory, or
s.r.f.) is associated with the surrounding objects, while another one
(molecular, or b.r.f.) is associated with the body. Accordingly, there are two
sets of operators of angular momentum – in the s.r.f. (left generators of the
rotation group $\hat{J}_{k}$) and in the b.r.f. (right generators of the
rotation group $\hat{I}_{k}$).
The maximal set of commuting operators in the space of functions on the group
$SU(2)\sim SO(3)$ consist of total angular momentum
$\hat{\mathbf{J}}^{2}=\hat{\mathbf{I}}^{2}$ and two projections: $\hat{J}_{3}$
in s.r.f. and $\hat{I}_{3}$ in b.r.f. A classification of rotator sates
$|j\,m\,k\>$ is made with the help of this set,
$\hat{\mathbf{J}}^{2}|J\,m\,k\>=j(j+1)|J\,m\,k\>,\quad\hat{J}_{3}|J\,m\,k\>=m|J\,m\,k\>,\quad\hat{I}_{3}|J\,m\,k\>=k|J\,m\,k\>.$
(27)
By virtue of the relation $\hat{\mathbf{J}}^{2}=\hat{\mathbf{I}}^{2}$ the left
and right irreps are labeled by the same $j$ and the operators $\hat{J}_{3}$
and $\hat{I}_{3}$ have the same spectrum, namely, their eigenvalues $m$ and
$k$ belong to the set $-j,-j+1,\ldots,j-1,j$. The operator $\hat{I}_{3}$,
distinguishing equivalent representations in the decomposition of the left GRR
of the rotation group into irreps, corresponds to an additive quantum number,
independent of a choice of the s.r.f. This number plays an important role in
the theory of molecular spectra [7, 8].
Let us now consider the groups of motions $M(D)$ and $M(D,1)$, including
rotations and translations. The generators of left rotations in this case
consist of two summands – the orbital and spin momenta,
$\hat{J}_{\mu\nu}=\hat{L}_{\mu\nu}+\hat{S}_{\mu\nu}$, whereas the generators
of right rotations depend only on $z$,
$\hat{J}^{R}_{{\underline{m}}{\underline{n}}}=\hat{S}^{R}_{{\underline{m}}{\underline{n}}}$.
The simplest example is a three-parameter group $M(2)$. Here, we deal only
with one operator of projection of the angular momentum,
$\hat{J}=\hat{L}+\hat{S}$; the operator of right projection coincides with the
operator of spin projection, $\hat{J}^{R}=-\hat{S}$. The maximal set of
commuting operators can be composed of the operators of momentum and spin:
$\hat{p}_{k},\hat{S}.$
For the group $M(3)$, the maximal set contains 6 commuting operators, which
can be chosen as (reduction $M(3)_{\mathrm{ext}}\supset T(3)_{\mathrm{ext}}$,
$M(3)_{\mathrm{int}}\supset\mathrm{Spin}(3)_{\mathrm{int}}\supset
U(1)_{\mathrm{int}}$)
$\hat{p}_{k},\;\hat{p}_{k}\hat{J}^{k}=\hat{p}_{k}\hat{S}^{k},\;\hat{S}_{k}\hat{S}^{k}=\hat{S}^{R}_{k}\hat{S}^{Rk},\;\hat{S}^{R}_{3}.$
(28)
Functions on the group $M(3,1)$ depend on 10 parameters, and, accordingly,
there are 10 commuting operators (two Casimir operators and two sets of four
operators, constructed from left (14) and right (15) generators), see (25).
For a fixed mass and spin, i.e., in the framework of a representation
determined by the Casimir operators of the Poincaré group, the spectra of left
and right generators prove to be the same. In a similar way, the Casimir
operators of the Lorentz group (24) are constructed from $\hat{S}^{\mu\nu}$ or
$\hat{S}^{{\underline{m}}{\underline{n}}}_{R}$ uniformly, and, therefore, the
spectrum of operators of left and right spin projections for fixed values of
$j_{1}$ and $j_{2}$ is the same. In particular, Casimir operators
$\hat{\mathbf{S}}^{2}=\sum(S_{k})^{2}$ and
$\hat{\mathbf{S}}_{R}^{2}=\sum(S_{k}^{R})^{2}$ of compact subgroups
$SU(2)_{\mathrm{ext}}\supset\mathrm{Spin}(3,1)_{\mathrm{ext}}$ and
$SU(2)_{\mathrm{int}}\supset\mathrm{Spin}(3,1)_{\mathrm{int}}$ with
eigenvalues $S(S+1)$ and $S_{R}(S_{R}+1)$ have the same spectrum; $S$ and
$S_{R}$ belong to the set $|j_{1}-j_{2}|,\,|j_{1}-j_{2}|+1,\dots,j_{1}+j_{2}$,
see (64). Note that “right” quantum numbers $S^{R},\;S_{3}^{R},\;B_{3}^{R}$
can be only integer for particles of integer spin and half-integer for
particles of half-integer spin.
Left generators of the Poincaré group $\hat{p}_{\mu}$ and $\hat{J}_{\mu\nu}$
are associated with additive quantum numbers – the momentum and total angular
momentum projections. Right generators $\hat{p}_{\mu}^{R}$ and
$\hat{J}_{\mu\nu}^{R}$, and, in particular, the operators $\hat{B}^{R}_{3}$
and $\hat{S}^{R}_{3}$ entering the maximal set also are associated with
additive quantum numbers.
Right generators commute with left ones and, consequently, with the
corresponding finite transformations. Therefore, under finite left
transformations (changes of the s.r.f.) the eigenfunctions of right generators
remain eigenfunctions with the same eigenvalues. In other words, right
generators determine internal quantum numbers that do not change under changes
of the s.r.f. They can be identified with charges. Indeed, charges are usually
understood as additive numbers that do not change under changes of the s.r.f.
(Lorentz transformations). Right generators from the commutative (Cartan)
subgroup satisfy such a definition.
Therefore, an orientable object is characterized by 10 quantum numbers – 6
numbers (momentum $p_{\mu}$, spin, helicity) are determined by the left
generators and Casimir operators, whereas $4$ numbers are determined by the
right generators (Lorentz characteristics $j_{1},j_{2}$ and two charges). In
comparison with the usual description of fields in Minkowski space
($4$-momentum, spin, spin projection, representation of the Lorentz group),
there appear two additional quantum numbers.
However, there is an essential difference between left and right generators:
whereas left generators correspond to external, exact, symmetries, right
generators correspond to internal symmetries which may be broken.
Let us turn to the example of a nonrelativistic rotator, described by the
quadratic Hamiltonian
$\hat{H}=\sum A_{k}(\hat{I}_{k})^{2},$ (29)
where $A_{k}$ are the moments of inertia. For a completely symmetric rotator
($A_{1}=A_{2}=A_{3}=A$), not only left transformations but also right ones are
symmetry transformations of Hamiltonian (29); the symmetry group is
$SO(3)_{\mathrm{ext}}\times SO(3)_{\mathrm{int}}$. In the axially symmetric
case, only the right rotations (with the generator $\hat{I}_{3}$) about the
axis $\boldsymbol{\xi}_{3}$ is a symmetry of the body; the symmetry group is
$SO(3)_{\mathrm{ext}}\times SO(2)_{\mathrm{int}}$. This symmetry corresponds
to the additive quantum number $k$ (see (27)). Finally, in the case of three
different momenta of inertia the body is asymmetric, and therefore the left
transformations with the generators $\hat{I}_{k}$ are not its symmetries; the
symmetry group is $SO(3)_{\mathrm{ext}}$.
Returning to the 3+1-dimensional case, we consider the operator
$\hat{p}_{\mu}\hat{\Gamma}^{\mu\underline{0}}$, where
$\Gamma^{\mu\underline{0}}$ are linear differential operators in $z$; this
operator is invariant under the transformations
$M(3,1)_{\mathrm{ext}}\times\mathop{{\rm Spin}}(3)_{\mathrm{int}}\times U(1)$
[1]. The equations for the eigenvalues of this operator
$\hat{p}_{\mu}\hat{\Gamma}^{\mu\underline{0}}f(x,z)=\varepsilon msf(x,z),$
(30)
where $\varepsilon=\pm 1$, in the subspaces $f(x,z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}})$
$f(x,{\underline{z}}{\vphantom{z}},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z)$, after a
separation of the spatial and orientation variables for spin 1/2 and 1
(corresponding to polynomials of 1st and 2nd degree with respect to $z$), turn
into the equations of Dirac and Duffin–Kemmer (see [1] for details).
The chirality operator $\hat{\Gamma}^{5}$, and also the operators
$\hat{{\mathbf{S}}}\hat{{\mathbf{B}}}$ and $\hat{B}_{3}^{R}$ from the set (25)
don’t commute with both space reflection $\hat{P}$ and
$\hat{p}_{\mu}\hat{\Gamma}^{\mu\underline{0}}$. Thus the operator
$\hat{B}_{3}^{R}$, as well as $\hat{\Gamma}^{5}$, corresponds to a broken
symmetry. In the subspace of functions $f(x,z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}})$
we have $\hat{\Gamma}^{5}=i\hat{B}^{R}_{3}$, whereas in the subspace
$f(x,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z,{\underline{z}}{\vphantom{z}})$
we have $\hat{\Gamma}^{5}=-i\hat{B}^{R}_{3}$. (Here, the multiplier $i$
originates from the fact that we consider non-unitary finite-dimensional
representation of the Lorentz group, for which the boost generators are anti-
hermitian.) Like chirality, $B^{R}_{3}$ is an additive conserved number only
for massless particles. Massive particles correspond to eigenfunctions of the
operator of space parity $\hat{P}=\hat{I}_{s}$ and the operator
$\hat{p}_{\mu}\hat{\Gamma}^{\mu\underline{0}}$, which do not commute with the
operators $\hat{\Gamma}^{5}$ and $\hat{B}^{R}_{3}$. Therefore, massive
particles, described by equations of the form (30), cannot have a definite
charge $B^{R}_{3}$ and chirality $\Gamma^{5}$.
The quantum number corresponding to the generator $\hat{S}^{R}_{3}$ must be
conserved also for massive Dirac particles, since $\hat{S}^{R}_{3}$ commutes
not only with all left generators of the Poincaré group, but also with the
operators $\hat{p}_{\mu}\hat{\Gamma}^{\mu\underline{0}}$ and $\hat{P}$.
The question arises whether the known elementary particles possess a conserved
quantum number corresponding to the operator $\hat{S}^{R}_{3}$. An answer to
this question, in fact, will also answer another question, whether one can
consider usual particles as ‘‘orientable objects’’ in the sense of the above
definition.
## 4 $S_{3}^{R}$ charge
One can attempt to associate the charge corresponding to the right generator
$\hat{S}_{3}^{R}$ of the Poincaré group to observable characteristics of
physical particles. Notice that although the “internal” quantum numbers
corresponding to right generators do not change under left transformations,
the discrete transformations (automorphisms of the Poincaré group) act on both
right and left generators (see [1] for details). The known behavior under
discrete transformations helps one to identify right characteristics with
properties of physical particles.
As to the operator $\hat{S}_{3}^{R}$:
1\. It corresponds to an additive (conserved) quantum number.
2\. It has integer eigenvalues for particles of integer spin and half-integer
eigenvalues for particles of half-integer spin.
3\. It does not change sign under space reflection.
4\. It changes sign under charge conjugation.
Then it follows that this number equals to zero for real neutral particle
(i.e., particle that coincides its own antiparticle); at the same time real
neutral particles with definite $S_{3}^{R}$ have to have integer spin.
If one considers $S_{3}^{R}$-charge of particles described by finite-
dimensional representations of the Lorentz group $T_{[j_{1},j_{2}]}$,
$j_{1}+j_{2}=s$, then for particles of spin $1/2$ two values are possible:
$1/2$ and $-1/2$; for particles of spin 1 three values are possible: $1,0$ and
$-1$. In particular, for a photon and $Z^{0}$-boson, as real neutral
particles, we have $S_{3}^{R}=0$.
Let us see which values of $S_{3}^{R}$ can be associated with particles with
wave functions $f(x,z)$ that are eigenfunctions of the operator
$\hat{S}_{3}^{R}$, $\hat{S}_{3}^{R}f(x,z)=S_{3}^{R}f(x,z)$. Since the sign of
$S_{3}^{R}$ changes under the charge conjugation, particles and antiparticles
must have $S_{3}^{R}$-charges of opposite signs. For definiteness, let an
electron $e^{-}$ has the $S_{3}^{R}$-charge $-\frac{1}{2}$; then a positron
$e^{+}$ has $S_{3}^{R}=\frac{1}{2}$. For $W^{-}$, as a charged particle, it is
natural to expect $S_{3}^{R}=\pm 1$ (the value $S_{3}^{R}=0$ is excluded by a
more detailed analysis, see below). Next, since $\tilde{\nu}_{e}$ only admits
the values $\pm\frac{1}{2}$, the reaction $W^{-}\rightarrow
e^{-}+\tilde{\nu}_{e}$ implies $S_{3}^{R}=-\frac{1}{2}$ for $\tilde{\nu}_{e}$
and $S_{3}^{R}=-1$ for $W^{-}$. Therefore, we have
$\begin{array}[]{rcccrcccc}\frac{1}{2}:&\nu_{e}&e^{+}&&1:&W^{+}&&0:&\gamma,Z^{0}\\\
-\frac{1}{2}:&e^{-}&\tilde{\nu}_{e}&&-1:&W^{-}&&&\end{array}$ (31)
Applying the same consideration to other families of fundamental fermions, we
find the following classification with respect to the sign of $S_{3}^{R}$
$\begin{array}[]{rccccccc}\frac{1}{2}:&\nu_{e}&\nu_{\mu}&\nu_{\tau}&&u&c&t\\\
-\frac{1}{2}:&e^{-}&\mu&\tau&&d&s&b\end{array}$ (32)
Therefore, the $S_{3}^{R}$-charge, whose sign changes under both
$\hat{C}\hat{P}\hat{T}$-transformation and charge conjugation $\hat{C}$,
distinguishes not only particles and antiparticles but also the “up-down”
components in doublets of elementary fermions. This charge is conserved in any
interactions, since the carriers of electromagnetic and strong interactions
are characterized by $S_{3}^{R}=0$, whereas we have already examined weak
charged currents.
As a consequence of (31) and (32), we find the following empirical expression
for $S_{3}^{R}$ in terms of other charges:
$S_{3}^{R}=\frac{L-B}{2}+Q,$ (33)
where $L,B,Q$ are the lepton, baryon and electric charges, respectively. This
formula relates the “right” charge $S_{3}^{R}$ with observable characteristics
of particles.
For the above-mentioned fundamental particles of spin 1 the charge $S_{3}^{R}$
coincides with the electric charge, whereas for spin $1/2$ particles it
coincides with the mean value of the electric charge corresponding to a lepton
or quark doublet.
We also note that for left particles and right antiparticles $S_{3}^{R}$
coincides with the projection of the weak isospin $T_{3}$.
Let us now consider the relation between “right” quantum number $S_{3}^{R}$
and spin. We have already noted that spectra of right and left spin operators
are not independent, in particular, $S_{3}^{R}$ can be only integer for
particles of integer spin and half-integer for particles of half-integer spin
$s$,
$(-1)^{2S_{3}^{R}}=(-1)^{2s},$
and due to (33) we have
$(-1)^{L-B+2Q}=(-1)^{2s}.$ (34)
In 1961 Michel and Lurçat [9] have noted that for all the known particles with
integer $B$ there holds the relation
$(-1)^{B+L}=(-1)^{2s},$ (35)
in other words, $B+L+2s$ is always even. Later, this observation resulted in
the concept of $R$-parity, being positive for all the known particles, and
defined as
$R=(-1)^{3(B-L)+2s}$ (36)
or $R=(-1)^{3B+L+2s}$, where the multiplier $3$ is introduced for an inclusion
of quarks into the analysis.
On the condition that the electric charge is integer, relation (35) is a
consequence of (33). Indeed, since the right projection $S_{3}^{R}$ and spin
$s$ must take integer and half-integer values simultaneously, both $(B\pm
L)/2$ and spin $s$ are integer and half-integer simultaneously. Furthermore,
for fractional $1/3$-multiple charges (quarks), we have
$(-1)^{2s}=(-1)^{2S_{3}^{R}}=(-1)^{6S_{3}^{R}}=(-1)^{3(L-B)}$, since $6Q$ is
even-valued. As a consequence, for all particles with integer charge $Q$ or
with $1/3$-multiple charge $Q$, $R$-parity is positive.
## 5 Spin 1/2: fermionic quadruplets
Consider irreps of $SL(2,C)_{\mathrm{int}}$, whose weight diagrams are
determined by eigenvalues of generators of the right projections of spin
$\hat{S}_{3}^{R}$ and $\hat{B}_{3}^{R}$.
Any finite-dimensional irrep occurs in the decomposition of left (or right)
GRR with the multiplicity equal to its dimension. For instance, in the case of
$SL(2,C)$ the irrep $T_{[1/2\;0]}$ occurs in the decomposition of the left GRR
twice: columns $(z^{1}\;z^{2})$ and
$({\underline{z}}{\vphantom{z}}^{1}\;{\underline{z}}{\vphantom{z}}^{2})$ carry
this representation; analogously, the irrep $T_{[1/2\;0]}$ also occurs in the
decomposition of the right GRR twice (rows
$(z^{1}\;{\underline{z}}{\vphantom{z}}^{1})$ and
$(z^{2}\;{\underline{z}}{\vphantom{z}}^{2})$).
Linear functions of coordinates $z$ on the Lorentz group describe spin $1/2$
particles. There are $4$ linearly independent functions that are transformed
differently under a change of a b.r.f.. These are
$z,{\underline{z}}{\vphantom{z}},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}$
(we have omitted the usual “laboratory” indices, since $z^{1}$ and $z^{2}$ are
transformed in the same way under the action of $SL(2,C)_{\mathrm{int}}$).
These functions can be arranged into linear combinations being eigenfunctions
of various operators. The states of spin $1/2$ with a definite charge
$S_{3}^{R}$ and a chirality correspond to the functions
$\begin{array}[]{lcc}&S_{3}^{R}=-1/2&S_{3}^{R}=1/2\\\
R\;(\Gamma^{5}=1/2):&e^{ipx}z^{\alpha}&e^{ipx}{\underline{z}}{\vphantom{z}}^{\alpha}\\\
L\;(\Gamma^{5}=-1/2):&e^{ipx}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}}&e^{ipx}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}\\\
&&\end{array}$ (37)
The states with a definite parity $\eta$ are eigenfunctions of the operator
$\hat{P}$, $\hat{P}f(x,{z})=\eta f(x,{z})$; in the rest frame at $p_{0}=\pm
m,$ we have
$\begin{array}[]{lcc}&S_{3}^{R}=-1/2&S_{3}^{R}=1/2\\\ \eta=1&e^{\pm
imx^{0}}(z^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}})&e^{\pm
imx^{0}}({\underline{z}}{\vphantom{z}}^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}})\\\
\eta=-1&e^{\pm imx^{0}}(z^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}})&e^{\pm
imx^{0}}({\underline{z}}{\vphantom{z}}^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}})\\\
&&\end{array}$ (38)
We note that functions from (38) are eigenfunctions for $\hat{\Gamma}^{00}$
and $\hat{p}_{0}$, and, therefore, they are solutions of the left-invariant
equation (30).
Assuming a symmetry violation under right transformations of the Poincaré
group, and retaining only the particle-antiparticle symmetry, we find that a
mixed picture becomes possible:
$\begin{array}[]{lcc}&S_{3}^{R}=-1/2&S_{3}^{R}=1/2\\\ &e^{\pm
imx^{0}}(z^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}})&e^{ipx}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}\\\
&e^{ipx}z^{\alpha}&e^{\pm
imx^{0}}({\underline{z}}{\vphantom{z}}^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}})\\\
&&\end{array}$ (39)
States from (37) correspond to the massless or ultra-relativistic case, when
one has two chiral left fermions (for example, $e_{L}^{-}$ and $\nu_{L}$) and
two chiral right antifermions ($e_{R}^{+}$ and $\tilde{\nu}_{R}$), see Fig.1a.
The massive electron and neutrino correspond to states from (38). In this
case, the electron $e^{-}$ and the neutrino $\nu$ have a positive internal
parity; the positron $e^{+}$ and the antineutrino $\tilde{\nu}$ have a
negative internal parity, see Fig.1b.
Finally, states from (39) can be considered as a good approximation within the
range of energies much larger than the neutrino mass. In this approximation,
the quadruplet of leptons
$e^{-},\,\tilde{e}^{+},\,\nu_{L},\,\tilde{\nu}_{R}\,$ contains an electron
with a positive internal parity, a positron with a negative internal parity; a
massless neutrino can be only a left one (chirality is negative) and the
antineutrino can be only a right one (chirality is positive). In addition, a
change of signs both of $S_{3}^{R}$ and of chirality or parity corresponds to
a transition to an antiparticle, whereas a change of sign alone leads to a
transition to a state which is not present in (39).
The quark quadruplet $u,\tilde{u},d,\tilde{d}$, where $u$ and $d$ are
characterized by the same internal parity $\eta=1$, and, according to (32), by
opposite signs of $S_{3}^{R}$, corresponds to (38). In addition, as should be
expected, particles and corresponding antiparticles are characterized by
opposite parity. However, in this case we also encounter a violation of the
symmetry with respect to right transformations of the Poincaré group –
components of $SL(2,C)_{\mathrm{int}}$-dublet have different mass.
This classification can be visualized in the form of a weight diagram of the
representation $T_{[1/2\;0]}\oplus T_{[0\;1/2]}$ of the group
$SL(2,C)_{\mathrm{int}}$.
Figure 1: The weight diagram of the representation $T_{[1/2\;0]}\oplus
T_{[0\;1/2]}$ of $SL(2,C)_{\mathrm{int}}$. The dotted line joins states
related by transformations of $SL(2,C)_{\mathrm{int}}$. a) States with
definite chirality, functions (37). b) States with definite parity $\eta$,
functions (38).$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z,\,\nu_{L}$$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}},\,e^{-}_{L}$$z,\,\tilde{\nu}_{R}$${\underline{z}}{\vphantom{z}},\,e^{+}_{R}$$S_{3}^{R}$$-iB_{3}^{R}$$a$
Note that besides the eigenfunctions of $\hat{S}_{3}^{R}$ one can also
construct states with definite charge parity,
$\hat{C}f(x,{z})=\eta_{c}f(x,{z})$ (which describe the Majorana neutrino), or
with $\hat{C}\hat{P}\hat{T}$-parity (the so-called ‘‘physical Majorana
neutrino’’, see [10, 11]).
Let us now consider functions corresponding to a massive particle, moving
along the axis $x^{3}$. They can be obtained from functions in the rest frame
(38), which are characterized by a certain internal parity, with the help of a
Lorentz transformation
$P=UP_{0}U^{\dagger},\quad Z=UZ_{0},\quad\hbox{where}\;\;P_{0}=\pm\mathop{{\rm
diag}}\\{m,m\\},\quad U=\mathop{{\rm diag}}\\{e^{a},e^{-a}\\}\in
SL(2,C)_{\mathrm{ext}},$
the sign of $P_{0}$ corresponds to the sign of $p_{0}$,
$p_{\mu}=k_{\mu}{\rm sign}\ p_{0},\quad k_{0}=m\cosh 2a,\quad k_{3}=m\sinh
2a,\quad e^{\pm a}=\sqrt{(k_{0}\pm k_{3})/m}.$ (40)
By applying these transformations to the state with $S_{3}^{R}=-1/2$, $\eta=1$
at $p_{0}>0$, we find
$f^{\prime}_{m,1/2}(x,z)=e^{i(k_{0}x^{0}+k_{3}x^{3})}\left[C_{1}(z^{1}e^{a}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{1}}e^{-a})+C_{2}(z^{2}e^{-a}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{2}}e^{a})\right],$
where the first term in the square brackets corresponds to $s_{3}=1/2$, and
the second term corresponds to $s_{3}=-1/2$. In the ultra-relativistic case
with a positive $a$ (i.e. with $k_{3}>0$) there remain only two components,
$f^{\prime}_{m,1/2}(x,{z})\approx
e^{i(k_{0}x^{0}+k_{3}x^{3})}\left(C_{1}z^{1}e^{a}-C_{2}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{2}}e^{a}\right).$
which are eigenfunctions of the helicity operator $\hat{p}\hat{S}$ with the
eigenvalues $p_{3}s_{3}=\frac{1}{2}k_{3}\mathop{{\rm sign}}s_{3}$; these
components are also eigenfunctions of the operator $\hat{\Gamma}^{5}$ with the
same sign. In a similar way, considering the case $a<0$ and other states from
(38), we conclude that in the ultra-relativistic limit with $p_{0}>0$ signs of
chirality $\hat{\Gamma}^{5}$ and helicity $\hat{p}\hat{S}$ are the same. We
stress, that the above conclusions derived for the ultra-relativistic case
coincide with the results obtained from the Dirac equation.
Consider now the states corresponding to spin $1/2$ particles from the
viewpoint of solutions of left-invariant RWE of first order in more detail.
Equation (30) for functions linear in $z$ splits into a pair of Dirac
equations for functions from subspaces $f(x,z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}})$
($S_{3}^{R}=-1/2$) and $f(x,{\underline{z}}{\vphantom{z}},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z)$ ($S_{3}^{R}=1/2$).
The sign of the mass term in these equations is $\varepsilon=\eta\mathop{{\rm
sign}}p_{0}\mathop{{\rm sign}}S_{3}^{R}$ (see [1, 12, 13] for details).
For spin $s=1/2$, eigenfunctions of the operator $\hat{S}_{3}^{R}$ and space
parity $\hat{P}=\hat{I}_{s}$ are $z^{\alpha}\pm\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{{\dot{\alpha}}}$,
${\underline{z}}{\vphantom{z}}^{\alpha}\pm\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}}$.
In the rest frame solutions of two mentioned above Dirac equations with
$\varepsilon=1$ have the form
$\displaystyle f_{1}(x,z)=e^{imx^{0}}C_{\alpha}(z^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{{\dot{\alpha}}})+e^{-imx^{0}}C^{\prime}_{\alpha}(z^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{{\dot{\alpha}}}),\quad
S_{3}^{R}=-1/2,$ (41) $\displaystyle
f_{2}(x,z)=e^{imx^{0}}D_{\alpha}({\underline{z}}{\vphantom{z}}^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}})+e^{-imx^{0}}D^{\prime}_{\alpha}({\underline{z}}{\vphantom{z}}^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}}),\quad
S_{3}^{R}=1/2.$ (42)
As is known, the free Dirac equation have solutions corresponding to a pair of
non-equivalent irreps of the improper Poincaré group with opposite signs of
$\eta$ and $p_{0}$. Consider the solution (41) of the first equation.
Assuming, as usual, that the wave-function of an antiparticle is a bispinor,
being charge-conjugated to a certain negative-frequency solution of the Dirac
equation [14], we find that the antiparticle is associated with the function
$e^{imx^{0}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{C}}}$\hss}C^{\prime}_{\alpha}(-1)^{\alpha}({\underline{z}}{\vphantom{z}}^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}})$,
being complex-conjugated to the negative-frequency part of solution (41)
($\hat{C}\hat{P}\hat{T}$-conjugation yields
$e^{imx^{0}}C^{\prime}_{\alpha}({\underline{z}}{\vphantom{z}}^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}}$).
This function is a solution of the equation (30) with the same
$\varepsilon=1$, see the first term of (42); however, it is characterized by
opposite signs of $\eta$ and $S_{3}^{R}$ ($\eta=-1$, $S_{3}^{R}=1/2$). Thus, a
particle (electron) and a antiparticle (positron) are described by positive-
frequency solutions of eq. (30) with $\varepsilon=1$ and opposite signs of
$S_{3}^{R}$ and $\eta$.
There remain two unused functions, $e^{imx^{0}}(z^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{{\dot{\alpha}}})$
and $e^{imx^{0}}({\underline{z}}{\vphantom{z}}^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}})$,
that correspond to a particle with a negative parity and an antiparticle with
a positive parity, which provides solutions of the Dirac equations with
$\varepsilon=-1$,
$\displaystyle f_{3}(x,z)=e^{imx^{0}}C_{\alpha}(z^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{{\dot{\alpha}}})+e^{-imx^{0}}C^{\prime}_{\alpha}(z^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{{\dot{\alpha}}}),\quad
S_{3}^{R}=-1/2,$ (43) $\displaystyle
f_{4}(x,z)=e^{imx^{0}}D_{\alpha}({\underline{z}}{\vphantom{z}}^{\alpha}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}})+e^{-imx^{0}}D^{\prime}_{\alpha}({\underline{z}}{\vphantom{z}}^{\alpha}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{{\dot{\alpha}}}),\quad
S_{3}^{R}=1/2.$ (44)
However, these functions with $p_{0}>0$ can’t describe electron or positron:
does not exist an electron with negative parity and a positron with positive
parity.
Consequently, it is natural to associate the remaining functions with another
particles, as was done above.
## 6 Spin 1
Spin $1$ particles are described by quadratic combinations of ${z}$, which are
transformed with respect to the representations of the Lorentz group
$T_{[j_{1}j_{2}]}$, $j_{1}+j_{2}=1$. These are a $6$-dimensional adjoint
representation $T_{[10]}\oplus T_{[01]}$ (two complex-conjugate matrices from
$SO(3,C)$) and a $4$-dimensional vector representation
$T_{[\frac{1}{2}\frac{1}{2}]}$ (matrix from $SO(3,1)$).
One can see that if the spin part of a wave function is transformed according
to the representation $T_{[s0]}$ or $T_{[0s]}$ then the eigenvalue of the
Casimir operator $W^{2}$ is equal to $-m^{2}s(s+1)$, i.e., $s$ is spin.
Therefore, all the states carrying the representation $T_{[10]}\oplus
T_{[01]}$ have spin $1$.
For this representation all $6$ states are characterized by chirality
$\Gamma^{5}=j_{1}-j_{2}=\pm 1$. The multiple weight $S_{3}^{R}=B_{3}^{R}=0$,
being in the center of the diagram (Fig.2a), corresponds to a pair of states.
Notice that states with a definite parity $V_{\eta=\pm 1}^{0}$, corresponding
to $S_{3}^{R}=0$ (Fig.2b), are also characterized by a definite charge parity,
and therefore they can describe real neutral particles.
Figure 2: The weight diagrams of the representation $T_{[1\;0]}\oplus
T_{[0\;1]}$ of $SL(2,C)_{\mathrm{int}}$, $\Gamma^{5}=\pm 1$. (a), $\eta=\pm 1$
(b). The dotted lines join states related by the transformations of
$SL(2,C)_{\mathrm{int}}$.$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}$$V^{-}_{L}$$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z$$zz$$V^{-}_{R}$${\underline{z}}{\vphantom{z}}{\underline{z}}{\vphantom{z}}$$V^{+}_{L}$$V^{+}_{R}$$z{\underline{z}}{\vphantom{z}}$$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}$$V^{0}_{L}$$V^{0}_{R}$$S_{3}^{R}$$-iB_{3}^{R}$$a$
By restoring laboratory indices, one can easily see that each point of the
weight diagram (Fig.2) corresponds to three states (according to the number of
possible spin projections), that transform equally under
$SL(2,C)_{\mathrm{int}}$, but differently under $SL(2,C)_{\mathrm{ext}}$. In
particular, for states with $S_{3}^{R}=0$ among four pairwise products three
functions correspond to spin $1$, namely
$z^{1}{\underline{z}}{\vphantom{z}}^{1},\;z^{2}{\underline{z}}{\vphantom{z}}^{2},\;z^{1}{\underline{z}}{\vphantom{z}}^{2}+z^{2}{\underline{z}}{\vphantom{z}}^{1},$
since, due to unimodularity, $\det
Z=z^{1}{\underline{z}}{\vphantom{z}}^{2}-z^{2}{\underline{z}}{\vphantom{z}}^{1}=1$
is a Lorentz scalar.
Making a reduction to the compact group $SU(2)$, we obtain two triplets: left
and right, corresponding to the diagonals on Fig.2a, or triplets with a fixed
parity, Fig.2b.
Consider now the representation $T_{[\frac{1}{2}\frac{1}{2}]}$ (Fig.3).
Figure 3: The weight diagrams of the representation
$T_{[\frac{1}{2}\;\frac{1}{2}]}$ of $SL(2,C)_{\mathrm{int}}$, $\Gamma^{5}=0$.
On Fig.a the dotted line joins states related by the transformations of
$SL(2,C)_{\mathrm{int}}$, On Fig.b the dotted line joins states related by the
transformations of $SU(2)\subset SL(2,C)_{\mathrm{int}}$.$z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}$${\underline{z}}{\vphantom{z}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z$$z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z$${\underline{z}}{\vphantom{z}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}$$S_{3}^{R}$$-iB_{3}^{R}$$a$
For this representation, all states have the chirality $\Gamma^{5}=0$. The
reduction to the compact subgroup $SU(2)$ gives a triplet and singlet, which
can be seen on Fig.3b. Besides, in contrast to $T_{[1\;0]}\oplus T_{[0\;1]}$,
for $T_{[\frac{1}{2}\frac{1}{2}]}$ possible values of spin are $1$ and $0$.
Thus, to describe spin $1$-particles, functions $f(x,{z})$, carrying
representation $T_{[\frac{1}{2}\frac{1}{2}]}$, must satisfy certain subsidiary
conditions. In particular, in the subspace of functions $f(x,z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}})$
at $\hat{\mathrm{p}}^{2}=m^{2}$ we have [12]
$\hat{W}^{2}=-m^{2}(j_{1}+j_{2})(j_{1}+j_{2}+1)+4\hat{p}_{\mu}q^{\mu}\hat{p}_{\nu}\hat{V}^{\nu},$
(45)
where
$q^{\mu}=\frac{1}{2}\sigma^{\mu}_{\;\;\alpha{\dot{\alpha}}}z^{\alpha}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{\alpha}}$
and
$\hat{V}^{\mu}=\frac{1}{2}\sigma^{\mu}_{\;\;\alpha{\dot{\alpha}}}\partial^{\alpha}{\underline{\partial}}{\vphantom{\partial}}^{{\dot{\alpha}}}$.
Consequently, in the case $s=j_{1}+j_{2}$ a necessary and sufficient condition
of spinorial irreducibility is given by
$\hat{p}_{\mu}q^{\mu}\hat{p}_{\nu}\hat{V}^{\nu}f(x,z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}})=0.$
(46)
For the representations $T_{[s\,0]}$ and $T_{[0\,s]}$, this condition is
fulfilled identically, since in this case $\hat{V}^{\mu}f(x,z)=0$. In the
general case, taking into account that in the momentum representation, the
action of the operator $q^{\mu}\hat{p}_{\mu}$ is reduced to multiplication by
a number, we arrive at alternative conditions,
$\displaystyle p_{\mu}q^{\mu}=0,$ (47)
$\displaystyle\hat{p}_{\nu}\hat{V}^{\nu}f(x,z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}})=0.$
(48)
In the first case, we have a space of functions of two $4$-vectors $p_{\mu}$,
$q_{\mu}$, which are subject to invariant constraints,
$p^{2}=m^{2},\quad p_{\mu}q^{\mu}=0,\quad q^{2}=0.$ (49)
In the rest frame, according to (49), we have $z^{1}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{1}+z^{2}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{2}=0$.
Such an approach to the construction of wave functions, describing elementary
particles, was suggested by Wigner in the article [15], where the discussion
was restricted to particles of integer spin and to real-valued $q_{\mu}$ with
the constraints $p^{2}=m^{2}$, $p_{\mu}q^{\mu}=0$, $q^{2}=-1$. Different
generalizations of the Wigner’s approach were considered in [16, 17, 18, 19,
20].
The second requirement (48) is only a condition on the functions $\phi(x)$ and
it does not concern the spinorial variables. Indeed, representing the function
$f(x,{z})$ in the form
$f(x,z)=\phi_{\alpha}^{\;\;{\dot{\beta}}}(x)z^{\alpha}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\beta}}=\Phi_{\mu}(x)q^{\mu},\qquad\Phi_{\mu}(x)=-\bar{\sigma}_{\mu}^{\;\;{\dot{\beta}}\alpha}\phi_{\alpha{\dot{\beta}}}(x),\quad$
(50)
we find [12] that $\Phi_{\mu}(x)$ obey equations for spin $1$-particles in the
Proca form
$(\hat{p}^{2}-m^{2})\Phi_{\mu}(x)=0,\qquad\hat{p}^{\mu}\Phi_{\mu}(x)=0.$ (51)
Above we, considering possible values of $S_{3}^{R}$ for $W^{\pm}$ bosons,
have excluded $S_{3}^{R}=0$. Here we see that all states of spin $1$ with
$j_{1}+j_{2}=1$ and $S_{3}^{R}=0$ are related to true neutral particles and
can’t be related with $W^{\pm}$.
Therefore, in the framework of the present theory, we arrive to two families
of particles of spin $1$: two triplets, corresponding to $T_{[1\;0]}\oplus
T_{[0\;1]}$ and a quadruplet corresponding to
$T_{[\frac{1}{2}\;\frac{1}{2}]}$, which (since a reduction to a compact
subgroup was done) is decomposed into a triplet with $\eta=1$ and a singlet
with $\eta=-1$. They are candidates to describe well-known particles of spin
$1$ – the triplet of intermediate vector bosons and the photon.
To give an exact answer, let us consider the weight diagram of $6$-dimensional
adjoint irrep of the Lorentz group $T_{[1\;0]}\oplus T_{[0\;1]}$ (Fig.2). The
multiple weight $S_{3}^{R}=0$ can be related with real neutral particles – the
photon $\gamma$ and the $Z^{0}$-boson, the weights with $S_{3}^{R}=1$ and
$S_{3}^{R}=-1$ can be related with $W^{+}$ and $W^{-}$ bosons. In addition,
each of the latter appears twice as $W_{L}^{\pm}$ and $W_{R}^{\pm}$ (linear
combinations correspond to states with a definite parity $W_{\eta=\pm
1}^{\pm}$).
As far as the $4$-dimensional representation $T_{[\frac{1}{2}\;\frac{1}{2}]}$
is concerned, the $W^{+}$ and $W^{-}$ bosons can be associated with states
having $S_{3}^{R}=\pm 1$, with the parity $\eta=1$, whereas the photon and
$Z^{0}$-boson can be associated with two states having the zero charge
$S_{3}^{R}$, see Fig.3b.
However, more detail consideration exclude the case $T_{[1\;0]}\oplus
T_{[0\;1]}$. In massless limit not only $S_{3}^{R}$, but also $B_{3}^{R}$ and
$\Gamma^{5}$ are conserved quantum numbers. Then, $e_{L}^{-}$ and
$\tilde{\nu}_{R}$ are characterized by $\Gamma^{5}=1/2$ and $\Gamma^{5}=-1/2$
(see Fig.1 and (37)), so for $W^{-}$ we have $\Gamma^{5}=0$. The latter is
fulfilled for $T_{[\frac{1}{2}\;\frac{1}{2}]}$, but not for $T_{[1\;0]}\oplus
T_{[0\;1]}$. Analogously, it is easy to see (Fig.1) that $e_{L}^{-}$ and
$\tilde{\nu}_{R}$ are characterized by opposite values of $B_{3}^{R}$, and
therefore the charged $W^{-}$ ($S_{3}^{R}=-1$) must have $B_{3}^{R}=0$, which
holds true for states with $S_{3}^{R}=-1$, described by the representation
$T_{[\frac{1}{2}\;\frac{1}{2}]}$ (Fig.3), but not by $T_{[1\;0]}\oplus
T_{[0\;1]}$ (Fig.2).
## 7 Quasiregular representations and spin description
(geometrical models of spinning particles)
The consideration of GRR of the Poincaré group ensures the possibility of
consistent description of particles with arbitrary spin by means of scalar
functions on $\mathcal{M}\times\mathrm{Spin}(3,1)$, where $\mathcal{M}$ is
Minkowski space. At the same time, for description of spinning particles it is
possible to use the spaces $\mathcal{M}\times L$, where $L$ is some
homogeneous space of the Lorentz group (one or two-sheeted hyperboloid, cone,
projective space and so on); see, for example, [21, 22, 23, 15, 16, 17, 18,
19, 20, 24, 25, 26]. In some papers fields on homogeneous spaces are
considered; in other papers such spaces are treated as phase spaces of some
classic mechanics, and the latter are treated as models of spinning
relativistic particles.
These spaces appear in the framework of the next group-theoretical scheme. Let
us consider the left quasiregular representation of the Poincaré group
$T(g)f(g_{0}K)=f(g^{-1}g_{0}K),\quad K\subset\mathrm{Spin}(3,1),$ (52)
and since $x$ is invariant under right rotations (see (13))
$g_{0}\leftrightarrow(X,Z),\quad g_{0}K\leftrightarrow(X,ZK).$
Therefore the relation (52) defines the representation of the Poincaré group
in the space of functions $f(x,zK)$ on
$\mathcal{M}\times(\mathrm{Spin}(3,1)/K).$ (53)
Generally speaking, in the space of scalar functions on $\mathrm{Spin}(3,1)/K$
one can realize only a part of irreps of the Lorentz group, and in the space
of scalar functions on $\mathcal{M}\times(\mathrm{Spin}(3,1)/K)$ one can
realize only a part of irreps of the Poincaré group. In particular, the case
$K=\mathrm{Spin}(3,1)$ corresponds to a scalar field on Minkowski space.
Thus the consideration of left quasiregular representations allows one to
construct a number of spin models classified by subgroups
$K\subseteq\mathrm{Spin}(3,1)$. However the dimension of the space $M(3,1)/K$
is reduced in comparison with $M(3,1)$ by the number of generators of the
group $K$, respectively the number of commuting operators and the number of
quantum numbers is reduced as well.
There exist $13$ homogeneous spaces $M(3,1)/K$, containing Minkowski space
[27, 21, 22]. We will consider $5$ such spaces which were used in constructing
different geometrical models.
Some homogeneous spaces related to the group $SL(2,C).$
| | Name and dimension
---
of space
| Elements and
---
Transformations
Subgroup $K_{i}$ | | Internal
---
numbers
1 | Complex affine plane | 4 | $\begin{array}[]{c}(z^{1},z^{2})\rightarrow\\\ (\alpha z^{1}\\!+\\!\gamma z^{2},\beta z^{1}\\!+\\!\delta z^{2})\end{array}$ | $\left(\begin{array}[]{cc}1&\zeta\\\ 0&1\end{array}\right)$ | $j_{1},j_{2}$
2 | Complex projective | | | |
line $\mathbb{C}P^{1}\sim S^{2}$ | 2 | | $z=\displaystyle\frac{z^{1}}{z^{2}}$, $z\rightarrow\frac{\alpha z+\gamma}{\beta z+\delta}$
---
$\left(\begin{array}[]{cc}\alpha&\beta\\\ 0&\alpha^{-1}\end{array}\right)$ | – |
3 | Lobachevskian 3-space | | | |
(positive sheet of | | | | |
$H_{1}^{1,3}$ hyperboloid) | 3 | $\begin{array}[]{c}Q\rightarrow UQU^{\dagger},\\\ \det Q=1\end{array}$ | | $\left(\begin{array}[]{cc}\alpha&\beta\\\ -\bar{\beta}&\bar{\alpha}\end{array}\right),$
---
$|\alpha|^{2}+|\beta|^{2}=1$
$j_{1}=j_{2},\;\eta$ |
4 | Imaginary | | | |
Lobachevskian 3-space | | | | |
($H_{-1}^{1,3}$ hyperboloid) | 3 | $\begin{array}[]{c}Q\rightarrow UQU^{\dagger},\\\ \det Q=-1\end{array}$ | | $\left(\begin{array}[]{cc}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{array}\right),$
---
$|\alpha|^{2}-|\beta|^{2}=1$
$j_{1}=j_{2},\;\eta$ |
5 | The cone $H_{0}^{1,3}$ | 3 | $\begin{array}[]{c}Q\rightarrow UQU^{\dagger},\\\ \det Q=0\end{array}$ | $\left(\begin{array}[]{cc}e^{-i\varphi}&0\\\ \zeta&e^{i\varphi}\end{array}\right)$ | $j_{1}=j_{2}$
Here $z^{1},z^{2}$, and
${\underline{z}}{\vphantom{z}}^{1},{\underline{z}}{\vphantom{z}}^{2}$ are
elements of the first and the second columns of the matrix $Z\in SL(2,C)$,
$2\times 2$ matrix $Q$ corresponds to $4$-vector $q^{\mu}$,
$Q=\left(\begin{array}[]{cc}q^{0}+q^{3}&q^{1}-iq^{2}\\\
q^{1}+iq^{2}&q^{0}-q^{3}\end{array}\right),\qquad
U=\left(\begin{array}[]{cc}\alpha&\beta\\\
\gamma&\delta\end{array}\right),\qquad\det U=1.$
The latter vectors have different expressions in terms of $z$ in different
cases. For a cone,
$q^{\mu}=\sigma^{\mu}_{\;\;\dot{\beta}\alpha}z^{\alpha}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{\dot{\beta}}$ [28];
in two other cases $q^{\mu}$ is expressed via the tetrads
$v^{\mu}_{\;\;{\underline{n}}}=\sigma^{\mu}_{\;\;\dot{\beta}\alpha}\sigma_{{\underline{n}}}^{\;\;{\underline{a}}\dot{\underline{a}}}z^{\beta}_{\;\;{\underline{b}}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{\beta}}_{\;\;\dot{\underline{a}}}$
[1], then we have $q^{\mu}=v_{\;\;\underline{0}}^{\mu}$ for the subgroup
$K_{4}=SU(2)$, different $SU(1,1)$-subgroups correspond to
$v_{\;\;\underline{1}}^{\mu},v_{\;\;\underline{2}}^{\mu},$
$v_{\;\;\underline{3}}^{\mu}$ or to their linear combinations.
Let us discuss quantum numbers that can label quantum states corresponding to
scalar functions
$f^{\prime}(y^{\prime})=f(y),\quad
f^{\prime}(y)=T(g)f(y)=f(g^{-1})y,\;y^{\prime}=gy,\quad y\in M(3,1)/K,$ (54)
defined on the above listed spaces.
1\. The scalar field $f(x^{\mu},z^{\alpha},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}})$ on
the $8$-dimensional space $M(3,1)/K_{1}$ (spinning space $SL(2,C)/K_{1}$ – a
complex affine plane) depends on elements $z^{\alpha}$ of the first column of
the matrix $Z$ and complex conjugates $\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}$.
Such a field was studied in [21, 22, 25]. It possesses four characteristics
related to the orientation variables $z$ (the spin, its projection, and a pair
$(j_{1},j_{2})$ that fixes irrep of the Lorentz subgroup). To a given irrep of
the Lorentz subgroup correspond homogeneous functions of the power $2j_{1}$ in
$z^{\alpha}$ and $2j_{2}$ in $\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}$. As
follows from (Appendix. Generators and weight diagrams of the Lorentz group),
eigenvalues of the generators $\hat{S}_{R}^{3}$ and $\hat{B}_{R}^{3}$ are
fixed, they are expressed in terms of $j_{1}$ and $j_{2}$,
$S_{R}^{3}=j_{2}-j_{1},\;iB_{R}^{3}=(j_{2}+j_{1}).$
In contrast to the case of functions on $M(3,1)$, the space $M(3,1)/K_{1}$ is
not invariant under space reflection,
$Z\overset{\hat{P}}{\rightarrow}(Z^{\dagger})^{-1}$ or $z^{\alpha}\to-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}}$,
$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}}\to
z^{\alpha}$ [1], and functions $f(x^{\mu},z^{\alpha},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}})$ of
elements of the first column convert to functions $f(x^{\mu},-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}_{\dot{\alpha}},{\underline{z}}{\vphantom{z}}^{\alpha})$
of elements of the second column. Thus, states with a given parity cannot be
described by scalar functions on $M(3,1)/K_{1}$.
2\. Let us consider a projective model. The $6$-dimensional space
$M(3,1)/K_{2}$ (the spinning space $SL(2,C)/K_{2}$ is a $2$-dimensional
sphere) is a space of the least dimensions which can provide a spin
description by one-component functions. Particle models in this space were
studied in detail in [19, 20]. A relation to the previous (spinor) model are
given by the relations
$\phi(x^{\mu},z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z)=f(x^{\mu},z,1,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z,1),\qquad
f(x^{\mu},z^{1},z^{2},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{1},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{2})=(z^{2})^{2j_{1}}(\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{2})^{2j_{2}}\phi(x^{\mu},z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z),$
where $z={z^{1}}/{z^{2}}$. Here, $z$ are transformed linear fractionally in
contrast to other models where $z$ are transformed linearly. It is easily to
see that the transformation lows of same functions $\phi(x^{\mu},z,\bar{z})$
under the Lorentz group $SL(2,C)_{\mathrm{ext}}$ depend on $j_{1},j_{2}$ (the
corresponding generators depend on $j_{1},j_{2}$),
$\phi(x^{\mu},z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z)\to({z^{2\prime}})^{2j_{1}}({\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{2\prime}})^{2j_{2}}\phi({x^{\mu}}^{\prime},z^{\prime},\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{\prime}).$ (55)
In particular, this means that the functions $\phi(x^{\mu},z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z)$ do not contain any
information about a Lorentz group representation. We note that transformations
(55) of the functions $\phi(x^{\mu},z,\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z)$ are not reduced to
an argument change, and such functions are not scalar ones with respect to the
definition (54).
Space reflection transform $z$ into
${\underline{z}}{\vphantom{z}}={\underline{z}}{\vphantom{z}}^{1}/{\underline{z}}{\vphantom{z}}^{2}$,
which means, as in the previous case, that states with a given inner parity
cannot be described by functions $\phi(x^{\mu},z,\bar{z})$.
3\. Vector models use functions $f(p_{\mu},q_{\mu})$ of $4$-momentum $p_{\mu}$
and a spinning variable $q_{\mu}$,
$\hat{S}_{\mu\nu}=i(q_{\mu}\partial{q^{\nu}}-q_{\nu}\partial{q^{\mu}}),\quad\hat{\mathbf{S}}\hat{\mathbf{B}}=0.$
(56)
Since the Casimir operator $\hat{\mathbf{S}}\hat{\mathbf{B}}$ of the Lorentz
group is zero, we have $j_{1}=j_{2}$. A reduction of a irrep
$T_{[j_{1},j_{1}]}$ of the Lorentz group to a compact rotation subgroup is
given by the equation
$\textstyle T_{[j_{1},j_{1}]}=\sum_{j=0}^{2j_{1}}T_{j}.$ (57)
Thus, the models correspond to particles with integer spins that are described
by the representation $T_{[j_{1},j_{1}]}$ of the Lorentz group.
A $4$-vector $q_{\mu}$ is given by point on the hyperboloid $H_{-1}^{1,3}$,
$H_{1}^{1,3}$ or on the cone $H_{0}^{1,3}$ (see the table), respectively
$q_{\mu}q^{\mu}=0,\pm 1$. The condition $p_{\mu}q^{\mu}=0$ can be used to
select states with a spin $s=2j_{1}$ maximal for a given irrep
$T_{[j_{1},j_{1}]}$.
Thus, we have a family of models with scalar functions $f(p_{\mu},q_{\mu})$
and constraints
$p_{\mu}p^{\mu}=m^{2},\qquad p_{\mu}q^{\mu}=0,\qquad q_{\mu}q^{\mu}=0,\pm 1.$
(58)
The spaces of functions $f(p_{\mu},q_{\mu})$ on the hyperboloids is invariant
under space reflection,
$q^{\mu}=v_{\;\;\underline{0}}^{\mu}\rightarrow-(-1)^{\delta_{\mu 0}}q^{\mu}$,
$q^{\mu}=v_{\;\;\underline{3}}^{\mu}\rightarrow(-1)^{\delta_{\mu 0}}q^{\mu}$,
and therefore, such spaces can serve to describe states with a definite inner
parity $\eta$. Functions on the cone depending on
$q^{\mu}=\sigma_{\;\;\dot{\beta}\alpha}^{\mu}z^{\alpha}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z^{\dot{\beta}}$ under
space reflection are converted to functions of
$\underline{q}^{\mu}=\sigma^{\mu}_{\;\;\dot{\beta}\alpha}{\underline{z}}{\vphantom{z}}^{\alpha}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{\beta}}$.
That is why one cannot construct such scalar functions corresponding to states
with a definite parity $\eta$.
Thus, scalar functions on the homogeneous spaces
$\mathcal{M}\times(\mathrm{Spin}(3,1)/K)$, $K\subset\mathrm{Spin}(3,1)$,
describe spinning particles, however, they correspond to states where a part
of inner (right) quantum numbers is fixed (i.e., they are expressed via other
quantum numbers) or are not defined at all. In this case, the number of
commuting operators and the number of quantum numbers that characterize the
field is reduced by the number of generators of the subgroup $K$.
## 8 Concluding remarks
Orientable objects are described by the field $f(x,z)$ on the Poincaré group.
Functions $f(x,z)$ depend on $10$ parameters and admit two kinds of
transformations – left (change of space-fixed reference frame, or Lorentz
transformations) and right (change of body-fixed reference frame). These
transformations form the direct product $M(3,1)_{\mathrm{ext}}\times
M(3,1)_{\mathrm{int}}$.
An orientable object is characterized by $10$ quantum numbers. $8$ of them
have a standard interpretation (these are the $4$-momentum $p^{\mu}$, spin
$s$, helicity, and representation $(j_{1},j_{2})$ of the Lorentz group).
Two additional quantum numbers $S_{3}^{R}$ and $B_{3}^{R}$ that correspond to
the generators $\hat{S}_{3}^{R}$ and $\hat{B}_{3}^{R}$ of the group
$SL(2,C)_{\mathrm{int}}\subset M(3,1)_{\mathrm{int}}$ can be interpreted as
some charges. Indeed, the charges are additive quantum numbers, being
independent of a choice of the laboratory reference frame. Generators of
$SL(2,C)_{\mathrm{int}}$ commute with the generators $M(3,1)_{\mathrm{ext}}$
(i.e., with the generators of the Lorentz transformations), and therefore
“right” quantum numbers $S_{3}^{R}$ and $B_{3}^{R}$ do not change under a
change of the laboratory reference frame.
The two additional “right” quantum numbers $S_{3}^{R}$ and $B_{3}^{R}$
characterizing orientable objects possess some properties with respect to
discrete transformations, and their possible values are related to the spin
value. It was noted that “right” quantum numbers can be (in fact, uniquely)
ascribed to all known elementary particles. Thus, we believe that the complete
and, therefore, more adequate description of elementary (spinning) particles
is achieved if one considers them as orientable objects and use the
corresponding relativistic classification theory developed in this work.
In spite of the fact that left and right transformations commute, the spectra
of left and right generators $\hat{S}_{3}^{R}$ and $\hat{B}_{3}^{R}$ are not
independent. In particular, the “right” charges $S_{3}^{R}$ and $B_{3}^{R}$
must be integer for particles with integer spin and half-integer for particles
with half-integer spin. Note that, if $S_{3}^{R}$ supposed to be a conserved
quantum number, then $B_{3}^{R}$ has a definite value only for states with a
definite chirality (but not parity, since $\hat{B}_{3}^{R}$ does not commute
with the parity operator), i.e., $B_{3}^{R}$ can be conserved only for
massless particles.
The classification of orientable objects yields the following properties in
the one-particle sector. For fermions of spin $1/2$, there are four states
(quadruplet, realized by the up/down components of a weak doublet and by their
antiparticles), distinguished by right generators of the Poincaré group: the
sign of $B_{3}^{R}$-charge (instead of which one can choose the sign of
chirality or internal parity) and by the sign of $S_{3}^{R}$-charge. Particles
of spin $1$ also form a quadruplet, whose quantum numbers coincide with those
of $W^{+},W^{+},W^{0},A^{0}$. Besides, potentially there are another $6$
states corresponding to the representations $T_{[10]}\oplus T_{[01]}$ of the
group $SL(2,C)_{\mathrm{int}}$.
Note, once again, that in contrast with the left (external) symmetries, the
right (internal) symmetries can be generally broken, and, respectively, states
related by these symmetries can have different characteristics (including
mass).
We have considered a description on a basis of finite-dimensional
representations of the Lorentz group (and the related finite-dimensional RWE).
In our following work, we hope to consider unitary infinite-dimensional
representations of the Lorentz group and the related equations of the Majorana
type, as well as the case of interaction.
Acknowledgements D.M.G. acknowledges the permanent support of FAPESP and CNPq.
## Appendix. Generators and weight diagrams of the Lorentz group
Besides the four-dimensional vector notation for spin operators (see
(16),(17)), it is also convenient to use a three-dimensional notation:
$\hat{S}_{k}=\frac{1}{2}\epsilon_{ijk}\hat{S}^{ij}$,
$\hat{B}_{k}=\hat{S}_{0k}$. In the space of functions on the group
$f(z^{\alpha},\,{\underline{z}}{\vphantom{z}}^{\alpha})$ (functions of the $4$
elements of a matrix $SL(2,C)$ (2)) a direct calculation yields for left and
right generators333For the sake of brevity, we have used the notation that we
applied in [1, 12], $z^{\alpha}=z_{\;\;\underline{1}}^{\alpha}$, $\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}=\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}^{\;\;\dot{\underline{2}}}$,
${\underline{z}}^{\alpha}=z_{\;\;\underline{2}}^{\alpha}$, $\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\underline{z}}}}$\hss}\underline{z}_{\dot{\alpha}}=\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{\alpha}}^{\;\;\dot{\underline{1}}}$.
[12]
$\displaystyle\hat{S}_{k}=\frac{1}{2}(z\sigma_{k}\partial_{z}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\sigma}}}$\hss}\sigma_{k}\partial_{\stackrel{{\scriptstyle*}}{{z}}\\!\\!{\vphantom{z}}}\,)+...\;,$
$\displaystyle\hat{B}_{k}=\frac{i}{2}(z\sigma_{k}\partial_{z}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\sigma}}}$\hss}\sigma_{k}\partial_{\stackrel{{\scriptstyle*}}{{z}}\\!\\!{\vphantom{z}}}\,)+...\;,\quad
z=(z^{1}\;z^{2}),\quad\partial_{z}=(\partial/\partial{z^{1}}\;\partial/\partial{z^{2}})^{T};$
(59) $\displaystyle\hat{S}_{k}^{R}=-\frac{1}{2}(\chi\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\sigma}}}$\hss}\sigma_{k}\partial_{\chi}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\chi}}}$\hss}\chi\sigma_{k}\partial_{\stackrel{{\scriptstyle*}}{{\chi}}\\!\\!{\vphantom{\chi}}}\,)+...\;,$
$\displaystyle\hat{B}_{k}^{R}=-\frac{i}{2}(\chi\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\sigma}}}$\hss}\sigma_{k}\partial_{\chi}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\chi}}}$\hss}\chi\sigma_{k}\partial_{\stackrel{{\scriptstyle*}}{{\chi}}\\!\\!{\vphantom{\chi}}}\,)+...\;,\quad\chi=(z^{1}\;{\underline{z}}{\vphantom{z}}^{1}),\quad\partial_{\chi}=(\partial/\partial{z^{1}}\;\partial/\partial{{\underline{z}}{\vphantom{z}}^{1}})^{T};$
(60)
The terms $...$ stand for analogous expressions obtained by the change $z\to
z^{\prime}=({\underline{z}}{\vphantom{z}}^{1}\;{\underline{z}}{\vphantom{z}}^{2})$,
$\chi\to\chi^{\prime}=(z^{2}\;{\underline{z}}{\vphantom{z}}^{2})$. In
particular,
$\hat{S}_{3}^{R}=\frac{1}{2}(-z\partial_{z}+{\underline{z}}{\vphantom{z}}\partial_{\underline{z}}{\vphantom{z}}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\partial_{\stackrel{{\scriptstyle*}}{{z}}\\!\\!{\vphantom{z}}}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}\partial_{\stackrel{{\scriptstyle*}}{{{\underline{z}}{\vphantom{z}}}}\\!\\!{\vphantom{{\underline{z}}{\vphantom{z}}}}}\,)\;,\qquad\hat{B}_{3}^{R}=\frac{i}{2}(-z\partial_{z}+{\underline{z}}{\vphantom{z}}\partial_{\underline{z}}{\vphantom{z}}-\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\partial_{\stackrel{{\scriptstyle*}}{{z}}\\!\\!{\vphantom{z}}}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}\partial_{\stackrel{{\scriptstyle*}}{{{\underline{z}}{\vphantom{z}}}}\\!\\!{\vphantom{{\underline{z}}{\vphantom{z}}}}}\,)\;.$
(61)
It is known that from $\hat{S}_{k}$ and $\hat{B}_{k}$ one can construct linear
combinations $\hat{M}_{k}$ and $\hat{\bar{M}}{\vphantom{M}}_{k}$,
$\displaystyle\hat{M}_{k}=\frac{1}{2}(\hat{S}_{k}-i\hat{B}_{k})=z\sigma_{k}\partial_{z}+{\underline{z}}{\vphantom{z}}\sigma_{k}\partial_{\underline{z}}{\vphantom{z}},\quad\hat{M}_{+}=z^{1}\partial/\partial{z^{2}},\quad\hat{M}_{-}=z^{2}\partial/\partial{z^{1}},$
$\displaystyle\hat{\bar{M}}{\vphantom{M}}_{k}=-\frac{1}{2}(\hat{S}_{k}+i\hat{B}_{k})=\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\sigma}}}$\hss}\sigma_{k}\partial_{\stackrel{{\scriptstyle*}}{{z}}\\!\\!{\vphantom{z}}}+\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{\sigma}}}$\hss}\sigma_{k}\partial_{\stackrel{{\scriptstyle*}}{{{\underline{z}}{\vphantom{z}}}}\\!\\!{\vphantom{{\underline{z}}{\vphantom{z}}}}}\,,\quad\hat{\bar{M}}{\vphantom{M}}_{+}=\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{1}}\partial/\partial{\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{2}}},\quad\hat{\bar{M}}{\vphantom{M}}_{-}=\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{2}}\partial/\partial{\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{{\underline{z}}{\vphantom{z}}}}}$\hss}{\underline{z}}{\vphantom{z}}^{\dot{1}}},$
(62)
such that $[\hat{M}_{i},\hat{\bar{M}}{\vphantom{M}}_{k}]=0$; in addition, for
unitary representations of the Lorentz group, as it follows from the condition
$\hat{S}_{k}^{\dagger}=\hat{S}_{k}$, $\hat{B}_{k}^{\dagger}=\hat{B}_{k}$, the
relation $\hat{M}_{k}^{\dagger}=\hat{\bar{M}}{\vphantom{M}}_{k}$ must be
fulfilled (for finite-dimensional non-unitary representations
$\hat{S}_{k}^{\dagger}=\hat{S}_{k}$, $\hat{B}_{k}^{\dagger}=-\hat{B}_{k}$ and
$\hat{M}_{k}^{\dagger}=-\hat{\bar{M}}{\vphantom{M}}_{k}$).
Taking into account the fact that the operators $\hat{M}_{k}$ and
$\hat{\bar{M}}{\vphantom{M}}_{k}$ satisfy commutation relations of the algebra
$su(2)$, we find the following relations for the spectra of the Casimir
operators of the Lorentz subgroup:
$\displaystyle\hat{\mathbf{S}}^{2}-\hat{\mathbf{B}}^{2}=2(\hat{\mathbf{M}}^{2}+\hat{\bar{\bf
M}}{\vphantom{M}}^{2})=2j_{1}(j_{1}+1)+2j_{2}(j_{2}+1)=-\frac{1}{2}(k^{2}-\rho^{2}-4),\quad$
$\displaystyle\hat{\mathbf{S}}\hat{\mathbf{B}}=-i(\hat{\mathbf{M}}^{2}-\hat{\bar{\bf
M}}{\vphantom{M}}^{2})=-i\left(j_{1}(j_{1}+1)-j_{2}(j_{2}+1)\right)=k\rho,\qquad$
$\displaystyle\hbox{where }\quad\rho=-i(j_{1}+j_{2}+1),\quad k=j_{1}-j_{2}.$
(63)
That is, the irreps of the Lorentz group $SL(2,C)$ are labeled by a pair of
numbers $[j_{1},j_{2}]$. It is convenient to label unitary infinite-
dimensional irreps by pairs of numbers $(k,\rho)$; in addition, the irreps
$(k,\rho)$ and $(-k,-\rho)$ are equivalent [4, 28].
For finite-dimensional and unitary infinite-dimensional irreps of the group
$SL(2,C)$, the formulae of reduction to the compact $SU(2)$-subgroup have the
respective form
$T_{[j_{1},j_{2}]}=\sum_{j=|j_{1}-j_{2}|}^{j_{1}+j_{2}}T_{j},\qquad
T_{(k,\rho)}=\sum_{j=k}^{\infty}T_{j},$ (64)
see [28].
The difference $j_{1}-j_{2}$ (the difference between the number of dotted and
undotted indices) can also be obtained as an eigenvalue of the chirality
operator $\hat{\Gamma}^{5}$ (20).
Representations of low dimensions have a simple realization. The two-
dimensional irreps $T_{[1/2\;0]}$ and $T_{[0\;1/2]}$, which induce the
transformations of spinors – these are complex-conjugate matrices from
SL(2,C), three-dimensional irreps $T_{[1\;0]}$ and $T_{[0\;1]}$ – complex-
conjugate matrices from $SO(3,C)$, and four-dimensional matrices
$T_{[1/2\;1/2]}$, which induce the transformations of 4-vectors – this is a
representation by real-valued matrices from $SO(3,1)$.
The weight diagrams of the representations $T_{[1/2\;0]}\oplus T_{[0\;1/2]}$
and $T_{[1\;0]}\oplus T_{[0\;1]}$ are given by the figure. In addition, the
axes $S_{3}$ and $-iB_{3}$ on which we indicate the eigenvalues of the
corresponding operators are rotated by the angle of $45^{\circ}$ with respect
to the axes $m_{1}$ and $m_{2}$, on which we indicate eigenvalues of the
operators $\hat{M}_{3}$ and $\hat{\bar{M}}{\vphantom{M}}_{k}$.
$T_{[1/2\;0]}\oplus T_{[0\;1/2]}$$S_{3}$$-iB_{3}$$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{2}}$$\hbox
to0.0pt{$\stackrel{{\scriptstyle*}}{{\phantom{z}}}$\hss}z_{\dot{1}}$$z^{2}$$z^{1}$$m_{1}$$m_{2}$
\put(200.0,0.0){\leavevmode}
## References
* [1] D.M. Gitman and A.L. Shelepin. Fields on the Poincaré group and quantum description of orientable objects. Eur. Phys. J. C, 61(1):111–139, 2009. arXiv:hep-th/0901.2537.
* [2] E.P. Wigner. On unitary representations of the inhomogeneous Lorentz group. Ann. Math., 40(1):149–204, 1939.
* [3] S. Coleman and J. Mandula. All possible symmetries of the $S$-matrix. Phys. Rev., 159(5):1251–1256, 1967.
* [4] A.O. Barut and R. Raczka. Theory of Group Representations and Applications. PWN, Warszawa, 1977.
* [5] E.P. Wigner. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, New York, 1959.
* [6] L.D. Landau and E.M. Lifschitz. Quantum Mechanics, volume 3 of Course of Theoretical Physics. Pergamon, Oxford, 1977.
* [7] L.S. Biedenharn and J.D. Louck. Angular Momentum in Quantum Physics. Addison-Wesley, Reading, Massachusetts, 1981.
* [8] R.N. Zare. Angular Momentum. Understanding Spatial Aspects in Chemistry and Physics. Wiley, New York, 1988.
* [9] F. Lurçat and L. Michel. Sur les relations entre charges et spin. Nuovo Cimento, 21(3):574–576, 1961.
* [10] B. Kayser and A.S. Goldhaber. CPT and CP properties of Majorana particles, and the consequences. Phys. Rev. D, 28(9):2341–2344, 1983.
* [11] B. Kayser, F. Gibrat-Debu, and F. Perrier. The Physics of Massive Neutrinos. World Scientific, Singapore, 1989.
* [12] D.M. Gitman and A.L. Shelepin. Fields on the Poincaré group: Arbitrary spin description and relativistic wave equations. Int. J. Theor. Phys., 40:603–684, 2001. arXiv:hep-th/0003146.
* [13] I.L. Buchbinder, D.M. Gitman, and A.L. Shelepin. Discrete symmetries as automorphisms of the proper Poincaré group. Int. J. Theor. Phys., 41(4):753–790, 2002. arXiv:hep-th/0010035.
* [14] V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii. Relativistic Quantum Theory. Pergamon, New York, 1971.
* [15] E.P. Wigner. In A. Salam, editor, Theoretical Physics, page 59, Trieste, 1963\. IAEA.
* [16] Y.S. Kim and E.P. Wigner. Cylindrical group and massless particles. J. Math. Phys., 28(5):1175–1179, 1987.
* [17] L.C. Biedenharn, H.W. Braden, P. Truini, and H Van Dam. Relativistic wavefunctions on spinor spaces. J. Phys. A, 21:3593–3610, 1988.
* [18] Z. Hasiewicz and P. Siemion. A bosonic model for particles with arbitrary spin. Int. J. Mod. Phys. A, 7(17):3979–3996, 1992.
* [19] S.M. Kuzenko, S.L. Lyakhovich, and A.Yu. Segal. A geometric model of the arbitrary spin massive particle. Int. J. Mod. Phys. A, 10(10):1529–1552, 1995.
* [20] S.L. Lyakhovich, A.Yu. Segal, and A.A. Sharapov. Universal model of a $D=4$ spining particle. Phys. Rev. D, 54(8):5223–5238, 1996.
* [21] H. Bacry and A. Kihlberg. Wavefunctions on homogeneous spaces. J. Math. Phys., 10(12):2132–2141, 1969.
* [22] A. Kihlberg. Fields on a homogeneous space of the Poincare group. Ann. Inst. Henri Poincaré, 13(1):57–76, 1970.
* [23] C.P. Boyer and G.N. Fleming. Quantum field theory on a seven-dimensional homogeneous space of the Poincaré group. J. Math. Phys., 15(7):1007–1024, 1974.
* [24] A.A. Deriglazov and D.M. Gitman. Classical description of spinning degrees of freedom of relativistic particles by means of commuting spinors. Mod. Phys. Lett. A, 14:709–720, 1999.
* [25] W. Drechsler. Geometro-stohastically quantized fields with internal spin variables. J. Math. Phys., 38(11):5531–5558, 1997.
* [26] V.V. Varlamov. Maxwell field on the Poincare group. Int. J. Mod. Phys. A, 20:4095–4112, 2005. arXiv:math-ph/0310051.
* [27] D. Finkelstein. Internal structure of spinning particles. Phys. Rev., 100(3):924–931, 1955.
* [28] I.M. Gel’fand, M.I. Graev, and N.Ya. Vilenkin. Generalized Functions, volume 5. Academic Press, New York, 1966.
|
arxiv-papers
| 2010-01-28T23:07:22 |
2024-09-04T02:49:08.094152
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D.M. Gitman, A.L. Shelepin",
"submitter": "Dmitry Gitman",
"url": "https://arxiv.org/abs/1001.5290"
}
|
1001.5344
|
# Long-range forces : atmospheric neutrino oscillation at a magnetized
detector
Abhijit Samanta111E-mail address: abhijit.samanta@gmail.com Ramakrishna
Mission Vivekananda University, Belur Math, Howrah 711 202, India
and
222past address where the work was initiated.Harish-Chandra Research
Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India
###### Abstract
Among the combinations $L_{e}-L_{\mu}$, $L_{e}-L_{\tau}$ and
$L_{\mu}-L_{\tau}$ any one can be gauged in anomaly free way with the standard
model gauge group. The masses of these gauge bosons can be so light that it
can induce long-range forces on the Earth due to the electrons in the Sun.
This type of forces can be constrained significantly from neutrino
oscillation. As the sign of the potential is opposite for neutrinos and
antineutrinos, a magnetized iron calorimeter detector (ICAL) would be able to
produce strong constraint on it. We have made conservative studies of these
long-range forces with atmospheric neutrinos at ICAL considering only the
muons of charge current interactions. We find stringent bounds on the
couplings $\alpha_{e\mu,e\tau}\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1.65\times 10^{-53}$
at 3$\sigma$ CL with an exposure of 1 Mton$\cdot$yr if there is no such force.
For nonzero input values of the couplings we find that the potential
$V_{e\mu}$ opposes and $V_{e\tau}$ helps to discriminate the mass hierarchy.
However, both potentials help significantly to discriminate the octant of
$\theta_{23}$. The explanation of the anomaly in recent MINOS data (the
difference of $\Delta m^{2}_{32}$ for neutrinos and antineutrinos), using
long-range force originated from the mixing of the gauge boson $Z^{\prime}$ of
$L_{\mu}-L_{\tau}$ with the standard model gauge boson $Z$, can be tested at
ICAL at more than 5$\sigma$ CL. We have also discussed how to disentangle this
from the solution with CPT violation using the seasonal change of the distance
between the Earth and the Sun.
neutrino oscillation, atmospheric neutrino, long-range force
###### pacs:
14.60.Pq
## I Introduction
The large hadron collider at CERN will probe the extensions of standard model
above the electroweak scale. On the other hand, some of the extensions below
the electroweak scale can be probed at the neutrino oscillation experiments.
The extensions below the electroweak scale introduce massless or nearly
massless gauge foot ; axion or Higgs bosons singletm ; tripletm ; hall ;
Chang:1989xm and they couple with matter very feebly and remain invisible.
These lead to the existence of new kind of forces either i) generating
deviations of gravitational law at short distances, or ii) predicting low mass
particles whose exchange will induce forces at long distances, generally
violating the equivalence principle Damour:1996xt ; fischbach . A number of
experiments have been searching for these new forces. The null results provide
bounds on particle physics models, gravitational physics, and even on
cosmological models fischbach ; Adelberger:2003zx ; Williams:1995nq .
The bounds on these couplings to baryon and/or lepton number Lee:1955vk can
be obtained from the testing of equivalence principle Eotvos:1922pb (the free
fall acceleration is same for all bodies independent of their chemical
content). In Okun:1995dn , the author has used this idea to establish a bound
on the strength of an hypothetical vectorial leptonic force and obtained the
bound on the “fine structure” constant $\alpha\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}10^{-49}.$ See
Dolgov:1999gk for a review. A comparable limit also comes from lunar laser
ranging Williams:1995nq , which measures the differential acceleration of the
Earth and the Moon towards the Sun.
However, one can extend the SM with an additional $U(1)$ gauge symmetry
without introducing any anomaly for one of the lepton flavor combinations:
$L_{e}-L_{\mu}$, $L_{\mu}-L_{\tau}$, and $L_{e}-L_{\tau}$. The masses of the
gauge bosons can be so light that the induced forces may have terrestrial
range. Then the electrons inside the Sun can induce forces on the Earth
surface depending on the lightness of the gauge boson. These forces on the
Earth may also be from supernova neutrinos, galactic electrons depending on
its ranges. These couple only to electron (and neutrino) density inside a
massive object. As a result, the acceleration experienced by an object depends
on its leptonic content and mass; and thus violates equivalence principle.
The long-range (LR) forces can play role in neutrino oscillation. The
relatively stronger bounds than those from testing of the equivalence
principle have been obtained from solar, atmospheric and supernova data
Joshipura:2003jh ; Bandyopadhyay:2006uh ; GonzalezGarcia:2006vp ;
Grifols:1993rs ; Grifols:2003gy 333There are plenty relic neutrinos and
antineutrinos in the universe ($\sim 50/cm^{3}$), which may screen the
leptonic charges of the of celestial bodies. However, it has been analyzed in
detail and shown that the screening is impossible Blinnikov:1995kp . The long-
range forces due to galactic electrons, which can affect the galactic rotation
curves if the range $R_{LR}\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}R_{\rm gal}$ (where
$R_{\rm gal}$ is the distance from the galactic center $\sim$ 10 kpc), is also
now very tightly constrained Bandyopadhyay:2006uh . Finally, all these results
are consistent with the existing bounds on violation of equivalence principle.
The sign of this potential is opposite for neutrinos and anti-neutrinos and
hence can lead to apparent differences in neutrino and anti-neutrino
oscillation probabilities without introducing CP or CPT violation. For
instance, the recently found discrepancy in the survival probabilities of
$\nu_{\mu}$s and $\bar{\nu}_{\mu}$s in the MINOS experiment pvahle has been
explained using the mixing of $Z^{\prime}$ boson of $L_{\mu}-L_{\tau}$
symmetry with the $Z$ boson of the SM model; and the required associated
parameters to explain this anomaly can be found in Heeck:2010pg .
We have studied the long-range forces with atmospheric neutrinos at the
magnetized iron calorimeter detector (ICAL) proposed at the India-based
neutrino Observatory (INO) Arumugam:2005nt , which can directly measure the
potential detecting separately $\nu_{\mu}$s and $\bar{\nu}_{\mu}$s. In this
article we have focussed on:
\- the bounds on the couplings of long-range forces,
\- how significantly long-range forces modify the sensitivity of the
measurement of oscillation parameters; particularly, the octant of 2-3 mixing
(sign of $\delta^{\rm oct}=\theta_{23}-45^{\circ}$) and the mass hierarchy
(sign of $\Delta m_{31}^{2}$),
\- test of the explanation of the MINOS data with long-range forces
Heeck:2010pg .
\- possibility to disentangle the effect of CPT and long-range forces 444The
advantages of atmospheric neutrinos to discriminate $CPT$ violation from $CP$
violation and nonstandard interactions have been discussed in Samanta:2010ce
..
We have also studied the changes of sensitivity to mass hierarchy and to
$\theta_{23}$-octant with true (input) $\theta_{23}$ values considering no
such forces (This was not studied in earlier works.).
The paper is organized as follows. We discuss the full three flavor neutrino
oscillation in matter with long-range potential in sec. II, the details of the
analysis method in sec. III, and the bounds on the couplings in Sec. IV. The
effects of the long-range potential on discrimination of mass hierarchy and on
$\theta_{23}$-octant is discussed in sec. V. Finally, the testing of the
explanation of the anomaly in MINOS data using long-range force is described
in Sec. VI. The discussion and conclusion are given in VII.
## II Oscillation in presence of long-range potential
The electrons inside the Sun generate a potential at the Earth by
Joshipura:2003jh :
$V_{e\mu,e\tau}=\alpha_{e\mu,e\tau}\frac{N_{e}}{R_{ES}}\approx 1.3\times
10^{-11}{\rm eV}\left(\alpha_{e\mu,e\tau}/10^{-50}\right),$ (1)
where, $\alpha_{e\mu,e\tau}=g^{2}_{e\mu,e\tau}/{4\pi}$; $g_{e\mu,e\tau}$ are
the gauge couplings of $L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$ symmetries.
$N_{e}\approx 10^{57}$ is the number of electrons inside the Sun sun and
$R_{ES}=7.6\times 10^{26}$ GeV-1, the distance between the Earth and the Sun.
In a three neutrino framework the neutrino flavor states
$|\nu_{\alpha}\rangle$, $\alpha=e,\mu,\tau$ can be expressed as linear
superpositions of the neutrino mass eigenstates $|\nu_{i}\rangle$, $i=1,2,3$
with masses $m_{i}$ :
$|\nu_{\alpha}\rangle=\sum_{i}U_{\alpha i}|\nu_{i}\rangle~{}.$ (2)
$U$ is the $3\times 3$ unitary matrix. The time evolution of the flavor states
is
$i\frac{\rm d}{\rm
dt}\left[\nu_{\alpha}\right]=\frac{1}{2E}UM_{\nu}^{2}U^{\dagger}\left[\nu_{\alpha}\right],$
(3)
where, $[\nu_{\alpha}]$ is the vector of flavor eigenstates, and
$\left[\nu_{\alpha}\right]^{T}=\left[|\nu_{e}\rangle,|\nu_{\mu}\rangle,|\nu_{\tau}\rangle\right]$.
The evolution equation in the presence of matter and long-range potential is
$\displaystyle i\frac{\rm d}{{\rm
d}t}\left[\nu_{\alpha}\right]=\frac{1}{2E}\left[UM_{\nu}^{2}U^{\dagger}+\left(\begin{array}[]{ccc}A&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right)+V_{LR}\right]\left[\nu_{\alpha}\right],$ (7)
where,
$\displaystyle V_{LR}=\left(\begin{array}[]{ccc}V_{e\mu}&0&0\\\
0&-V_{e\mu}&0\\\ 0&0&0\end{array}\right){\rm
or}\left(\begin{array}[]{ccc}V_{e\tau}&0&0\\\ 0&0&0\\\
0&0&-V_{e\tau}\end{array}\right).$ (14)
The matter term $A=2\sqrt{2}G_{F}n_{e}E=7.63\times 10^{-5}~{}{\rm
eV}^{2}~{}\rho({\rm gm/cc})~{}E({\rm GeV})~{}\hbox{eV}^{2}.$ Here, $G_{F}$,
$n_{e}$ and $\rho$ are the Fermi constant, the electron number density and the
matter density of the medium, respectively. The evolution equation for
antineutrinos has the reversed sign for $A$, $V_{LR}$ and the phase $\delta$.
We have numerically solved full three flavor oscillation in presence of matter
and long-range forces.
However, to understand the bounds it is easy to consider two flavor $\mu-\tau$
oscillation, (which has been studied in Joshipura:2003jh to constrain the
long-range forces using atmospheric neutrino data of Super-Kamiokande
experiment Super-K ). To understand the effect of 1-3 mixing, one needs to
consider the changes of the effective oscillation parameters due to $V_{LR}$.
The $\mu-\tau$ oscillation in presence of $V_{e\tau}$ is governed by the
evolution equation
$\displaystyle i\frac{d}{dt}\begin{pmatrix}\nu_{\mu}\\\ \nu_{\tau}\
\end{pmatrix}=$ $\displaystyle\begin{pmatrix}-\frac{\Delta m_{32}^{2}}{4E}\cos
2\theta_{23}&\frac{\Delta m_{32}^{2}}{4E}\sin 2\theta_{23}\\\ \frac{\Delta
m_{32}^{2}}{4E}\sin 2\theta_{23}&\frac{\Delta m_{32}^{2}}{4E}\cos
2\theta_{23}-V_{e\tau}\ \end{pmatrix}\begin{pmatrix}\nu_{\mu}\\\ \nu_{\tau}\
\end{pmatrix}$ (15)
Then, the survival probability of $\nu_{\mu}$
$P_{\mu\mu}=1-\sin^{2}2\tilde{\theta}_{23}~{}\sin^{2}\frac{\Delta\tilde{m}_{23}^{2}L}{4E},$
(16)
where, $L$ is the neutrino flight path length. The effective mixing angle
$\tilde{\theta}_{23}$ and $\Delta\tilde{m}_{32}^{2}$ are related with their
vacuum quantities by the relations
$\sin^{2}2\tilde{\theta}_{23}=\frac{Sin^{2}2\theta_{23}}{\left[(\xi_{e\tau}-\cos
2\theta_{23})^{2}+\sin^{2}2\theta_{23}\right]}$ (17)
and
$\Delta\tilde{m}_{23}^{2}=\Delta m_{23}^{2}\left[(\xi_{e\tau}-\cos
2\theta_{23})^{2}+\sin^{2}2\theta_{23})^{1/2}\right];$ (18)
where, $\xi_{e\tau}\equiv\frac{2V_{e\tau}E}{\Delta m_{32}^{2}}$. The potential
$V_{e\tau}$ and the corresponding $\xi$ change sign for $\bar{\nu}$.
Similarly, for $L_{e}-L_{\mu}$ gauge symmetry the survival probability can be
obtained and they satisfy
$P_{\mu\mu}(V_{e\tau})=P_{\bar{\mu}\bar{\mu}}(-V_{e\tau})=P_{\mu\mu}(-V_{e\mu})=P_{\bar{\mu}\bar{\mu}}(V_{e\mu})$
(19)
## III The $\chi^{2}$ analysis
To evaluate the potential of ICAL with atmospheric neutrinos we generate
events by NUANCE-v3 Casper:2002sd and consider only the muon energy and its
direction (directly measurable quantities) of the events for a conservative
estimation. The energy and angular resolutions of the muons at ICAL are very
high: 4-10% for energy and 4-12% for zenith angle, which are obtained from
GEANT geant simulation. The ranges are due to different energies and
different angles with respect to the iron layers. These uncertainties are very
negligible compared to the uncertainties in reconstructed neutrinos due to
kinematics of the scattering processes.
The major uncertainty arises from the particles produced in the event other
than muon. One might expect that consideration of the hadrons for neutrino
energy $\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}2$ GeV will improve the sensitivities substantially.
But, in Samanta:2010xm , it has been found that there is a very marginal
improvement on measurement of $\theta_{23}$ and a small improvement on $\Delta
m_{31}^{2}$: $\delta(\Delta m_{31}^{2})=0.02\times 10^{-3}$eV2. The fact is
that the total hadron energy is carried out by multiple low energy hadrons.
The average energy per hadron is $\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}$ 1 GeV and the
average number of hadrons per event is $\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}$ 2\. The energy
resolution of the hadrons at this energy is $\sim$ 80%, and consequently, the
neutrino energy resolutions do not improve significantly after adding the
hadrons 555The detection of the neutral current events may also be possible,
which have no directional information, but the energy dependence of the
oscillation averaged over all directions can contribute to the total
$\chi^{2}$ in the sensitivity studies..
The $\chi^{2}$ is calculated according to the Poisson probability distribution
with flat uncertainties of the oscillation parameters. The term due to the
contribution of prior information of the oscillation parameters measured by
other experiments is not added to $\chi^{2}$ to examine solely the performance
of ICAL. The data have been binned in cells of equal size in the $\log_{10}E$
\- $L^{0.4}$ plane, where $L=2R\cos\theta_{Z}$. The choice of binning is
motivated by pattern of the oscillation probability
$P(\nu_{\mu}\rightarrow\nu_{\mu})$ in the $L-E$ plane Samanta:2008ag . The
distance between two consecutive oscillation peaks driven by $\Delta
m^{2}_{23}$ increases (decreases) as one goes to lower $L$ ($E$) values for a
given $E$ ($L$). The binning of $L$ has been optimized to get better
sensitivity to the oscillation parameters. To maintain $\chi^{2}/d.o.f\approx
1$ for Monte Carlo simulation study, number of events should be $>4$ per cell
Samanta:2008af (as large number of cells at high energies have number of
events less than 4 or even zero and they increase the $\chi^{2}/d.o.f$
substantially beyond 1). If the number is less than 4 (which happens in the
high energy bins), we combine events from the nearest cells.
The migration of the number of events from true neutrino energy and zenith
angle cells to muon energy and zenith angle cells is made using exact energy-
angle correlated 2-dimensional resolution functions Samanta:2006sj . For each
set of oscillation parameters, we integrate the oscillated atmospheric
neutrino flux folding with the cross section, the exposure time, the target
mass, the efficiency and the two dimensional energy-angle correlated exact
resolution functions to obtain the predicted data in each cell in $L-E$ plane
for the $\chi^{2}$ analysis. We use the charge current cross section of
Nuance-v3 Casper:2002sd and the Honda flux in 3-dimensional scheme
Honda:2006qj . The number of bins and resolution functions have been optimized
in Samanta:2008af . Both theoretical (fit values) and experimental (true
values) data for $\chi^{2}$ analysis have been generated in the same way by
migrating number of events from neutrino to muon energy and zenith angle bins
using the resolution functions Samanta:2009qw .
The systematic uncertainties of the atmospheric neutrino flux are crucial for
determination of the oscillation parameters. We have divided them into two
categories: (i) the overall flux normalization uncertainties which are
independent of the energy and zenith angle, and (ii) the spectral tilt
uncertainties which depend on $E$ and $\theta_{\rm z}$.
The flux with uncertainties included can be written as
$\displaystyle\Phi(E,\theta_{Z})=\Phi_{0}(E)\left[1+\delta_{E}\log_{10}\frac{E}{E_{0}}\right]$
$\displaystyle\times\left[1+\delta_{Z}(|\cos\theta_{Z}|-0.5)\right]\times\left[1+\delta_{f_{N}}\right]$
For $E<1$ GeV we take the energy dependent uncertainty $\delta_{E}=15\%$ and
$E_{0}=1$ GeV and for $E>10$ GeV, $\delta_{E}=5\%$ and $E_{0}=10$ GeV. The
overall flux uncertainty as a function of zenith angle is parametrized by
$\delta_{Z}$. According to Honda:2006qj we use $\delta_{Z}=4\%$, which leads
to 2% vertical/horizontal flux uncertainty. We take the overall flux
normalization uncertainty $\delta_{f_{N}}=10\%$ and the overall neutrino
cross-section uncertainty $\delta_{\sigma}=10\%$.
We evaluate the $\chi^{2}$ for both normal hierarchy (NH) and inverted
hierarchy (IH) with $\nu$s and $\bar{\nu}$s separately for a given set of
oscillation parameters. Then we find the total $\chi^{2}$
($=\chi^{2}_{\nu}+\chi^{2}_{\bar{\nu}}$).
We have set the inputs of $|\Delta m_{32}^{2}|=2.5\times 10^{-3}$eV2, and
$\delta_{CP}=0$. We marginalize $\chi^{2}$ over $\Delta
m_{32}^{2},~{}\theta_{23},~{}\theta_{13}$ and $\alpha$. We have chosen the
range of $\Delta m_{32}^{2}=2.0-3.0\times 10^{-3}$eV2,
$\theta_{23}=37^{\circ}-54^{\circ},$ $\theta_{13}=0^{\circ}-12.5^{\circ}$, and
$\alpha_{e\mu,e\tau}=0-3\times 10^{-53}$. The solar parameters are fixed at
their best-fit values: $\Delta m_{21}^{2}=7.67\times 10^{-5}$eV2 and
$\sin^{2}\theta_{12}=0.312$ Fogli:2008ig . The effect of $\Delta m_{21}^{2}$
comes in sub-leading order in the oscillation probability for atmospheric
neutrinos when $E\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}$ GeV and it is very negligible.
Figure 1: The upper bounds of $\alpha_{e\mu}$ and $\alpha_{e\tau}$ for true
(input) $\alpha=0$ in the left pannel; and the corresponding upper as well as
lower bounds for input $\alpha=1.1538\times 10^{-53}$ in the right pannel,
respectively.
Figure 2: The sensitivities to mass hierarchy for different true (input)
$\theta_{23}$ values for input IH (left) and input NH (right), respectively.
For each case we have considered true (input) values of $\alpha_{LR}=0$,
$\alpha_{e\mu}=1.1538\times 10^{-53}$, and $\alpha_{e\tau}=1.1538\times
10^{-53}$, respectively.
Figure 3: The sensitivities to deviation of 2-3 mixing from maximal mixing as
well as its octant for both true (input) IH (left) and NH (right). For each
case we have considered $\alpha_{LR}=0$, $\alpha_{e\mu}=1.1538\times
10^{-53}$, $\alpha_{e\tau}=1.1538\times 10^{-53}$, respectively.
In this analysis we have considered an exposure of 1 Mton.year (which is 10
years run of 100 kTon) of ICAL and energy range 0.8 - 15 GeV.
## IV Bounds on couplings
For atmospheric neutrinos at the magnetized ICAL the bounds on $\alpha_{e\mu}$
and $\alpha_{e\tau}$ are shown in the left pannel in Fig. 1 and its strength
to constrain them with upper as well as lower bounds are shown in the right
pannel in Fig. 1 with a typical nonzero input (true) value. We have checked
that the bounds do not change significantly for different combinations of true
values of oscillation parameters.
The bounds $\alpha_{e\mu}\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}5.5\times 10^{-52}$
and $\alpha_{e\tau}\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}6.4\times 10^{-52}$ at
90% CL have been obtained from present atmospheric neutrino oscillation data
Joshipura:2003jh . Considering both atmospheric and solar data the bounds are
$\alpha_{e\mu}\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}3.4\times 10^{-53}$ and $\alpha_{e\tau}\mathrel{\hbox
to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}2.5\times 10^{-53}$ at $3\sigma$ CL
Bandyopadhyay:2006uh . The better precision and smaller value by a few order
of magnitude of solar mass squared difference than the atmospheric one play
here the main role to make it significantly stringent.
The bounds are relatively stronger at the magnetized detector:
$\alpha_{e\mu,e\tau}\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1.65\times 10^{-53}$
at 3$\sigma$ CL. For atmospheric neutrinos it comes mainly due to the high
precision of $\Delta m_{32}^{2}$ and this can be understood quite well from
simple two flavor survival probability of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ in
vacuum with effective $\Delta m_{32}^{2}$ and $\theta_{23}$ for long-range
potentials. This becomes possible due to the fact that the 1-3 mixing effect
is sub-leading (as $\theta_{13}$ $<11.38^{\circ}$ Schwetz:2011qt ) and it can
be neglected at this moment for simplicity. When the potential $V_{LR}$ comes
to the play, it tries to change the effective value of $\Delta m_{31}^{2}$
(see Eq. 18) and becomes tightly constrained. The sign $V_{LR}$ is opposite
for $\nu$ and $\bar{\nu}$, and ICAL can detect them separately. This makes the
bounds more tighter at ICAL and lessens the difference in bounds between
$\alpha_{e\mu}$ and $\alpha_{e\tau}$, while the difference is substantially
large at non-magnetized detectors.
## V Effects on 2-3 sector
### V.1 Determination of mass hierarchy
In Fig. 2 we show the sensitivity to mass hierarchy for different true values
of $\theta_{23}$. We show it for three cases: assuming no potential for long-
range forces ($\alpha_{LR}=0$) and with a benchmark input (true) value for
$\alpha_{e\mu}$ and $\alpha_{e\tau}$, respectively.
For all three cases the sensitivity increases as one goes to the higher
$\theta_{23}$ values and it is a general feature for both cases with and
without the potential for long-range force. This can be understood from the
$\nu_{\mu}$ flux at the detector considering the effect of 1-3 mixing. Here we
assume no potential for long-range force. The ratio of the $\nu_{\mu}$ flux at
the detector (oscillated) and at the source (original) Samanta:2010xm :
$\frac{F_{\mu}}{F_{\mu}^{0}}\approx K(\sin
2\theta_{23})-f(\theta_{23})\left(1-\frac{1}{r}\right)P_{A}(\theta_{13}),$
(20)
where, $K(\sin 2\theta_{23})$ is an even function of the deviation (symmetric
with respect to change of the octant), and $P_{A}(\theta_{13})$ is a function
of $\theta_{13}$ only, and
$f(\theta_{23})\equiv\left(s_{23}^{4}-\frac{s^{2}_{23}}{r}\right)$ (21)
which increases quickly with $\theta_{23}$, so that for $r=3-4$,
$f(\theta_{23}=40^{\circ})\ll f(\theta_{23}>50^{\circ})$. Therefore, for
$\theta_{23}<45^{\circ}$ the flux $F_{\mu}$ has much weaker dependence on
$\theta_{13}$ than for $\theta_{23}>45^{\circ}$. This is reflected in the
sensitivity to mass hierarchy as the lower limit of $\theta_{13}$ has been
taken zero during marginalization and no prior contribution from other future
experiments has been considered. However, in future the lower limit will be
known from other experiments (as very recently T2K puts a lower bound
Abe:2011sj ); then the $\theta_{23}$-dependence will be less and the
sensitivity to mass hierarchy will also be substantially improved
Samanta:2009qw .
The potential $V_{e\mu}$ opposes, while $V_{e\tau}$ helps to determine the
hierarchy (see Fig. 2). As the 1-3 mixing is small
($\theta_{13}<11.2^{\circ}$), the potential $V_{e\tau}$ decreases the
effective value of $\Delta m_{31}^{2}$. Consequently, the resonant energy
decreases (see Fig. 2 of Samanta:2006sj ), where the atmospheric neutrino flux
is relatively large and it helps in determination of hierarchy. On the other
hand, as 2-3 mixing is large, $V_{e\mu}$ increases the effective value of
$\Delta m_{31}^{2}$ for $\theta_{23}>45^{\circ}$ (see Eq. 18). In spite of an
enhancement in 1-3 mixing (which happens for both potentials) the final
sensitivity decreases for $V_{e\mu}$ due to relatively low statistics at the
resonant zones. These happen for $\nu$ with NH and for $\bar{\nu}$ with IH.
### V.2 Determination of octant of $\theta_{23}$
The sensitivity to 2-3 mixing, mainly the deviation from its maximal mixing
($\delta=\theta_{23}-45^{\circ}$) and the octant (sign of $\delta$) are shown
in Fig. 3 for three cases: i) $\alpha_{LR}=0$, ii) $\alpha_{e\mu}=1.1538\times
10^{-53}$, and iii) $\alpha_{e\tau}=1.1538\times 10^{-53}$, respectively. It
is important to note here that octant discrimination is significantly improved
in presence of long-range potentials. This is due to the fact that effective
1-3 mixing is always larger for both potentials. For a given hierarchy the
change of the sensitivity with the change of true (input) octant depends
mainly on the magnitude of $\delta^{\rm eff}={\theta_{23}}^{\rm
eff}-45^{\circ}$, which again strongly depends on the potentials (see Eq. 17).
The change of the effective value of $\Delta m_{31}^{2}$ works here
subdominantly (while it was a dominating factor for determination of the mass
hierarchy). Now, for $V_{e\tau}$ with NH, sensitivity to octant determination
is better for $\theta_{23}>45^{\circ}$ than $\theta_{23}<45^{\circ}$ as
$\delta^{\rm eff}$ is increased due to $V_{e\tau}$ for neutrinos and the flux
is two times higher for neutrinos than antineutrinos. This is opposite for
$V_{e\mu}$: octant determination is better for $\theta_{23}<45^{\circ}$ than
$\theta_{23}>45^{\circ}$. Similarly, the results with IH can be understood
using antineutrinos considering the symmetries in Eq. 19.
## VI MINOS anomaly
The anomaly in recent MINOS data (difference in measured values of 2-3 mass
splittings and mixing angles for $\nu$ and $\bar{\nu}$ pvahle ), which implies
CPT violation or signal for non-standard interactions, has been explained by
long-range potential due to $L_{\mu}-L_{\tau}$ gauge symmetry Heeck:2010pg .
The Sun does not contain any $\mu$ or $\tau$ and hence there is no direct
bound from neutrino oscillation. However, the potential can be induced
indirectly on the Earth by the mixing of the gauge boson $Z^{\prime}$ of
$L_{\mu}-L_{\tau}$ with the standard model $Z$ boson Heeck:2010pg :
$V_{\mu\tau}=3.6\times 10^{-14}{\rm
eV}\left(\frac{\alpha_{\mu\tau}}{10^{-50}}\right)$ (22)
The sign changes for $\bar{\nu}$. From the analysis of atmospheric neutrino
data at ICAL with an exposure of 1 Mton$\cdot$yr, we find the bound on the
coupling $\alpha_{\mu\tau}\geq 3.2(4.2)\times 10^{-51}$ at 3(5)$\sigma$ CL and
the explanation of MINOS data Heeck:2010pg can be tested at more than
5$\sigma$ CL.
## VII Discussion and Conclusion
In this paper, we have estimated the conservative bounds on long-range forces
with atmospheric neutrinos at ICAL: $\alpha_{e\mu,e\tau}\mathrel{\hbox
to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}1.65\times 10^{-53}$ at 3$\sigma$ CL. This bounds are
significantly stronger than the present bounds.
The CPT violation and the long-range forces, which are the candidates for
solution of recent anomaly in MINOS data, can be discriminated with
atmospheric neutrinos at ICAL. The distance between the Sun and the Earth
varies and the difference between aphelion and perihelion is about 3%, which
causes seasonal variation of the long-range forces. But, there should not be
any such change for CPT violation.
Assuming one by one nonzero input value of these couplings it is found that
while the potential $V_{e\mu}$ opposes; $V_{e\tau}$ helps to discriminate the
mass hierarchy. However, both potentials help to discriminate octant of
$\theta_{23}$. The anomaly of MINOS data, which has been explained using long-
range force potential, can be tested at ICAL with more that 5$\sigma$ CL.
Acknowledgements: This work was started at Harish-Chandra Research Institute
and has been supported partly by the Neutrino Physics projects of this
institute. The use of cluster computational facility installed by the funds of
this project is gratefully acknowledged.
## References
* (1) R. Foot, Mod. Phys. Lett. A 6, 527 (1991); X.-G. He, G. C. Joshi, H. Lew and R. R. Volkas, Phys. Rev D 43, 22 (1991).
* (2) M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104, 199 (1981); J. E. Kim, Phys. Rev. Lett., 43 103 (1979).
* (3) Y. Chikashige, R. N. Mohapatra and R. D. Peccei, Phys. Lett. B 98, 265 (1981).
* (4) G. B. Gelmini and M. Roncadelli, Phys. Lett. B 99, 411 (1981).
* (5) L. J. Hall and S. J. Oliver, Nucl. Phys. Proc. Suppl. 137, 269 (2004) [arXiv:hep-ph/0409276] and references therein.
* (6) D. Chang, W. Y. Keung and P. B. Pal, Phys. Rev. D 42, 630 (1990).
* (7) T. Damour, Class. Quant. Grav. 13, A33 (1996) [arXiv:gr-qc/9606080].
* (8) Fischbach E, Gillies G T, Krause D E, Schwan J G and Talmadge C 1992 Metrologia 29 213
* (9) E. G. Adelberger, B. R. Heckel and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003) [arXiv:hep-ph/0307284].
* (10) J. G. Williams, X. X. Newhall and J. O. Dickey, Phys. Rev. D 53, 6730 (1996).
* (11) T. D. Lee and C. N. Yang, Phys. Rev. 98 (1955) 1501.
* (12) R. V. Eotvos, D. Pekar, E. Fekete, Annalen Phys. 68, 11-66 (1922).
* (13) L. Okun, Phys. Lett. B 382, 389 (1996) [arXiv:hep-ph/9512436].
* (14) A. D. Dolgov, Phys. Rept. 320, 1 (1999).
* (15) A. S. Joshipura and S. Mohanty, Phys. Lett. B 584, 103 (2004) [arXiv:hep-ph/0310210].
* (16) A. Bandyopadhyay, A. Dighe and A. S. Joshipura, Phys. Rev. D 75, 093005 (2007) [arXiv:hep-ph/0610263].
* (17) M. C. Gonzalez-Garcia, P. C. de Holanda, E. Masso and R. Zukanovich Funchal, JCAP 0701, 005 (2007) [arXiv:hep-ph/0609094].
* (18) J. A. Grifols, E. Masso and S. Peris, Astropart. Phys. 2, 161 (1994).
* (19) J. A. Grifols and E. Masso, Phys. Lett. B 579, 123 (2004) [arXiv:hep-ph/0311141].
* (20) S. I. Blinnikov, A. D. Dolgov, L. B. Okun and M. B. Voloshin, Nucl. Phys. B 458, 52 (1996) [arXiv:hep-ph/9505444].
* (21) Talk given by P. Vahle at 24th International Conference On Neutrino Physics And Astrophysics (Neutrino 2010), June 14, 2010.
* (22) J. Heeck and W. Rodejohann, arXiv:1007.2655 [hep-ph]; H. Davoudiasl, H. -S. Lee, W. J. Marciano, [arXiv:1102.5352 [hep-ph]].
* (23) V. Arumugam et al. [INO Collaboration], [ See, http://www.imsc.res.in/ ino ]
* (24) A. Samanta, Phys. Lett. B693, 296-300 (2010). [arXiv:1005.4851 [hep-ph]].
* (25) John N. Bahcall, Neutrino Astrophysics (Cambridge Univ. Press,Cambridge, England, 1989).
* (26) Y. Fukuda et al. Phys Rev Lett. 82, 2644 (1999); Y. Fukuda et al. Phys Lett . B388, 397 (1999);Y. Fukuda et al. Phys Rev Lett. 85, 3999 (2000).
* (27) D. Casper, Nucl. Phys. Proc. Suppl. 112, 161 (2002) [arXiv:hep-ph/0208030].
* (28) http://geant4.web.cern.ch/geant4/
* (29) A. Samanta, A. Y. .Smirnov, JHEP 1107, 048 (2011). [arXiv:1012.0360 [hep-ph]].
* (30) A. Samanta, Phys. Rev. D 79, 053011 (2009) [arXiv:0812.4640 [hep-ph]].
* (31) A. Samanta, Phys. Rev. D 80, 113003 (2009) [arXiv:0812.4639 [hep-ph]].
* (32) A. Samanta, Phys. Lett. B 673, 37 (2009) [arXiv:hep-ph/0610196].
* (33) M. Honda, T. Kajita, K. Kasahara, S. Midorikawa and T. Sanuki, Phys. Rev. D 75, 043006 (2007) [arXiv:astro-ph/0611418].
* (34) A. Samanta, Phys. Rev. D81, 037302 (2010). [arXiv:0907.3540 [hep-ph]].
* (35) G. L. Fogli et al., Phys. Rev. D 78, 033010 (2008) [arXiv:0805.2517 [hep-ph]].
* (36) T. Schwetz, M. Tortola, J. W. F. Valle, New J. Phys. 13, 063004 (2011). [arXiv:1103.0734 [hep-ph]].
* (37) K. Abe et al. [ T2K Collaboration ], [arXiv:1106.2822 [hep-ex]].
|
arxiv-papers
| 2010-01-29T13:21:26 |
2024-09-04T02:49:08.103114
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhijit Samanta",
"submitter": "Abhijit Samanta",
"url": "https://arxiv.org/abs/1001.5344"
}
|
1001.5428
|
# Search for Extra-dimensions in a single-jet and missing energy channel at
CMS experiment
CMS Collaboration
E-mail
###### Abstract:
A possible solution to the hierarchy problem is the presence of extra spatial
dimensions beyond the three ones which are known from our everyday experience.
The phenomenological ADD model of large extra-dimensions predicts a missing
transverse energy plus a single-jet signature. This contribution addresses the
sensitivity of the CMS detector at the LHC pp collider to parameters of this
model, focusing on the conditions expected for second half of 2010 running
($\sqrt{s}=10$ , O(100) ). It is shown that a significant improvement of the
existing limits can be obtained in such an early stage.
This contributions outlines the analysis procedures for the search of large
extra dimensions in the missing transverse energy plus a single-jet channel,
using the Compact Muon Solenoid (CMS) detector [1]. The description of the
detector performance and the simulated samples of events correspond to what
expected for 10 center-of-mass energy and integrated luminosity up to 200 .
Full details of the analysis can be found in Ref. [2].
The phenomenological ADD model [3] aims to solve the hierarchy problem between
the electroweak and Planck scales by introducing a number $\delta$ of extra
spatial dimensions, which in the simplest scenario are compactified over a
torus and all have the same radius $R$. The fundamental scale $M_{D}$ is
related to the effective $4$-dimensional Planck scale $M_{Pl}$ according to
the formula $M_{Pl}^{2}\sim M_{D}^{\delta+2}R^{\delta}$. Current experimental
constraints allow a scenario with $\delta\geq 2$, corresponding to extra-
dimensions sizes below $5\cdot 10^{-2}$ mm if the fundamental scale $M_{D}$ is
of the order of TeV. Searches in both the jet+$E_{T}^{miss}\,$ and the
$\gamma+E_{T}^{miss}\,$ have been performed by CDF [4], D0 [5], and LEP
experiments [6]. The best 95% confidence limits on $M_{D}$ are 1.40(1.04)for
the extra dimensions scenario with $\delta=2(4)$.
This study is focused on the production of a graviton $G$ balanced by a
energetic hadronic jet via the $q\bar{q}\rightarrow gG$, $qg\rightarrow qG$,
and $gg\rightarrow gG$ processes. The new physics signature is a high-
transverse-momentum ($p_{T}>100\div 200$ GeV ) jet in the central region of
the detector, recoiling back-to-back in the transverse plane with a
$E_{T}^{miss}\,$ of similar magnitude.
The Standard Model process $Z(\nu\nu)+$jets leads to invisible energy
recoiling against jets and is described by the same signature as the signal,
thus the contribution from this “irreducible” background needs to be estimated
in the best possible way. Other important background sources are $W+$jets with
a leptonic $W$ decay (if the charged lepton is not reconstructed), QCD di-jets
(when one or more jets are mismeasured and a significant amount of
$E_{T}^{miss}$ is produced), and top-pair and single-top quark production,
especially for events with few of collimated jets where leptons are not
identified.
The ADD-model signal has been produced with the SHERPA Monte Carlo generator
[7], in different samples with $M_{D}$ ranging from 1 to 3 and $\delta$ from 2
to 6. The transverse momentum of the outcoming parton was required harder than
150 GeV . A set of background processes were generated with a sample size
corresponding to an integrated luminosity larger than 200 , then processed by
a full simulation of the detector. The hadronization and fragmentation of
quarks and gluons (along with the underlying event) were performed using
PYTHIA 6.409 [8] and the CTEQ61L Parton Density Functions (PDF) [9] were used.
With these production parameters, the signal cross sections at leading-order
ranges from 279 pb (for $M_{D}=$1, $\delta=2$) to 0.58 pb (for $M_{D}=$3,
$\delta=6$).
The analysis procedure is based on a set of cuts aimed to maximize the ratio
of number of signal events over the square root of number of background
events. At trigger level, a single jet stream is exploited, requiring at least
1 jet with $p_{T}>70(110)$ GeV at Level 1 (High Level Trigger). As shown in
Fig. 1$(a)$, the signal leads to a long tail in the distribution of the
vectorial sum of jets transverse momenta ($MHT$), hence a cut $MHT>$250 GeV
was imposed at the pre-selection level. In order to reduce the impact of jets
not coming from hard interaction, only jets with transverse momenta larger
than 50 GeV within $|\eta|<3$ are considered. To clean the events from
isolated lepton contamination, along with electrons and photons misidentified
as jets, the fraction of jet energy collected by the electromagnetic
calorimeter over the total energy is required to be lower than 0.9 and
isolated tracks (having less than 10% of $p_{T}$ in a ${0.02<\Delta R<0.35}$
cone) are removed. The leading jet is required to have $p_{T}>200$ GeV and
$|\eta|<$1.7. A veto against events with more than two jets and a number of
angular cuts $\Delta\phi(\textrm{jet\,1},MHT)>2.8$ and
$\Delta\phi(\textrm{jet\,2},MHT)>0.5$ complete the selection. The missing
$H_{T}$ distribution for signal and background after all selections is shown
in Fig. 1$(b)$. The signal shows up as an excess of events in addition to the
dominant background $Z(\nu\nu)+$jets.
Figure 1: $(a)$ $MHT$ distributions for ADD signal ($M_{D}=2$ , $\delta=2$)
and relevant backgrounds before any selection, after 200 . $(b)$ Missing
$H_{T}$ distribution after all selections are applied. Histograms are overlaid
and number of events correspond to 200 .
The most important systematic uncertainties from theory come from the cross
section sensitivity to the renormalization and factorization scale
(${}^{+7.5\%}_{-6.7\%}$), and from uncertainties on the parton density
function (${}^{+11.5\%}_{-9.5\%}$). Uncertainties associated to energy and
angular jet resolution (assumed to be 10% and 0.1 rad, respectively) and jet
calibration (shifted by $\pm 10\%$) turn to be the most relevant instrumental
effects. Their relative shift from the value with no systematic effect depends
on the ADD points, ranging from 10% to 16%. The value of instantaneous
luminosity, that is assumed to have a $\pm 10\%$ uncertainty, was
incorporated.
The irreducible background of $Z(\nu\nu)+$jets (“invisible Z”) can be deduced
from samples of events containing a high-$p_{T}$ $W$ boson decaying
leptonically. The selection defining the control region has been kept the same
of signal region, except that a single isolated muon with $p_{T}>20$ GeV and
$|\eta|<$2.4 was required. This procedure allowed to reproduce the same
kinematic region that was designed for the signal, but having a muon for which
the hypothesis of coming from $W$ is highly probable. This sample was cleaned
from the processes with at least one well-isolated muon passing the selection,
then corrected by the ratio between $W+$jets and $Z+$jets production cross
sections and muon reconstruction and isolation efficiency. The number of
invisible $Z$ events in the signal region was found to be $N(Z(\nu\nu)+{\rm
jets})^{Sign}=163\pm 22\,{\rm(stat)}\pm 13{\rm(syst)}\pm
17{\rm(MC)}$111Notation $MC$ refers to the counting error due to the limited
number of generated Monte Carlo events., to be compared with $N(Z(\nu\nu)+{\rm
jets})^{MC}=182\pm 13\,{\rm(stat)}$.
The same region can be used to measure the $W(l\nu)+$jets contribution in the
signal region. All the background estimates are consistent with the result
from simulations.
The total background can be estimated as $N_{B}=243\pm 23\,{\rm(stat)}\pm
13\,{\rm(syst)}$ events after 200 of integrated luminosity. The 95% C.L. limit
was found by scanning the parameter space to minimize the negative Log
Likelihood [11]. When different $M_{D}$, $\delta$ are interpolated, the
exclusion plot in Fig. 2$(a)$ can be derived. The amount of data needed for a
$5\sigma$ discovery was also calculated, using an estimator based on Profile
Likelihood. When results from different signal points are interpolated, the
plot in Fig. 2$(b)$ is obtained, that represents the sensitivity for a
discovery after 200 .
Figure 2: $(a)$: Exclusion plot at 95% C.L., showing the minimum luminosity
necessary to exclude a given value of $M_{D}$. $(b)$: Discovery potential of
the analysis as a function of $M_{D}$ and $\delta$ after 200 . The horizontal
thick lines correspond to 3$\sigma$ and 5$\sigma$ significance level. In both
cases, sensitivity is plotted for two different extra dimension scenarios.
These results indicate that the current exclusion limits from Tevatron
experiments can be matched at LHC with the first physics run. Exclusion limits
at 95% for $M_{D}=3$ , $\delta=2$, $M_{D}=2$ , $\delta=4$ can be reached after
only 11 and 5.0 , respectively; also early discoveries for $\delta=2(4)$
scenarios are possible, if $M_{D}$ is below 3.1(2.3) , respectively.
## References
* [1] CMS Collaboration, CERN-LHCC 2006/001 (2006)
* [2] CMS Collaboration, CMS-PAS-EXO-09-013
* [3] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429, 263
* [4] CDF Collaboration, FERMILAB-PUB-08-247-E, Submitted to Phys. Rev. Lett.
* [5] D0 Collaboration, Phys. Rev. Lett. 101, 011601 (2008)
* [6] DELPHI Collaboration, Eur. Phys. J. C 38, 395 (2005)
* [7] T. Gleisberg, S. Hoeche, F. Krauss et al. JHEP 0402, 056 (2004)
* [8] T. Sjöstrand et al. Comput. Phys. Commun. 135 238 (2001)
* [9] J. Pumplin et al., JHEP 07, 012 (2002)
* [10] W.-M. Yao et al. [Particle Data Group], Journ. of Phys. G 33, 1 (2006)
* [11] R. D. Cousins, J. T. Linnemann, J. Tucker, NIM A595 480, (2008)
|
arxiv-papers
| 2010-01-29T16:10:16 |
2024-09-04T02:49:08.109934
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leonardo Benucci (for the CMS Collaboration)",
"submitter": "Leonardo Benucci Dr.",
"url": "https://arxiv.org/abs/1001.5428"
}
|
1002.0084
|
# Systematic design study of all-optical delay line based on Brillouin
scattering enhanced cascade coupled ring resonators
Myungjun Lee1, Michael E. Gehm1,2, and Mark A. Neifeld1,2 1 Department of
Electrical Computer Engineering, University of Arizona, Tucson, AZ 85721-0104,
USA 2 College of Optical Sciences, University of Arizona, Tucson, AZ
85721-0104, USA mjlee76@email.arizona.edu
###### Abstract
We present a technique to improve the slow-light performance of a side-coupled
spaced sequence of resonators (SCISSOR) combined with a stimulated Brillouin
scattering (SBS) gain medium in optical fiber. We evaluate device performance
of SCISSOR-only and SCISSOR + SBS systems for different numbers of cascaded
resonators from 1 to 70 using two different data fidelity metrics including
eye-opening and mutual information. A practical system design is demonstrated
by analyzing its performance in terms of fractional delay, power transmission,
and data fidelity. We observe that the results from the two metrics are in
good agreement. Based on system optimization under practical resource and
fidelity constraints, the SCISSOR consisting of 70 cascaded resonators
provides a fractional delay of $\sim$ 8 with 22 dB attenuation at a signal bit
rate of 10 Gbps. The combined optimal SCISSOR (with 70 resonators) + SBS
system provides a improved fractional delay up to $\sim$ 17 with unit power
transmission under the same constraints.
Keywords: slow-light, optical delay line, stimulated Brillouin scattering
(SBS), ring resonator, mutual information, non-linear optics, group delay,
group velocity dispersion
## 1 Introduction
Tunable all-optical delay systems that dynamically manipulate the group
velocity of light have received a great deal of attention for optical
information processing applications such as data buffering and
synchronization. Various slow-light devices, including those based on
electromagnetically induced transparency (EIT) in atomic vapor, stimulated
Brillouin and Raman scattering (SBS and SRS) in optical fiber, and photonic
structures in dielectric material, have been explored as potential
realizations of a practical delay system [1–10].
As for on-chip approaches, coupled resonators or photonic crystals are
promising techniques that would allow easy integration with other electronics
or optical components. Many recent demonstrations of coupled resonator optical
waveguides (CROW) and side coupled integrated spaced sequence of resonators
(SCISSOR) have been designed and fabricated in compact sizes ($\sim$ 10
$\mu\textrm{m}^{2}$ ) and with the possibility of dynamic delay control and
large delay-bandwidth product [2–4]. A more recent analysis from Otey et al
shows that cascaded resonators can even capture light pulses (i.e., stopped
light) by completely compressing the system bandwidth and that the captured
pulse can then be released [5].
A large fractional delay (equivalent to the delay-bandwidth product) can be
achieved by a chain of resonators. Unfortunately, these devices suffer a
fundamental trade-off between transmission loss and delay, which potentially
limits the use of large numbers of resonators. For example, a CROW consisting
of 6 ring resonators demonstrated continuously controllable fractional delay
up to 3 at a signal bit rate (BR) of 10 Gbps and a bit error rate (BER) of
$10^{-9}$ [2]. Its transmission loss, however, is 3 dB (i.e., 0.5 dB/ring) and
therefore the use of any additional resonator will increase the BER higher
than $10^{-9}$. For comparison, Xia et al have demonstrated a chain of 56
cascaded micro-ring resonators in a side-coupled configuration using a
silicon-on-insulator waveguide. They achieved a large fractional delay ($\sim$
5) at a BR of 10 Gbps and BER of $10^{-4}$ [4]. This high BER is a direct
consequence of the 22 dB transmission loss of the device resulting from the
use of the large number of rings.
One possible way of preserving acceptable output signal quality without
sacrificing the delay performance is to use a Brillouin amplifier. An SBS
gain-based delay system could provide significant signal amplification and its
tunable gain bandwidth could be increased up to 25 GHz, which allows high
speed data transmission [11, 12]. In addition, an optimal SBS gain system
would provide additional fractional delay of up to 3 [13]. Therefore,
combining this system with cascaded resonators or other photonic resonance
structures seems like a promising method for compensating their respective
disadvantages while increasing maximum fractional delay [14–16].
In general, large slow light delay is accompanied by substantial group
velocity dispersion (GVD) that manifests itself as signal distortion. The
presence of higher-order GVD terms lead to changes in the pulse shape. Under
such a condition, we require a metric to quantitatively measure the output
data quality along with the delay. A common measure of communications
performance for the propagation of a pulse train is an eye-diagram. Eye-
diagrams are useful for estimating signal distortion via the maximum eye-
opening; and its location represents the delay [16–19]. Neifeld and Lee have
presented an alternative metric that uses Shannon information to estimate the
information capacity and information delay in the presence of noise [20, 21].
In this paper, we utilize these two metrics to evaluate SCISSOR, SBS, and
SCISSOR + SBS under practical resource and fidelity constraints. By jointly
optimizing the system parameters of the SCISSOR + SBS system, we determine the
maximum fidelity-constrained fractional delay at a BR of 10 Gbps.
## 2 Data fidelity metric
The important quantities to consider for evaluating slow-light system
performance are the fractional delay and received data fidelity. When a single
pulse or a pulse sequence propagates through the dispersive media, it
undergoes GVD. There are several metrics including the pulse broadening factor
[22, 23], amplitude and phase distortion [10], eye-opening [16–19], and mutual
information [20] that have been introduced to quantify the slow-light
performance. In what follows, we consider the eye-opening and information-
theoretic metrics.
### 2.1 Eye-opening metric
An eye-diagram is used to visualize the shape of communications waveforms and
it is generated by repetitively superimposing subsequent traces of a given
data stream over a fixed time interval. The eye-opening (EO) is the maximum
difference between the minimum value of “ones” and the maximum value of
“zeros” at the bit center. The data distortion (D) can be quantified using the
eye-opening and it is defined as
$D=1-\max(EO).$ (1)
If a pulse sequence passes through a dispersive medium, the output signal
could be broadened or distorted, and then D will increase due to the increased
intersymbol interference (ISI). Note that distortion has a monotonic
relationship with BER and D = 0.35 indicates a corresponding BER $\simeq$
10-9, resulting in reliable communication [15, 19].
An eye-opening based delay can be calculated by the time difference $T_{EO}$
between the input and output eye center defined when the EO is maximal. The
fractional eye-opening delay (EOD) is defined as the time delay divided by the
input pulsewidth $T_{p}$, that is, EOD = $T_{EO}/T_{p}$.
### 2.2 Information theoretic metric
Information theory was first explored by Shannon and information rate has
become a standard method to characterize the quality of a communication
channel [24, 25]. Recently, an information theoretic metric was introduced
using the mutual information between the slow light input and output signals,
in order to measure the information based delay (ID) and information
throughput (IT) [20]. The IT-metric in this paper is based on the channel
model displayed in figure 1(a). Figure 1(b) shows the examples of 3 bits
output signal propagated through an arbitrary slow light channel, which may
include effects of delay, distortion, and noise. The input X is a binary-
valued sequence and it is modulated via on-off keying (OOK). The slow-light
delay system is represented by the channel operator HSL, where HSL could
represent any kind of delay device. The mutual information (MI) represents the
quantity of transmitted data, and estimates how much input information about X
is known when the output Y is observed. Thus, the MI can be defined as
$I(X;Y)=H(X)-H(X|Y)$, where $H(X)$ is the entropy of the discrete input X,
representing the a priori uncertainty, and $H(X|Y)$ is the conditional entropy
after the output is observed [24, 25]. We assume that the output signal Y is
corrupted by additive white Gaussian noise (AWGN) with zero mean and variance
$\sigma^{2}$. We also assume that the elements $x_{i}$ of a specific n-bit
input sequence X are independent and identically-distributed (IID), leading to
a prior probability $p(x_{i})$ = (1/2)n. Under these assumptions, the MI can
be written as:
$I(X;Y)=n+\int\sum_{i=1}^{M}p(x_{i})p(Y|x_{i})\log_{2}\frac{p(Y|x_{i})p(x_{i})}{\sum_{j=1}^{M}p(x_{j})p(Y|x_{j})}dY,$
(2)
where n is the number of input bits, $\textit{M}=2^{\textit{n}}$ is the number
of possible n-bit input sequences, and $p(x_{i},Y)$ is the joint probability
density function (PDF) of $x_{i}$ and Y. The integral over Y in equation (2)
can be solved by the Monte Carlo simulation with important sampling. Here,
$p(Y|x_{i})$ is the PDF of Y conditioned on $x_{i}$ that is expressed by the
Gaussian PDF:
$p(Y|x_{i})\simeq\frac{1}{(2\pi\sigma^{2})^{nL}}\exp(-\frac{1}{2\sigma^{2}}|Y-H_{SL}x_{i}|^{2}),$
(3)
where L is the number of simulation samples used to represent a single
Gaussian pulse.
Note that the concept of delay is not easily captured within $I(X;Y)$. In
order to apply I(X;Y) to the analysis of slow light systems, we impose a
window structure, which confines an input pulse sequence within a finite
duration window [20, 26]. With this approach, we can compute the MI between X
and only that part of Y contained within the output window (OW) as a function
of window offset [20, 21]. Here we use a simple example to describe the IT-
metric, let us first consider an ideal distortion free delay device with
$\sigma^{2}$ = 0, as shown in figure 2. The 3 bits of Gaussian pulses with a
50% return-to-zero (RZ) modulation format serve as an input, and the Gaussian
pulse is defined to have field amplitude $E(t)$ = exp$(-(t/T_{HW})^{2})$,
where $T_{HW}$ = $T_{b}$/2 is the bit half-width at 1/$e^{2}$ intensity and
$T_{b}$ is the bit period. The 50% RZ modulation denotes that a logical one is
represented by a half-bit wide pulse, therefore, $T_{p}$ = $T_{b}$/2. In order
to compute the MI for this example of the three bit transmission, we consider
all 8 possible states (M = 8), as shown in figure 2(a). We assume that the
input bit period $T_{b}$ = 100 ps and the value of delay $T_{D}$ = 400 ps. The
input signal is fitted within an input window (IW), and then we can compute
the MI between input X within the IW and only part of Y contained in the OW
for many different OW locations in figures 2(a) and (b). We observe the values
of MI = 3 bit and 1 bit for the two candidate output windows (OW1 and OW2) at
two different values of window offset = 400 ps and 600 ps, respectively, as
shown in figure 2(b). For this example, when the window offset is the same as
the delay $T_{D}$, all the input signal information can be transferred without
loss caused by distortion, noise, and energy leaking outside the window. Thus,
the peak value of I(X;Y) represents the amount of information that can be
transmitted through the slow light channel; while the location of this peak
provides an information-theoretic measure of delay. Therefore, we define the
peak-height as the information throughput (IT) and the peak location as the
information delay (ID) of the SL device, where the normalized IT is
$IT=\frac{\max\\{I(X;Y)\\}}{\textmd{n-bits}},$ (4)
and this definition will be used throughout the remainder of the paper.
## 3 Ring resonators
### 3.1 Single resonator
For a coupled ring resonator, as shown in figure 3, the output fields can be
related to the input fields through a complex amplitude transfer function
$H_{\mathrm{Ring}}(\omega)=\frac{E_{2}(\omega)}{E_{1}(\omega)}=\frac{k-a\
\mathrm{exp}(i\phi(\omega))}{1-ka\ \mathrm{exp}(i\phi(\omega))},$ (5)
where $E_{i}$ is the complex field amplitudes, k is the self-coupling
coefficient ($k^{2}=1-\rho^{2}$), $\rho$ is a cross-coupling coefficient, a =
exp(-$\alpha L_{R}$/2) is the round trip amplitude loss of the resonator,
$L_{R}$ is the ring circumference, and $\alpha$ is the total attenuation
coefficient which includes all sources of loss such as material absorption,
bending loss, and scattering loss from waveguide roughness [3, 4]. The round-
trip phase shift $\phi(\omega)$ in the ring can be represented by
$\phi(\omega)$ = 2$\pi n_{R}L_{R}$ $(\omega-\omega_{0})$/c, where $n_{R}$ is
the effective index of the ring, c is the speed of light, and $\omega_{0}$ is
the resonance angular frequency.
The phase response $\Phi_{\mathrm{Ring}}(\omega)$ of the transfer function is
obtained by the relation $H_{\mathrm{Ring}}(\omega)$ =
$|H_{\mathrm{Ring}}(\omega)|$exp(j$\Phi_{\mathrm{Ring}}(\omega)$) and it is
given in terms of k, a, and $\phi(\omega)$ as follows:
$\Phi_{\mathrm{Ring}}(\omega)=\pi+\phi(\omega)+\tan^{-1}\Big{[}\frac{k\
\mathrm{sin}\phi(\omega)}{a-k\
\mathrm{cos}\phi(\omega)}\Big{]}+\tan^{-1}\Big{[}\frac{ka\
\mathrm{sin}\phi(\omega)}{1-ka\ \mathrm{cos}\phi(\omega)}\Big{]}.$ (6)
Next, we consider the group delay which is a direct consequence of the amount
of phase shift in equation (6) within the filter passband. It is defined as
the negative derivative of the phase of the transfer function with respect to
the angular frequency:
$\displaystyle\tau_{\mathrm{Ring}}=-\frac{d\Phi_{\mathrm{Ring}}(\omega)}{d\omega}$
$\displaystyle=-\frac{n_{R}L_{R}}{c}+\frac{k(k-a\
\cos\phi(\omega))}{a^{2}-2ka\
\cos\phi(\omega)+k^{2}}+\frac{ka(ka-\cos\phi(\omega))}{1-2ka\
\cos\phi(\omega)+k^{2}a^{2}}.$ (7)
Equations (6) and (7) explain the behavior of the propagated light through the
resonator. At resonance, for $\textit{k}<\textit{a}$, the ring and the
waveguide are overcoupled and the phase shift increases rapidly as a function
of angular frequency, leading to pulse delay [6, 7, 8, 9]. On the other hand,
for $\textit{k}>\textit{a}$, they are undercoupled and the phase shift
decreases rapidly as a function of angular frequency, resulting in pulse
advancement. Critical coupling occurs when $\textit{k}=\textit{a}$. Here, the
transmission becomes zero at the resonance frequency as the round trip loss of
the ring is exactly the same as the fractional loss through the resonance
coupling [27]. In our design study, we are particularly interested in pulse
delay, and thus all candidate systems use overcoupled resonators.
In figure 4, we depict the resonator characteristics for four different values
of k = 0.8, 0.9, 0.95, and 0.97 with a practical value of attenuation
coefficient $\alpha$ = 1 cm-1 and $L_{R}$ = 150 $\mu$m. Using the numerical
simulations based on equations (5) - (7), we calculate and plot the
transmission, phase shift, and group delay spectra in figures 4(a), 4(b), and
4(c), respectively. Corresponding Gaussian input and delayed output pulses are
shown in figure 4(d), where the input pulsewidth is $T_{p}$ = 50 ps. A silicon
waveguide is assumed, thus an effective refractive index $n_{R}$ = 3.0 is
used. In figure 4(a), as k approaches the critical value of the round trip
loss a from below, the full width at half depth (FWHD) of the resonator
transmission function becomes narrower and deeper. This leads to the slope of
the phase shift becoming larger, as shown in figure 4(b), and therefore a
larger group delay is achieved for the larger value of k = 0.97 in figure
4(c). However, the maximum achievable pulse delay is limited by the tradeoff
between the group delay and pulse distortion, causing oscillation at the pulse
rising edge, as shown in figure 4(d).
Now, let us consider a pulse train at a BR of 10 Gbps rather than a single
pulse, where BR = 1/$T_{b}$. Figures 5(a), 5(b), and 5(c) present the EO-
delay, distortion, and power throughput (PT), respectively, for a resonator as
a function of k and $L_{R}$. We define the PT as the ratio of the propagated
output signal power to the input signal power in the resonator. This numerical
simulation is performed by propagating a 127-bit pseudo-random Gaussian pulse
train with 50% RZ modulation format at a BR of 10 Gbps. Our computation covers
a range of k from 0.94 to 0.99 and $L_{R}$ from 10 $\mu$m to 250 $\mu$m and
these ranges are chosen to observe the ring resonator characteristics. It is
interesting to note that we observe both the slow and fast light regimes to
the left and right sides, respectively, of the critical coupling line (green
dashed), in figure 5(a). To increase the delay one can increase k or LR, but
both distortion and energy loss increase at the same time. For a given value
of the maximum distortion constraint (e.g. D $\simeq$ 0.35) in figure 5(b), we
can find many k and $L_{R}$ pairs that provide the same values of distortion-
constrained EO-delay $\simeq$ 0.76 with corresponding PT $\simeq$ 0.67, as
observed in black dotted lines in figures 5(a) and 5(c). Therefore, we will
focus on varying k while keeping a fixed practical value of $L_{R}$ = 150
$\mu$m.
### 3.2 SCISSOR
We now consider a SCISSOR, as shown in figure 6. It is assumed that the
SCISSOR has multiple identical rings and its transfer function
$E_{\mathrm{out}}(\omega)/E_{\mathrm{in}}(\omega)=H_{\mathrm{SCISSOR}}(\omega)=(H_{\mathrm{Ring}}(\omega))^{N}$,
where N is the number of resonators. Figure 7 presents the characteristics of
the SCISSOR for four different numbers of rings (N = 1, 3, 5, and 8) with
$L_{R}$ = 150 $\mu$m, $\alpha$ = 1 cm-1, and k = 0.85. Because the phase shift
at resonance for multiple resonators is additive, the magnitude of the total
group delay from a summation of the delays of all individual ring resonators
increases as a function of N. The FWHD of SCISSOR transmission resonance also
becomes wider than that of a single ring with the resonance transmission
approaching zero. As a result, output pulse power decreases and the output
pulse shape becomes more distorted from the its original input shape as
SCISSOR length N increases, shown in figure 7(d).
## 4 Optimal System Design Study
In this section, we explore optimal system designs for SCISSOR, SBS, and
SCISSOR + SBS. Our approach is to maximize the delay performance under
practical system resource constraints while maintaining constant data fidelity
[14, 17, 18] .
### 4.1 SCISSOR
We use EO and IT metrics, as described in Section 2, to evaluate the SCISSOR
structure. Figure 8 describes the results of the computations summarizing (1)
the EO-delay with associated D and (2) the information theoretic delay with
associated IT as a function of N, where three different noise strengths of
$\sigma^{2}$= 0.2, 0.3, and 0.4 are used for the IT computation. The EO-based
results presented in this paper are based on propagating a 127 bit pseudo-
random pulse train with a RZ modulation format at a BR = 10 Gbps. For
information-based results, we have utilized 8 bit input sequences with the
same modulation format and BR, and therefore a total 256 ( M = 28 states in
equation (2) ) possible bit patterns are considered. For each input pattern,
we use 106 noise samples to obtain reliable results by using a Monte Carlo
technique. As expected we found that increasing N increases both the EOD and
ID at the cost of increased distortion. As a result, the normalized IT values
decrease. We observe both EOD and ID yield similar delay values, as shown in
figure 8(a). From figure 8(b), we see that IT decreases faster for higher
noise strength with increasing N, thus the fidelity of information
transmission decreases with increasing $\sigma^{2}$ because the decreased
signal to noise ratio (SNR) causes information to be lost. Based on D and IT
results for the SCISSOR with N = 4, distortion of D = 0.342 is measured, while
three different values of IT = 0.943, 0.873, and 0.812 are computed with
corresponding AWGN levels of $\sigma^{2}$ = 0.2, 0.3, and 0.4, respectively,
as shown in figure 8(b). For the given specific noise level of $\sigma^{2}$ =
0.3, one can use at most 4 cascaded resonators while simultaneously
maintaining more than 87% (i.e., IT $\geq$ 0.87 or approximately 7 out of 8
bits) of the transmitted information. Therefore, we take the IT constraint IT
$\geq$ 0.87 for $\sigma^{2}$ = 0.3, to correspond with distortion constraint D
$\leq$ 0.35.
Next, under the two signal quality constraints (IT $\geq$ 0.87 and D $\leq$
0.35), we optimize k to maximize EOD and ID for three different attenuation
coefficients $\alpha$ = 0, 1, and 3 cm-1 as a function of N from 1 to 70. The
optimal SCISSOR characteristics using both IT and EO metrics are presented in
figure 9. We note that the results from the two different metrics provide
similar trends. When N = 70, the maximum fractional delays of approximately
10, 8, and 4 are achieved for $\alpha$ = 0, 1, and 3 cm-1, respectively, at
the BR = 10 Gbps. As N increases, both fidelity-constrained-EOD and ID
increases, however, early delay saturation for the highest attenuation value
is observed in figure 9(a). As N increases, k must decrease in order to
increase the effective FWHD so that the system can satisfy the D and IT
constraints. For the same reason, optimal k at higher attenuation is smaller
than that at smaller attenuation, as shown in figure 9(c). Note that these
trends are also explained by the group delay relation of equation (7). The
transferred energy in a lossy SCISSOR decreases exponentially because of the
induced loss, shown in figure 9(b), and thus, the transmission losses of the N
= 70 SCISSOR become around 22 dB and 43 dB for $\alpha$ = 1 and 3 cm-1,
respectively. This is what would limit the delay performance and reduce the
data fidelity of such a system. Inevitably, we must conclude that an
amplification process is required for the SCISSOR. As mentioned earlier, an
SBS gain medium is a good choice for both increasing the delay performance and
the signal amplification by combining it with the SCISSOR.
### 4.2 Broadband SBS
Slow light via the stimulated Brillouin scattering process has previously been
demonstrated for tunable delay in optical fiber [10, 23, 28]. The SBS process
is a nonlinear interaction between a strong pump wave and a weak probe wave
that is mediated by an acoustic wave. The acoustic wave generated from this
interaction scatters photons to the probe wave, shifting the scattered light
downward to the Stokes frequency $\omega_{s}=\omega_{p}-\Omega_{B}$, where
$\Omega_{B}$ is the Brillouin frequency shift in optical fiber. As a result,
the Stokes field experiences strong gain at $\omega_{s}$. For a typical single
mode fiber, the Brillouin frequency shift $\Omega_{B}$ is $\sim$ 10 GHz and
the Brillouin linewidth $\Gamma$ is $\sim$ 40 MHz near the communication
wavelength of 1550 nm. However, this narrow bandwidth limits the achievable
data rate to only several megabits per second. Much of the recent research in
the SBS slow-light community focuses on broadening the available SBS
bandwidth, and several techniques have been experimentally demonstrated that
accommodate a GHz data rate. A primary technique is direct modulation of a
Gaussian noise source, generated by an arbitrary waveform generator. Gain
bandwidths of up to 25 GHz have been experimentally demonstrated [11, 13, 12].
Under the small signal approximation, the input field $E(0,\omega$) will be
amplified at the fiber output according to $E(L_{f},\omega$) =
$E(0,\omega)H_{\mathrm{SBS}}(\omega)$, with the SBS transfer function
$H_{\mathrm{SBS}}(\omega)$ = exp($k(\omega)L_{f}$). Here, $L_{f}$ represents
the fiber length and $k(\omega$) is the complex wave vector. For the pump
broadened SBS, $k(\omega$) = $P_{p}(\omega)\otimes g_{B}(\omega)$ can be
obtained by convolving the pump spectrum $P_{p}(\omega)$ with the Lorentzian
gain profile
$g_{B}(\omega)=g_{0}/[1-j((\omega-\omega_{\mathrm{s}})/(\Gamma/2))]$, where
$g_{0}$ is the line-center gain coefficient. Pant et al showed that a super
Gaussian function provides a good approximation of the optimal pump profile
$P_{p}(\omega)=(x_{1}/x_{2})$exp$[-(\omega-((\omega_{s}+\Omega_{B}))/x_{2})^{2x_{3}}]$,
where the parameters $x_{1}$, $x_{2}$, and $x_{3}$ define pump peak power,
pump width, and pump shape (i.e. $x_{3}$=1 is Gaussian and $x_{3}\gg$ 1
becomes nearly rectangular) [18]. In the next subsection, these three
parameters will be optimized subject to the fidelity constraint and the
maximum SBS gain constraint, G = max{$k(\omega)L_{f}$} $\leq$ 10\. When
$\omega=\omega_{0}$, the line-center gain of the broadband SBS is defined by
$\textit{G}\simeq g_{0}x_{1}\Gamma\pi L_{f}/2Ax_{2}$, where A is the mode
area. This gain constraint is imposed to avoid the nonlinear amplifier
behavior and the maximum available gain G = 10 is conservative value as
compare to the Brillouin gain threshold of $\sim$ 25 [28].
### 4.3 SBS + SCISSOR
Recall the results presented in figure 9, from which we proposed the utility
of a joint SCISSOR + SBS system. To demonstrate more explicitly the advantages
of this combined slow-light device, we present a practical system design,
analyzing its performance in terms of several important factors such as FD,
PT, D, and IT in this section. The transfer function for such a device is
given by $H(\omega)$ = $H_{\mathrm{SCISSOR}}(\omega)\times
H_{\mathrm{SBS}}(\omega)$, and its real and imaginary parts at resonance are
related to the gain and the refractive index profiles through the Kramers-
Kronig relation. Figure 10 shows the normalized transmission spectra for
individual and combined systems along with the spectrum of a 128 bit pseudo-
random RZ sequence at BR = 10 Gbps. We assumed the signal carrier frequency
and the SCISSOR resonance frequency $\omega_{0}$ are the same as the SBS
Stokes frequency $\omega_{s}$. To better understand the impact of using this
combined system, we look at the input and output eye-diagrams after
propagating through the combined transmission spectrum, as shown in figure 10,
for several different SBS gain values of G = 0, 1, 5, and 10. These results
are shown in figure 11. For simplicity, we first consider the SCISSOR with N =
1 and note that the combined system with SBS G = 0 is a resonator-only system.
It is known that the Gaussian pulse propagating through the ring resonator
undergoes dispersion effects that can cause oscillations at the pulse rising
and/or trailing edges mainly due to the cubic-GVD, as shown in figure 11(b)
[29]. On the other hand, in the SBS system, the output pulse undergoes
distortion in the form of pulse broadening mainly due to the quadratic-GVD.
Note that distortion management techniques for the SBS system basically
suppress the quadratic-GVD term as demonstrated by Stenner et al [10]. By
comparing figures 11(b) and 11(d), the fractional EOD for the SCISSOR only and
SCISSOR + SBS (G = 10) are 0.61 and 1.88 respectively, therefore, it is
clearly observed that the SCISSOR + SBS combination not only improves delay
performance, but also suppresses the pulse oscillation in the pulse trailing
edge arising from the resonator. Although the SBS process also introduces the
pulse broadening, it is not significant in this example. In addition,
combining SBS + SCISSOR provides additional benefits in terms of delay and PT
improvement. Therefore, the combined system provide $\sim$ 3.1 times larger
delay with only a small sacrifice of $\sim$ 1.2 times eye-closing when
compared to the SCISSOR-only system.
Figure 12 shows the summary of the optimal results for the resonator loss a =
1 cm-1 as a function of N = 1 - 70 for two candidate systems: SCISSOR-only and
SCISSOR + SBS systems under data fidelity constraints (IT $\geq$ 0.87 and D
$\leq$ 0.35). We observe that results via both metrics agree well. In general,
the maximum fidelity-constrained delays gradually increase, while optimal SBS
gain G and SCISSOR coupling coefficient k decrease. The gain and k must be
chosen effectively to achieve the maximum delay performance under the IT and D
limit. Therefore, for the region of N $<$ 7, the maximum gain can remain
constant, whereas k decreases. However, any further increase in N requires a
decrease in SBS gain as shown in figure 12(c). We know that the presence of
loss causes a nonnegligible decrease in PT for increasing N as shown in figure
12(b). The results, however, indicate that the combined system can
significantly improve the PT and the delay performance. Even for large number
of rings (N = 70) the combined system can achieve unit power transmission
ratio. The optimal design curves presented in figures 12(a) - (d) represent
bounds on the performance of our proposed delay devices subject to real-world
operating and fidelity constraints. In summary, the proposed technique enables
a maximum fractional delay of $\sim$ 17, which is $\sim$ 2.1 times the maximum
SCISSOR-only delay, with unit power transmission using a cascade of 70 ring
resonators combined with an SBS gain medium and can overcome Khurgin’s
fundamental limit for the fractional delay for the SCISSOR, in which N $>$ 100
resonators are required for fractional delay of 10 [30].
## 5 Conclusion
We have presented a practical system design for increasing the fractional
delay while maintaining high data fidelity by combining SBS and SCISSOR. We
have employed two different fidelity metrics (EO-metric and IT-metric) to
evaluate the slow-light system performance subject to real-world resource
constraints. By jointly optimizing the system parameters, the combined SBS +
SCISSOR system can provide larger delay and improved power throughput compare
to the SCISSOR-only system. We have shown that the maximum fidelity
constrained-delay of $\sim$ 17 for SBS + SCISSOR can be achieved with an unit
power transmission at a bit rate of 10 Gbps.
We gratefully acknowledge the financial support of the DARPA/DSO Slow-Light
Program.
## References
## References
* [1] Hau L V, Harris S E, Dutton Z and Behroozi C H 1999 Nature 397 594
* [2] Morichetti F, Melloni A, Ferrari C and Martinelli M 2008 Opt. Express 16 8395
* [3] Melloni A, Morichetti F, Ferrari C, and Martinelli M 2008 Opt. Lett. 33 2389
* [4] Xia F, Sekaric L and Vlasov Y 2007 Nat. Photonics 1, 65
* [5] Otey C R, Povinelli M L and Fan S 2008 J. Lightwave Technol. 26 3784
* [6] Heebner J E, Boyd R W and Park Q 2002 J. Opt. Soc. Am. B 19 722
* [7] Heebner J E, Wong V, Schweinsberg A, Boyd R W and Jackson D J 2004 IEEE J. Quantum Electron. 40 726
* [8] Govindan V and Blair S 2008 J. Opt. Soc. Am. B 25 C23
* [9] Lenz G, Eggleton B J, Madsen C K and Slusher R E 2008 IEEE J. Quantum Electron. 37 525
* [10] Stenner M D, Neifeld M A, Zhu Z, Dawes A M C and Gauthier D J 2005 Opt. Express 13 9995
* [11] Song K Y and Hotate K 2007 Opt. Lett. 32 217
* [12] Zhu Z, Dawes A M C, Gauthier D J, Zhang L and Willner A E 2007 J. Lightwave Technol. 25 201
* [13] Cabrera-Granado E, Calderon O G, Melle S and Gauthier D J 2008 Opt. Express 16 16032
* [14] Lee M, Pant R and Neifeld M A 2008 Appl. Opt. 47 6404
* [15] Lee M, Pant R, Stenner M D and Neifeld M A 2008 Opt. Commun. 281 2975
* [16] Shi Z and Boyd R W 2009 Phys. Rev. A 79 013805
* [17] Pant R, Stenner M D, Neifeld M A, Shi Z, Boyd R W and Gauthier D J 2007 Appl. Opt. 46 6513
* [18] Pant R, Stenner M D, Neifeld M A and Gauthier D J 2008 Opt. Express 16 2764
* [19] Downie J D 2005 J. Lightwave Technol. 23 2031
* [20] Neifeld M A and Lee M 2008 J. Opt. Soc. Am. B 25, C31
* [21] Neifeld M A and Lee M 2008 Slow and Fast Light (Optical Society of America, 2008) SWA5
* [22] Chin S, Gonzalez-Herraez M and Thevenaz L 2009 Opt. Express 17 21910
* [23] Chin S, Gonzalez-Herraez M and Thevenaz L 2008 Opt. Express 16 12181
* [24] Shannon C E 1948 Bell System Technical J. 27 379
* [25] Shannon C E 1948 Bell System Technical J. 27 623
* [26] Stenner M D and Neifeld M A 2008 Opt. Express 16 651
* [27] Schwelb O 2004 J. Lightwave Technol. 22 1380
* [28] Zhu Z, Gauthier D J, Okawachi Y, Sharping J E, Gaeta A L, Boyd R W and Willner A E 2005 J. Opt. Soc. Am. B 22 2378
* [29] Madsen C K and Zhao J H 1999 Optical filter design and analysis; A signal processing approach (Wiley)
* [30] Khurgin J B and Morton P A 2009 Opt. Lett. 34 2655
## Figure captions
Figure 1: (a) Channel model of a slow-light delay device. OOK GP generates a
train of Gaussian pulses (GP) modulated via on off keying (OOK). (b) Example
of slow-light output pulses including effects of delay, distortion, and noise.
Top figures – without distortion, middle figures – moderate distortion, and
bottom figures – large distortion causes ISI. Figure 2: Application of the
information theoretic analysis to an ideal delay device for the 3 bits
transmission (8 possible states). (a) Example signals at the input (dashed)
and output (solid) of the slow light operator. The input window (IW) along
with two candidate output windows (OW) are shown. (b) Mutual information as a
function of output window offset. Note that IT represents the information
throughput and ID represents the information delay. Figure 3: System
parameters for a single microring resonator. Figure 4: Lossy resonator
characteristics for four values of k = 0.8, 0.9, 0.95, and 0.97. (a)
Transmission spectra, (b) phase spectra, (c) group delay, and (d) Gaussian
input and delayed output pulses. Figure 5: (Color online) Single resonator
characteristics. (a) Fractional delay (FD), (b) distortion (D), and (c) power
transmission (PT) as a function of k and $L_{R}$. The green-dashed line in (a)
represents the location of the critical coupling. The black-dotted line in (b)
represents D = 0.35, and corresponding distortion-constrained FD and PT are
also represented by same black-dotted lines in figures (a) and (c),
respectively. SL represents the slow light region and FL represents the fast
light region. Figure 6: Side-coupled integral spaced sequence of resonators
(SCISSOR). Figure 7: SCISSOR characteristics for four different number of
resonators N = 1, 3, 5, and 8. (a) Transmission spectra, (b) phase spectra,
(c) group delay, and (d) Gaussian input and delayed output pulses. Figure 8:
Summary of EO and IT results for SCISSOR with N= 1 - 6. (a) EOD and ID for
three different values of noise strength. (b) IT on the left axis and
distortion on the right axis. The system parameters of $L_{R}$ = 150 $\mu$m
and k = 0.915 are used. Figure 9: Optimal results of (a) EO-constrained
fractional EOD and IT-constrained fractional ID, (b) power throughput, (c)
self-coupling coefficient k on the left axis and $L_{R}$ = 150 $\mu$m on the
right axis, and (d) Distortion on the left axis and IT on the right axis as a
function of N Figure 10: Transmission spectra (log-scale) for SBS, SCISSOR,
and SCISSOR+SBS along with input pulse spectrum at a BR = 10 Gbps. SBS
parameters of gain = 10, gain bandwidth = 10 GHz, and super-Gaussian factor =
2 and SCISSOR parameters of N = 1, k = 0.965, $\alpha$ = 1 cm-1, and $L_{R}$ =
150 $\mu$m are used. Figure 11: Input and output eye-diagrams vs. SBS gain.
(a) Input and output eye-diagrams for (b) SCISSOR-only (SBS G = 0), (c)
SCISSOR + SBS (G = 1), (d) SCISSOR + SBS (G = 5), and (e) SCISSOR + SBS (G =
10). Double-side arrows represent $T_{EO}$. Note that N = 1. Figure 12:
Optimal results of (a) EO-constrained fractional EOD and IT-constrained
fractional ID, (b) power throughput (dB), (c) SBS gain, and (d) ring
parameters k on the left axis and $L_{R}$ on the right axis as a function of N
= 1-70. Unit transmission is defined as PT = 1 (i.e. PT = 0 dB)
## Figure captions
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:
Figure 9:
Figure 10:
Figure 11:
Figure 12:
|
arxiv-papers
| 2010-01-30T19:59:41 |
2024-09-04T02:49:08.118468
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Myungjun Lee, Michael E. Gehm, and Mark A. Neifeld",
"submitter": "Myungjun Lee",
"url": "https://arxiv.org/abs/1002.0084"
}
|
1002.0280
|
# Continuous variable entanglement distillation of Non-Gaussian Mixed States
Ruifang Dong1,2,∗, Mikael Lassen1,2, Joel Heersink1,3, Christoph Marquardt1,3,
Radim Filip4, Gerd Leuchs1,3 and Ulrik L. Andersen2 1Max Planck Institute for
the Science of Light, Günther-Scharowsky-Str. 1/Bau 24, 91058 Erlangen,
Germany
2Department of Physics, Technical University of Denmark, Building 309, 2800
Lyngby, Denmark
3Institute of Optics, Information and Photonics, Friedrich-Alexander-
University Erlangen-Nuremberg, Str. 7/B2, 91058 Erlangen, Germany
4Department of Optics, Palacký University, 17. listopadu 50, 77200 Olomouc,
Czech Republic
∗rdon@fysik.dtu.dk
###### Abstract
Many different quantum information communication protocols such as
teleportation, dense coding and entanglement based quantum key distribution
are based on the faithful transmission of entanglement between distant
location in an optical network. The distribution of entanglement in such a
network is however hampered by loss and noise that is inherent in all
practical quantum channels. Thus, to enable faithful transmission one must
resort to the protocol of entanglement distillation. In this paper we present
a detailed theoretical analysis and an experimental realization of continuous
variable entanglement distillation in a channel that is inflicted by different
kinds of non-Gaussian noise. The continuous variable entangled states are
generated by exploiting the third order non-linearity in optical fibers, and
the states are sent through a free-space laboratory channel in which the
losses are altered to simulate a free-space atmospheric channel with varying
losses. We use linear optical components, homodyne measurements and classical
communication to distill the entanglement, and we find that by using this
method the entanglement can be probabilistically increased for some specific
non-Gaussian noise channels.
###### pacs:
42.50.Lc,42.50.Dv,42.81.Dp,42.65.Dr
## I Introduction
Quantum communication is a promising platform for sending secret messages with
absolute security and developing new low energy optical communication systems
gisin02.rmp . Such quantum communication protocols require the faithful
transmission of pure quantum states over very long distances. Heretofore,
significant experimental progress has been achieved in free space and fiber
based quantum cryptography where communication over more than 100 km have been
demonstrated dixon08.oe ; Manderbach07.prl ; ursin07.natphy . The
implementation of quantum communication systems over even larger distances -
as will be the case for transatlantic or deep space communication - can be
carried out by using quantum teleportation. However, it requires that the two
communicating parties share highly entangled states. One is therefore faced
with the technologically difficult problem of distributing highly entangled
states over long distances. The most serious problem in such a transmission is
the unavoidable coupling with the environment which leads to losses and
decoherence of the entangled states.
Losses and decoherence can be overcome by the use of entanglement
distillation, which is the protocol of extracting from an ensemble of less
entangled states a subset of states with a higher degree of entanglement
bennett96.prl . Distillation is therefore a purifying protocol that selects
highly entangled pure states from a mixture that is a result of noisy
transmission. This protocol has been experimentally demonstrated for qubit
systems exploiting a posteriori generated polarization entangled states
kwiat01.nat ; zhao01.pra ; yamamoto03.nat ; pan03.nat ; zhao03.prl . Common
for these implementations of entanglement distillation is their relative
experimental simplicity; only simple linear optical components such as beam
splitters and phase shifters are used to recover the entanglement. This
inherent simplicity of the distillation setups arises from the non-Gaussian
nature of the Wigner function of the entangled states. It has however been
proved that in case the Wigner functions of the entangled states are Gaussian,
entanglement distillation cannot be performed by linear optical components,
homodyne detection and classical communication eisert02.prl ; fiurasek02.prl ;
giedke02.pra . This is a very important result since it tells us that standard
continuous variable entanglement generated by e.g. a second-order or a small
third-order non-linearity cannot be distilled by simple means as these kinds
of entangled states are described by Gaussian Wigner functions.
Several avenues around the no-go distillation theorem have been proposed. The
first idea to increase the amount of CV entanglement was put forward by
Opatrný et al. opatrny00.pra . They suggested to subtract a single photon from
each of the modes of a two-mode squeezed state using weakly reflecting beam
splitters and single photon counters, and thereby conditionally prepare a non-
Gaussian state which eventually could increase the fidelity of CV quantum
teleportation. This protocol was further elaborated upon by Cochrane et al.
Cochrane2002.pra and Olivares et al. Olivares2003.pra . Other approaches
relying on strong cross Kerr nonlinearities were suggested by Duan et al.
duan00b.prl ; duan2000.pra and Fiurášek et al. fiurasek2003.pra . The usage
of such non-Gaussian operations results in non-Gaussian entangled states. To
get back to the Gaussian regime, it has been suggested to use a Gaussification
protocol based on simple linear optical components and vacuum projective
measurements (which can be implemented by either avalanche photodiodes or
homodyne detection) browne03.pra . Distillation including the Gaussification
protocol was first considered for pure states by Browne et al. browne03.pra
but later extended to the more relevant case of mixed states by Eisert et al.
eisert03.pra .
Due to the experimental complexity of the above mentioned proposals, the
experimental demonstration of Gaussian entanglement distillation has remained
a challenge. A first step towards the demonstration of Gaussian entanglement
distillation was presented in ref. ourjoumtsev07.prl where a modified version
of the scheme by Opatrný et al. opatrny00.pra was implemented: single photons
were subtracted from one of the two modes of a Gaussian entangled state using
a nonlocal joint measurement and as a result, an increase of entanglement was
observed. Recently, the full scheme of Opartny et al. was demonstrated by
Takahashi et al. takahashi09.quant . They observed a gain of entanglement by
means of conditional local subtraction of a single photon or two photons from
a two-mode Gaussian state. Furthermore they confirmed that two-photon
subtraction also improves Gaussian-like entanglement.
In the work mentioned above, only Gaussian noise has been considered as for
example associated with constant attenuation. Gaussian noise is however not
the only kind of noise occurring in information channels. E.g. if the
magnitude or phase of the transmission coefficient of a channel is
fluctuating, the resulting transmitted state is a non-Gaussian mixed state.
Because of the non-Gaussianity of the transmitted state, the aforementioned
no-go theorem does not apply and thus Gaussian transformations suffice to
distill the state. Actually, such a non-Gaussian mixture of Gaussian states
can be distilled and Gaussified using an approach fiurasek07.pra related to
the one suggested in ref. browne03.pra . Alternatively, it is also possible to
distill and Gaussify non-Gaussian states using a simpler single-copy scheme
which is not relying on interference but is based on a weak measurement of the
corrupted states and heralding of the remaining state dong08.nat . Such an
approach has been also suggested for cat state purification suzuki06.pra ,
coherent state purification wittmann08.pra and squeezed state distillation
heersink06.prl .
The distillation of Gaussian entanglement corrupted by non-Gaussian noise was
recently experimentally demonstrated in two different laboratories. More
specifically, it was demonstrated that by employing simple linear optical
components, homodyne detection and feedforward, it is possible to extract more
entanglement out of a less entangled state that has been affected by
attenuation noise dong08.nat or phase noise hage08.nat .
In this paper we elaborate on the work in ref. dong08.nat , largely extending
the theoretical discussion on the characterization of non-Gaussian
entanglement and, on the experimental side, testing our distillation protocol
in new attenuation channels.
The paper is organized as follows: In Section II, the entanglement
distillation protocol utilized in our experiment is fully discussed. In
Section III the experimental setup for realization of the entanglement
distillation is described, and the experimental results are shown and
discussed in Section IV.
## II The entanglement distillation protocol
The basic scheme of entanglement distillation is illustrated in Fig. 2. The
primary goal is to efficiently distribute bipartite entanglement between two
sites in a communication network to be used for e.g. teleportation or quantum
key distribution. Suppose the two-mode entangled state (also known as
Einstein-Podolsky-Rosen state (EPR)) is produced at site A. One of the modes
is kept at A’s site while the second mode is sent through a noisy quantum
channel. As a result of this noise, the entangled state will be corrupted and
the entanglement is degraded. The idea is then to recover the entanglement
using local operations at the two sites and classical communication between
the sites. To enable distillation, it is however required to generate and
subsequently distribute a large ensemble of highly entangled states. After
transmission, the ensemble transforms into a set of less entangled states from
which one can distill out a smaller set of higher entangled states.
A notable difference between our distillation approach and the schemes
proposed in Refs. browne03.pra ; fiurasek07.pra is that our procedure relies
on single copies of distributed entangled states whereas the protocols in Ref.
browne03.pra ; fiurasek07.pra are based on at least two copies. The multi
copy approach relies on very precise interference between the copies, thus
rendering this protocol rather difficult. One disadvantage of the single copy
approach is the fact that the entangled state is inevitably polluted with a
small amount of vacuum noise in the distillation machine. This pollution can,
however, be reduced if one is willing to trade it for a lower success rate.
Before describing the details of the experimental demonstration, we wish to
address the question on how to evaluate the protocol. The entanglement after
distillation must be appropriately evaluated and shown to be larger than the
entanglement before distillation to ensure a successful demonstration. One way
of verifying the success of distillation is to fully characterise the input
and output states using quantum tomography and then subsequently calculate an
entanglement monotone such as the logarithmic negativity. However, in the
experiment presented in this paper (as well as many other experiments on
continuous variable entanglement) we only measured the covariance matrix as
such measurements are easier to implement. The question that we would like to
address in the following is whether it is possible to verify the success of
distillation based on the covariance matrix of a non-Gaussian state.
### II.1 Entanglement evaluation
In order to quantify the performance of the distillation protocol, the amount
of distillable entanglement before and after distillation ought to be
computed. It is however not known how to quantify the degree of distillable
entanglement of non-Gaussian mixed states bennett96.prl ; rains99.pra1 ;
rains99.pra2 . Therefore, as an alternative to the quantification of the
distillation protocol, one could try to estimate qualitatively whether
distillation has taken place by comparing computable bounds on distillable
entanglement before and after distillation. First we will have a closer look
at such bounds.
#### II.1.1 Upper and lower bounds on distillable entanglement
Although it is unknown how to find the amount of distillable entanglement of
non-Gaussian mixed states, we can easily find the upper and lower bounds by
computing the logarithmic negativity and the conditional entropy, respectively
vidal02.pra ; lower2 ; lower1 . These bounds can be found before and after
distillation, and the success of the distillation protocol can be
unambiguously proved by comparing these entanglement intervals: If the
entanglement interval is shifted towards higher entanglement and is not
overlapping with the interval before the distillation, the distillation has
proved successful. In other words, distillation has been performed if the
lower bound after the protocol is larger than the upper bound before. This is
illustrated in Fig. 1(a).
It has been proved that for any state, the log-negativity,
$LN(\rho)\equiv\log_{2}\left(2\mathcal{N}+1\right)=\log_{2}\left\|\rho^{T_{A}}\right\|_{1}.$
(1)
is an upper bound on the distillable entanglement; $E_{D}<LN(\rho)$
vidal02.pra . Here $\rho$ is the density matrix of the state,
$||\rho^{T_{A}}||$ is the trace norm of the partial transpose of the state
with respect to subsystem A, and the negativity is defined as
$\mathcal{N}(\rho)\equiv\frac{\left\|\rho^{T_{A}}\right\|_{1}-1}{2}.$ (2)
The negativity corresponds to the absolute value of the sum of the negative
eigenvalues of $\rho^{T_{A}}$ and it vanishes for non-entangled states.
In our experiment we were not able to measure the density matrix and thus
compute the exact value of the negativity. We therefore use another (more
strict) upper bound that is experimentally easier to estimate. As the
negativity is a convex function we have
$\mathcal{N}(\sum_{i}p_{i}\rho_{i})\leq\sum_{i}p_{i}\mathcal{N}(\rho_{i}).$
(3)
where $\rho_{i}$ denotes the $i$th hermitian component in the mixed state, and
$p_{i}$ is the weight for the $i$th component with $p_{i}\geq 0$ and
$\sum_{i}p_{i}=1$. Using this result we can find an upper bound on the log-
negativity for mixed states:
$LN(\sum_{i}p_{i}\rho_{i})\leq\log_{2}\left(1+2\sum_{i}p_{i}\mathcal{N}(\rho_{i})\right).$
(4)
This upper bound for the log-negativity will later be used to compute an upper
bound for the distillable entanglement.
Another entanglement monotone is the conditional entropy. In contrast to the
log-negativity, the conditional entropy yields a lower bound on the
distillable entanglement: $E_{D}>S(\tilde{\rho}_{A})-S(\tilde{\rho})$ lower1 ;
lower2 , where $\tilde{\rho}$ is the density matrix corresponding to Gaussian
approximation of the state and $\tilde{\rho}_{A}$ is the reduced density
matrix with respect to system A. The entropies of the states can be calculated
from the covariance matrix, $\mathbf{CM}$, using
$\displaystyle S(\tilde{\rho}_{A})$ $\displaystyle=$ $\displaystyle
f(\det\mathbf{A}),$ $\displaystyle S(\tilde{\rho})$ $\displaystyle=$
$\displaystyle\sum_{i}f(\mu_{i}),$ $\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\frac{x+1}{2}\log_{2}(\frac{x+1}{2})-\frac{x-1}{2}\log_{2}(\frac{x-1}{2}),$
(5)
where
$\mu_{1,2}=\sqrt{\frac{\gamma\pm\sqrt{\gamma^{2}-4\det\mathbf{CM}}}{2}},$ (6)
are the symplectic eigenvalues of the covariance density matrix and
$\gamma=\det\mathbf{A}+\det\mathbf{B}+2\det\mathbf{C}$. Here $\mathbf{A}$,
$\mathbf{B}$, and $\mathbf{C}$ are submatrices of the covariance matrix:
$\mathbf{CM}=\\{\mathbf{A},\mathbf{C};\mathbf{C}^{T},\mathbf{B}\\}$.
It is important to note that this lower bound is very sensitive to excess
noise of the two-mode squeezed state. Even for a small amount of excess noise,
the lower bound approaches zero and thus is not very useful. This is
illustrated in Fig.1(b) which shows the distillable entanglement intervals
before and after distillation of a noisy entangled state. Although the
distillation protocol might remove the non-Gaussian noise of the state, the
Gaussian noise of the state persists, and thus the entropy (that is the lower
bound on distillable entanglement) will remain very low even after
distillation. This results in an overlap between the two entanglement
intervals and thus the comparison of computable entanglement bounds fails to
witness the action of distillation in terms of distillable entanglement.
Figure 1: Schematic demonstration of entanglement distillation of non-Gaussian
mixed states. In figure (a), the distillation with a pure state is illustrated
via the shift of the entanglement interval composed by the upper and lower
bounds on distillable entanglement before and after the distillation protocol.
Figure (b) shows the distillation with mixed states, the lower bound of which
does not manifest increase even for a small excess noise in the state.
#### II.1.2 Logarithmic negativity
In our experiment, the entangled states possess a large amount of Gaussian
excess noise and thus the prescribed method is insufficient to prove the act
of entanglement distillation using distillable entanglement as a measure.
However, in certain cases we can use the logarithmic negativity as a measure
to witness the act of entanglement distillation even though we only have
access to the covariance matrix as we will explain in the following.
First we note that in general, the Gaussian logarithmic negativity is an
insufficient measure of entanglement distillation of non-Gaussian states as
this measure only yields an upper bound, and with upper bounds of LN both
before and after distillation a conclusion cannot be drawn. However, if the
state after distillation is perfectly Gaussified its Gaussian LN becomes the
exact LN, and if this exact value of LN is larger than the upper bound of LN
before distillation (computed from (eqn. 4)), one may successfully prove the
action of entanglement distillation entirely from the covariances matrices.
This condition will be used for some of the experiments presented in this
paper. More specifically, we will use this approach for testing entanglement
distillation in a binary transmission channel. For other transmission channels
investigated in this paper, the state will not be perfectly Gaussified in the
distillation process and the approach cannot be applied. For such cases,
however, we will resort to evaluations of the Gaussian part of the state in
terms of Gaussian entanglement.
#### II.1.3 Gaussian entanglement
In addition to an increase in distillable entanglement and logarithmic
negativity, the protocol can also be evaluated in terms of its Gaussian
entanglement. Although the Gaussian entanglement is not accounting for the
entanglement of the entire state (but only considers the second moments), it
is quite useful as it directly yields the amount of entanglement useful for
Gaussian protocols, a prominent example being teleportation of Gaussian
states.
In a Gaussian approximation, the state can be described by the covariance
matrix $\mathbf{CM}$ walls94.book . The logarithmic negativity ($LN$) can then
be found as
$LN=-\log_{2}\nu_{min}.$ (7)
where $\nu_{min}$ is the smallest symplectic eigenvalue of the partial
transposed covariance matrix. The symplectic eigenvalues can be calculated
from the covariance matrix using
$\nu_{1,2}=\sqrt{\frac{\delta\pm\sqrt{\delta^{2}-4\det\mathbf{CM}}}{2}}$ (8)
where $\delta=\det\mathbf{A}+\det\mathbf{B}-2\det\mathbf{C}$, $\mathbf{A}$,
$\mathbf{B}$ and $\mathbf{C}$ represent the submatrices in the correlation
matrix vidal02.pra . Then by finding the smallest eigenvalue of the covariance
matrix and inserting it in eqn. (7), a measure of the Gaussian entanglement of
the state can be found.
### II.2 Theory of our protocol
Figure 2: Schematics of the entanglement distillation protocol. A weak
measurement on beam B is diagnosing the state and subsequently used to herald
the highly entangled components of the state.
We now undertake our experimental setup a theoretical treatment in light of
the results of the previous section.
The schematic of our protocol is shown in Fig. 2. The two-mode squeezed or
entangled state is produced by mixing two squeezed Gaussian states at a beam
splitter. The squeezed states are assumed to be identical with variances
$V_{S}$ and $V_{A}$ along the squeezed and anti-squeezed quadratures,
respectively. The beam splitter has a transmittivity of $T_{S}$ and a
reflectivity of $R_{S}=1-T_{S}$. One mode (beam A) from the entangled pair is
given to Alice and the other part (beam B) is transmitted through a fading
channel. The loss in the fading channel is characterized by the transmission
factor $0\leq\eta(t)\leq 1$ which fluctuates randomly. The probability
distribution of the fluctuating attenuation can be divided into $N$ different
slots each associated with a sub-channel with a constant attenuation. The
transmission of sub-channel $i$ is $\eta_{i}$ and it occurs with the
probability $p_{i}$ so that $\sum^{N}_{i=1}p_{i}=1$. For a particular $i$th
sub-channel with transmission of $\eta_{i}$, the transmitted state is Gaussian
and can be fully characterised by the covariance matrix $\mathbf{CM}_{i}$:
$\displaystyle\mathbf{CM}_{i}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\mathbf{A}_{i}&\mathbf{C}_{i}\\\
\mathbf{C}^{T}_{i}&\mathbf{B}_{i}\end{array}\right),\quad\mathbf{A}_{i}=\left(\begin{array}[]{cc}V_{AX,i}&0\\\
0&V_{BX,i}\end{array}\right),$ (13) $\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}V_{AP,i}&0\\\
0&V_{BP,i}\end{array}\right),\quad\mathbf{C}=\left(\begin{array}[]{cc}C_{X,i}&0\\\
0&C_{P,i}\end{array}\right).$ (18)
where the elements are given by:
$\displaystyle V_{AX,i}$ $\displaystyle=$ $\displaystyle
T_{S}V_{S}+R_{S}V_{A},$ $\displaystyle V_{BX,i}$ $\displaystyle=$
$\displaystyle\eta_{i}(T_{S}V_{A}+R_{S}V_{S})+(1-\eta_{i}),$ $\displaystyle
V_{AP,i}$ $\displaystyle=$ $\displaystyle T_{S}V_{A}+R_{S}V_{S},$
$\displaystyle V_{BP,i}$ $\displaystyle=$
$\displaystyle\eta_{i}(T_{S}V_{S}+R_{S}V_{A})+(1-\eta_{i}),$ $\displaystyle
C_{X,i}$ $\displaystyle=$ $\displaystyle-
C_{P,i}=\sqrt{\eta_{i}}\sqrt{R_{S}T_{S}}(V_{A}-V_{S}).$ (19)
Then according to eqn. (4) we can find an upper bound for the log-negativity
of the state after transmission in the fluctuating channel
(using$||\rho_{i}^{T}||=1/\nu_{min,i}$):
$\displaystyle
LN(\sum_{i}p_{i}\rho_{i})\leq\log_{2}\sum_{i}(p_{i}/\nu_{min,i}).$ (20)
where $\nu_{min,i}$ corresponds to the smallest symplectic eigenvalue of the
$i$th partial transposed covariance matrix. This means that the right hand
side of this expression is also an upper bound on the distillable entanglement
of the non-Gaussian noisy state. Therefore, to truly prove that the
entanglement has increased, this bound must in principle be surpassed.
We now consider the Gaussian entanglement of our states using the Wigner
function formalism. The Wigner function of the total state and the $i$th state
can be described as
$\displaystyle W(\mathbf{X},\mathbf{P})$ $\displaystyle=$
$\displaystyle\sum_{i}^{N}p_{i}W_{i}(\mathbf{X},\mathbf{P}),$ $\displaystyle
W_{i}(\mathbf{X},\mathbf{P})$ $\displaystyle=$
$\displaystyle\frac{\exp\left(-\mathbf{X}\mathbf{V}_{X,i}^{-1}\mathbf{X}^{T}-\mathbf{P}\mathbf{V}_{P,i}^{-1}\mathbf{P}^{T}\right)}{4\pi^{2}\sqrt{\mbox{det}\mathbf{V}_{X,i}\;\mbox{det}\mathbf{V}_{P,i}}},$
(21)
where $\mathbf{X}=(x_{A},x_{B})$ and $\mathbf{P}=(p_{A},p_{B})$.
$\mathbf{V}_{X,i}$ and $\mathbf{V}_{P,i}$ are given by
$\displaystyle\mathbf{V}_{X,i}=\left(\begin{array}[]{cc}V_{AX,i}&C_{X,i}\\\
C_{X,i}&V_{BX,i}\end{array}\right),\;\mathbf{V}_{P,i}=\left(\begin{array}[]{cc}V_{AP,i}&C_{P,i}\\\
C_{P,i}&V_{BP,i}\end{array}\right).$ (26)
From the Wigner function the second moments of the quadratures can be
calculated through integration:
$\displaystyle\left\langle\hat{Z}\hat{Y}\right\rangle=\int
dx_{A}dx_{B}dp_{A}dp_{B}zyW(x_{A},x_{B},p_{A},p_{B})$
$\displaystyle=\sum_{i}p_{i}\int
dx_{A}dx_{B}dp_{A}dp_{B}zyW_{i}(x_{A},x_{B},p_{A},p_{B}),$ (27)
where $\hat{Z},\hat{Y}=\hat{X}_{A},\hat{X}_{B},\hat{P}_{A},\hat{P}_{B}$. As
the first moments of the vacuum squeezed states in both the quadratures are
zero, the variances directly correspond to the second moments. Therefore, the
elements of the total covariance matrix are simply the convex sum of the
symmetrical moments (II.2):
$\langle\hat{Z}\hat{Y}\rangle=\sum_{i}p_{i}\langle\hat{Z}\hat{Y}\rangle_{i}.$
(28)
Since all the moments (II.2) are just linear combinations of the transmission
factors $\eta_{i}$ and $\sqrt{\eta_{i}}$, the covariance matrix of the mixed
state has the following elements:
$\displaystyle V_{AX}$ $\displaystyle=$ $\displaystyle T_{S}V_{S}+R_{S}V_{A},$
$\displaystyle V_{BX}$ $\displaystyle=$
$\displaystyle\left<\eta\right>(T_{S}V_{A}+R_{S}V_{S})+(1-\left<\eta\right>),$
$\displaystyle V_{AP}$ $\displaystyle=$ $\displaystyle T_{S}V_{A}+R_{S}V_{S},$
$\displaystyle V_{BP}$ $\displaystyle=$
$\displaystyle\left<\eta\right>(T_{S}V_{S}+R_{S}V_{A})+(1-\left<\eta\right>),$
$\displaystyle C_{X}$ $\displaystyle=$ $\displaystyle-
C_{P}=\left<\sqrt{\eta}\right>\sqrt{R_{S}T_{S}}(V_{A}-V_{S}).$ (29)
where the symbol $\left<\cdot\right>$ denotes averaging over the fluctuating
attenuations. Comparing this set of equations with the set in (II.2)
associated with the second moments for the single sub-channels, we see that
the attenuation coefficient $\eta$ is replaced by the averaged attenuation
$\left<\eta\right>$, and $\sqrt{\eta}$ is replaced by
$\left<\sqrt{\eta}\right>$. It is interesting to note that if the attenuation
factor is constant (which means that the transmitted state will remain
Gaussian) there will always be some, although small, amount of Gaussian
entanglement left in the state. On the other hand, if the attenuation factor
is statistically fluctuating as in our case, the Gaussian entanglement of the
non-Gaussian state will rapidly degrade and eventually completely disappear.
To implement entanglement distillation, a part of the beam B is extracted by a
tap beam splitter with transmittivity $T$. A single quadrature is measured
(for example the amplitude quadrature, $\hat{X}_{t}$) and based on the
measurement outcome the remaining state is probabilistically heralded; it is
either kept or discarded depending on whether the measurement outcome is above
or below the threshold value $x_{th}$. The conditioned Wigner function of the
output signal state after the distillation is
$\displaystyle W_{p}(x_{A},p_{A},x^{\prime}_{B},p^{\prime}_{B})=$
$\displaystyle\int_{x_{th}}^{\infty}dx_{t}\int_{-\infty}^{\infty}dp_{t}\sum_{i=1}^{N}p_{i}W_{i}(x_{A},p_{A},x_{B},p_{B})W_{0}(x_{v},p_{v}).$
where $x_{B}=\sqrt{T}x^{\prime}_{B}-\sqrt{1-T}x_{t}$,
$p_{B}=\sqrt{T}p^{\prime}_{B}+\sqrt{1-T}p_{t}$,
$x_{v}=\sqrt{T}x_{t}-\sqrt{1-T}x^{\prime}_{B}$ and
$p_{v}=\sqrt{1-T}p_{t}+\sqrt{T}p^{\prime}_{B}$, the Wigner function
$W_{0}(x_{v},p_{v})$ represents the vacuum mode entering the asymmetric tap
beam splitter. After integration, the Wigner function can be written as
$\displaystyle W_{p}(x_{A},p_{A},x^{\prime}_{B},p^{\prime}_{B})=$ (31)
$\displaystyle\frac{1}{P_{S}}\sum_{i}p_{i}W^{\prime}_{X,i}(x_{A},x^{\prime}_{B};x_{th})\times
W^{\prime}_{P,i}(p_{A},p^{\prime}_{B}).$
This is a product mixture of two non-Gaussian states which should be compared
to the state before distillation which was a mixture of Gaussian states.
$P_{S}$ is the total probability of success.
The $\hat{X}$ related elements of the covariance matrix can be calculated from
this Wigner function directly by computing the symmetrically ordered moments:
$\displaystyle\langle X_{A}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X^{\prime}_{A}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X^{\prime}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X^{\prime}_{B}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X^{2}_{A}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X_{A}^{{}^{\prime}2}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X^{{}^{\prime}2}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X_{B}^{{}^{\prime}2}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X_{A}X^{\prime}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X_{A}X^{\prime}_{B}\rangle_{i}^{P}}{P_{S}}.$ (32)
with
$\displaystyle\langle X_{A}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{C_{X,i}\sqrt{R}}{\sqrt{2\pi
V^{\prime}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right),$
$\displaystyle\langle X^{\prime}_{B}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{TR}(V_{BXi}-1)}{\sqrt{2\pi
V^{\prime}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right),$
$\displaystyle\langle X^{2}_{A}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{RC_{X,i}^{2}x_{th}}{\sqrt{2\pi
V^{{}^{\prime}3}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right)+$
$\displaystyle\frac{V_{AX,i}}{2}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right],$
$\displaystyle\langle X^{{}^{\prime}2}_{B}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{RT(V^{\prime}_{DX,i}-1)^{2}x_{th}}{\sqrt{2\pi
V^{{}^{\prime}3}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right)+$
$\displaystyle\frac{RT(V_{BX,i}-1)^{2}+V_{BX,i}}{2V^{\prime}_{DXi}}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right],$
$\displaystyle\langle X_{A}X^{\prime}_{B}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{T}R(V^{\prime}_{DX,i}-1)C_{Xi}}{\sqrt{2\pi
V^{{}^{\prime}3}_{DXi}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right)+$
(33)
$\displaystyle\frac{\sqrt{T}C_{X,i}V^{\prime}_{DX,i}}{2}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right].$
where $V^{\prime}_{DX,i}=RV_{BX,i}+T$ is the output variance of the detected
mode and
$P_{S,i}=\frac{1}{2}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right]$
(34)
is the success probability of distilling the $\i$-th constituent of the mixed
state. The total probability of success $P_{S}$ is then given by
$P_{S}=\sum_{i}p_{i}P_{S,i}$.
Since the first moments of the $\hat{P}$ quadrature are vanishingly small, the
$\hat{P}$ related elements of the covariances matrix are directly given by
$\displaystyle\langle P^{2}_{A}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{1}{P_{S}}\sum_{i}p_{i}P_{S,i}V_{AP,i},$
$\displaystyle\langle P^{{}^{\prime}2}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{1}{P_{S}}\sum_{i}p_{i}P_{S,i}(TV_{BP,i}+R),$
$\displaystyle\langle P_{A}P^{\prime}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{T}}{P_{S}}\sum_{i}p_{i}P_{S,i}C_{P,i}.$
The covariance matrix $\mathbf{CM}^{P}$ can then be constructed from these
elements. This covariance matrix fully characterizes the Gaussian part of the
state and thus yields the Gaussian log-negativity by using eqn. (7).
As we discussed in Section II.1, to successfully demonstrate entanglement
distillation of non-Gaussian states, the upper bound on distillable
entanglement before distillation must be surpassed by the lower bound on
distillable entanglement after the distillation (see also Fig. 1). Due to the
fragility of the lower bound, this can be only achieved for almost pure states
as mentioned above. A theoretical demonstration is given in Fig. 3. Here we
consider the transmission of entanglement in a channel which is randomly
blocked: The entangled state is perfectly transmitted with the probability
$p_{1}=0.2$ and completely erased with the probability $p_{2}=0.8$. We assume
the two squeezed states which produce entanglement have variances along the
squeezed quadrature as $V_{S}=0.1$, the entangling beam splitter is symmetric
($T_{S}=50\%$) and the tap beam splitter has a transmission of $T=0.7$. The
distillation with a pure entangled state ($V_{A}=1/V_{S}$) as well as a mixed
state ($V_{A}=1/V_{S}+10$) are investigated as a function of the success
probability, and shown in Fig. 3 (a) and (b) respectively. Following the
theory of section II.1.1, we calculate the upper bound on distillable
entanglement of the non-Gaussian state before distillation as shown by the
bold straight lines in Fig. 3. The lower bounds on distillable entanglement
after distillation are computed and shown in Fig. 3 by the dashed lines. We
see that the proof of entanglement distillation of non-Gaussian states already
fails for a mixed state with a small amount of excess noise. The upper bounds
on Gaussian entanglement after distillation are also computed and shown in
Fig. 3 by solid lines. As the success probability reduces, the Gaussian
entanglement increases. Furthermore, when it surpasses the upper bound on
distillable entanglement before distillation (bold solid lines) at a certain
low success probability, the distilled state is Gaussified as well and thus we
can justify a Gaussian state in the entanglement measure.
Figure 3: (color online) Theoretical simulations of distillable entanglement
of non-Gaussian mixed states as a function of success probability. The two
plots are corresponding to two different purities of the entangled input
states. In figure (a), the distillation with initially pure state is plotted
($V_{A}=1/V_{S}$), while figure (b) shows the distillation with initially
mixed states ($V_{A}=1/V_{S}+10$). The other parameters are taken as:
$V_{S}=0.1$, $V_{N}=1$, $T_{S}=0.5$, $T=0.7$, $\eta_{1}=1$, $\eta_{2}=0$,
$p_{1}=0.2$, $p_{2}=1-p_{1}$. In both plots, the lower dashed line shows the
lower bound on distillable entanglement, and the upper solid line is the upper
bound on Gaussian entanglement. The bold straight line is the upper bound of
the non-Gaussian distillable entanglement before distillation.
## III Experimental realization
Figure 4: Schematics of the experimental setup for the preparation,
distillation and verification of the distillation of entanglement from a non-
Gaussian mixture of polarization entangled states.
The experimental realization of the distillation of corrupted entangled states
consists of three parts as schematically illustrated in Fig. 4: the
preparation, distillation and verification. In the following we describe each
part.
### III.1 Generation of polarisation squeezing and entanglement
The generation of polarization squeezed beams serves as the first step for the
demonstration of entanglement distillation. Here we exploit the Kerr
nonlinearity of silica fibers experienced by ultrashort laser pulses for the
generation of quadrature squeezed states. Fig. 5 depicts the setup for the
generation of a polarization squeezed beam. A pulsed (140 fs) Cr4+:YAG laser
at a wavelength of 1500 nm and a repetition rate of 163 MHz is used to pump a
polarization-maintaining fiber. Two linearly polarized light pulses with
identical intensities are traveling in single pass along the orthogonal
polarization axes ($x$ and $y$) of the fiber. Two quadrature squeezed states,
the squeezed quadrature of which are skewed by $\theta_{\mathrm{sq}}$ from the
amplitude direction, are thereby independently generated. After the fiber the
emerging pulses are overlapped with a $\pi/2$ relative phase difference. The
relative phase difference is achieved using a birefringence pre-compensation,
an unbalanced Michelson-like interferometer heersink03.pra ; heersink05.ol ;
dong08.ol ; fiorentino01.pra . This is controlled by a feedback locking loop
based on a $S_{2}$ measurement of a small portion ($\leq$0.1%) of the fiber
output. The measured error signal is fed back to the piezo-electric element of
the pre-compensation via a PI controller, so that the $S_{2}$ parameter of the
output mode vanishes. This results in a circularly polarized beam at the fiber
output ($\langle\hat{S}_{1}\rangle=\langle\hat{S}_{2}\rangle=0$,
$\langle\hat{S}_{3}\rangle=\langle\hat{S}_{0}\rangle=\alpha^{2}$). The
corresponding Stokes operator uncertainty relations are reduced to a single
nontrivial one in the so-called $\hat{S_{1}}-\hat{S_{2}}$ dark plane:
$\Delta^{2}\hat{S}_{\theta}\,\Delta^{2}\hat{S}_{\theta+\pi/2}\geq|\langle\hat{S}_{3}\rangle|^{2}$,
where $\hat{S}(\theta)=\cos(\theta)\hat{S}_{1}+\sin(\theta)\hat{S}_{2}$
denotes a general Stokes parameter rotated by $\theta$ in the dark
$\hat{S_{1}}-\hat{S_{2}}$ plane with $\langle\hat{S}_{\theta}\rangle=0$.
Therefore, polarization squeezing occurs if
$\Delta^{2}\hat{S}_{\theta}<|\langle\hat{S}_{3}\rangle|=\alpha^{2}$, in which
$\Delta^{2}\hat{S}_{\theta}$ can be directly measured in a Stokes measurement
heersink05.ol . As the noise of Stokes parameters $\hat{S}_{\theta}$ is linked
to the quadrature noise of the Kerr squeezed modes in the same angle
($\Delta^{2}\hat{S}_{\theta}\approx\alpha(\delta\hat{X}_{x,\theta}-\delta\hat{X}_{y,\theta})/\sqrt{2}\approx\alpha^{2}\Delta^{2}\hat{X}_{\theta}$
heersink05.ol ), the squeezed Stokes operator is
$\hat{S}(\theta_{\mathrm{sq}})$ and the orthogonal, anti-squeezed Stokes
operator is $\hat{S}(\theta_{\mathrm{sq}}+\pi/2)$. Due to the equivalence
between the polarization squeezing and vacuum squeezing josse03.prl , we
utilize the conjugate quadratures $\hat{X}$ and $\hat{P}$ to denote the
polarization squeezed and anti-squeezed Stokes operators.
Figure 5: Setup for the generation of polarisation squeezing. The fiber is a
13.2 meters long polarization-maintaining 3M FS-PM-7811 fiber with a mode
field diameter of 5.7 $\mu$m and a beat length of 1.67 mm. The interferometer
in front of the fiber introduces a phase shift $\delta\phi$ between the two
orthogonally polarised pulses to pre-compensate for the birefringence.
$\lambda/4$, $\lambda/2$: quarter–, half–wave plates, PBS: polarising beam
splitter. PZT: Piezo-electric element.
To generate polarization entanglement two identical polarization-maintaining
fibers are used. Two polarization squeezed beams, labeled A and B, are then
generated. By balancing the transmitted optical power of the two fibers, the
two resultant polarization squeezed beams have identical squeezing angles,
squeezing and anti-squeezing properties. The two polarization squeezed beams
are then interfered on a 50/50 beam splitter (Fig. 4) with the interference
visibility aligned to be $>98\%$. The relative phase between the two input
beams is locked to $\pi/2$ so that the two output beams after the beam
splitter have equal intensity and are maximally entangled. The two entangled
outputs remain circularly polarised, thus the quantum correlations between
them are lying in the dark $\hat{S}_{1}-\hat{S}_{2}$ plane with the signatures
$\hat{S}_{\mathrm{A}}(\theta_{sq})+\hat{S}_{\mathrm{B}}(\theta_{sq})\rightarrow
0$ and
$\hat{S}_{\mathrm{A}}(\theta_{sq}+\pi/2)-\hat{S}_{\mathrm{B}}(\theta_{sq}+\pi/2)\rightarrow
0$ (or $\hat{X}_{\mathrm{A}}+\hat{X}_{\mathrm{B}}\rightarrow 0$ and
$\hat{P}_{\mathrm{A}}-\hat{P}_{\mathrm{B}}\rightarrow 0$).
### III.2 Preparation of a non-Gaussian mixed state
The preparation of a non-Gaussian mixture of polarization entangled states is
implemented by transmitting one of the entangled beams, e.g. beam B, through a
controllable neutral density filter (ND). The filter is implemented to produce
a lossy channel with $N=45$ different transmittance levels, ranging from 0.1
to 1 in steps of 0.9/44. The entangled beam is then transmitted through the
lossy channel with 45 realizations. Combining all these realizations a non-
Gaussian mixed state, such as the one described by eqn. (II.2), is achieved,
with the probabilities $p_{i}$ all being identical. However, after the
measurement we can select a certain probability envelope function to give the
different channels pre-specified probability weights. With this technique we
can easily implement different transmission scenarios (see e.g. Fig. 9-1, Fig.
10-1, and Fig. 11-1). As a result of the lossy transmission, the Gaussian
entanglement between the two beams A and B are degraded or completely lost.
### III.3 Entanglement distillation
The distillation operation consists of a measurement of $\hat{X}$ on a small
portion of the mixed entangled beam. This is implemented by tapping 7% of beam
B after the ND filter using a beam splitter. The measurement is followed by a
probabilistic heralding process where the remaining state is kept or
discarded, conditioned on the measurement outcomes: e.g. if the outcome of the
weak measurement is larger than the threshold value, $X_{th}$, then the state
is kept. Note that the signal heralding process could in principle be
implemented electro-optically to generate a freely propagating distilled
signal state. However, to avoid such complications, our conditioning is
instead based on digital data post-selection using a verification measurement
on the conjugate quadratures $\hat{X}$ and $\hat{P}$ of the beams A and B.
### III.4 The tap and verification measurement
The tap and verification measurement are accomplished simultaneously by three
independent Stokes measurement apparatuses. Each measurement apparatus
consists of a half-wave plate and a polarizing beam splitter (PBS). Since the
light beam is circularly ($S_{3}$) polarized, a rotation of the half-wave
plate enables the measurement of different Stokes parameters lying in the
’dark’ polarization plane. For the tap measurement the half-wave plate is
always set at the angle corresponding to $\hat{X}$ in the ’dark’ polarization
plane. Via the verification measurement setup, the Gaussian properties of the
entangled states are characterized by measuring the entries of the covariance
matrix. By generating near symmetric states and choosing a proper reference
frame, we assume that the intra-correlations (such as
$\langle\hat{X}_{A}\hat{P}_{A}\rangle$) are zero. The measurements of these
entries are accomplished by applying polarization measurements of beam A and B
with both the half-wave plates set to the angle corresponding to either
$\hat{X}$ or $\hat{P}$ in the ’dark’ polarization plane. The outputs of the
PBS are detected by identical pairs of balanced photo-detectors based on 98%
quantum efficiency InGaAs PIN-photodiodes and with an incorporated low-pass
filter in order to avoid ac saturation due to the laser repetition
oscillation. The detected AC photocurrents are passively pairwise subtracted
and subsequently down-mixed at 17 MHz, low-pass filtered (1.9 MHz), and
amplified (FEMTO DHPVA-100) before being oversampled by a 16-bit A/D card
(Gage CompuScope 1610) at the rate of $10^{7}$ samples per second. The time
series data are then low-passed with a digital top-hat filter with a bandwidth
of 1 MHz. After these data processing steps, the noise statistics of the
Stokes parameters are characterized at 17 MHz relative to the optical field
carrier frequency ($\approx$200 THz) with a bandwidth of 1 MHz. The signal is
sampled around this sideband to avoid classical noise present in the frequency
band around the carrier schiller96.prl . For each polarization measurement,
the detected photocurrent noise of beam A and B and the tap beam were
simultaneously sampled for $2.4\times 10^{8}$ times, thus the self and cross
correlations between the data set of A and B could thereby easily be
characterized. The covariance matrix was subsequently determined and the log-
negativity was calculated according to eqn. (7).
## IV Experimental results
For perfect transmission (corresponding to no loss in beam B), the marginal
distributions of the entangled beams, A and B, along the quadrature $X$ and
$P$ are plotted in Fig. 6. In Fig. 6(a) the procedure of realizing the
different noisy channels is shown. The sampled data of different attenuation
channels is concatenated according to the different weights of the
transmission probabilities. These samples then provide the measurement data
for the distillation procedure. From Fig. 6(a) and Fig. 6 (b) we can see that,
each individual mode exhibits a large excess noise (measured fluctuation $>17$
dB). However the joint measurements on the entangled beams A and B exhibit
less noise fluctuation than the shot noise reference, as shown in Fig. 6(c).
The observed two-mode squeezing between beam A and B is $-2.6\pm 0.3$ dB and
$-2.4\pm 0.3$ dB for $\hat{X}$ and $\hat{P}$, respectively. From the
determined covariance matrix we compute the log-negativity to be $0.76\pm
0.08$.
Figure 6: (color online). Experimentally measured marginal distributions
associated with the (a) $X$ and $P$ of beam A, (b) $X$ and $P$ of beam B and
(c) the joint measurements $X_{A}+X_{B}$ and $P_{A}-P_{B}$. The black and red
curves are the distributions for shot noise and the quadrature on measurement,
respectively.
To experimentally demonstrate the distillation of entanglement out of non-
Gaussian noise, three different lossy channels are considered: the discrete
erasure channel, where the transmission randomly alternates between two
different levels, and two semi-continuous channels, where the transmittance
alternates between 45 different levels with specified probability amplitudes.
The probability distributions of the transmittance for the discrete channel
and the continuous channels are shown in Fig. 9-1, Fig. 10-1, and Fig. 11-1,
respectively.
### IV.1 The discrete lossy channel
The discrete erasure channel alternates between full (100%) transmission and
25% transmission at a probability of $0.5$. Each realisation is concatenated
to each other with identical weights. The concatenation procedure yields the
same statistical values as true randomly varying data. After transmission the
resulting state is a mixture of a highly entangled state and a weakly
entangled state. In the inset of Fig. 7, we show marginal distributions
illustrating the single beam statistics of the individual components of the
mixture. The statistics of beam B is seen to be contaminated with the
attenuated entangled state thus producing non-Gaussian statistics. For this
state we measure the correlations in $\hat{X}$ and $\hat{P}$ to be above the
shot noise level by $5.5\pm 0.3$ dB and $5.6\pm 0.3$ dB, respectively, and the
Gaussian LN to be $-1.63\pm 0.02$. The Gaussian entanglement is completely
lost as a result of the introduction of such time-dependent loss. This is in
stark contrast to the scenario where only stationary loss (corresponding to
Gaussian loss) is inflicting the entangled states. In that case, a certain
degree of Gaussian entanglement will always survive, although it will be small
for high loss levels.
Figure 7: (color online). Experimentally measured marginal distributions
illustrating the effect of distillation. (a) Example of concatenated sampled
data and the resulting marginal distribution for the amplitude quadrature in
the tap measurement. The vertical line indicates the threshold value chosen
for this realization. (b) Marginal distributions associated with the
measurements of $X$ and $P$ of beam B (two left figures) and (c) the joint
measurements $X_{A}+X_{B}$ and $P_{A}-P_{B}$ (two right figures). The black,
blue and red curves are the distributions for shot noise, the mixed state
before distillation and after distillation, respectively. Inset: phase-space
representation of the non-Gaussian mixed state and the post-selection
procedure used in the measurements. The black vertical line indicates the
threshold value.
The state is then fed into the distiller and we perform homodyne measurements
on beam A, beam B and the tap beam simultaneously. By measuring $\hat{X}$ in
the tap we construct the distribution shown by the red curve in the left hand
side of Fig. 7(a). The data trace of the mixed tap signal is plotted
accordingly on the right hand side. The measurements of $\hat{X}$ and
$\hat{P}$ of the signal entangled states were recorded as well. For
simplicity, we only show the distribution for the beam B (in Fig.
7(b-1),(b-2)) and the joint distribution of beam A and B (in Fig.
7(c-1),(c-2)). The blue (dashed) and red curves denote the distributions
before and after the post-selection process, respectively. From the blue
curves shown in Fig. 7, we can see that the entanglement between A and B is
lost due to the non-Gaussian noise. Performing postselection on this data by
conditioning it on the tap measurement outcome (denoted by $X_{th}=9.0$), we
observe a recovery of the entanglement. That is, the correlated distribution
of the signal turns out to be narrower than that of the shot noise (as shown
by red curves in Fig. 7(c).
Using the data shown in Fig. 7, a tomographic reconstruction of the covariance
matrices of the distilled entangled state was carried out. From these data we
determined the most significant eight of the ten independent parameters of the
covariance matrix, namely the variances of four quadratures
$\hat{X}_{A}$,$\hat{X}_{B}$, $\hat{P}_{A}$, $\hat{P}_{B}$ and covariances
between all pairs of quadratures of the entangled beams A and B. As mentioned
before, the intra-correlations were ignored. The resulting covariance matrices
are plotted in Fig. 8 for ten different postseletion threshold values from
$X_{th}=0.0$ to $X_{th}=9.0$ with a step of 1.0. With increasing postselection
threshold the distillation becomes stronger, as shown by the reduction of the
quadrature variances of $\hat{X}_{A}$, $\hat{X}_{B}$ and the increase of the
quadrature variances of $\hat{P}_{A}$, $\hat{P}_{B}$. Moreover, the reduction
(or increase) of the covariances $C(\hat{X}_{A},\hat{X}_{B})$
($C(\hat{P}_{A},\hat{P}_{B})$) was shown slightly slower. Consequently, the
entanglement of the two modes A and B was enhanced by the distillation.
Figure 8: (color online). Reconstructed covariance matrices of distilled
entangled states. The brown segmented plane shows the region for the
individual elements in the covariance matrix. The sub-bars represent the
results of our distillation protocol for 10 different threshold values
postseletion threshold values from $X_{th}=0.0$ to $X_{th}=9.0$ with a step of
1.0.
Furthermore, the distilled entanglement, or log-negativity, was investigated
as a function of the success probability, as shown in Fig. 9 by black open
circles. The error bars of the distilled log-negativity depend on two
contributions: First the measurement error, which is mainly associated with
the finite resolution of the A/D converter and noise of the electronic
amplifiers. This is considered by estimating the experimental error for all
the elements of the covariance matrix as ’0.03’. The measurement error for the
LN can be simulated by a Monte-Carlo model. Second, the statistical error is
due to the finite measurement time and the postselection process. It is
considered by adding a scaled term $\sqrt{2/(N-1)}$, where $N$ denotes the
number of postselected data frieden83.book . The probability distributions of
the two superimposed states in the mixture after distillation are shown for
different postselection thresholds, corresponding to
$X_{th}=0.0,2.0,4.0,6.0,9.0$, labeled by 1-5 in order. The plots explicitly
show the effect of the distillation protocol, when the postselection threshold
increases, the Gaussian LN increases, ultimately approaching the LN of the
input entanglement without losses. The probability distribution tends to a
single valued distribution, therefore the mixture of the two Gaussian
entangled states reduces to a single highly entangled Gaussian state, thus
demonstrating the act of Gaussification. However, the amount of distilled
data, or success probability, decreases, causing an increase in the
statistical error on the distillable entanglement. Based on the experimental
parameters, a theoretical simulation is plotted by the red curve and shows a
very good agreement with the experimental results.
Figure 9: (color online). Experimental and theoretical results outlining the
distillation of an entangled state from a discrete lossy channel. The
experimental results are marked by circles and the theoretical prediction is
plotted by the red solid line. The bound for Gaussian entanglement is given by
the blue line, and the upper bound for total entanglement before distillation
is given by the black dashed line. Both bounds are surpassed by the
experimental data. The weight of the two constituents in the mixed state after
distillation for various threshold values is also experimentally investigated
and shown in the plots labeled by 1-5. The error bars of the log-negativity
represent the standard deviations.
To further investigate whether the total entanglement is increased after
distillation, we compute the upper bound for the LN before distillation and
verify that this bound can be surpassed by the Gaussian LN after distillation.
The upper bound of LN without the Gaussian approximation is computable from
the LN of each Gaussian state in the mixture vidal02.pra , and we find
$LN_{upper}=0.49$, which is shown in Fig. 9 by the dashed black line. We see
that for a success probability around $10^{-4}$ the Gaussian LN crosses the
upper bound for entanglement. Since the state at this point is perfectly
Gaussified we may conclude that the total entanglement of the state has indeed
increased as a result of the distillation. Fig. 9-5 gives another explicit
explanation by showing that the probability contribution from the 75%
attenuated data reaches 0 when the post selection threshold is set to
$X_{th}=9.0$, which corresponds to the distilled entanglement of
$LN^{P}_{S}=0.67\pm 0.08$ with a success probability $P_{S}=1.69\times
10^{-5}$. On the other hand, from Fig. 9-3 and Fig. 9-4, we see that even a
small contribution from the 75% attenuated data will reduce the useful
entanglement for Gaussian operations.
### IV.2 The continuous lossy channel
We now generalize the lossy channels to have a continuously transmittance
distribution. The channel transmittance distribution is simulated by taking 45
different transmission levels as opposed to the two levels in the previous
section. In Ref. dong08.nat we reported a channel whose transmittance is
given by an exponentially decaying function with a long tail of low
transmittances, which simulates a short-term free-space optical communications
channel where atmospheric turbulence causes scattering and beam pointing noise
book . We showed that the entanglement available for Gaussian operations can
be successfully distilled from $-0.11\pm 0.05$ to $0.39\pm 0.07$ with a
success probability of $1.66\times 10^{-5}$. However, in practical scenarios
for a transmission channel, the highest transmittance level may not have the
biggest weight in the probability distribution and therefore the distributed
peak may be displaced from the 100% transmittance level. Further, there might
be more than one peak in the probability distribution diagram. For instance,
due to some strong beam pointing noise another distributed peak will appear in
the area of low transmittance levels. In the following we will test the
performance of the distillation protocol for two different transmittance
distributions. First, when the mixed state has a peak of the transmittance
distribution which is displaced from 1 to 0.8 (Fig. 10-1). Second, when we
incorporate a second peak which is located around the transmittance level of
0.3 (Fig. 11-1).
Figure 10: (color online). Experimental and theoretical results outlining the
distillation of an entangled state from a simulated continuous lossy channel
in which the peak of the transmittance distribution is displaced from 1 to
0.8. The experimental results are marked by circles and the theoretical
prediction by the red solid curve. The evolution of the weights of the various
constituents in the mixed state as the threshold value is changed is shown in
the figures labeled 1-5. The error bars of the log-negativity represent the
standard deviations.
As shown in Fig. 10, after propagation through the one-peak displaced channel
the Gaussian LN of the mixed state is found to be $-0.50\pm 0.04$, which is
below the bound for available entanglement(shown by the solid blue line) and
substantially lower than the original value of $0.76\pm 0.08$. The state is
subsequently distilled and the change in the Gaussian LN as the threshold
value increases (and the success probability decreases) was investigated both
experimentally (black open circles) and theoretically (red curve). The
evolution of the mixture is directly visualized in the series of probability
distributions in Fig. 10-1 to 10-5 corresponding to the postselection
thresholds $X_{th}$=0.0, 3.0, 5.0, 7.0, 9.0 respectively. We see that the
distribution weights of the low transmittance levels is gradually reduced,
while the weights of the high transmittance levels is increased as the post-
selection process becomes more and more restrictive by increasing the
threshold value. E.g. for $X_{th}=9$ the probabilities associated with
transmission levels lower than 0.7 are decreased from 20% before distillation
to 1.4% and the probability for transmission levels higher than 0.7
transmission are increased to 98.6% as opposed to 80% before distillation. It
is thus clear from these figures that the highly entangled states in the
mixture have larger weight after distillation, and the corresponding Gaussian
LN after distillation rises to $0.19\pm 0.06$ with the success probability of
$5.16\times 10^{-5}$.
Figure 11: (color online). Experimental and theoretical results outlining the
distillation of an entangled state from a simulated continuous lossy channel
in which the transmittance levels are distributed as such that there are two
peaks at both high transmittance levels (0.8) and low transmittance levels
(0.3). The experimental results are marked by circles and the theoretical
prediction by the red solid curve. The evolution of the weights of the various
constituents in the mixed state as the threshold value is changed is shown in
the figures labeled 1-5. The error bars of the log-negativity represent
standard deviations.
We now turn to investigate the distillation after propagation through the two-
peak displaced channel as shown in Fig. 11-1. Before distillation the Gaussian
LN of the mixed state is found to be $-1.13\pm 0.02$. Likewise, the relation
between the distilled Gaussian LN and the success probability was investigated
both experimentally and theoretically. The results are shown in Fig. 11 by
black open circles and the red curve, respectively. Through the probability
distribution plots in Fig. 11-1 to 11-5, the evolution of the mixture
corresponding to different choices of postselection thresholds ($X_{th}$=0.0,
3.0, 5.0, 7.0, 9.0 respectively) was illustrated with the same trend that we
see on the distillation after the one-peak displaced channel. For $X_{th}=9$
the probabilities associated with transmission levels lower than 0.7 are
decreased from 48% before distillation to 1.6% and the probability for
transmission levels higher than 0.7 transmission are increased to 98.4% as
opposed to 52% before distillation, and the corresponding Gaussian LN after
distillation reaches $0.19\pm 0.06$ with the success probability of
$3.39\times 10^{-5}$.
After having shown the successful entanglement distillation on different
distributions of non-Gaussian noise, we should note that the successful
entanglement distillation depends on the transmittance distribution of the
lossy channel. For some distributions, the success probability for distilling
available entanglement for Gaussian operations will be extremely small or not
be possible. For example, after a channel with the transmittance uniformally
distributed, the Gaussian log-negativity $LN_{S}=-1.26\pm 0.02$ before
distillation will only be increased to $-0.76\pm 0.03$ with a success
probability of $1.32\times 10^{-5}$. In general more uniform transmittance
distributions turned out to be more difficult for the distillation procedure.
Distributions with high probabilities for high transmission levels and
pronounced tails and peaks at low transmission levels (as would be expected in
atmospheric channels) are more suited.
## V Summary
In summary, we have proposed a simple method of distilling entanglement from
single copies of quantum states that have undergone attenuation in a lossy
channel with varying transmission. Simply by implementing a weak measurement
based on a beam splitter and a homodyne detector, it is possible to distill a
set of highly entangled states from a larger set of unentangled states if the
mixed state is non-Gaussian. The protocol was successfully demonstrated for a
discrete erasure channel where the transmittance alternates between 2 levels
and two semi-continuous transmission channels where the transmission levels
span 45 levels with specified distributions, respectively. We show that the
degree of Gaussian entanglement (which is relevant for Gaussian information
processing) was substantially increased by the action of distillation.
Moreover, we proved experimentally that the total entanglement was indeed
increased for the discrete channel. We found that the successful entanglement
distillation depends on the transmittance distribution of the lossy channel.
The demonstration of a distillation protocol for non-Gaussian noise provides a
crucial step towards the construction of a quantum repeater for transmitting
continuous variables quantum states over long distances in channels inflicted
by non-Gaussian noise.
## References
* (1)
### References
* (2) N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145(2002).
* (3) T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity,A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
* (4) R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, Nat. Phys. 3, 481 (2007).
* (5) A. R. Dixon, Z. L. Yuan, J.F. Sharpe, and A. J. Shields, Opt. Express 16, 18790 (2008).
* (6) A. Buchleitner, C. Viviescas, and M. Tiersch, Entanglement and decoherence. Springer, Berlin, 2009.
* (7) C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996).
* (8) P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, Nature 409, 1014 (2001).
* (9) Z. Zhao, J.-W. Pan, and M.-S. Zhan, Phys. Rev. A 64, 014301 (2001).
* (10) T. Yamamoto, M. Koashi, S. Kaya Özdemir, and N. Imoto, Nature 421, 343 (2003).
* (11) J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, Nature, 423, 417 (2003).
* (12) Z. Zhao, T. Yang, Y.-A. Chen, A.-N. Zhang, and J.-W. Pan, Phys. Rev. Lett. 90, 207901 (2003).
* (13) J. Eisert, S. Scheel, and M. B. Plenio, Phys. Rev. Lett. 89, 137903 (2002).
* (14) J. Fiurášek, Phys. Rev. Lett. 89, 137904 (2002).
* (15) G. Giedke and J. I. Cirac, Phys. Rev. A 66, 032316 (2002).
* (16) T. Opatrný, G. Kuriziki, and D.-G. Welsch, Phys. Rev. A 61, 032302 (2000).
* (17) P.T. Cochrane, T.C. Ralph and G.J. Milburn, Phys. Rev. A 65, 062306 (2002).
* (18) S. Olivares, M.G.A. Paris, and R. Bonifacio, Phys. Rev. A 67, 032314 (2003).
* (19) L.M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000).
* (20) L.M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Phys. Rev. Lett., 84, 4002 (2000).
* (21) L.M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Phys. Rev. A 62, 032304 (2000).
* (22) J. Fiurášek, L. Mista and R. Filip, Phys. Rev. A 67, 022304 (2003).
* (23) D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, Phys. Rev. A 67, 062320 (2003).
* (24) J. Eisert, D.E. Browne, S. Scheel, and M.B. Plenio, Ann. Phys. 311, 431 (2004).
* (25) A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 98, 030502 (2007).
* (26) H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, arXiv: quant-ph/0907.2159, 2009.
* (27) J. Fiurášek, L. Mista Jr., R. Filip, and R. Schnabel, Phys. Rev. A 75, 050302 (2007).
* (28) R. Dong, M. Lassen, J. Heersink, Ch. Marquardt, R. Filip, Gerd Leuchs, and U. L. Andersen, Nat. Phys. 4, 919 (2008).
* (29) S. Suzuki, M. Takeoka, M. Sasaki, U. L. Andersen, and F. Kannari, Phys. Rev. A 73, 042304 (2006).
* (30) Ch. Wittmann, D. Elser, U.L. Andersen, R. Filip, P. Marek and G. Leuchs, Phys. Rev. A 78, 032315 (2008).
* (31) J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, Phys. Rev. Lett. 96, 253601 (2006).
* (32) B. Hage, A. Samblowski, J. DiGuglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, Nat. Phys. 4, 915 (2008).
* (33) E. M. Rains, Phys. Rev. A 60, 173 (1999).
* (34) E. M. Rains, Phys. Rev. A 60, 179 (1999).
* (35) G. Vidal, and R. F. Werner, Phys. Rev. A 65, 032314 (2002).
* (36) J. Eisert and M. M. Wolf, arXiv: quant-ph/0505151, 2005.
* (37) M. M. Wolf, G. Geidke, J. I. Cirac, Phys. Rev. Lett. 96, 080502 (2006).
* (38) D. F. Walls and G. J. Milburn, Quantum Optics. Springer-Verlag, Berlin, 1994.
* (39) J. Heersink, T. Gaber, S. Lorenz, O. Glöckl, N. Korolkova, and G. Leuchs, Phys. Rev. A 68, 013815 (2003).
* (40) J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, Opt. Lett. 30, 1192 (2005).
* (41) R. Dong, J. Heersink, J. F. Corney, P. D. Drummond, U. L. Andersen, and G. Leuchs, Opt. Lett. 33, 116 (2008).
* (42) M. Fiorentino, J. E. Sharping, and P. Kumar, Phys. Rev. A 64, 031801(R) (2001).
* (43) V. Josse, A. Dantan, L. Vernac, A. Bramati, M. Pinard, and E. Giacobino, Phys. Rev. Lett. 91, 103601 (2003).
* (44) R. Dong, J. Heersink, J. Yoshikawa, Oliver Glöckl, U. L. Andersen, and G. Leuchs, New J. Phys. 9, 410 (2007).
* (45) S. Schiller, G. Breitenbach, S. F. Pereira, T. Müller, and J. Mlynek, Phys. Rev. Lett. 77, 2933 (1996).
* (46) B. R. Frieden, Probability, Statistical Optics and Data Testing. Springer, Berlin, 1983.
* (47) A. K. Majumdar, and J. C. Ricklin, Free-Space Laser Communications, Springer, New York, 2008.
### Acknowledgments
This work was supported by the EU project COMPAS (no.212008), the Deutsche
Forschungsgesellschaft and the Danish Agency for Science Technology and
Innovation (no. 274-07-0509). ML acknowledges support from the Alexander von
Humboldt Foundation and RF acknowledges MSM 6198959213, LC 06007 of Czech
Ministry of Education and 202/07/J040 of GACR.
|
arxiv-papers
| 2010-02-01T16:41:13 |
2024-09-04T02:49:08.126427
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruifang Dong, Mikael Lassen, Joel Heersink, Christoph Marquardt, Radim\n Filip, Gerd Leuchs and Ulrik L. Andersen",
"submitter": "Ruifang Dong",
"url": "https://arxiv.org/abs/1002.0280"
}
|
1002.0311
|
# A novel approach to Isoscaling: the role of the order parameter
$m=\frac{N-Z}{A}$
M. Huang Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou,
730000,China. Graduate University of Chinese Academy of Sciences, Beijing,
100049, China. Z. Chen Cyclotron Institute, Texas A$\&$M University, College
Station, Texas 77843 Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou, 730000,China. S. Kowalski Institute of Physics, Silesia
University, Katowice, Poland. R. Wada Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843 T. Keutgen FNRS and IPN, Université
Catholique de Louvain, B-1348 Louvain-Neuve, Belgium. K. Hagel Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843 J. Wang
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou,
730000,China. L. Qin Cyclotron Institute, Texas A$\&$M University, College
Station, Texas 77843 J.B. Natowitz Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843 T. Materna Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843 P.K. Sahu Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843 M.Barbui
Cyclotron Institute, Texas A$\&$M University, College Station, Texas 77843
C.Bottosso Cyclotron Institute, Texas A$\&$M University, College Station,
Texas 77843 M.R.D.Rodrigues Cyclotron Institute, Texas A$\&$M University,
College Station, Texas 77843 A. Bonasera bonasera@lns.infn.it Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843 Laboratori
Nazionali del Sud, INFN,via Santa Sofia, 62, 95123 Catania, Italy
###### Abstract
Isoscaling is derived within a recently proposed modified Fisher model where
the free energy near the critical point is described by the Landau $O(m^{6})$
theory. In this model $m=\frac{N-Z}{A}$ is the order parameter, a consequence
of (one of) the symmetries of the nuclear Hamiltonian. Within this framework
we show that isoscaling depends mainly on this order parameter through the
’external (conjugate) field’ H. The external field is just given by the
difference in chemical potentials of the neutrons and protons of the two
sources. To distinguish from previously employed isoscaling relationships,
this approach is dubbed: $m-scaling$. We discuss the relationship between this
framework and the standard isoscaling formalism and point out some substantial
differences in interpretation of experimental results which might result.
These should be investigated further both theoretically and experimentally.
###### pacs:
21.65.Ef, 24.10.-i, 24.10.Pa,25.70.Gh, 25.70.Pq
## INTRODUCTION
In near Fermi energy heavy ion collisions fragments are copiously produced.
The mass distributions of these fragments often exhibit a power law behavior
Minich82 ; bon00 . The isotopic distribution of these fragments is governed by
the free energy at the density and temperature of the emitting system. We have
recently discussed the isotope production in terms of the Modified Fisher
Model Bonasera08 . The experimental results exhibit a dependence on an order
parameter $m=\frac{N-Z}{A}$. This analysis and the often reported observation
of isoscaling for products of two similar reactions with different neutron to
proton ratios, N/Z Xu00 ; Tsang01 ; Botvina02 ; Ono03 ; Bonasera08 make it
clear that terms in the free energy which are sensitive to the difference in
neutron and proton concentrations are very important in the fragment formation
process. Indeed, isoscaling analyses based on the comparison of isotope yields
from excited systems of similar temperatures and Z have been employed to
obtain information on the symmetry energy and its density dependence Xu00 ;
Tsang01 ; Botvina02 ; Ono03 ; chen . The ratio of the isotope yields, R12,
between two similar reaction systems with different $N/A$ ratios can be
expressed by the following isoscaling relation Xu00 ; Tsang01 :
$R_{12}(N,Z)=Cexp(\alpha N+\beta Z),$ (1)
where the isoscaling parameters, $\alpha=(\mu_{n}^{1}-\mu_{n}^{2})/T$ and
$\beta=(\mu_{p}^{1}-\mu_{p}^{2})/T$, represent the differences of the neutron
( or proton) chemical potentials between systems 1 and 2, divided by the
temperature. C is a constant.
In terms of a modified Fisher model description the experimental yield of an
isotope with N neutrons and Z protons can be written as Minich82 ; Bonasera08
; bon00 :
$\displaystyle Y(N,Z)=Y_{0}A^{-\tau}exp\\{-[G(N,Z)$
$\displaystyle-\mu_{n}N-\mu_{p}Z]/T\\},$ (2)
where Y0 is a constant, G(N,Z) is the nuclear free energy at the time of the
fragment formation, $\mu_{n}$ and $\mu_{p}$ are the neutron and proton
chemical potentials, and T is the temperature of the emitting source. The
factor, $A^{-\tau}$, originates from the entropy of the fragment Minich82 .
Notice that in the ratio of yields employed in an isoscaling analysis the
power law term from Eq.(2) will cancel out.
In a grand canonical treatment the relationship between the isoscaling may be
expressed as Botvina02 ; Ono03 :
$\alpha(Z)=4C_{sym}\Delta(Z_{s}/{A_{s}})^{2}/T,$ (3)
where
$\Delta(Z_{s}/{A_{s}})^{2}={(Z_{s}/{A_{s}})^{2}}_{1}-{(Z_{s}/{A_{s}})^{2}}_{2}$,
$C_{sym}$ is the symmetry free energy and T is the temperature. The symmetry
free energy is presumed to be density dependent. In a similar fashion
$\beta(N)$ can be expressed as :
$\beta(N)=4C_{sym}\Delta(N_{s}/{A_{s}})^{2}/T,$ (4)
where
$\Delta(N_{s}/{A_{s}})^{2}={(N_{s}/{A_{s}})^{2}}_{1}-{(N_{s}/{A_{s}})^{2}}_{2}$.
This suggests that for two systems at temperature T in which the symmetry
energy is the dominant factor in determining the yield ratios of the emitted
fragments, the ratio $\beta(N)/\alpha(Z)$ can be expressed as
$\eta=\frac{\beta(N)}{\alpha(Z)}=-\frac{(N_{s}/A_{s})^{2}_{1}-(N_{s}/A_{s})^{2}_{2}}{(Z_{s}/A_{s})^{2}_{1}-(Z_{s}/A_{s})^{2}_{2}},$
(5)
This ratio will approach -1 when systems of very similar N/Z are considered
but in general can be quite different from -1.
It is well known that nucleon forces are isospin invariant, because of this we
expect that, in the absence of spontaneous symmetry breaking,
$(\mu_{n}^{1}-\mu_{n}^{2})=(\mu_{p}^{1}-\mu_{p}^{2})$ and hence
$\alpha=-\beta.$ (6)
In a manner similar to the case for mirror nuclei, at low excitation energies
we can expect this invariance to be broken by Coulomb energy contributions.
However in fragmentation reactions occurring near the critical point of the
liquid-gas phase change the nuclear symmetry should be restored because of the
invariance of the nuclear Hamiltonian when $m\rightarrow-m$. In a separate
paper discussing the analysis of the same data set used in this paper, a clear
fragment Z (or N) dependence of $\alpha(Z)$ (or $\beta(N)$) has been reported
chen ; huang10 . ¿From the detailed comparisons to the dynamical model (AMD)
calculation followed by the statistical decay code (Gemini), this Z dependence
is attributed to the statistical secondary decay process of the excited
fragments after they are formed at freezeout of the emitting source. A
significant modification of the isoscaling parameters is also suggested during
the cooling process. Similar results have been also reported in ref. Botvina02
. In that work it is also concluded that secondary decay effects play a
significant role in determining the observed ratio. Their conclusion is based
upon the application of an SMM model description to the experimental data
within a statistical model description of fragmentation. In this paper we
suggest that there may be an essential relatioship between the scaling
parameter and N/Z of the emitting system, which is related to the restoration
of symmetry near the critical point of the emitting source and which is
sustained in the experimental observables during the cooling process and
experimentally manifested in the isoscaling parameters.
## I Experiments and Analysis
Using high resolution detector telescopes with excellent isotope
identification capabilities, we recently studied a number of heavy ion
reactions to determine relative yields for production of a wide range of
isotopes chen ; huang10 . The experiment was performed at the K-500
superconducting cyclotron facility at Texas A$\&$M University. 40 A MeV
64Zn,70Zn and 64Ni beams irradiated 58Ni,64Ni, 112Sn,124Sn, 197Au, and 232Th
targets. Intermediate mass fragments (IMFs) were detected by a detector
telescope placed at 20 degrees relative to the beam direction. The telescope
consisted of four Si detectors. Each Si detector had an area of 5cm $\times$
5cm. The thicknesses are 129, 300, 1000 and 1000 $\mu$m. Using the $\Delta
E-E$ technique we were typically able to identify 6-8 isotopes for a given Z
up to Z=18 with energy thresholds of 4-10 A MeV. More details of the analysis
are contained in ref. chen ; huang10 .
Isoscaling analyses were carried out for all possible combinations of these
reactions. Eighteen different reactions are considered here and therefore more
than 150 combinations are studied. Data for each atomic number were
independently fit to extract the isoscaling parameter $\alpha(Z)$. $\beta(N)$
values were also extracted for each neutron number. For some systems the
extracted $\alpha(Z)$ parameter shows a steady decrease as Z increases. The
$\beta(N)$ parameter generally showsa much smaller variation with increasing
N, and has the opposite sign. A clear correlation between them, i.e.
$\alpha(Z)\sim-\beta(N)$ for the equivalent number of nucleons, $N=Z$, has
also been observed as suggested in the introduction(see Eq.(6)). In Fig.1, the
extracted isoscaling parameters for the case of Z = N = 7 are shown as a
typical example. Similar correlations are also observed for other selections
of Z and N values if Z = N.
Figure 1: (a) $\beta(N)$ vs $\alpha(Z)$ is plotted for N=Z=7. Line indicates
the locus $\beta(N)=-\alpha(Z)$; (b)ratio of $\beta(N)/\alpha(Z)$ data to
theoretical Eq.(5) (open squares); (c)$\beta(N)/\alpha(Z)$ vs. $\alpha(Z)$
from data (open triangles); (d)the analytical prediction Eq.(5) (open circles)
to compare with data.
In part a of Figure 1 we have plotted $\beta(N)$ vs $\alpha(Z)$. As seen in
the figure, the relation $\alpha(Z)\sim-\beta(N)$ is observed for
$\alpha(Z)\leq 0.5$ and may deviate slightly at the larger $\alpha(Z)$ values.
Those larger values are associated with the largest N/A values for the
compound system. In the bottom part of the figure, these values on the left
are compared to predictions of Eq.(5) on the right with the assumptions that
Z/A is that of the compound system. We note that the experimental values tend
to be significantly closer to -1 than the calculated values. Except at the low
experimental values of $\alpha(Z)$ where the scatter is significant, the
experimental values for $\beta(N)/\alpha(Z)$ are about $20\%$ lower in the
absolute value than the model values as indicated by the ratio of these two
quantities also plotted in the middle part of the figure. In order to see the
system dependence of $\alpha(Z)$ and $\beta(N)$ values, these values are
plotted for separate groups of fissility values in Fig.2. The fissility is
defined as $X=\frac{Z_{s}^{2}}{A_{s}}$, where $Z_{s}$ and $A_{s}$ are the
charges and masses of the source which we assume to be the compound nucleus
for simplicity. We can define a combined fissility parameter between reactions
(1) and (2) as $\Sigma X=X_{1}+X_{2}$. Larger (absolute) values of $\alpha(Z)$
and $\beta(N)$ correspond to large values of the $\Sigma X$ parameter. In the
figure $\alpha(Z)$ and $\beta(N)$ values are separately plotted for four
different ranges of fissility group for the same data set used in Fig.1. We
see no systematic correlation with the fissility parameters in the deviation
from $\alpha(Z)=-\beta(N)$, which might be suggestive of the fact that the
Coulomb force is not so effective for breaking the invoked invariance of the
Nuclear Hamiltonian. It would be very interesting to see if Coulomb effects
become more important for heavy colliding nuclei such as $U+U$. One should
note that a similar result is observed for $\alpha(Z)$ and $\beta(N)$ in other
IMFs, when Z and N are the same. One can also use $\alpha$ and $\beta$ values
averaged over a range of atomic (or neutron) number, though in this case the
averaged numbers depend on the somewhat range selected.
Figure 2: $\beta(N)$ vs $\alpha(Z)$ with $6\leq Z\leq 13$ for different ranges
of the fissility parameter. $\Sigma X\leq 60.9$ (top left), $60.9<\Sigma X\leq
72.2$ (top right), $72.2<\Sigma X\leq 83.4$ (bottom left) and $83.4<\Sigma
X\leq 94.7$ (bottom right).
While the trend in Figs.1 and 2 is interesting, it is important to note that
when the neutron and proton concentrations of the initial excited source are
different, two well established trends can act to shift the balance toward
symmetric matter and hence to bring the absolute values of the observed
$\alpha(Z)$ and $\beta(N)$ parameters closer together. The first is the
distillation effect in which early emission of particles favors neutron
emission over proton emission serot ; maria . As a result of this early
emission the fragmenting system will tend to have a higher symmetry than the
initial system. The second is secondary decay of initially excited fragments
Marie98 ; Hudan03 which favors a shift toward the evaporation attractor line
Charity88 . Thus even if the comparison of primary fragment yields would lead
to a significant difference in the two isoscaling parameters the subsequent
decay can reduce this difference.
## $m$-SCALING
Pursuing the question of phase transitions, we note that we have previously
discussed some of the present yield data within the Landau free energy
description Bonasera08 . In this approach the ratio of the free energy (per
particle) to the temperature is written in terms of an expansion:
$\frac{F}{T}=\frac{1}{2}am^{2}+\frac{1}{4}bm^{4}+\frac{1}{6}cm^{6}-m\frac{H}{T},$
(7)
where $m$ is the order parameter, $H$ is its conjugate variable and $a-c$ are
fitting parameters. In our case $m=(I/A)$. Notice that the free energy that we
have indicated with F includes the chemical potential of neutrons and protons
i.e. $AF(m,T)=[G(N,Z)-\mu_{n}N-\mu_{p}Z]$ (compare to Eq.(2)).
We observe that the free energy is even in the exchange of $m\rightarrow-m$
reflecting the invariance of the nuclear forces when exchanging N and Z. This
symmetry is violated by the conjugate field $H$ which arises when the source
is asymmetric in the chemical composition. We stress that correctly m and H
are related to each other through the relation $m=-\frac{\delta F}{\delta H}$.
An immediate consequence of the application of the Landau expression of Eq.(7)
in the Modified Fisher Model is that it brings a scaling law for m=0 isotopes.
Since F(m=0,T)=0, for any T, the yield in Eq.(2) is given as
$\displaystyle{Y(N,Z)=Y_{0}A^{-\tau},}$ (8)
for all reactions.
Figure 3: Yield ratio of m=0 isotopes vs A. The yield is normalized to that of
${}^{12}C$. Data from all 18 reaction systems studied in this experiment.
(top) even-even isotopes. (middle) odd-odd isotopes are plotted. (bottom)
pairing corrected yield, Y(N,Z)/exp($\delta$), are plotted for all m=0
isotopes. Lines in each figure are linear fitted ones. $\tau$ values are 3.3,
2.2 and 2.8 from the top to bottom. $a_{p}/T$ =2.2 is used in the bottom.
In Fig.3, yield ratios for m=0 isotopes are separately plotted as a function
of A for even-even (top) and odd-odd (middle) isotopes for all 18 reactions
studied here. In order to eliminate the effect of the constant in Eq.(2),
which are slightly different in each reaction system, the yield is normalized
to that of the 12C in each reaction. As seen in the figure, the yields from
the different reactions are indeed scaled well with A up to A $\sim$ 30 when
even-even and odd-odd isotopes are plotted separately. One should note that
data points for a given A represent all 18 different reactions in the figure.
The slope difference between even-even and odd-odd isotopes can be naturally
attributed to the pairing effect. However a large pairing effect is expected
only at a low temperature, because it is related to the shell effect. On the
other hand the emitting sources of these isotopes are expected to be at a high
temperature. Ricciardi et al. have given a possible explanation for this
observation Ricciardi04 ; Ricciardi05 . According model simulations which they
have performed the experimentally observed pairing effect is attributed to the
last chance particle decay of the excited fragments during their cooling. This
hypothesis is also supported by our model simulations presented in a separate
paper huang10 . In order to take into account the pairing effect, data for
even-even and odd-odd isotopes were simultaneously fitted by the following
equation,
$\displaystyle Y(N,Z)=Y_{0}A^{\tau}exp(\delta/T),$ (9)
$\displaystyle\delta(N,Z)=\left\\{\begin{array}[]{ll}a_{p}/A^{1/2}&(\textrm{odd-
odd})\\\ 0&(\textrm{even-odd})\\\ -~{}a_{p}/A^{1/2}&(\textrm{even-
even}).\end{array}\right.$ (13)
and the parameters $\tau$ and $a_{p}/T$ values was extracted. Using these
extracted parameters, the experimental yield was divided by the exponent in
Eq.(9) as factor. The results are plotted in the bottom of the figure for all
isotopes with m=0. The extracted $\tau$ value is 2.8 which is larger than the
normal critical exponent 2.3. This difference may reflect either the
temperature of the emitting source is below the critical temperature or that
the secondary decay processes modify the value.
Because of the symmetries of the free energy when we take the ratio between
two different systems, $presumably$ at the same temperature $T$ and density
$\rho$, all $even$ order terms in $m$ cancel out while the $odd$ terms remain.
Those terms depend on the $external$ field $H/T$. Taking the ratio between two
systems as in Eq.(2) we easily obtain:
$R_{12}(m)=Cexp(\Delta H/TmA),$ (14)
where $\Delta H/T=H_{1}/T-H_{2}/T$. We can fix the constant C by dividing each
experimental yield by the ${}^{12}C$ yield following in ref. Bonasera08 . The
goal is to get $C\rightarrow 1$ for reasons that will become clear below.
Comparing the latter equation with Eq.(1) we obtain: $\Delta H/TmA=\alpha
N+\beta Z$ i.e. $\alpha=-\beta=\frac{\Delta H}{T}$. As shown in Fig.1, for the
comparison for isotopes of a given Z with isotones having N equal to that Z,
this relation appears to be satisfied. The relation is valid more in general,
and in fact we could write the chemical potentials of neutrons and protons as:
$\mu_{n}N+\mu_{p}Z=\mu A+HmA,$ (15)
from this relation it follows that:
$2H=\mu_{n}-\mu_{p};2\mu=\mu_{n}+\mu_{p}.$ (16)
All these relations show that if $m$ is an order parameter then
$\alpha=-\beta=\Delta H/T$.
Figure 4: Experimental ratios vs m for isotopes with $6\leq Z\leq 13$ for (a)
$\frac{{}^{64}Ni^{232}Th}{{}^{70}Zn^{197}Au}$,
(b)$\frac{{}^{64}Ni^{112}Sn}{{}^{64}Ni^{58}Ni}$,
(c)$\frac{{}^{64}Ni^{124}Sn}{{}^{64}Ni^{64}Ni}$,
(d)$\frac{{}^{64}Ni^{197}Au}{{}^{64}Ni^{112}Sn}$,
(e)$\frac{{}^{64}Ni^{124}Sn}{{}^{70}Zn^{58}Ni}$ and
(f)$\frac{{}^{64}Ni^{124}Sn}{{}^{70}Zn^{112}Sn}$ respectively at $40MeV/A$.
The lines are the results of a linear fit according to Eq.(17).
The external field is given by the difference of chemical potentials between
neutrons and protons of the emitting system as expected. ¿From Eq.(14) we can
obtain the difference between the free energies (or alternatively the external
fields) as:
$\frac{-ln(R_{12}(m))}{A}=\Delta H/Tm+constant.$ (17)
Thus a plot of $-ln(R)/A$ versus m should give a linear relation whose slope
is given by $\Delta H/T$. This linear relation is demonstrated in Figs.1 and 2
where such a plot is obtained for different colliding systems for the isotopes
in the selected range of Z. In thatgiven range $\alpha(Z)$ increases about 50%
on average chen ; huang10 . As discussed in references chen ; huang10 , the
observed fragment Z (or N) dependence of the isoscaling parameters is mainly
established during the statistical cooling of the excited fragments. In fact
it has been demonstrated that $\alpha(Z)$ parameter extracted from the primary
fragments of the AMD simulations shows no significant fragment Z dependence.
It should be noted that it is important to normalize the distribution ( for
instance to ${}^{12}C$ ) as we have done in order that the normalizing
constant in front of the yield in Eq.(14) is one. If not this will carry a
$\frac{1}{A}$ term which might violate the scaling. Overall the scaling is
satisfied for this set of data as seen in Fig.4. Compared to ’traditional’
isoscaling where a fit is performed for each detected charge $Z$ (or each $N$)
we see that all the data collapse into one curve.
We can further elucidate the role of the external field $H/T$ writing the
Landau expansion and ’shifting’ the order parameter by $m_{s}$ which is the
position of the minimum of the free energy. Such a position depends on the
neutron to proton concentration of the source Bonasera08 . Thus
$\frac{F}{T}=\frac{1}{2}a(m-m_{s})^{2}+\frac{1}{4}b(m-m_{s})^{4}+\frac{1}{6}c(m-m_{s})^{6}.$
(18)
Comparing to Eq.(7) we easily obtain
$\frac{H}{T}=(a+bm^{2}+cm^{4})m_{s}+(\frac{1}{2}\frac{a}{m}+\frac{3}{2}bm+\frac{5}{2}cm^{3})m_{s}^{2}+O(m_{s}^{4})...,$
(19)
thus $H$ depends on the source isospin concentration though the parameter
$a,b,c$ which are terms of the free energy. We stress that these terms refer
to the free energy and $not$ to the internal symmetry energy. If b and c are
of comparable magnitude to parameter a, then taking terms of a, Eq.(19) can be
further simplified as
$\frac{H}{T}=-am_{s}+\frac{1}{2}a\frac{m_{s}^{2}}{m}+O(m_{s}^{4})...,$ (20)
## II Reconciliation of the two approaches
Standard isoscaling results have been derived under a general grand canonical
approach Ono03 ; Tsang01 ; Botvina02 . The Landau approach should be
equivalent to it under certain conditions. Experimentally the b and c values
have not been established because all isotopes identified in the present data
have m $<$ 0.5 except for nucleons. In the case that b and c are of comparable
magnitude to parameter a, which is assumed in the derivation of eq.(20), we
easily obtain:
$\frac{\Delta H}{T}mA=a\Delta
m_{s}(N-Z)-\frac{1}{2}a(m^{2}_{s1}-m^{2}_{s2})A=\alpha N+\beta Z$ (21)
which introduces a volume term. Equating similar terms we get:
$\alpha=a\Delta m_{s}-\frac{1}{2}a(m^{2}_{s1}-m^{2}_{s2})$ (22)
where $\Delta m_{s}=m_{s1}-m_{s2}$. It is straightforward to demonstrate the
equivalence of the last equation to eq.(3) derived from the grand canonical
approach. In particular we get:
$\alpha+\beta=-a(m^{2}_{s1}-m^{2}_{s2})$ (23)
which shows that the two approaches are equivalent and that $m$ is an order
parameter if $\alpha+\beta=0$ i.e. neglecting $O(m_{s}^{2})$ terms in the
external field. In figure (5) we plot $\alpha+\beta$ vs.
$m^{2}_{s1}-m^{2}_{s2}$, unfortunately the error bars are rather large but we
can see a systematic deviation from zero as expected from eq.(23) for large
differences in concentration.
Figure 5: $\alpha(Z)+\beta(N)$ vs the difference (solid circles) in
concentration for the two reaction systems for the case of Z=N=7. The dotted
line is the result of a linear fit.
This indicates that, at the level of sensitivity so far acheived with data of
this type the presence of higher order terms in m is difficult to quantify.
Thus, within the error bars, $m$ could be considered an order parameter when
relatively neutron (or proton) rich sources are considered. In particular,
phase transitions in finite systems could be studied using the same language
of macroscopic systems i.e., ’turning on and off’ an external field Bonasera08
.
## SUMMARY
In conclusion, in this paper we have discussed scaling of ratios of yields
from different colliding systems under similar physical control parameters,
i.e. density and temperature. A careful and precise determination of isotopic
yields is needed in order to see the features of the system near the phase
transition. There is an order parameter, m, given by the difference in neutron
and proton concentrations of the detected fragments which leads to an expected
isoscaling relation, a direct consequence of the restored symmetry of the
nuclear Hamiltonian when exchanging neutrons with protons. The data suggest
that the Coulomb field may not significantly violate such a symmetry. The
existence of m-scaling might be a signature for near criticality of the
fragmenting system. Other properties of the ’rich’ nuclear Hamiltonian, such
as pairing, appear to result in small violations of the scaling. This is an
interesting physical aspect which deserves further and deep investigation both
theoretically and experimentally. Also, it would be interesting to search for
m-scaling violations in heavily charged colliding systems such as U+U. The
absence of a violation in these cases would suggest that densities and
deformations of the fragments are such that the effect of Coulomb is
significantly reduced. Studies of the other extreme case of very exotic
colliding systems would be also be valuable to probe the effects of high
’external’ field on the phase transition. The atomic nucleus constitutes a
formidable laboratory to test our knowledge and understanding of phase
transitions in a finite system and offers a unique possibility for different
quantum aspects similar to other bosons and fermion mixtures.
A major consideration in the interpretation of the results presented in this
paper is the effect of the secondary decay process. In the experiments excited
fragments cool down to the ground state before they are detected. The
reconstruction of the primary fragments from the experimentally observed IMFs
and associated particles is not straightforward, since multiple excited
primary fragments may be simultaneously produced in multifragmentation
reactions making the unambiguous identification of the primary fragment
distribution difficult. Indeed a major goal of the experiments from which the
present isoscaling data are taken was to employ fragment-particle correlation
measurements to reconstruct the primary fragment distribution. The correlation
data are still being analyzed Wada05 .
###### Acknowledgements.
This work is supported by the U.S. Department of Energy and the Robert A.
Welch Foundation under grant A0330. One of us(Z. Chen) also thanks the “100
Persons Project” of the Chinese Academy of Sciences for the support.
## References
* (1) R. W. Minich et al., Phys. Lett. B118, 458 (1982).
* (2) A. Bonasera et al., Rivista Nuovo Cimento, 23 (2000) 1.
* (3) A. Bonasera et al., Phys. Rev. Lett. 101, 122702 (2008) and in preparation.
* (4) H. S. Xu et al., Phys. Rev. Lett. 85, 4, (2000).
* (5) M. B. Tsang et al., Phys. Rev. C64, 054615 (2001).
* (6) A.S. Botvina, O. V. Lozhkin and W. Trautmann, Phys. Rev. C65, 044610 (2002).
* (7) A. Ono et al., Phys. Rev. C68, 051601(R) (2003).
* (8) Z.Chen et al., arXiv:1002.0319 [nucl-ex] 1Feb2010.
* (9) M.Huang et al., arXiv:1001.3621 [nucl-ex] 22Jan2010.
* (10) A. Ono, et al., Phys. Rev. C70, 041604(R) (2004).
* (11) M. V. Ricciardi et al., Nucl. Phys. A733 (2004) 299.
* (12) M. V. Ricciardi et al., Nucl. Phys. A749 (2005) 122c.
* (13) H. Muller and B. D. Serot, Phys. Rev. C52 (1995) 2072.
* (14) V.Baran et al., Phys.Rep.410(2005)335. Eur. Phys. J.A30,203 (2006). Cyclotron Institute, Texas A&M University, (2005), II-3, unpublished. DataTables 46, 1 (1990).
* (15) N. Marie et al., Phys. Rev. C58, 256 (1998).
* (16) S. Hudan et al., Phys. Rev. C76, 064613 (2003) 53, 501 (2004).
* (17) R. J. Charity et al., Nucl. Phys. A483, 371 (1988).
* (18) K. Huang, Statistical Mechanics, second edition, ch.16-17, J. Wiley and Sons, New York, 1987.
* (19) R.Wada et al., annual report of the Cyclotron Institute, Texas A$\&$M University, (2005), II-3, unpublished. One can find the article in the web page :http://cyclotron.tamu.edu/publications.html.
|
arxiv-papers
| 2010-02-01T20:03:40 |
2024-09-04T02:49:08.133341
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Huang, Z. Chen, S. Kowalski, R. Wada, T. Keutgen, K.Hagel, J. Wang,\n L. Qin, J.B. Natowitz, T. Materna, P.K. Sahu, M.Barbui, C.Bottosso,\n M.R.D.Rodrigues, and A. Bonasera",
"submitter": "Meirong Huang",
"url": "https://arxiv.org/abs/1002.0311"
}
|
1002.0319
|
# Isocaling and the Symmetry Energy in the Multifragmentation Regime of Heavy
Ion Collisions
Z. Chen Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou,
730000,China. S. Kowalski Institute of Physics, Silesia University,
Katowice, Poland. M. Huang Cyclotron Institute, Texas A$\&$M University,
College Station, Texas 77843 Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou, 730000,China. Graduate University of Chinese Academy of
Sciences, Beijing, 100049, China. R. Wada wada@comp.tamu.edu Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843 T. Keutgen
FNRS and IPN, Université Catholique de Louvain, B-1348 Louvain-Neuve, Belgium.
K. Hagel Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843 J. Wang Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, 730000,China. L. Qin Cyclotron Institute, Texas A$\&$M University,
College Station, Texas 77843 J.B. Natowitz Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843 T. Materna Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843 P.K. Sahu Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843 A. Bonasera
Cyclotron Institute, Texas A$\&$M University, College Station, Texas 77843
Laboratori Nazionali del Sud, INFN,via Santa Sofia, 62, 95123 Catania, Italy
###### Abstract
The ratio of the symmetry energy coefficient to temperature, $a_{sym}/T$, in
Fermi energy heavy ion collisions, has been experimentally extracted as a
function of the fragment atomic number using isoscaling parameters and the
variance of the isotope distributions. The extracted values have been compared
to the results of calculations made with an Antisymmetrized Molecular Dynamics
(AMD) model employing a statistical decay code to account for deexcitation of
excited primary fragments. The experimental values are in good agreement with
the values calculated but are significantly different from those
characterizing the yields of the primary AMD fragments.
###### pacs:
21.65.Ef, 24.10.-i, 24.10.Pa,25.70.Gh, 25.70.Pq
## I. INTRODUCTION
In Fermi energy heavy ion collisions, fragments are copiously produced. The
mass distributions of these fragments exhibit a power low behavior which has
been discussed in terms of the Modified Fisher Model Minich82 ; Bonasera08 .
The isotope distributions of these fragments play a key role in these
analyses. Theoretical studies indicate that the isotope formation is governed
by the free energy at the density and temperature of the emitting system. The
experimental observation of isoscaling for two similar reactions with
different neutron to proton ratios, N/Z, demonstrates that the free energies
and therefore the yields of the fragments are also closely related to the N/Z
of the emitting system Xu00 ; Tsang01 ; Botvina02 ; Ono03 . Thus the
experimental yield of isotope with N neutrons and Z protons can be given by
Minich82 ; Bonasera08 ; Albergo85 ; Tsang01 :
$\displaystyle Y(N,Z)=Y_{0}F(N,Z)A^{-\tau}exp\\{-[G(N,Z)$
$\displaystyle-\mu_{n}N-\mu_{p}Z]/T\\}$ (1)
where $Y_{0}$ is a constant and G(N,Z) is the nuclear free energy at the time
of the fragment formation. $\mu_{n}$ and $\mu_{p}$ are the chemical potential
of neutron and proton, and T is the temperature of the emitting source. The
factor F(N,Z) is the correction factor for the feeding from the statistical
decay processes. The factor, $A^{-\tau}$, originates from the entropy of the
fragment Minich82 . The symmetry energy term in the free energy, G(N,Z), is
usually expressed as:
$E_{sym}=a_{sym}(N-Z)^{2}/A$ (2)
where A=N+Z and $a_{sym}$ is the symmetry energy coefficient which depends on
the nuclear density $\rho$ and the temperature T of the emitting source.
From Eq.(1) $R_{12}$, the ratio of the isotope yields for two similar reaction
systems with different N/Z ratios, can be written as:
$R_{12}(N,Z)=Cexp(\alpha N+\beta Z)$ (3)
This relation is known as the isoscaling relation. The isoscaling parameters,
$\alpha=(\mu_{n}^{1}-\mu_{n}^{2})/T$ and $\beta=(\mu_{p}^{1}-\mu_{p}^{2})/T$)
are the differences of the neutron or proton chemical potentials between the
systems 1 and 2, divided by the temperature. C is a constant. System 1 is
normally taken as the more neutron rich of the two.
As discussed in refs. Botvina02 ; Moretto08 ; Ono04_1 , the isoscaling
parameters and the symmetry energy coefficient are closely related. For a
multifragmentation regime, as pointed out in ref. Ono04_1 , this relation is
given by:
$\alpha(Z)=4a_{sym}\Delta(Z/\bar{A})^{2}/T$ (4)
where $\Delta(Z/\bar{A})^{2}={(Z/\bar{A})^{2}}_{1}-{(Z/\bar{A})^{2}}_{2}$ for
the two reaction systems and $\bar{A}$ is the average mass number of isotopes
for a given Z.
There are two issues for the determination of the $a_{sym}$ values in Eq.(4).
One is the source temperature T. Since the beginning of the experimental study
of heavy ion collisions in the multifragmentation regime, significant efforts
have been made to extract the source temperature, but different methods of
temperature extraction can lead to different results and uncertainties still
remain Kelic06 . Another issue is the effect of the secondary decay process,
expressed by F(N,Z) in Eq.(1). In experiments fragments have typically cooled
down to the ground state before they are detected. Indeed, in previous works,
excitation energies of the primary fragments have been evaluated by studying
the associated light charged particle multiplicities Marie98 ; Hudan03 . Such
data raise the question of the degree of confidence for the experimentally
extracted symmetry energy coefficient, in which this important effect is not
properly corrected. In fact in a separate paper using the same data set
presented here we have demonstrated that the secondary decay processes
significantly effect the isobaric yield ratios and the experimentally
extracted symmetry energy coefficient Huang10 . In this paper, we focus on the
relation between the ratio $a_{sym}/T$ and isoscaling parameters and between
the ratio and the widths of the isotope distributions. The experimentally
extracted values of $a_{sym}/T$ extracted from both observables are compared
to those extracted from the model calculations.
## II. EXPERIMENT
The experiment was performed at the K-500 superconducting cyclotron facility
at Texas A$\&$M University. 64,70Zn and 64Ni beams were used to irradiate
58,64Ni, 112,124Sn, 197Au and 232Th targets at 40 A MeV. Intermediate mass
fragments (IMFs) were detected by a detector telescope placed at 20∘. The
telescope consisted of four Si detectors. Each Si detector was 5cm x 5cm. The
nominal thicknesses were 129, 300, 1000, 1000 $\mu$m. All Si detectors were
segmented into four sections and each quadrant had a 5∘ opening angle in polar
and azimuthal angles. Therefore the energies of the fragments were measured at
two polar angles of the quadrant detector, namely $\theta$ = 17.5∘ $\pm$ 2.5∘
and $\theta$ = 22.5∘ $\pm$ 2.5∘. Typically 6-8 isotopes for a given atomic
numbe up to Z=18 were clearly identified with the energy threshold of 4-10 A
MeV, using the $\Delta$E-E technique for any two consecutive detectors. The
$\Delta$E-E spectrum was linearized empirically. Mass identification of the
isotopes were made using a range-energy table Hubert90 . In the analysis code,
isotopes are identified by a parameter $Z_{Real}$. For the isotope with A=2Z,
$Z_{Real}$ = Z is assigned and other isotopes are identified by interpolation
between them. Typical $Z_{Real}$ spectra are shown in Fig.1. The energy
spectrum of each isotope was extracted by gating the isotope in a 2D plot of
$Z_{Real}$ vs energy. The yields of light charged particles (LCPs) in
coincidence with IMFs were also measured using 16 single crystal CsI(Tl)
detectors of 3cm thickness set around the target. The light output from each
detector was read by a photo multiplier tube. The pulse shape discrimination
method was used to identify p, d, t, h and $\alpha$ particles. The energy
calibration for these particles were performed using Si detectors (50 -300
$\mu$m) in front of the CsI detectors in separate runs.
The yield of each isotope was evaluated, using a moving source fit. For LCPs,
three sources (projectile-like(PLF), nucleon-nucleon-like(NN) and target-like
(TLF)) were used. The NN-like sources have source velocities of about a half
of the beam velocity. The parameters are searched globally for all 16 angles.
For IMFs, since the energy spectra were measured only at the two angles of the
quadrant detector, the spectra were parameterized using a single NN-source.
Using a source with a smeared source velocity around half the beam velocity,
the fitting parameters were first determined from the spectrum summed over all
isotopes for a given Z, assuming A=2Z. Then all extracted parameters except
for the normalizing yield parameter were used for the individual isotopes.
This procedure was based on the assumption that, when the spectrum is plotted
in energy per nucleon, the shape of the energy spectrum is the same for all
isotopes for a given Z. Indeed the observed energy spectra of isotopes are
well reproduced by this method. For IMFs, a further correction was made for
the background. As seen in Fig.1, the isotopes away from the stability line,
such as 10C and 29Mg, have a very small yields and the background contribution
is significant. In order to evaluate the background contribution to the
extracted yield from the source fit, a two Gaussian fit to each isotope
combined with a linear background was used. The fits are shown in Fig.1. Each
peak consists of two Gaussians. The second Gaussian (about 10% of the height
of the first one) is added to reproduce the shape of the valley between two
isotopes. This component is attributed to the reactions of the isotope in the
Si detector. The centroid of the Gaussians was set to the value calculated
from the range-energy table within a small margin. The final yield of an
isotope was determined by correcting the yield evaluated from the moving
source fit by the ratio between the two Gaussian yields and the linear
background. Rather large errors ( $\sim\pm 10\%$) are assigned for the
multiplicity of the NN source for IMFs, originating from the source fit
besides the background estimation. The errors from the source fit are
evaluated from the different assumptions of the parameter set for the source
velocity and temperature.
Figure 1: Typical linearized isotope spectra are shown for Z=6 (upper figure)
and 12 (lower) cases. The number at the top of each peak is the mass number
assigned. Linear back ground is assumed from valley to valley for a given Z.
Each Gaussian indicates the yield of the isotope above the back ground.
## III. ISOSCALING
At the top of Fig. 2, the yield ratio of Eq.(3) for the reactions 64Ni + 124Sn
and 64Ni + 58Ni is plotted as a function of N. The $\alpha$ parameter is
determined by individual fits to yield ratios for isotopes with a given Z. The
extracted values are plotted in the bottom of Fig. 2. As seen in the figure,
the extracted $\alpha(Z)$ parameter shows a steady decrease as Z increases for
Z $\geq$ 4\. The $\beta(N)$ parameter generally shows much similar variation
with increasing N, and has the opposite sign. Hereafter $\alpha$ and $\beta$
are denoted as $\alpha(Z)$ and $\beta(N)$. Isoscaling parameters, $\alpha(Z)$
and $\beta(Z)$, have been evaluated for all possible combinations between two
reactions. For 18 different reactions are considered here. More than 150
different combinations have been studied.
Figure 2: (upper) $\alpha(Z)$ values as a function of Z for 64Ni + 124Sn and
64Ni + 58Ni. From the left to right each lines correspond to Z=1 to Z=18.
(lower) The extracted $\alpha(Z)$ values are plotted as a function of Z.
In Fig. 3 the extracted $\alpha(Z)$ values are plotted as a function of
$\Delta(Z/\bar{A})^{2}$ for Z=6 and Z=12. Each data point represents a
combination of two reactions. As seen in the figure, the $\alpha(Z)$ values
are linearly related to $\Delta(Z/\bar{A})^{2}$. The slope associated with
this relationship increases gradually as Z increases. The correlations have
been fit by a linear function for each Z and the slope values, which
correspond to the value 4$a_{sym}$/T in Eq.(4), have been extracted.
In Fig. 4 the extracted values of $a_{sym}$/T are plotted as a function of Z
and shown by solid circles. A clear trend is observed for the parameter,
$a_{sym}$/T. The value increases from 4 to 14 as Z increases from 4 to 15. The
value for Z=3 is much larger than Z=4. This is partially caused by the isotope
distribution. Since 5Li is unstable and decays before arriving to the
detector, $\bar{A}$ deviates from the actual centroid of the isotope
distribution. An attempt has been made to fit the distribution by a Gaussian
function and determine $\bar{A}$ as the centroid value. This procedure makes
$a_{sym}$/T around 3 for Z=3, but the uncertainty is significant, especially
for neutron deficient systems. Therefore in the plot the experimental
Z/$\bar{A}$ value appears without correction.
Figure 3: $\alpha(Z)$ values as a function of $\Delta(Z/A)^{2}$ for Z=6
(upper) and Z=12 (lower). The lines are the results of a linear fit. Figure 4:
Experimental $a_{sym}(\rho,T)/T$ values extracted from Eq.(4) are shown by
solid circles as a function of Z. Open circles show calculated
$a_{sym}(\rho,T)/T$ values for the fragments yields in AMD-Gemini
calculations. Squares are experimental results of $\zeta(Z)$ discussed in the
section IV.
## IV. SYMMETRY ENERGY AND VARIANCE OF THE ISOTOPE DISTRIBUTION
The multiplicity distributions of the isotopes for a given Z show a quadratic
distribution when they are plotted on a logarithmic scale. Since the symmetry
energy term is the only term proportional to (N-Z)2 in the free energy, this
suggests that the variance of the distributions is closely related to the
symmetry energy coefficient. In order to explore the relation between the
symmetry energy term in the free energy and the variance of the isotope
distribution, Ono et al. introduced a generalized function K(N,Z) for the free
energy in ref. Ono04_1 as given below.
$K(N,Z)=\sum_{i=1}^{n}w_{i}(N,Z)[-lnY_{i}(N,Z)+\alpha_{i}(Z)N+\gamma_{i}(Z)]$
(5)
Here i represents each reaction. The summation is taken over i for the
different N/Z reaction systems in order to get isotope multiplicity
distribution in a wide range from proton rich to neutron rich isotopes. The
average weights, wi(N,Z), are determined by minimizing the statistical errors
in K(N,Z) for a given (N,Z). The isoscaling parameter, $\alpha_{i}(Z)$, is the
isoscaling parameter value evaluated in the previous section. For each Z the
parameters, $\gamma_{i}$(Z), are determined by optimizing the agreement of the
quantities [-lnY(N,Z) + $\alpha_{i}(Z)$ \+ $\gamma_{i}$(Z)] from different
reactions. A typical K(N,Z) distribution from the experiment is shown in Fig.
5. The isotope distributions for a given Z exhibit a smooth quadratic
distribution and they can be fit by a function:
$K(N,Z)=\xi(Z)N+\eta(Z)+\zeta(Z)(N-Z)^{2}/A$ (6)
Where $\xi(Z)$, $\eta$(Z), $\zeta$(Z) are the fitting parameters. As one can
see the functional form, $\zeta$(Z) is related to the symmetry energy
coefficient given in Eq.(4) as:
$\zeta(Z)=a_{sym}/T$ (7)
In Fig. 4, the values of $\zeta$(Z) extracted using this technique are shown
by solid squares. The values are generally about 1 or 2 units smaller than the
$a_{sym}$ /T values evaluated in the previous section (solid circles), but the
general trend is in good agreement. The difference in the extracted values for
Z=3 originate from the different determination of the average value of A. In
the analysis in this section the average masses of these isotopes are
determined by the centroid of the quadratic distributions.
Figure 5: K(N,Z) distribution from five reaction systems, 64Ni+58Ni,
64Ni,112Sn, 197Au, 232Th.
## V. COMPARISONS WITH MODEL SIMULATIONS
The experimentally detected fragments are the final products of the reaction.
Excited primary fragments will have cooled down by statistical decay before
they arrive in the detector. Excitation energies of the primary fragments have
been evaluated experimentally by measuring the light charged particle
multiplicities in coincidence with the fragments Marie98 ; Hudan03 . Typical
excitation energies of 2 to 3 MeV/nucleon have been derived. In order to study
the effect of the secondary decay process on the experimentally extracted
ratio, $a_{sym}$/T, the simulation codes of an Antisymmetrized Molecular
Dynamics (AMD) model Ono96 ; Ono99 ; Hudan06 and a statistical decay code,
Gemini Charity88 , have been used. These codes have often been used to study
the fragment production in Fermi energy heavy ion reactions and the global
features of the experimental results have been well reproduced Ono02 ; Wada98
; Ono04_2 ; Wada00 ; Wada04 ; Hudan06 . Since the AMD calculation requires a
lot of CPU time, only two of the experimental reaction systems have been
studied. The systems examined are 64Zn+112Sn and 64Ni+124Sn at 40 AMeV. All
calculations shown in this paper have been performed in a newly installed
computer cluster in the Cyclotron Institute Wada09 . In order to obtain yields
of the final products, the deexcitation of primary fragments formed at 300
fm/c was followed using the Gemini code until they cooled to the ground state.
Using the same analysis described in the section III, the scaling parameters,
$\alpha(Z)$ and $\beta(Z)$ were then extracted from the simulated events as a
function of Z. The average mass number of the isotopes for a given Z was also
evaluated from the calculations. The calculated variation of $a_{sym}$/T is
shown by open circles in Fig. 4. The values are typically one to two units
higher than the experimental values (open circles) but exhibit an essentially
identical trend. For the primary fragments at t = 300 fm/c the same analysis
has been made. These fragments are identified using a coalescence technique in
phase space. The evaluated symmetry coefficients using $R_{c}$ = 5 are shown
in Fig. 6. $R_{c}$=5 corresponds to a radius of 5 fm in configuration space.
For the primary fragments the calculated values are close to the experimental
values observed for Z $\leq$6 but, in contrast to the experimental results,
they remain more or less constant for all Z. The rather flat distribution of
$a_{sym}$/T values over the entire range of Z is consistent with the picture
of the origin of the primary fragments from a common emitting source with a
given density and temperature. These comparisons indicate that the $a_{sym}$/T
values are significantly modified by the secondary process for Z $>$ 4
Figure 6: Comparisons between the experimentally extracted symmetry energy
coefficient to temperature ratio and those from AMD. Solid circles show the
experimental values determined from Eq.(4) and open circles show results from
of AMD-Gemini calculations. Closed squares indicate results for the primary
fragments with Rc=5.
## VI. SUMMARY
The symmetry energy coefficient to temperature ratio, $a_{sym}$/T , as a
function of Z has been extracted from experimental data in two different ways,
namely from isoscaling parameters and from the variance of the observed
isotope distributions. The results from the two techniques are in reasonable
agreement. Experimental values increase from $\sim$ 4 to 12 as Z increases
from 4 to 15. The values and trends observed for the final fragments are well
reproduced by the AMD plus Gemini model simulations. However these values are
significantly different from the values extracted for the primary fragments,
especially for Z $>$ 4, suggesting that the derivation of the observed ratio
of symmetry energy coefficient to temperature is significantly perturbed by
the secondary decay processes.
###### Acknowledgements.
We thank the staff of the Texas A$\&$M Cyclotron facility for their support
during the experiment. We thank L. Sobotka for letting us to use his spherical
scattering chamber. We also thank A. Ono and R. Charity for letting us to use
their calculation codes. This work is supported by the U.S. Department of
Energy under Grant No. DE-FG03-93ER40773 and the Robert A. Welch Foundation
under Grant A0330. One of us(Z. Chen) also thanks the “100 Persons Project” of
the Chinese Academy of Sciences for the support.
## References
* (1) R. W. Minich et al., Phys. Lett. B118, 458 (1982).
* (2) A. Bonasera et al., Phys. Rev. Lett. 101, 122702 (2008).
* (3) H. S. Xu et al., Phys. Rev. Lett. 85, 4, (2000).
* (4) M. B. Tsang et al., Phys. Rev. C64, 054615 (2001).
* (5) A.S. Botvina, O. V. Lozhkin and W. Trautmann, Phys. Rev. C65, 044610 (2002).
* (6) A. Ono et al., Phys. Rec. C68, 051601(R) (2003).
* (7) S. Albergo et al., Nuovo Cimento, A89, 1 (1985).
* (8) L. G. Moretto, C. O. Dorso, J. B. Elliott, and L. Phair, Phys. Rev. C77, 037603 (2008).
* (9) A. Ono, et al., Phys. Rev. C70, 041604(R) (2004).
* (10) A. Keli$\acute{c}$, J. B. Natowitz and K. H.Schmidt, Eur. Phys. J.A30,203 (2006).
* (11) N. Marie et al., Phys. Rev. C58, 256 (1998).
* (12) S. Hudan et al., Phys. Rev. C76, 064613 (2003)
* (13) M.Huang et al., arXiv:1001.3621 [nucl-ex] 22Jan2010.
* (14) R.Wada et al., annual report of the Cyclotron Institute, Texas A&M University, (2005), II-3, unpublished.
* (15) F. Hubert, R. Bimbot and H. Gauvin, At. Data Nucl. Data Tables 46, 1 (1990).
* (16) A. Ono and H. Horiuchi, Phys. Rev C53, 2958 (1996).
* (17) A. Ono, Phys Rev C59, 853 (1999)
* (18) A. Ono and H. Horiuchi, Prog. Part. Nucl. Phys. 53, 501 (2004).
* (19) R. J. Charity et al., Nucl. Phys. A483, 371 (1988).
* (20) A. Ono, et al., Phys. Rev. C66, 014603 (2002).
* (21) R. Wada et al., Phys. Lett. B422, 6, (1998).
* (22) R. Wada et al., Phys. Rev. C62, 034601 (2000).
* (23) R. Wada et al., Phys. Rev. C69, 044610 (2004)
* (24) S. Hudan, R. T. de Souza and A. Ono, Phys. Rev. C73, 054602 (2006).
* (25) R.Wada et al., annual report of the Cyclotron Institute, Texas A&M University, (2009), unpublished.
|
arxiv-papers
| 2010-02-01T19:50:27 |
2024-09-04T02:49:08.138758
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z. Chen, S. Kowalski, M. Huang, R. Wada, T. Keutgen, K. Hagel, J.Wang,\n L. Qin, J.B. Natowitz, T. Materna, P.K. Sahu, and A. Bonasera",
"submitter": "Meirong Huang",
"url": "https://arxiv.org/abs/1002.0319"
}
|
1002.0556
|
# UCAC3: Astrometric Reductions
Charlie T. Finch, Norbert Zacharias, Gary L. Wycoff finch@usno.navy.mil U.S.
Naval Observatory, Washington DC 20392–5420
###### Abstract
Presented here are the details of the astrometric reductions from the $x,y$
data to mean Right Ascension (RA), Declination (Dec) coordinates of the third
U.S. Naval Observatory (USNO) CCD Astrograph Catalog (UCAC3). For these new
reductions we used over 216,000 CCD exposures. The Two-Micron All-Sky Survey
(2MASS) data are extensively used to probe for coordinate and coma-like
systematic errors in UCAC data mainly caused by the poor charge transfer
efficiency (CTE) of the 4K CCD. Errors up to about 200 mas have been corrected
using complex look-up tables handling multiple dependencies derived from the
residuals. Similarly, field distortions and sub-pixel phase errors have also
been evaluated using the residuals with respect to 2MASS. The overall
magnitude equation is derived from UCAC calibration field observations alone,
independent of external catalogs. Systematic errors of positions at UCAC
observing epoch as presented in UCAC3 are better corrected than in the
previous catalogs for most stars. The Tycho-2 catalog is used to obtain final
positions on the International Celestial Reference Frame (ICRF). Residuals of
the Tycho-2 reference stars show a small magnitude equation (depending on
declination zone) that might be inherent in the Tycho-2 catalog.
astrometry — catalogs — methods: data analysis
## 1 INTRODUCTION
The U.S. Naval Observatory (USNO) CCD Astrograph Catalog (UCAC) project began
observations in the Southern Hemisphere at Cerro Tololo Interamerican
Observatory (CTIO) in January 1998. In October 2000 the first U.S. Naval
Observatory CCD Astrograph catalog (UCAC1) (Zacharias et al., 2000) was
published covering about 80% of the southern sky with positions and
preliminary proper motions for over 27 million stars. The Astrograph was moved
to the Naval Observatory Flagstaff Station (NOFS) in October 2001 to complete
the Northern Hemisphere observing after 2/3 of the sky were completed from
CTIO. The second USNO CCD Astrograph catalog (UCAC2) (Zacharias et al., 2004)
was released in July 2003 with the same level of completeness as in UCAC1, but
with improved reduction techniques, early epoch plates for improved proper
motions and extended sky coverage. All sky coverage for UCAC observations were
completed in October 2004.
The third USNO CCD Astrograph Catalog (UCAC3) (Zacharias et al., 2010) is the
first all-sky data release in the UCAC series, containing about 100 million
entries with a slightly fainter limiting magnitude than in previous versions.
The magnitude range for UCAC3 is roughly 8.0 to 16.3 mag in the UCAC bandpass
(579-642 nm, hereafter UCAC magnitude). A detailed introduction into UCAC3
with comparisons to other catalogs and warnings for the users are given in
(Zacharias et al., 2010). Any user is also urged to read the extensive
“readme” file provided with the DVD or on-line release.
The UCAC3 is based on a complete re-reduction of the pixel data aiming at more
completeness with the inclusion of double star fitting, problem case
investigations and the slightly deeper limiting magnitude (Zacharias, 2010).
The final positions are based on the Tycho-2 (Høg et al., 2000) reference
frame as in UCAC2. However, Two-Micron All Sky Survey (2MASS) (Skrutskie et
al., 2006) residuals are used to probe for systematic errors in astrometric
reductions as will be explained below.
In UCAC1 and UCAC2 it was shown that the 4K CCD in the astrograph camera has a
relatively poor charge transfer efficiency (CTE), leading to coma-like
systematic errors in the uncorrected stellar positions. The effect is seen
mainly in the $x$-axis (right ascension), which is the direction of the fast
readout of charge, while the $y$-axis (declination) shows a much smaller
effect. In UCAC1 a simple empirical approach was used to correct for this
effect with the basic assumption that the effect was linear along the $x$-axis
and no assumption for a dependency on magnitude. For UCAC2 the empirical
approach was extended with a more complex model as a function of $x,y$, and
instrumental magnitude derived from flip observations of calibration fields.
Flip observations are obtained by observing the same field with the telescope
on one side of the pier (east or west) then repeat with the telescope on the
other side. Thus two images of the same area in the sky are obtained which are
rotated by $180^{\circ}$ with respect to each other.
In both previous UCAC catalogs sub-pixel phase and field distortion errors
have also been investigated. For UCAC1 the pixel phase was modeled with an
empirical function showing an amplitude on the order of 12 mas resembling a
sine-function. For UCAC2 the pixel phase errors were investigated further and
found to also be a function of the full width at half maximum (FWHM) of the
stellar image profiles. The field distortion pattern was modeled in both
previous reductions by binning reference star residuals from individual frames
used in the reductions.
The new pixel data reductions up to $x,y$ data are described in (Zacharias,
2010). This paper describes the reductions following the $x,y$ data up to the
CCD-based mean RA, Dec positions. Early epoch data and procedures to derive
proper motions are presented in the UCAC3 release paper (Zacharias et al.,
2010), while details about an important part of the early epoch data, the
Southern Proper Motion (SPM) data will be given elsewhere (Girard et al.,
2010).
## 2 INPUT DATA
For the astrometric reductions of UCAC3 as described in this paper, two sets
of input data are needed: the $x,y$ data from selected CCD observations, and
reference stars. Here two different reference star catalogs are used for
different purposes.
### 2.1 CCD Frame Selection
Out of the about 278,000 UCAC frames ever taken, all applicable survey frames,
calibration field, and minor planet exposures are selected for the reductions
presented here. Poor quality frames are excluded. Quality criteria include
limiting magnitude, internal fit precision of high S/N stellar images, and
mean image elongation. About 15% of the observations are qualified as “poor”,
see also (Zacharias et al., 2000). All frames which pass those quality
criteria are included. Then the all-sky completeness from the selected data
are examined and frames which almost meet the quality criteria are included as
needed to provide a complete all-sky coverage.
The final UCAC3 catalog contains mainly the survey frame data. The minor
planet observations will be published separately, while the calibration field
observations are only used in the first reduction steps to derive corrections
to systematic errors. A summary of the CCD observations is given in Table 1.
The frames taken along the path of Pluto are included here from a
collaboration with L. Young (SWRI) for occultation predictions.
A total of about 50 fields in the sky observed at about 10 to 30 different
epochs with multiple exposures each time were used as astrometric
calibrations. These typically low galactic latitude fields around ICRF
targets, equatorial calibration fields, and open clusters were observed about
10 to 30 times during the project and with the telescope flipped between
orientations east and west of the pier to provide a reversal of RA, Dec
orientation in $x,y$ space. Data from these fields are utilized to derive
certain systematic error corrections (see below). However, these frames are
not included in the final UCAC3 reductions based on the Tycho-2 reference
catalog.
Frames around extragalactic link sources taken with the astrograph at times of
deep CCD observations (mostly with the KPNO and CTIO 0.9 m telescopes) are not
included here either. These data are still under investigation and a separate
paper is in preparation regarding the optical link to ICRF quasars.
### 2.2 $x,y$ Data
For all astrometric reductions described below only the $x,y$ results from
pixel data profile fit model five (symmetric Lorentz profile with pre-set
$\alpha,\beta$ shape parameters) are used. This fit model has five free
parameters per single star image (background, amplitude, center $x,y$, width
of profile), similar to the more familiar two-dimensional Gauss model.
However, the Lorentz profile matches the observed point-spread function (PSF)
of our data significantly better than the Gaussian model. For more details and
explicit PSF model functions see the UCAC3 pixel reduction paper (Zacharias,
2010).
Double star fits of blended images are based on the same Lorentz image profile
model, however three more free parameters are used for the two center
coordinates and the amplitude, respectively, of the secondary component. Both
the primary and secondary component are fit simultaneously. A single width of
the profiles for both components of a double star and a single background
level parameter is used in this fit.
The $x,y$ data files also contain internal errors on the center coordinates
derived from the least-square fits, two instrumental magnitudes based on the
profile fit model and a real aperture photometry, respectively, and several
auxiliary flags from the raw pixel reduction step. For more information about
the raw data reductions see the UCAC3 pixel reduction paper (Zacharias, 2010).
### 2.3 Reference Stars
The 2MASS catalog does not provide proper motions. However, the UCAC and 2MASS
observations were made almost at the same epoch and the 2MASS positions are of
high quality with random errors of about 70 mas per coordinate for well
exposed stars and small systematic errors (Zacharias et al., 2006). Due to the
deep limiting magnitude and high density the 2MASS catalog is an excellent
tool to probe UCAC data for systematic errors on a statistical basis, allowing
to stack up many residuals as a function of a large number of parameters, as
will be described below. Table 2 lists the number of available observations of
reference stars, each providing a residual along RA ($x$) and Dec ($y$).
For this purpose a subset of the 2MASS point source catalog is constructed.
Stars are selected from the Naval Observatory Merged Astrometric Dataset
(NOMAD) using the 2MASS identifier and imposing a limit of V or R $\leq$ 16.5
to select 116,247,341 2MASS stars. For these, the original 2MASS positions are
retrieved and matched with UCAC observations on a frame-by-frame basis.
Proper motions of this reference star catalog are taken out of the NOMAD
catalog. Even if these are not very accurate for faint stars, they bridge the
few years of epoch difference to UCAC data to avoid large-scale biases from
galactic dynamics. For most stars in the southern hemisphere, the epoch
difference between 2MASS and UCAC is $\sim\pm$ 1 yr, while it gradually
increases for stars at higher declinations. Near the north celestial pole the
epoch difference is about four years.
The 2MASS data are used only to derive most of the systematic error
corrections of the UCAC $x,y$ data. As with UCAC2, the Tycho-2 catalog is used
as the only source for reference stars in a final astrometric reduction to
obtain UCAC3 positions on the International Celestial Reference Frame (ICRF).
Only stars from the Tycho-2 catalog indicated as having good astrometry have
been used for this reduction. Tycho-2 proper motions have been used to
propagate the positions to the epoch of the individual UCAC $x,y$
observations.
Both reference star catalogs are sorted by declination and stored in binary
direct access files. For each catalog a match to the $x,y$ data are performed
to produce cross-reference output files per CCD frame containing the record
numbers from the $x,y$ direct access data and the reference star catalog, as
well as the magnitude of the star, B$-$V color, and astrometry flag (for
Tycho-2). This scheme allows a fast runtime for the many passes through all
the data to perform the astrometric reductions without the need for a match of
reference stars each time.
## 3 ASTROMETRIC REDUCTIONS
For the astrometric reductions in UCAC3 extensive systematic error
investigations are performed. The largest systematic error is caused by the
poor charge transfer efficiency (CTE) of the detector, followed by geometric
field angle distortions. An iterative approach was adopted utilizing 2MASS
residuals to determine each of these effects in turn as described below. The
idea here is to use an empirical approach as done in the past, but investigate
further dependencies to better model the effects seen in the 4K CCD pixel
data. Preliminary results were presented earlier (Finch et al., 2009). A pure
magnitude equation is derived from internal calibration observations only.
Finally the position shifts as function of the sub-pixel location of an image
in the focal plane (the sub-pixel phase error) is derived, however, the $x,y$
data used for that are the original measures before applying the other
corrections.
Systematic position offsets as a function of color and differential color
refraction due to Earth’s atmosphere are small (about 5 mas) due to the narrow
UCAC spectral bandpass. No such corrections are applied for the UCAC3 catalog.
### 3.1 Preliminary Field Distortion Pattern
In the first step of the astrometric reductions, the CTE caused systematic
errors (coma-like) are approximated by a linear model as a function of
magnitude and $x$ pixel coordinate, similar to the CTE modeling of UCAC1. This
takes out about 80% of the CTE effect. The pre-corrected $x,y$ data are then
used in a preliminary astrometric reduction with 2MASS reference stars to
produce an approximate field distortion pattern (FDP), i.e. systematic errors
purely based on the $x,y$ location of a stellar image on the area of the
detector. Although the coma-like corrections for the CTE effect should not
bias the purely geometric FDP corrections, the scatter in the data is largely
reduced by correcting for most of the CTE effect first. This leads to a more
accurate FDP as would be generated without applying any CTE corrections.
Maximal systematic errors in the FDP are about 24 mas, with typical
corrections in the 5 to 10 mas range. This preliminary FDP is then applied to
the otherwise uncorrected $x,y$ data to analyze the CTE effect from scratch in
the following step. The same reasoning applies here; with good FDP correction
in hand the scatter of the residuals is reduced to accurately probe the
systematic errors induced by the poor CTE.
### 3.2 CTE Effect
When the CCD reads out an image, the electrons are transferred from
pixel$-$to$-$pixel until the charge reaches the output register. The low CTE
of the 4K CCD causes charge to be left behind as this transfer occurs, leading
to slightly asymmetric images. The amount of asymmetry and the derived $x,y$
center of stellar images with respect to the unaffected position depend on the
$x$ pixel location, the brightness of the star, the length of the exposure,
and other factors.
As found here and in previous UCAC reductions the CTE effect is by far the
most substantial systematic error seen in the raw UCAC $x,y$ data amounting up
to about 200 mas. The effect is predominantly seen in the $x$-axis along right
ascension and increases from no effect near $x=0$ pixel to a maximum near
$x=4094$ pixel, as evident in Figure 1. A similar effect is also seen in the
$y$-axis, but to a lesser degree because of a slower charge transfer in the
direction of declination.
For the reductions, the UCAC frames are separated into individual data sets
depending on observation site (CTIO or NOFS) and telescope orientation (east
or west). Over 216,000 frames are used for the reductions, see Table 1. Most
frames at CTIO were observed with the telescope west of the pier, while at
NOFS the regular observing was performed east of the pier.
Because the instrument had to be disassembled for the move from CTIO to NOFS
the camera alignment and other instrument properties are slightly different at
both sites. This explains the slightly different patterns for systematic error
corrections found for the two sites. However, the 2MASS reference star catalog
is dense enough to provide a sufficient number of residuals for statistically
significant results even on relatively small sub-sets of the UCAC data.
For UCAC3 we found that the CTE systematics show a dependence on FWHM,
brightness of the star, exposure time, location on the chip ($x,y$), camera
orientation (east, west) and observing site (CTIO, NOFS). For example as
Figures 1 and 2 show the CTE caused different systematic position offsets for
the $x$ coordinate as a function of magnitude for 20 and 150 sec exposures,
respectively.
The residuals with respect to the 2MASS reference stars are split into two
major data sets for investigating the CTE effect, CTIO west and NOFS east. A
plotting program is then used to display and determine empirical corrections
to create a complex look-up table. The table is split up into four different
FWHM bins keeping a roughly equal number of frames per bin, with half step
magnitude bins ranging from 8 to 16 magnitude for all standard exposure times
of 20$-$200 seconds duration, and 5 bins along the $x,y$ axes. Each data set
is evaluated and look-up tables are created to correct for the residuals, (see
sample, Table 3).
For each of the main data sets, CTIO west and NOFS east, tables are created
separately for the $x$ and $y$ axes. After testing we found that these tables
could also be used for the data of the other configurations, (CTIO east and
NOFS west) corresponding to the observation site. After applying corrections
and re-running the residuals the correction tables are continuously updated
until the residuals flattened out. The largest correction from the residuals
for CTIO is 204 mas in the $x$-axis and $-$32 mas in the $y$-axis. For NOFS
the largest correction from the residuals is 216 mas in the $x$-axis and $-$67
mas in the $y$-axis.
### 3.3 Pure Magnitude Equation
The 2MASS positions are uncorrelated to the $x,y$ pixel coordinates of the
UCAC observations. Thus any systematic errors seen in the 2MASS$-$UCAC
residuals as a function of UCAC $x,y$ and also as a function of magnitude
times these coordinates (coma terms) are inherent in the UCAC data and can be
corrected with the above procedure.
However, this is not the case for a pure magnitude dependent systematic error,
which could originate in either or both catalogs. With the systematic error
corrections applied to the UCAC $x,y$ data as described above any possible
pure magnitude equation from 2MASS was transferred into the UCAC positions.
Different catalogs have typically different magnitude equations. As an example
Figure 3 shows the residuals as function of magnitude from a reduction of UCAC
data with Tycho-2 reference stars, after applying all systematic error
corrections based on the 2MASS reductions. This clearly shows a difference in
magnitude equation between 2MASS (= UCAC system at this point) and Tycho-2
without knowing what the error free positions might be.
The flip observations of calibration fields are used to determine the overall,
pure magnitude dependent systematic errors in the UCAC data independent of any
external reference star catalog. With these observations the same field in the
sky has been observed with sets of frames rotated by 180∘ with respect to each
other. Any $x,y$ coordinate offset as function of magnitude shows up in the
residuals of a transformation of east versus west frames $x,y$ data. We
derived overall magnitude equation slope terms for long and short exposure
frames and both sites separately. These are then applied globally in the final
astrometric reductions.
After applying all corrections a small magnitude term of a few mas/mag is
still seen in the residuals of the Tycho-2 reductions of UCAC data as shown in
the UCAC3 release paper (Zacharias et al., 2010). This indicates such a
magnitude equation is present in the Tycho-2 catalog itself, which is found to
vary as a function of declination zone as one would expect.
It would be preferable to derive all systematic error corrections from
internal calibration observations. However, the flip observations alone do not
allow us to do so because of the degeneracy between a pure magnitude equation
and coma-like terms. Only after correcting the UCAC $x,y$ data for coordinate
dependent (including coma) terms do the flip observations allow for a unique
solution of the magnitude equation. Of course the assumption here is a
constant magnitude equation, not changing from exposure to exposure. This
assumption can not easily be made for photographic astrometry, which is
affected by a highly non-linear detector, but should hold better for CCD data.
### 3.4 Field Distortions
Field distortion patterns (FDPs) are derived for UCAC3 by binning thousands of
reference star residuals of individual CCD frames using the same procedure as
in the previous UCAC reductions. These reductions are performed on the CTE
corrected data but without applying the preliminary FDP used before.
From deriving FDPs of various subsets of the data we found that the FDP is
almost constant except for a small difference depending on observing site
(CTIO or NOFS). Residuals are created using all good survey frames with
respect to 2MASS reference stars split into CTIO west and NOFS east data sets.
The data using opposite camera orientations than used for most of the
observations (i.e. CTIO east and NOFS west) did not have significantly
different field distortions so only two correction tables are used for the
final reductions. FDP corrections for frames taken with the camera orientation
not used frequently are applied by rotating the FDP of the data with the large
amount of observations at that site.
Figure 4 (top) shows the FDP for CTIO west with vectors up to 23 mas in
length. The FDP for NOFS east is slightly different with vectors up to 24 mas
at some bins. In Figure 4 (bottom), we show the difference (CTIO west$-$NOFS
east) of the two data sets. Although these differences are small (below 6 mas)
they are systematic and well determined. Therefore we choose to use a separate
FDP map to correct the data of each site.
### 3.5 Subpixel Phase Errors
After the above mentioned systematic errors have been corrected the 2MASS
reference star catalog is used again to generate residuals from all applicable
UCAC survey frames. Residuals are analyzed as a function of the original $x$
and $y$ pixel coordinate fraction (sub-pixel phase) before other corrections
are applied. Various sub-sets of the data are looked at. Systematic errors are
found to be a function of the FWHM of the image profiles. Figure 5 gives some
examples. The results are found to be independent of exposure time, as
expected. However, a slight difference between the CTIO and NOFS data is
found.
The amplitude of the sub-pixel phase dependent systematic errors in the star
positions is shown in Figure 6, here for the corrections to the $x$
coordinate; those for the $y$ coordinate are somewhat smaller. All sub-pixel
phase systematic corrections are smaller than what was found in UCAC2. This is
a consequence of using an image profile model in UCAC3 (Lorentz profile) which
better fits the true PSF than was the case for UCAC2 (Gauss model).
However, the function of the sub-pixel phase systematic errors are more
complex in the UCAC3 than UCAC2 data, where a simple sine and cosine term were
sufficient. For the UCAC3 data we had to expand to three sine and three cosine
terms in order to fit the sub-pixel phase errors sufficiently well (Figure 5).
These six parameters are determined separately for 12 sets of data binned by
FWHM (from 1.5 to 3.0 pixels), and split by NOFS and CTIO data. Calibration
tables are then generated for equal steps along FWHM by interpolation and
$x,y$ corrections applied, separately for each coordinate, based on these
tables.
## 4 MEAN POSITIONS
Positions of all detected objects are obtained frame-by-frame from a final
astrometric reduction with the Tycho-2 reference star catalog and correcting
raw $x,y$ data first for the sub-pixel phase errors, then for systematic
errors as a function of $x,y$ (FDP), then for mixed terms of coordinate with
magnitude (CTE effect), and finally for a pure magnitude equation, as
explained above. Apparent places and refraction are corrected rigorously using
the Software for Analyzing Astrometric CCD (SAAC) code (Winter, 1999), which
also utilizes the Naval Observatory Vector Astrometry Subroutines (NOVAS) code
(Kaplan, 1989)111$http://aa.usno.navy.mil/software/novas/novas\\_info.php$.
The thus obtained positions are on the ICRF at individual epoch of each CCD
frame (between 1998 and 2004) and are output to FPOS (final position) files.
Previously identified and flagged observations of minor planets and high
proper motion stars are output to separate files. All other individual
positions are output by declination zones and then sorted by declination.
Weighted mean positions are calculated from the individual images of each
star, generating a running star number, MPOS (mean position file) on the fly.
Over 139 million objects are identified at this step.
All MPOS entries are then matched with early epoch star catalogs and another,
more comprehensive 2MASS extract containing about 338 million objects. These
2MASS stars are selected directly from the 2MASS point source catalog without
going through NOMAD. The R magnitude was estimated based on the 2MASS near-
infrared J$-$K color and stars with R $\leq$ 17.0 or J $\leq$ 15.5 are
selected. The unique identifier for stars matched across catalogs is the MPOS
star number. Individual early epoch positions are output together with MPOS
entries (CCD epoch observations) and sorted by MPOS number. Weighted proper
motions and mean positions are then calculated to obtain the UCAC3 release
catalog data (Zacharias et al., 2010). Objects which did not have either a
reasonable proper motion determined or could not be matched with 2MASS are
dropped at this point. Only these compiled catalog mean positions and proper
motions are published in UCAC3, not the MPOS or FPOS data, which likely will
be made available for the final UCAC4 release after further updates.
## 5 COMPARISON WITH HIPPARCOS
A total of 1510 Hipparcos stars in the 8 to 12 magnitude range are randomly
selected (about 300 all sky per magnitude interval) and flagged in the UCAC
data. These stars are not used as reference stars in test reductions using
Tycho-2 reference stars. Thus the obtained positions are field star positions
from UCAC observations independent of the Hipparcos and Tycho catalog
positions. Individual UCAC observed positions are then compared to the
original (ESA, 1997) and new Hipparcos reductions (van Leeuwen, 2007) at the
epoch of UCAC observations, using Hipparcos mean positions, proper motions and
parallaxes.
After excluding outliers ($\geq$ 200 mas position difference in either
coordinate), RMS values over observations in each bin are calculated (Table
4). Similarly sorting all observations by magnitude or color, respectively and
binning over 100 observations lead to the plots shown in Figure 7 and 8.
The expected position errors from UCAC3 and Hipparcos data at the epoch of our
UCAC observations are also presented in Table 4 together with the expected RMS
of the combined error and the ratio of expected to observed scatter,
separately for each coordinate. In all cases the observed errors (from the
scatter of the UCAC3$-$Hipparcos position differences) is slightly smaller
than the expected errors as calculated from the combined formal errors for the
same observations, thus at least some of these are overestimated (see also
discussion section below).
UCAC position differences of those sampled Hipparcos stars do not show
systematic errors as a function of magnitude or color exceeding about 10 mas
over the range sampled. Plots with respect to the original or new Hipparcos
catalog are almost identical.
However, 23 Hipparcos stars (1.5% of this sample) show very large differences
(between 300 and 600 mas in either coordinate) when comparing the new
reduction Hipparcos positions with the UCAC3 positions at UCAC epoch. A
similar number of stars is found when comparing with the original Hipparcos
Catalogue; however, for not exactly the same stars. All possible combinations
of inconsistencies between the two Hipparcos solutions and UCAC data are
found, with two of the three positions or all three separated by several
standard errors of their internal errors.
## 6 DISCUSSION
The use of an image profile model better matching the actual PSF than a
Gaussian model is essential for the astrometric reductions of blended images.
A better matching model also does reduce the amplitude of the sub-pixel phase
error and is advisable to be used when no such corrections are being applied
to the data. In particular, a comparison of Figure 6 with a similar figure of
the UCAC2 paper show that with a Gaussian model and 2.0 pixel/FWHM sampling
the pixel phase error has an amplitude of 11 mas, while with the image profile
model 5 as used in UCAC3 this amplitude is only about 6 mas.
The use of such a PSF profile model (still with the same number of fit
parameters per star as the traditional Gaussian model) allows to neglect
positional errors as function of sub-pixel phase completely for a sampling of
about 2.5 pixel/FWHM or larger without the need to investigate possible other
dependencies of this systematic error as a function of other things. However,
for single stars and with calibration data in hand to correct for the position
offsets caused by a sub-pixel phase dependency, the use of a more
sophisticated image profile model than a Gaussian might not have an apparent
advantage for astrometric reductions.
The slight difference in amplitude of the sub-pixel phase corrections between
CTIO and NOFS data is surprising. It could be caused by a slightly different,
observed PSF between the two sites, even for the same seeing (FWHM). Whether
this is caused by differences in the instrument, guiding or atmosphere is
currently not known.
The small systematic errors of UCAC based positions of randomly selected
Hipparcos stars confirms the good correction of UCAC3 epoch positions as
function of magnitude and color, at least for the 8 to 12 magnitude range.
With UCAC3 positions agreeing with Hipparcos data the magnitude equation seen
in residuals with respect to Tycho-2 is an indication for such small
systematic errors in the Tycho-2 catalog. These are likely introduced through
the proper motions, thus the early epoch, ground-based data, as also indicated
in the UCAC3 release paper.
The random errors in the observed UCAC$-$Hipparcos position differences are
even slightly smaller than the expected, combined formal errors. For the new
Hipparcos reductions the difference is only a few percent, while for the
comparison with the original Hipparcos Catalogue data the observed errors are
about 10% smaller than expected. This indicates a slightly overestimated error
in the original Hipparcos Catalogue proper motions. The formal position errors
for the individual UCAC observations do include the formal image profile fit
error, and the conventional plate adjustment error propagation. The weighting
scheme used in this individual CCD frame least-square adjustments also include
an estimated error contribution from the turbulence in the atmosphere, scaled
by the exposure time. The mismatch between the actual PSF and the image
profile model can lead to an overestimation of the center position errors,
particularly for stars as bright as this sample of Hipparcos stars, which
would explain the slightly smaller than expected scatter in the Hipparcos to
UCAC position differences. The exclusion of outliers at an arbitrary limit of
200 mas could be another possible explanation. At the faint end of Hipparcos
(11th magnitude) the Hipparcos catalog positions are of comparable precision
to typical mean UCAC positions (based on 4 images) at their about 2000 epoch.
The next step after UCAC, the USNO Robotic Astrometric Telescope (URAT)
program (Zacharias, 2008) to begin in 2010 thus will likely be capable of
improving proper motions of individual Hipparcos stars significantly.
The entire UCAC team is thanked for making this all-sky survey a reality. For
more detailed information about “who is who” in the UCAC project the reader is
referred to the readme file and UCAC3 release paper. The California Institute
of Technology is acknowledged for the pgplot software. More information about
this project is available at
http://www.usno.navy.mil/usno/astrometry/.
## References
* ESA (1997) The Hipparcos Catalogue, European Space Agency (ESA) 1997, publication SP1200
* Finch et al. (2009) Finch, C. T., Zacharias, N., Girard, T., Wycoff, G., & Zacharias, M. I. 2009, Bulletin of the American Astronomical Society, 41, 427
* Girard et al. (2010) Girard, T. et al. 2010, paper about SPM4 (in preparation)
* Høg et al. (2000) Høg, E., et al. 2000, A&A, 355, L27
* Kaplan (1989) Kaplan, G. 1989, AJ, 97, 1197,
* Skrutskie et al. (2006) Skrutskie, M. F., et al. 2006, AJ, 131, 1163
* van Leeuwen (2007) van Leeuwen, F. 2007, Astrophysics and Space Science Library, Vol. 350, Springer, New York
* Winter (1999) Winter, L. 1999, Ph.D. thesis, Hamburg Observatory, Germany
* Zacharias et al. (2000) Zacharias, N., Zacharias, M. I., & Rafferty, T. J. 2000, AJ, 118, 2503 (UCAC1 paper)
* Zacharias et al. (2004) Zacharias, N., Urban, S. E., Zacharias, M. I., Wycoff, G. L., Hall, D. M., Monet, D. G., & Rafferty, T. J. 2004, AJ, 127, 3043 (UCAC2 paper)
* Zacharias et al. (2006) Zacharias, N., McCallon, H. L., Kopan, E., & Cutri, R. M. 2006, in Proceedings of JD16 of IAU GA 2003, Eds. R.Gaume, D.McCarthy, J.Souchay, USNO Washington DC
* Zacharias (2008) Zacharias, N. 2008, in Proc. IAU Symp. 248, Eds. Wenjing Jin, Imants Platais, & Michael A.C. Perryman, Cambridge Univ.Press p.310
* Zacharias et al. (2010) Zacharias, N. et al. submitted to AJ (UCAC3 release paper)
* Zacharias (2010) Zacharias, N. 2010, in preparation for AJ (UCAC3 pixel red.paper)
Figure 1: CTIO west residuals in $x$ with respect to 2MASS reference stars as
a function of UCAC model magnitude using frames taken at 20 second exposures
(short). The top plot shows the residuals for low $x$ near pixel 1 and the
bottom plot for high $x$ near pixel 4094. Each dot represents the mean over
1000 residuals. Figure 2: CTIO west residuals in $x$ with respect to 2MASS
reference stars as a function of UCAC model magnitude using frames taken at
150 second exposures (long). The top plot shows the residuals for low $x$ near
pixel 1 and the bottom plot for high $x$ near pixel 4094. Each dot represents
the mean over 1000 residuals. Figure 3: Residuals in $x$ (top) and $y$
(bottom) for CTIO west with respect to Tycho-2 reference stars as a function
of UCAC model magnitude. Each dot represents the mean over 3000 residuals.
Figure 4: Field distortion pattern (FDP) plot of 2MASS stacked residuals for
CTIO west (top) and the difference between vectors of CTIO west and NOFS east
(bottom). The scaling of the vectors is 10,000 which makes the largest
corrections for CTIO west (top) 23 mas and the largest difference vector
(bottom) 6 mas. Figure 5: CTIO west residuals in $x$ with respect to 2MASS
reference stars as a function of sub-pixel phase. The top and bottom plot show
residuals with an average FWHM of 1.54 and 2.11 pixel respectively. Each dot
represents the mean over 5000 residuals. The fitted curve is from a least-
squares adjustment using a model with a total of six Fourier terms. Figure 6:
Amplitude of the sub-pixel phase dependent positional correction as a function
of image profile width (FWHM) for the CCD astrograph frames taken at CTIO
(filled) and NOFS (open circles), respectively. Figure 7: Position differences
UCAC$-$Hipparcos (new reductions) as a function of V magnitude of a random
sample of Hipparcos stars reduced as field stars in UCAC processing. Each dot
is the mean of 100 individual UCAC observations. Figure 8: Position
differences UCAC$-$Hipparcos (new reductions) as a function of B$-$V color of
a random sample of Hipparcos stars reduced as field stars in UCAC processing.
Each dot is the mean of 100 individual UCAC observations.
Table 1: Summary of UCAC frames aaFirst row number represents the total number of UCAC frames while the second row number gives the number of frames used in the final UCAC3 reduction. Site/orientation | number of | number of | number of | number of
---|---|---|---|---
| Calibration Frames | Survey Frames | Minor Planet Frames | Pluto Frames
CTIO east | 1582 | 5 | 14 | 0
| 0 | 3 | 14 | 0
CTIO west | 1583 | 163460 | 828 | 10
| 0 | 155143 | 796 | 0
NOFS east | 2452 | 66940 | 1340 | 84
| 0 | 58523 | 1156 | 74
NOFS west | 1525 | 2580 | 32 | 0
| 0 | 2397 | 28 | 0
Total | 7142 | 232985 | 2214 | 94
| 0 | 216066 | 2068 | 74
Table 2: Number of reference star observations used for reductions | number of | number of
---|---|---
site/orientation | 2MASS star | Tycho-2 star
| observations | observations
CTIO east | 8772812 | 1600
CTIO west | 203915168 | 9683015
NOFS east | 58493251 | 4071859
NOFS west | 7856501 | 186557
Total | 279037732 | 13943031
Table 3: Example CTIO west CTE lookup table for frames taken at 20 second exposures with corrections given in mas | 8.0 | 8.5 | 9.0 | 9.5 | 10.0 | 10.5 | 11.0 | 11.5 | 12.0 | 12.5 | 13.0 | 13.5 | 14.0 | 14.5 | 15.0 | 15.5 | 16.0
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag | mag
$xbin1$ | 147 | 148 | 143 | 134 | 119 | 108 | 95 | 81 | 66 | 49 | 33 | 18 | -8 | -44 | -69 | -76 | -86
$xbin2$ | 116 | 111 | 115 | 111 | 105 | 90 | 80 | 69 | 53 | 42 | 27 | 18 | -4 | -32 | -52 | -59 | -74
$xbin3$ | 101 | 92 | 95 | 85 | 76 | 67 | 60 | 44 | 41 | 33 | 24 | 14 | 2 | -17 | -31 | -39 | -48
$xbin4$ | 68 | 61 | 61 | 59 | 56 | 43 | 38 | 33 | 29 | 23 | 19 | 11 | 2 | -10 | -17 | -21 | -30
$xbin5$ | 52 | 47 | 42 | 41 | 43 | 37 | 32 | 28 | 25 | 23 | 17 | 13 | 11 | 3 | 1 | 0 | -10
Table 4: RMS position differences and expected errors of observed UCAC3 $-$ Hipparcos positions. | | difference | Hipparcos err | UCAC3 err | combined err | ratio
---|---|---|---|---|---|---
number | Vmag range | RA Dec | RA Dec | RA Dec | RA Dec | RA Dec
observ. | mag | mas mas | mas mas | mas mas | mas mas | diff/c.err
Hipparcos (original)
423 | 08.0 $-$08.5 | 41.9046.3 | 10.7008.8 | 46.9047.1 | 48.1047.9 | 0.8700.97
717 | 08.5 $-$09.0 | 45.2041.9 | 12.7011.7 | 47.8048.0 | 49.4049.4 | 0.9200.85
802 | 09.0 $-$ 10.0 | 43.6044.6 | 21.8018.3 | 47.3047.6 | 52.1051.0 | 0.8400.88
1227 | 10.0 $-$ 11.0 | 45.6045.5 | 28.3022.5 | 43.3043.9 | 51.7049.3 | 0.8800.92
978 | 11.0 $-$ 99.0 | 52.8048.2 | 42.7035.7 | 42.2042.7 | 60.0055.6 | 0.8800.87
Hipparcos (new reduction)
423 | 08.0 $-$08.5 | 42.3045.8 | 08.9007.3 | 46.9047.1 | 47.7047.6 | 0.8900.96
722 | 08.5 $-$09.0 | 46.4044.2 | 12.0010.0 | 47.8048.0 | 49.3049.1 | 0.9400.90
801 | 09.0 $-$ 10.0 | 45.3042.6 | 14.0011.6 | 47.3047.5 | 49.3048.9 | 0.9200.87
1238 | 10.0 $-$ 11.0 | 47.4044.1 | 20.8016.9 | 43.4043.9 | 48.1047.1 | 0.9900.94
993 | 11.0 $-$ 99.0 | 52.3049.9 | 36.7027.5 | 42.2042.8 | 56.0050.9 | 0.9300.98
|
arxiv-papers
| 2010-02-02T18:01:12 |
2024-09-04T02:49:08.149866
|
{
"license": "Public Domain",
"authors": "Charlie T. Finch, Norbert Zacharias, Gary L. Wycoff",
"submitter": "Charlie Finch",
"url": "https://arxiv.org/abs/1002.0556"
}
|
1002.0567
|
# A New Approximation to the Normal Distribution Quantile Function
Paul M. Voutier
###### Abstract
We present a new approximation to the normal distribution quantile function.
It has a similar form to the approximation of Beasley and Springer [3],
providing a maximum absolute error of less than $2.5\cdot 10^{-5}$. This is
less accurate than [3], but still sufficient for many applications. However it
is faster than [3]. This is its primary benefit, which can be crucial to many
applications, including in financial markets.
## 1 Introduction
The use of the inverse of the CDF for a probability distribution, also known
as the quantile function, is widespread in statistical modelling (see, for
example, [5, 7]).
During recent work, the need arose for a fast and reasonably accurate
approximation to the normal distribution quantile function, $N^{-1}(x)$.
Accuracy similar to the approximation in Equation 26.2.23 of [1] was
sufficient (max absolute error less than $4.5\cdot 10^{-4}$). But speed was
crucial.
The approximation of Beasley and Springer [3], along with related
approximations such as Acklam’s [2], provides improvements in terms of both
accuracy and speed.
Both the Acklam and the Beasley-Springer approximations are based on the same
ideas:
(1) consider narrow tails separately from a wide central area
(2) use a rational function of $x$ to approximate $N^{-1}(x)$ in this wide
central area (avoiding expensive operations like $\log$ and sqrt)
(3) take advantage of the fact that $N^{-1}(x-1/2)$ is an odd function.
The second and third ideas suggest that for the central region, we consider
rational approximations of the form
$(x-1/2)F((x-1/2)^{2}),$
where $F$ is a rational function. The approximations of Acklam, Beasley-
Springer, and others for the central region are of this form.
The Beasley-Springer approximation for the central region is sometimes called
a $(3,4)$ scheme, since the numerator of $F$ is cubic in $(x-1/2)^{2}$ and the
denominator of $F$ is of degree $4$ in $(x-1/2)^{2}$. Similarly, the Acklam
approximation is called a $(5,5)$ scheme.
## 2 New Approximations
For increased speed, here we consider a $(2,2)$ scheme for the central region
and a $(3,2)$ scheme for the tails.
We chose the boundaries between the central region and the tails to be at
$0.0465$ and $0.9535$, since with the above schemes and boundaries the maximum
absolute error in both regions was nearly the same and both slightly less than
$2.5\cdot 10^{-5}$.
### 2.1 Central Region
#### 2.1.1 $0.0465\leq p\leq 0.9535$
Put $q=p-0.5$ and let $r=q^{2}$. For $0.0465\leq p\leq 0.9535$, define
$f_{central}(p)=q\frac{\displaystyle a_{2}r^{2}+a_{1}r+a_{0}}{\displaystyle
r^{2}+b_{1}r+b_{0}}=q\left(a_{2}+\frac{\displaystyle
a_{1}^{\prime}r+a_{0}^{\prime}}{\displaystyle r^{2}+b_{1}r+b_{0}}\right)$
where
$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle 0.389422403767615,$
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle-1.699385796345221,$
$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle 1.246899760652504,$
$\displaystyle a_{0}^{\prime}$ $\displaystyle=$ $\displaystyle
0.195740115269792,$ $\displaystyle a_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-0.652871358365296,$ $\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle 0.155331081623168,$ $\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle-0.839293158122257.$
The benefit of the second expression is that we save one multiplication by
using it. Similarly, normalising the denominator so that the leading
coefficient is $1$, rather than the constant coefficient as some authors do,
also saves another multiplication.
There are 12 points of maximum error (also known as alternating points) in the
interval $[0.0465,0.9535]$:
$(p,err_{abs})$ | $(p,err_{abs})$
---|---
$(0.046500,2.494327\cdot 10^{-5})$ | $(0.592289,2.494326\cdot 10^{-5})$
$(0.054264,2.494331\cdot 10^{-5})$ | $(0.752182,2.494327\cdot 10^{-5})$
$(0.081621,2.494328\cdot 10^{-5})$ | $(0.859308,2.494323\cdot 10^{-5})$
$(0.140694,2.494323\cdot 10^{-5})$ | $(0.918381,2.494328\cdot 10^{-5})$
$(0.247820,2.494327\cdot 10^{-5})$ | $(0.945738,2.494331\cdot 10^{-5})$
$(0.407712,2.494326\cdot 10^{-5})$ | $(0.945350,2.494327\cdot 10^{-5})$
From the theorems of Chebyshev and de la Vallée Poussin (see [4, Section
5.5]), it follows that $f_{central}(p)$ is essentially the best possible
rational approximation of $(2,2)$ scheme.
For comparison, the maximum absolute error of the “central” approximation in
[3] is under $1.85\cdot 10^{-9}$.
This approximation was found using the minimax function within the numapprox
package of Maple:
Digits:=60:with(numapprox):
uBnd:=0.4535^2:
minimax(x->inverseCDFCentralRatApprox(x),0..uBnd,[2,2],x->sqrt(x));
where
inverseCDFCentralRatApprox(x) is the function $N^{-1}(\sqrt{x}+1/2)/\sqrt{x}$,
uBnd is the range we want the approximation over,
$[2,2]$ specifies that we want the degree of both the numerator and the
denominator to be $2$, and
$\sqrt{x}$ is the weight function we use, since we want to get the best
approximation to $N^{-1}(\sqrt{x}+1/2)$ rather than
$N^{-1}(\sqrt{x}+1/2)/\sqrt{x}$.
We tried other values of uBnd near $0.4535$, but the smallest maximum absolute
error was found with this particular value.
#### 2.1.2 $0.025\leq p\leq 0.975$
The use of an even wider central region may be preferred, as this can provide
further performance gains by reducing the expensive log and sqrt operations
required for the tails.
We give one such example here (found as above using Maple, but with
uBnd=0.475).
Put $q=p-0.5$ and let $r=q^{2}$. For $0.025\leq p\leq 0.975$, define
$f_{central}(p)=q\left(a_{2}+\frac{\displaystyle a_{1}r+a_{0}}{\displaystyle
r^{2}+b_{1}r+b_{0}}\right)$
where
$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle 0.151015505647689,$
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle-.5303572634357367,$
$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle 1.365020122861334,$
$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle 0.132089632343748,$
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle-.7607324991323768.$
The maximum absolute error for this approximation is less than $1.16\cdot
10^{-4}$ which occurs near $p=0.9692$. While this error is much larger than
the error in the previous section, it is still well smaller than the maximum
error for the Abramowitz-Stegun approximation ($4.5\cdot 10^{-4}$).
### 2.2 Tails
#### 2.2.1 $e^{-37^{2}/2}<p<0.0465$
For $5.3\ldots\cdot 10^{-298}=e^{-37^{2}/2}<p<0.0465$, put
$r=\sqrt{\log(1/p^{2})}$ and define
$f_{tail}(p)=\frac{c_{3}r^{3}+c_{2}r^{2}+c_{1}r+c_{0}}{r^{2}+d_{1}r+d_{0}}=c_{3}r+c_{2}^{\prime}+\frac{c_{1}^{\prime}r+c_{0}^{\prime}}{r^{2}+d_{1}r+d_{0}}.$
where
$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle 16.896201479841517652,$
$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle-2.793522347562718412,$
$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle-8.731478129786263127,$
$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle-1.000182518730158122,$
$\displaystyle c_{0}^{\prime}$ $\displaystyle=$ $\displaystyle
16.682320830719986527,$ $\displaystyle c_{1}^{\prime}$ $\displaystyle=$
$\displaystyle 4.120411523939115059,$ $\displaystyle c_{2}^{\prime}$
$\displaystyle=$ $\displaystyle 0.029814187308200211,$ $\displaystyle d_{0}$
$\displaystyle=$ $\displaystyle 7.173787663925508066,$ $\displaystyle d_{1}$
$\displaystyle=$ $\displaystyle 8.759693508958633869.$
As with the “central” approximation, this approximation was also found using
the minimax function within the numapprox package of Maple:
Digits:=60:with(numapprox):
v:=0.0465:
uBnd:=0.4535^2:
minimax(y->inverseCDF(exp(-y*y/2)), sqrt(log(1/v^2))..37, [3,2]);
Note that since we are approximating $N^{-1}(x)$ itself here, we do not
include a weight function in the arguments of the minimax function and so the
default weight function $1$ is used.
The maximum absolute error in this case is less than $2.458\cdot 10^{-5}$.
#### 2.2.2 $0.9535<p<1-e^{-37^{2}/2}$
Due to the symmetry of $N^{-1}(p)$ about $p=1/2$, we approximate $N^{-1}(p)$
by $-f_{tail}(1-p)$ (note that here $r=\sqrt{\log(1/(1-p)^{2})}$).
## 3 Abramowitz and Stegun Approximations
Having found the above new approximations, we turned our attention to the
approximations in Equations 26.2.22 and 26.2.23 of [1]. As those authors note,
these approximations are from [6]. In particular, Sheets 67 and 68 on pages
191–192 of [6].
If we restrict our attention to ranges like $e^{-37^{2}/2}<p<1-e^{-37^{2}/2}$
(this includes almost the entire IEEE-754 range of representable real
numbers), then we can improve on the approximations of Abramowitz and Stegun.
For example, in this range, we can replace Equation 26.2.23 of [1] with
$x_{p}=t-\frac{c_{2}t^{2}+c_{1}t+c_{0}}{d_{3}t^{3}+d_{2}t^{2}+d_{1}t+1}+\epsilon(p),$
where $|\epsilon(p)|<8\cdot 10^{-5}$ and
$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle 2.653962002601684482,$
$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle 1.561533700212080345,$
$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle 0.061146735765196993,$
$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle 1.904875182836498708,$
$\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle 0.454055536444233510,$
$\displaystyle d_{3}$ $\displaystyle=$ $\displaystyle 0.009547745327068945.$
This is over five times more accurate than the approximation in [1]. However,
as one increases the range even closer to $0$ and $1$, the max absolute
increases until we obtain Equation 26.2.23 of [1]. The near-best possible
nature of Equation 26.2.23 is illustrated by the graph in Sheet 68 of [6]
showing that Chebyshev’s theorem nearly holds for this approximation.
Note also that this approximation shows the justification for the use of
$\sqrt{\log(1/p^{2})}$ in these tail approximations. As $p\rightarrow 0$,
$N^{-1}(p)$ approaches $-\sqrt{\log(1/p^{2})}$ plus a quantity that approaches
$0$ as $p$ does.
## 4 Performance
Using Java (JDK $1.6.0\\_17$), we coded the following approximations in order
to compare their performance.
$\bullet$ the Abramowitz-Stegun approximation (AS in the table below)
$\bullet$ the Beasley-Springer approximation (BS in the table below)
$\bullet$ the approximation from Section 2 using the central region
approximation in Section 2.1.1 (Rat22A in the table below)
$\bullet$ the approximation from Section 2 using the central region
approximation in Section 2.1.2 (Rat22B in the table below).
In each case, we calculated the approximation 200,000 times for each $p$ from
$0.001$ to $0.999$ with $0.001$ as our step size. These calculations were done
on a Dell Inspiron 1525, running Windows Vista and using an Intel Core 2 Duo
T5800 2.00 GHz CPU. The times in milliseconds for each approximation are given
in the table below.
method | time(ms)
---|---
AS | 25,210
BS | 10,212
Rat22A | 8052
Rat22B | 6649
As one would expect, the new approximations given here are faster than the
currently known ones. The comparison between Rat22A and Rat22B is also
interesting, as it shows the impact of the calculation of the log and sqrt
operations. Although these operations only need to be performed for a small
subset of all values of $p$, reducing the number of these operations by just
under 50% reduced the CPU time required by nearly 20%.
## References
* [1] M. Abramowitz and I.E. Stegun (ed.), Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. National Bureau of Standards, Washington (1964).
* [2] P. J. Acklam, `http://home.online.no/~pjacklam/notes/invnorm/`.
* [3] J. D. Beasley and S. G. Springer, The percentage points of the Normal Distribution, Applied Statistics. 26 (1977), 118–121.
* [4] E. W. Cheney, Introduction to Approximation Theory, AMS Chelsea Publishing, Providence (1982).
* [5] W. G. Gilchrist, Statistical Modelling with Quantile Functions, Chapman & Hall, London (2000).
* [6] C. Hastings, Approximations for Digital Computers. Princeton University Press, Princeton (1955).
* [7] P. Jaeckel, Monte Carlo Methods in Finance. Wiley, Chichester (2002).
Paul Voutier
London, UK
paul.voutier@gmail.com
|
arxiv-papers
| 2010-02-02T20:31:52 |
2024-09-04T02:49:08.155355
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paul M. Voutier",
"submitter": "Paul Voutier",
"url": "https://arxiv.org/abs/1002.0567"
}
|
1002.0648
|
# Neutrino oscillations in Kerr-Newman space-time
Jun Ren1111Email: renjun@hebut.edu.cn and Cheng-Min Zhang2 1 School of
Science, Hebei University of Technology, 300130, Tianjin, China
2 National Astronomical Observatories, Chinese Academy of Sciences, 100012
Beijing, China
###### Abstract
The mass neutrino oscillation in Kerr-Newman(K-N) space-time is studied in the
plane $\theta=\theta_{0}$, and the general equations of oscillation phases are
given. The effect of the rotation and electric charge on the phase is
presented. Then, we consider three special cases: (1) The neutrinos travel
along the geodesics with the angular momentum $L=aE$ in the equatorial plane.
(2) The neutrinos travel along the geodesics with $L=0$ in the equatorial
plane. (3) The neutrinos travel along the radial geodesics at the direction
$\theta=0$. At last, we calculate the proper oscillation length in the K-N
space time. The effect of the gravitational field on the oscillation length is
embodied in the gravitational red shift factor. When the neutrino travels out
of the gravitational field, the blue shift of the oscillation length takes
place. We discussed the variation of the oscillation length influenced by the
gravitational field strength, the rotation $a^{2}$ and charge $Q$.
PACS number: 95.30.Sf, 14.60.Pq
Keywords: neutrino oscillation; Kerr-Newman space-time; oscillation length
## I Introduction
Mass neutrino mixing and oscillations were proposed by Pontecorvopon , and
Mikheyev, Smirnov and Wolfenstein (MSW for short) explored the effect of
transformation of one neutrino flavor into another in a medium with varying
densityMikheyev ; wolf . Recently, the consideration of the mass neutrino
oscillations has been a hot topic. There have been many theoreticalponte ;
bilenky2 ; nuss ; kay ; giunti3 ; rich ; kiers ; mann4 and experimentalfuk ;
apo ; egu ; ahn ; ahm ; mic ; aha studies about the neutrino oscillations.
Then, the neutrino oscillations in the flat space time were extended to the
cases in the curved space-timeahlu ; pir ; null ; fuller ; zhangcm ; zhangcm4
; pereira ; wang1 . Neutrino oscillation experiments were also considered to
test the equivalence principlemann3 . Calculating the phase along the geodesic
line will produce a factor 2 in the high energy limit, compared with the value
along the null line, which often exists in the flat and Schwarzschild space-
timezhangcm ; zhangcm4 ; 12 ; lipkin ; lipkin2 ; grossman . This issue of the
factor 2 is due to the difference between the time-like and null geodesics.
Furthermore, some alternative mechanisms have been proposed to account for the
gravitational effect on the neutrino oscillationalter ; mann1 ; mann2 . The
inertial effects on neutrino oscillations and neutrino oscillations in non-
inertial frames were also called attentionlamb1 ; lamb2 . As a further
theoretical exploration, neutrino oscillations in space-time with both
curvature and torsion19 ; 20 ; 21 have been studied.
In recent years, the researches about the neutrino oscillation have been made
new progress. A further mechanism to generate pulsar kicks, which was based on
the spin flavor conversion of neutrinos propagating in a gravitational field,
and the neutrino geometrical optics in gravitational field (in particular in a
Lense-Thirring background), have been proposed by Lambiaselamb3 ; lamb4 . Some
publications were centered on the theoretical study and experimental
measurement of the mixing angle $\theta_{13}$lyc ; rat ; eby . And CP
violation in neutrino oscillations were considered by some authorskli ; sch ;
gav ; alt . In addition, Cuesta and Lambiase studied the neutrino mass
spectrumlamb . Akhmedov, Maltoni and Smirnov presented the neutrino
oscillograms for different oscillation channels and discussed the effects of
non-vanishing 1-2 mixingsmir .
In this paper, we extend the mass neutrino oscillation work from Schwarzschild
space-time to Kerr-Newman space-time, since the Kerr-Newman metric is rather
important in black hole physics, where a most generally stationary solution
with axial symmetry has been existingruffini . For the reason of simplicity,
we confine our treatment in two generation neutrinos (electron and muon). We
give the general equations of the oscillation phases along the arbitrary null
and the time-like geodesics, respectively, in the equal $\theta$ plane,
$\theta=\theta_{0}$. The phase along the geodesic will also produce a factor
of $2$ in the K-N space-time, $\Phi(geod)=2\Phi(null)$, in the high energy
limit. In our derivation we have not assumed a weak field approximation.
We discuss three spacial cases. Firstly, the oscillation phases along the
geodesics with $L=aE$ are considered in the equatorial plane. $E$ is the
energy per unit mass of the particle. $L$ and $a$ are the angular momentum per
unit mass of the particle and K-N space-time, respectively. The geodesics with
$L=aE$ in K-N space-time play the same roles as the radial geodesics in the
Schwarzschild and in the Reissner-Nordstrom geometry. In this case, the phases
both along the null geodesic and the time-like geodesic are similar in form to
the phases along the radial geodesics (null and time-like) in the
Schwarzschild space-time. Secondly, we calculate the oscillation phases along
the geodesics with $L=0$ in the equatorial plane. This kind of geodesics is
also important in K-N space-time. In the Schwarzschild space-time with non-
rotating spherically symmetry, particles with $L=0$ can propagate along the
radial geodesics. But in K-N space-time, because of dragging effect, the
coordinate $\varphi$ must change if a particle with $L=0$ travels along the
geodesics. Thirdly, the phases along the radial geodesics at the direction
$\theta=0$ are given. Only at the poles $\theta=0$ and $\theta=\pi$, the
ergosphere coincides with the event horizon. At the direction $\theta=0$, the
effects of the rotation of the space-time on the oscillation length are found
to be more than those in the other directions.
At last, we calculate the proper oscillation length in the K-N space time. The
oscillation length is proportional to the local energy (local measurement),
$E^{loc}=E/\sqrt{g_{00}}$, of the neutrino, where $E$ is a constant along the
geodesic. The decrease in the local energy leads to the decrease in the
oscillation length as the neutrino travels out of the gravitational field. So,
the blue shift of the oscillation length occurs, which is unlike the case of
the gravitational red shift for light signal. In the equatorial plane in K-N
space-time, the rotation have no contribution to the oscillation length
because $g_{00}$ has nothing to do with the rotating parameter $a$ in this
plane. The rotation $a^{2}$ of the gravitational field shortens the
oscillation length in other equal $\theta$ plane, compared with the length in
R-N space time. We also give that the length varies according to $\theta$. And
charge $Q$ shortens it too, compared with the Kerr space-time case. But, the
gravitational field lengthens it, compared with the case in flat space-time.
In this paper, we take the neutrino as a spin-less particle to go along the
geodesic because the spin and the curvature coupling has a little contribution
to the geodesic derivationaud . Moreover, the neutrino is a high energy
particle, so we do not think the neutrino spin has more contribution to the
geodesic.
The paper is organized as the follows. In Sec.2, we briefly review the
standard treatment of neutrino oscillation in the flat space-time. In Sec.3,
we give the general expressions of the oscillation phases along the null and
time-like geodesics in arbitrary equal $\theta=\theta_{0}$ plane. In Sec.4, we
discuss the neutrino phase in three special cases. In Sec.5, we discuss the
proper oscillation length in K-N space-time. At last, the conclusion and
discussion are given. Throughout the paper, the units $G=c=\hbar=1$ and
$\eta_{\mu\nu}=diag(+1,-1,-1,-1)$ are used.
## II the standard treatment of neutrino oscillation in flat space-time
In a standard treatment, the flavor eigenstate $\mid\nu_{\alpha}\rangle$ is a
superposition of the mass eigenstates $\mid\nu_{k}\rangle$, i.e.null ; fuller
$|\nu_{\alpha}\rangle=\sum_{k}U_{\alpha k}exp[-i\Phi_{k}]|\nu_{k}\rangle,$ (1)
where
$\Phi_{k}=E_{k}t-\vec{p}_{k}\cdot\vec{x},(k=1,2),$ (2)
and the matrix elements $U_{\alpha k}$ comprise the transformation between the
flavor and mass bases. $E_{k}$ and $\vec{p}_{k}$ are the energy and momentum
of the mass eigenstates $\mid\nu_{k}\rangle$, and they are different for
different mass eigenstates. If the neutrino produced at a space-time point
$A(t_{A},\vec{x}_{A})$ and detected at $B(t_{B},\vec{x}_{B})$, the expression
for the phase in Eq.(2), which is coordinate independent and suitable for
application in a curved space-time, isnull ; stodolsky
$\Phi_{k}=\int_{A}^{B}p^{(k)}_{\mu}dx^{\mu},$ (3)
where
$p^{(k)}_{\mu}=m_{k}g_{\mu\nu}\frac{dx^{\nu}}{ds},$ (4)
is the canonical conjugate momentum to the coordinate $x^{\mu}$ and $m_{k}$ is
the rest mass corresponding to mass eigenstate $|\nu_{k}\rangle$. $g_{\mu\nu}$
and $s$ are metric tensor and an affine parameter, respectively.
The following assumptions are often applied in the standard treatmentponte :
(1) The mass eigenstates are taken to be the energy eigenstates, with a common
energy; (2) up to $O(m/E)$, there is the approximation $E\gg m$; (3) a
massless trajectory is assumed, which means that the neutrino travels along
the null trajectory. In the case of two neutrinos mixing $\nu_{e}-\nu_{\mu}$,
we can write
$\nu_{e}=cos\theta\nu_{1}+sin\theta\nu_{2},\nu_{\mu}=-sin\theta\nu_{1}+cos\theta\nu_{2}.$
(5)
Here $\theta$ is the vacuum mixing angle. The oscillation probability that the
neutrino produced as $|\nu_{e}\rangle$ is detected as $|\nu_{\mu}\rangle$
isboehm
$\emph{P}(\nu_{e}\rightarrow\nu_{\mu})=|\langle\nu_{e}|\nu_{\mu}(x,t)\rangle|^{2}=sin^{2}(2\theta)sin^{2}(\frac{\Phi_{kj}}{2}),$
(6)
where, $\Phi_{kj}=\Phi_{k}-\Phi_{j}$, is the phase shift. From the standard
treatment of the neutrino oscillationnull ; fuller ; zhangcm , the standard
result for the phase is
$\Phi_{k}\simeq\frac{m^{2}_{k}}{2E_{0}}|\vec{x}_{B}-\vec{x}_{A}|.$ (7)
Here $E_{0}$ is the energy for a massless neutrino. So, the phase shift
responsible for the oscillation is given by
$\Phi_{kj}\simeq\frac{\Delta m^{2}_{kj}}{2E_{0}}|\vec{x}_{B}-\vec{x}_{A}|,$
(8)
where $\Delta m^{2}_{kj}=m^{2}_{k}-m^{2}_{j}$.
## III neutrino oscillation phase along the null and the time-like geodesic
in the plane $\theta=\theta_{0}$
In this section, we study the neutrino oscillation in equal
$\theta=\theta_{0}$ surface. The line element of K-N space time takes the form
$ds^{2}=g_{00}dt^{2}+g_{11}dr^{2}+g_{22}d\theta^{2}+g_{33}d\varphi^{2}+2g_{03}dtd\varphi.$
(9)
The relevant components of the canonical momentum of the $k^{th}$ massive
neutrino in Eq.(4) are
$\displaystyle p_{t}^{(k)}$ $\displaystyle=$ $\displaystyle
p_{0}^{(k)}=m_{k}g_{00}\dot{t}+m_{k}g_{03}\dot{\varphi};$ $\displaystyle
p_{r}^{(k)}$ $\displaystyle=$ $\displaystyle m_{k}g_{11}\dot{r};$
$\displaystyle p_{\varphi}^{(k)}$ $\displaystyle=$ $\displaystyle
m_{k}g_{33}\dot{\varphi}+m_{k}g_{03}\dot{t},$ (10)
where
$\dot{t}=\frac{dt}{ds},\dot{r}=\frac{dr}{ds},\dot{\varphi}=\frac{d\varphi}{ds}$.
Because the metric tensor components do not depend on the coordinate $t$ and
$\varphi$, their canonical momenta $p_{t}^{(k)}$ and $p_{\varphi}^{(k)}$ are
constant along the trajectory. In fact, the momentum $p_{0}^{(k)}$conjugate to
$t$ is the asymptotic energy of the neutrino at $r=\infty$. It is stressed
that it is the covariant energy $p_{0}$ (not $p^{0}$) the constant of motion.
Otherwise the ambiguous definition of the energy will lead to the confusion in
understanding the neutrino oscillation.
The phase along the null geodesic from point A to point B is given by null ;
zhangcm ; stodolsky
$\displaystyle\Phi_{k}^{null}$ $\displaystyle=$
$\displaystyle\int_{A}^{B}p_{\mu}^{(k)}dx^{\mu}=\int_{A}^{B}(p_{0}^{(k)}dt+p_{\varphi}^{(k)}d\varphi+p_{r}^{(k)}dr)$
(11) $\displaystyle=$
$\displaystyle\int_{A}^{B}(p_{0}^{(k)}\frac{dt}{dr}+p_{\varphi}^{(k)}\frac{d\varphi}{dr}+p_{r}^{(k)})dr.$
We can obtain the following relations which are useful in the calculation
$\displaystyle g^{00}$ $\displaystyle=$ $\displaystyle-\frac{g_{33}}{\Delta
sin^{2}\theta},g^{33}=-\frac{g_{00}}{\Delta sin^{2}\theta},$ $\displaystyle
g^{03}$ $\displaystyle=$ $\displaystyle\frac{g_{03}}{\Delta
sin^{2}\theta},g^{2}_{03}-g_{00}g_{33}=\Delta sin^{2}\theta,$ (12)
where $\Delta=r^{2}-2Mr+a^{2}+Q^{2}$. Solving the equation (10) for $\dot{t}$
and $\dot{\varphi}$, we obtain
$\displaystyle\dot{t}=-\frac{g_{33}E_{k}+g_{03}L_{k}}{\Delta
sin^{2}\theta_{0}},\dot{\varphi}=\frac{g_{03}E_{k}+g_{00}L_{k}}{\Delta
sin^{2}\theta_{0}},$ (13)
where $E_{k}=\frac{p_{0}^{(k)}}{m_{k}}$ and
$L_{k}=-\frac{{}_{p_{\varphi}^{(k)}}}{m_{k}}$ are the energy and angular
momentum per unit mass, respectively.
In the standard treatment of the neutrino oscillation, the neutrino is usually
supposed to travel along the nullnull ; fuller ; ponte ; zuber ; boehm ;
bahcall . Following the standard treatment, we will calculate the phase along
the light-ray trajectory from $A$ to $B$.
The lagrangian appropriate to motions in the plane (for which $\dot{\theta}=0$
and $\theta=$ a constant $=\theta_{0}$) ischand
$\mathscr{L}=\frac{1}{2}(g_{00}\dot{t}^{2}+2g_{03}\dot{t}\dot{\varphi}+g_{11}\dot{r}+g_{33}\dot{\varphi}^{2}).$
(14)
The Hamiltonian is given by
$\mathscr{H}=E_{k}\dot{t}-L_{k}\dot{\varphi}+\frac{p_{r}^{(k)}}{m_{k}}\dot{r}-\mathscr{L}.$
(15)
Because of the independence of the Hamiltonian on $t$, we can deduce that
$2\mathscr{H}=E_{k}\dot{t}-L_{k}\dot{\varphi}+\frac{p_{r}^{(k)}}{m_{k}}\dot{r}=\delta_{1}=constant.$
(16)
Without loss generality, we can set, $\delta_{1}=1$ for time-like geodesics,
$\delta_{1}=0$ for null geodesics. Substituting (13) into (16) and setting
$\delta_{1}=0$ for null geodesics, we have the radial equation
$g_{11}\dot{r}^{2}=\frac{g_{33}E^{2}_{k}+2g_{03}E_{k}L_{k}+g_{00}L^{2}_{k}}{sin^{2}\theta_{0}\Delta}.$
(17)
We define a new function
$V(r)=g_{33}E^{2}_{k}+2g_{03}E_{k}L_{k}+g_{00}L^{2}_{k}.$ (18)
The different $V(r)$ determines the phase of the different trajectory. From
(17), we get
$\dot{r}=\frac{\sqrt{-V}}{\rho sin\theta_{0}},$ (19)
where $\rho^{2}=r^{2}+a^{2}cos^{2}\theta_{0}$. So, the equations governing $t$
and $\varphi$ are
$\frac{dt}{dr}=-\frac{\rho(g_{33}E_{k}+g_{03}L_{k})}{\Delta
sin\theta_{0}\sqrt{-V}},\frac{d\varphi}{dr}=\frac{\rho(g_{03}E_{k}+g_{00}L_{k})}{\Delta
sin\theta_{0}\sqrt{-V}}.$ (20)
The mass-shell condition isnull
$m^{2}_{k}=g_{\mu\nu}p^{(k)\mu}p^{(k)\nu}=p_{\mu}^{(k)}p^{(k)\mu}=p_{0}^{(k)}p^{(k)0}+p_{\varphi}^{(k)}p^{(k)\varphi}+p_{r}^{(k)}p^{(k)r}.$
(21)
Substituting
$p^{(k)0}=g^{00}p_{0}^{(k)}+g^{03}p_{\varphi}^{(k)},p^{(k)\varphi}=g^{33}p_{\varphi}^{(k)}+g^{30}p_{0}^{(k)}$
and (10) into the equation of the mass-sell condition(21), we obtain
$p^{(k)r}=\frac{m_{k}\sqrt{-V-sin^{2}\theta_{0}\Delta}}{\rho sin\theta_{0}}.$
(22)
In the process of calculation, the relations (12) are used. Applying the
relativistic condition $p_{0}^{k}\gg m_{k}$, we have the relation
$p^{(k)r}\simeq\frac{m_{k}}{\rho
sin\theta_{0}}(\sqrt{-V}-\frac{sin^{2}\theta_{0}\Delta}{2\sqrt{-V}}).$ (23)
Adopting (20) and $p^{(k)r}$, the phase along the null geodesics (11) is
approximated by
$\Phi^{null}_{k}\simeq\int_{A}^{B}\frac{m_{k}\rho
sin\theta_{0}dr}{2\sqrt{-V}}.$ (24)
The phase (24) is a general result. The different function $V(r)$ corresponds
to the different motion and determines the different phase consequently. If
$a=0$, we can obtain the oscillation phase in the Reissner-Nordstorm space-
time; if $Q=0$, the Kerr space-time case is given. If $a=0,Q=0$, the function
$V(r)$ reduces to
$V(r)=-r^{2}sin^{2}\theta_{0}E^{2}_{k}.$ (25)
The phase (24) becomes to
$\Phi^{null}_{k}=\int_{A}^{B}\frac{m^{2}_{k}}{2p_{0}^{(k)}}dr=\frac{m^{2}_{k}}{2p_{0}^{(k)}}(r_{B}-r_{A}).$
(26)
This is just the phase in Schwarzschild space-timefuller ; null .
The velocity of an extremely relativistic neutrino is nearly the speed of
light. In the standard treatment, the neutrino is supposed to travel along the
null lineponte ; null ; fuller ; zuber ; boehm ; bahcall . Despite of this,
the propagation difference between a massive neutrino and a photon can have
important consequences and this tiny derivation becomes important for the
understanding of the neutrino oscillation. Therefore, for more general
situations, we start to calculate the phase along the time-like geodesic. The
factor of $2$ will be obtained, when compared the time-like geodesic phase
with the null geodesic phase in the high energy limit. The classical orbit is
defined to a planenull ; zhangcm , $\theta=\theta_{0}$, $d\theta=0$. The phase
along the time-like geodesic iszhangcm ; 12 ; ahlu ; stodolsky
$\Phi_{k}^{geod}=\int_{A}^{B}p_{\mu}^{(k)}dx^{\mu}=\int_{A}^{B}(p_{0}^{(k)}\frac{dt}{dr}+p_{\varphi}^{(k)}\frac{d\varphi}{dr}+p_{r}^{(k)})dr.$
(27)
For time-like geodesic, $\delta_{1}=1$, equation (16) becomes
$E_{k}\dot{t}-L_{k}\dot{\varphi}+\frac{p_{r}^{(k)}}{m_{k}}\dot{r}=1,$ (28)
while the equations for $\dot{t}$ and $\dot{\varphi}$ (13) are the same for
time-like geodesicschand . Substituting $\dot{t}$ and $\dot{\varphi}$, we have
$\frac{ds}{dr}=\frac{1}{\dot{r}}=\frac{\sqrt{-g_{11}}}{\sqrt{-1-\frac{V}{\Delta
sin^{2}\theta_{0}}}}.$ (29)
So, we obtain the equations for $dt/dr$ and $d\varphi/dr$ for time-like
geodesics
$\displaystyle\frac{dt}{dr}=-\frac{\sqrt{-g_{11}}}{\Delta\sin^{2}\theta_{0}}\frac{(g_{33}E_{k}+g_{03}L_{k})}{\sqrt{-1-\frac{V}{\Delta
sin^{2}\theta_{0}}}},\frac{d\varphi}{dr}=\frac{\sqrt{-g_{11}}}{\Delta\sin^{2}\theta_{0}}\frac{(g_{00}L_{k}+g_{03}E_{k})}{\sqrt{-1-\frac{V}{\Delta
sin^{2}\theta_{0}}}}.$ (30)
According to mass shell condition, $p_{r}^{(k)}$ is given by
$p_{r}^{(k)}=-m_{k}\sqrt{-g_{11}}\sqrt{-1-\frac{V}{\Delta\sin^{2}\theta_{0}}}.$
(31)
Thus, the phase along the time like geodesic is
$\Phi_{k}^{geod}=\int_{A}^{B}\frac{m_{k}\sqrt{-g_{11}}dr}{\sqrt{-1-\frac{V}{\Delta\sin^{2}\theta_{0}}}}.$
(32)
If the high energy limit is taken into account, Eq. (32) reduces to
$\Phi_{k}^{geod}\simeq\int_{A}^{B}\frac{m_{k}\rho\sin\theta_{0}dr}{\sqrt{-V}}=2\Phi_{k}^{null}.$
(33)
It is often noted that the factor $2$ of the neutrino phase calculations
exists in the flat space-timelipkin ; lipkin2 and in the Schwarzschild space-
timezhangcm ; zhangcm4 ; 12 , which is believed to be the difference between
the null geodesic and the time like geodesic. The neutrino phase induced by
the null condition, as in the standard treatment, comes from the 4-momentum
$p^{\mu}$ defined along the time-like geodesic, and the equation (17)
governing $\dot{r}$ to the null geodesic. If the 4-momentum defined along the
null geodesic was instead used to compute the null phase, we would obtain zero
because of the null condition. When we calculate the phase along the time-like
geodesic, $\dot{r}$ in (28) is defined to the time-like geodesic. It is the
difference producing the factor $2$. It can be proved that the neutrino phase
along the null is the half of the value along the time like geodesic in the
high energy limit in a general curved space-time(see APPENDIX A in
literaturezhangcm ).
## IV three special cases
### IV.1 Oscillation phases along the geodesics with $L_{k}=aE_{k}$ in the
equatorial plane
It is very important that the geodesic is described in the equatorial plane
$\theta=\pi/2$ in the K-N space-time. The geodesics with $L_{k}=aE_{k}$ play
the same roles as the radial geodesics in the Schwarzschild and in the
Reissner-Nordstrom geometry. In this case, for null geodesic $\dot{t}$,
$\dot{\varphi}$ and $\dot{r}$ reduce to
$\dot{t}=\frac{r^{2}+a^{2}}{\Delta}E_{k};\dot{\varphi}=\frac{a}{\Delta}E_{k};\dot{r}=\pm
E_{k}.$ (34)
These equations in fact define the shear-free null-congruences which we use
for constructing a null basis for a description of the K-N space-time in a
Newman-Penrose formalismchand . The function $V(r)$ for null geodesic becomes
to,
$V(r)=-r^{2}E^{2}_{k}.$ (35)
So, the phase along the null is
$\Phi^{null}_{k}\simeq\int_{A}^{B}\frac{m_{k}\rho
sin\theta_{0}dr}{2\sqrt{-V}}=\int_{A}^{B}\frac{m_{k}}{2E_{k}}dr=\frac{m^{2}_{k}}{2p_{0}^{(k)}}(r_{B}-r_{A}),$
(36)
which appears the same form as that of the Schwarzschild space-time radial
oscillation case.
We now turn to a consideration of the time-like geodesic case. The equations
for $\dot{t},\dot{\varphi}$ are the same as for the null geodesics, while
$\dot{r}$ becomes to
$\dot{r}=\sqrt{E_{k}^{2}+\frac{1}{g_{11}}}.$ (37)
Substituting $L_{k}=aE_{k}$ into (32), we obtain the phase along the time-like
geodesic
$\Phi_{k}^{geod}=\int_{A}^{B}\frac{m_{k}dr}{[(\frac{p_{0}^{(k)}}{m_{k}})^{2}+\frac{1}{g_{11}}]^{1/2}}.$
(38)
Compared with the phase along the radial time-like geodesic in the
Schwarzschild space-timezhangcm ,
$\Phi_{k}^{geod}(Sch)=\int_{A}^{B}\frac{m_{k}dr}{\sqrt{(\frac{p_{0}^{(k)}}{m_{k}})^{2}-g_{00}}}=\int_{A}^{B}\frac{m_{k}dr}{\sqrt{(\frac{p_{0}^{(k)}}{m_{k}})^{2}+\frac{1}{g_{11}}}},$
(39)
we find that the oscillation phase with $L_{k}=aE_{k}$ in K-N space-time has
the similar form as the phase along the radial in Schwarzschild space-time.
Substituting $g_{11}=-\frac{r^{2}}{\Delta}$ into equation (38), we have
$\Phi_{k}^{geod}=\int_{A}^{B}\frac{m_{k}dr}{\sqrt{b+\frac{2M}{r}-\frac{a^{2}+Q^{2}}{r^{2}}}},$
(40)
where $b=(\frac{p_{0}^{(k)}}{m_{k}})^{2}-1$. Equation (40)can be integrated
directly to give
$\displaystyle\Phi_{k}^{geod}$ $\displaystyle=$
$\displaystyle\frac{m_{k}}{b}\sqrt{br^{2}_{B}+2Mr_{B}-a^{2}-Q^{2}}-\frac{m_{k}}{b}\sqrt{br^{2}_{A}+2Mr_{A}-a^{2}-Q^{2}}$
(41) $\displaystyle-$
$\displaystyle\frac{Mm_{k}}{b^{3/2}}ln\frac{br_{B}+M+\sqrt{b(br^{2}_{B}+2Mr_{B}-a^{2}-Q^{2})}}{br_{A}+M+\sqrt{b(br^{2}_{A}+2Mr_{A}-a^{2}-Q^{2})}}.$
Eq.(41) shows the effects of rotation $a^{2}$ on the oscillation phase.
If $a=0$, we can obtain $\dot{t},\dot{\varphi},\dot{r}$ along the radial null-
geodesics in the equatorial plane in Reissner-Nordstrom space-time
$\dot{t}=\frac{r^{2}}{r^{2}-2Mr+Q^{2}},\dot{\varphi}=0,\dot{r}=\pm E.$ (42)
Therefore, the phases along the radial null and time-like geodesic in
Reissner-Nordstrom space-time are given by, respectively
$\displaystyle\Phi^{null}_{k}(RN)$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{k}}{2p_{0}^{(k)}}(r_{B}-r_{A}),$
$\displaystyle\Phi_{k}^{geod}(RN)$ $\displaystyle=$
$\displaystyle\int_{A}^{B}\frac{m_{k}dr}{\sqrt{b+\frac{2M}{r}-\frac{Q^{2}}{r^{2}}}}.$
(43)
Letting $a=0$ in (41), the integral of equation (43) is given.
### IV.2 Oscillation phases along the geodesics with $L=0$ in the equatorial
plane
The geodesics with $L_{k}=0$ is another important class of geodesics in K-N
space-time. If the coordinate $t$ and $\varphi$ has a relation
$d\varphi/dt=-g_{03}/g_{33}$, the canonical momentum $p_{\varphi}^{(k)}$ in
(10) vanishes. The corresponding $\dot{t}$, $\dot{\varphi}$ and $\dot{r}$ for
null geodesic are
$\dot{t}=-\frac{g_{33}}{\Delta}E_{k};\dot{\varphi}=\frac{g_{03}}{\Delta}E_{k};\dot{r}=\frac{\sqrt{-g_{33}}}{r}E_{k}.$
(44)
And $\dot{r}$ for time-like geodesics is
$\dot{r}=\sqrt{\frac{1+g_{33}E^{2}_{k}/\Delta}{g_{11}}}.$ (45)
Substituting $L_{k}=0$ into (24) and (32), the phases along the null and time-
like geodesic are given by, respectively
$\displaystyle\Phi^{null}_{k}$ $\displaystyle=$
$\displaystyle\int_{A}^{B}\frac{m^{2}_{k}}{2p_{0}^{(k)}}\frac{rdr}{\sqrt{-g_{33}}}=\int_{A}^{B}\frac{m^{2}_{k}}{2p_{0}^{(k)}}\sqrt{-g_{11}\widetilde{g_{00}}}dr,$
(46) $\displaystyle\Phi_{k}^{geod}$ $\displaystyle=$
$\displaystyle\int_{A}^{B}\frac{\sqrt{-g_{11}\widetilde{g_{00}}}m_{k}dr}{\sqrt{(\frac{p_{0}^{(k)}}{m_{k}})^{2}-\widetilde{g_{00}}}}.$
(47)
where $\widetilde{g_{00}}=g_{00}-g_{03}^{2}/g_{33}$. It is difficult to
integrate (46) and (47) directly. We can work out them by expanding as $a^{2}$
when $a^{2}$ is a small quantity.
### IV.3 Oscillation phase along the radial geodesic at $\theta=0$
Unlike in the Schwarzschild and in the Reissner-Nordstrom space-time, the
event horizon does not coincide with the ergosphere where $g_{00}$ vanishes in
K-N space-time. This is an important feature which distinguishes the K-N
space-time from the others. The ergosphere that is a stationary limit surface
coincides with the event horizon only at the poles $\theta=0$ and
$\theta=\pi$. The phase along the null geodesic in the direction $\theta=0$
can be written as
$\displaystyle\Phi_{k}^{null}$ $\displaystyle=$
$\displaystyle\int_{A}^{B}\frac{m_{k}\rho\sin\theta_{0}dr}{2\sqrt{-V}}$ (48)
$\displaystyle=$
$\displaystyle\int_{A}^{B}\frac{m_{k}\rho\sin\theta_{0}dr}{2\sqrt{r^{2}+a^{2}+\frac{a^{2}}{\rho^{2}}(2Mr-Q^{2})\sin\theta^{2}_{0}}\sin\theta_{0}}.$
Substituting $\theta_{0}=0$, the equation (48) becomes
$\Phi_{k}^{null}=\int_{A}^{B}\frac{m^{2}_{k}}{2p_{0}^{(k)}}dr=\frac{m^{2}_{k}}{2p_{0}^{(k)}}(r_{B}-r_{A}).$
(49)
By similar calculation, the phase along the time-like geodesics at $\theta=0$
is given by
$\Phi_{k}^{geod}=\int_{A}^{B}\frac{m_{k}dr}{(b+\frac{2Mr-Q^{2}}{r^{2}+a^{2}})^{1/2}},$
(50)
where $b=(\frac{p_{0}^{(k)}}{m_{k}})^{2}-1$.
## V proper oscillation length
The propagation of a neutrino is over its proper distance , but $dr$ in (24)
is only a coordinate. The proper distance can be written aslandau
$dl=\sqrt{(\frac{g_{0\mu}g_{0\nu}}{g_{00}}-g_{\mu\nu})dx^{\mu}dx^{\nu}}.$ (51)
In K-N space-time, we have
$dl=\sqrt{-g_{11}dr^{2}+(\frac{g_{03}^{2}}{g_{00}}-g_{33})d\varphi^{2}}.$ (52)
Substituting $\frac{d\varphi}{dr}$, we obtain
$dr=\frac{\sqrt{-g_{00}V}}{\sqrt{\Delta}E_{k}sin\theta_{0}}dl.$ (53)
In order to discuss conveniently, we adopt the differential form of (24)
$d\Phi_{k}^{null}=\frac{m_{k}\rho sin\theta_{0}dr}{2\sqrt{-V}}.$ (54)
Substituting (53), we have
$d\Phi_{k}^{null}=\frac{m_{k}^{2}}{2p_{0}^{(k)}}\sqrt{g_{00}}dl.$ (55)
It is assumed that the mass eigenstates are taken to be the energy
eigenstates, with a common energy in the standard treatment. The equal energy
assumption is considered to be correct by some authorsgrossman ; lipkin ;
stodolsky2 and studied carefully in paperszhangcm4 ; giunti2 ; leo . In
addition, it is adopted widely in many literatures, for examplenull ; fuller ;
zhangcm ; giunti . $p_{0}$ will represent the common energy of different mass
eigenstates. In fact, the condition of equal momentum is also adopted to study
the neutrino oscillation. In the flat space-time, both conditions (the equal
energy and the equal momentum) present practically the same neutrino
oscillation resultszhangcm4 . There are conditions of time translation
invariance and space translation invariance in the flat space-time. So, energy
conservation and momentum conservation of a free particle are right. In the
curved (stationary) space-time, the energy of a particle is conserved along
the geodesic due to the existence of a time-like killing vector field.
However, the canonical conjugate momentum to $r$, $p_{r}$ is not conserved
because $(\frac{\partial}{\partial r})^{a}$ is not killing in the curved
(stationary) space-time. Consequently, it is very difficult to study neutrino
oscillation if the condition of equal momentum is adopted in curved space-
time. In this section, our discussion is on the base of the results in the
standard treatment which the phase is calculated along the null. Then, the
phase shift which determines the oscillation is
$d\Phi^{null}_{kj}=d\Phi^{null}_{k}-d\Phi^{null}_{j}=\frac{\Delta
m_{k}^{2}}{2p_{0}}\sqrt{g_{00}}dl,$ (56)
where $\Delta m^{2}_{kj}=m^{2}_{k}-m^{2}_{j}$. The equation (56) can be
rewritten as
$\frac{dl}{d(\frac{\Phi^{null}_{kj}}{2\pi})}=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{g_{00}}}=\frac{4\pi p_{0}^{loc}}{\Delta
m^{2}_{kj}}.$ (57)
The term $\frac{4\pi p_{0}}{\Delta m^{2}_{kj}\sqrt{g_{00}}}$ in (57) can be
interpreted as oscillation length $L_{OSC}$ (which is defined by the proper
distance as the phase shift $\Phi^{null}_{kj}$ changing $2\pi$) measured by
the observer at rest at a position $r$, and $p_{0}^{loc}=p_{0}/\sqrt{g_{00}}$
is the local energy. As $r\rightarrow\infty$, $p_{0}^{loc}$ approaches to the
energy $p_{0}$ measured by the observer at infinity. $\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}$ is the oscillation length in the flat space-time. Equation (57)
is universal significance in curved space-time. In fact, $\sqrt{g_{00}}$ is
the gravitational red shift factor which shows the effect of the gravitational
field on the oscillation length. Consider two static observers $O$ (the radial
coordinate $r$) and $O^{\prime}$ (the radial coordinate $r^{\prime}$). The
oscillation length measured by $O$ and by $O^{\prime}$ is, respectively
$L_{OSC}(r)=\frac{4\pi p_{0}}{\Delta m^{2}_{kj}}\frac{1}{\sqrt{g_{00}(r)}},\\\
L_{OSC}(r^{\prime})=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{g_{00}(r^{\prime})}}.$ (58)
We can obtain the relation
$\frac{L_{OSC}(r^{\prime})}{L_{OSC}(r)}=\frac{\sqrt{g_{00}(r)}}{\sqrt{g_{00}(r^{\prime})}}.$
(59)
If $r^{\prime}>r$, we have $L_{OSC}(r^{\prime})<L_{OSC}(r)$ and blue shift
occurs. Physically, the oscillation length is proportional to the local energy
of the neutrino. When the neutrino travels out of the gravitational field, the
local energy decreases. Consequently, the neutrino oscillation length
decreases and blue shift takes place. From equation (57), the oscillation
length increases in the gravitation field because of $0<g_{00}<1$ out of the
the infinite red shift surface. The effect of the gravitational blue shift on
the oscillation length may have the possible observable effect from
experiments. In the Schwarzschild space-time, $g_{00}=1-2M/r$, we have
$L_{OSC}(Sch)=\frac{4\pi p_{0}}{\Delta m^{2}_{kj}}\frac{1}{\sqrt{1-2M/r}}.$
(60)
In order to study the influence of Charge on the neutrino oscillation, we
consider the oscillation length in the Reissner-Nordstrom space-time
$L_{OSC}(RN)=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{g_{00}}}=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}}.$ (61)
Compared with the case in the Schwarzschild space-time, the oscillation length
decreases due to the influence of charge $Q$.
The metric component $g_{00}$ in the K-N space-time is
$g_{00}=1-\frac{2Mr-Q^{2}}{\rho^{2}},$ (62)
where $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$. In the equatorial plane, there is,
$g_{00}=1-2M/r+Q^{2}/r^{2}$, which is the same as $g_{00}$ in the Reissner-
Nordstrom space-time. Thus, it is concluded that the neutrino oscillation
length along the geodesics in the equatorial plane in the K-N space-time is
identical to that in the Reissner-Nordstrom space-time and the rotating
parameter $a^{2}$ does not work in this plane. Therefore, we have to select
other plane $\theta=\theta_{0}\neq\pi/2$ to highlight the effect of rotation
on the oscillation length. In the plane $\theta=\theta_{0}$, the oscillation
length can be written as
$L_{OSC}(K-N)=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{1-\frac{2Mr-Q^{2}}{r^{2}+a^{2}\cos^{2}\theta_{0}}}}$
(63)
It is obvious that the oscillation length decreases too because of the
rotation of the gravitational field compared with that in R-N space-time.
Letting $Q=0$ in (63), the oscillation length in Kerr space-time is given by
$L_{OSC}(Kerr)=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{1-\frac{2Mr}{r^{2}+a^{2}\cos^{2}\theta_{0}}}}.$ (64)
Comparing with (63), the charge $Q$ shortens the oscillation length. We can
obtain that the oscillation length varies with $\theta_{0}$ by
$\frac{d}{d\theta_{0}}L_{OSC}(K-N)=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}(g_{00})^{3/2}\frac{2Mr-Q^{2}}{(r^{2}+a^{2}\cos^{2}\theta_{0})^{2}}a^{2}\sin\theta_{0}\cos\theta_{0}.$
(65)
In K-N space-time, we conclude that the oscillation length increases with
$\theta$ within $0<\theta<\pi/2$, and it becomes maximum in the equatorial
plane. Then, it decreases with $\theta$ within $\pi/2<\theta<\pi$. At the
direction $\theta=0$ and $\theta=\pi$, the oscillation length occurs minimum,
$L_{OSC}(K-N)=\frac{4\pi p_{0}}{\Delta
m^{2}_{kj}}\frac{1}{\sqrt{1-\frac{2Mr-Q^{2}}{r^{2}+a^{2}}}}.$ (66)
In summary, the gravitational field lengthens oscillation length; both the
rotation $a^{2}$ and the charge $Q$ shorten the oscillation length.
## VI conclusion and discussion
In this paper, we have given the phase of mass neutrino propagating along the
null and the time like geodesic in the gravitational field of a rotating
symmetric and charged object, which is described by Kerr-Newman metric. Most
astrophysical bodies in universe have rotation and charge generally. Thus the
work about the neutrino oscillation in the K-N space time is important and
meaningful for the black hole astrophysics. We work out the general formula of
oscillation phase on the equal $\theta=\theta_{0}$ plane with the generality.
The phase along the geodesic is the double of that along the null in the high
energy limit, which is the same in the cases in flat and Schwarzschild space-
time. By setting $\theta=\pi/2$, the phases in the equatorial plane are given.
As $a=0$ or $Q=0$, we obtain the phases in the R-N space-time or in the Kerr
space-time. Moreover, we study three special cases in K-N space-time:
geodesics with $L=aE$; geodesics with $L=0$; radial geodesics at $\theta=0$.
Among them, the geodesics with $L=aE$ have the same importance as the radial
geodesics in the Schwarzschild and in the R-N geometry. The phases obtained
are very similar in form to the cases along the radial geodesics in the
Schwarzschild and in the R-N space-time.
In Sec.5, the proper oscillation length in the K-N space time is studied in
detail. We find that oscillation length in curved space-time is proportional
to the local energy, which is regraded as the neutrino ”climbs out of the
gravitational potential well”. So, the blue shift occurs. Then, the effects of
rotation and charge of the space-time on the oscillation length are given.
Because of the correction of the gravitation field, the oscillation length
increases, compared with the flat space time case. However, both the rotation
$a^{2}$ and the charge $Q$ shorten the oscillation length. It is noted, the
rotation has null contribution to the length in the equatorial plane in K-N
space-time, because red shift factor is independence of $a^{2}$ in this plane.
Finally, we remark that our result exists generality, which can be exploited
to study the neutrino oscillation near the rotating compact star, neutron star
or black hole.
## References
* (1) Pontecorvo B 1957 Mesonium and antimesonium _Zh. Eksp. Theor. Fiz._ 33 549
* (2) Mikheyev S P and Smirnov A Y 1986 Resonant amplification of oscillations in matter and solar-neutrino spectroscopy _Il Nuovo Cimento C_ 9 17
* (3) Wolfenstein L 1978 Neutrino oscillations in matter _Phys. Rev. D_ 17 2369
* (4) Bilenky S and Petcov S T 1987 Massive neutrinos and neutrino oscillations _Rev. Mod. Phys._ 59 671
* (5) Bilenky B, Giunti C and Grimus W 1999 Phenomenology of neutrino oscillations _Progress in Particle and Nuclear Physics_ 43 1 (hep-th/9812360)
* (6) Nussinov S 1976 Solar neutrinos and neutrino mixing _Phys. Lett. B_ 63 201
* (7) Kayser B 1981 On the quantum mechanics of neutrino oscillation _Phys. Rev. D_ 24 110
* (8) Giunti C, KimC W and Lee U W 1991 When do neutrinos really oscillate? Quantum mechanics of neutrino oscillations _Phys. Rev. D_ 44 3635
* (9) Rich J 1993 Quantum mechanics of neutrino oscillations _Phys. Rev. D_ 48 4318
* (10) Kiers K, Nussinov S and Weiss N 1996 Coherence effects in neutrino oscillations _Phys. Rev. D_ 53 537 (hep-ph/9506271)
* (11) Sarkar U and Mann R B 1988 Neutrino Masses in Superstring Theories with Intermediate Scales _International Journal of Modern Physics A_ 3 2165
* (12) Fukuda Y et al(Super-K) 1998 Evidence for Oscillation of Atmospheric Neutrinos _Phys. Rev. Lett_. 81 1562
* (13) Apollonio et al(CHOOZ) M 1999 Limits on neutrino oscillations from the CHOOZ experiment _Phys. Lett. B_ 466 415
* (14) Eguchi K et al(KamLAND) 2003 First Results from KamLAND: Evidence for Reactor Antineutrino Disappearance _Phys. Rev. Lett_. 90, 021802
* (15) Ahn M H et al(K2K) 2003 Indications of Neutrino Oscillation in a 250 km Long-Baseline Experiment _Phys. Rev. Lett_. 90, 041801; 2006 Measurement of neutrino oscillation by the K2K experiment _Phys. Rev. D_ 74 072003
* (16) Ahmed S N et al(SNO) 2004 Measurement of the Total Active 8B Solar Neutrino Flux at the Sudbury Neutrino Observatory with Enhanced Neutral Current Sensitivity _Phys. Rev. Lett_. 92 181301
* (17) Michael D G et al(MINOS) 2006 Observation of Muon Neutrino Disappearance with the MINOS Detectors in the NuMI Neutrino Beam _Phys. Rev. Lett_. 97 191801
* (18) Aharmin B et al(SNO) 2008 Independent Measurement of the Total Active 8B Solar Neutrino Flux Using an Array of 3He Proportional Counters at the Sudbury Neutrino Observatory _Phys. Rev. Lett._ 101 111301
* (19) Ahluwalia D V and Burgard C 1996 Gravitationally Induced Neutrino-Oscillation Phases _Gen. Relativ. Gravit._ 28 1161 (arXiv:gr-qc/9603008)
* (20) Piriz D D, Roy M and Wudka J 1996 Neutrino oscillations in strong gravitational fields _Phys. Rev. D_ 54 1587 (hep-ph/9604403)
* (21) Fornengo N, Giunti C M, Kim C M and Song J 1997 Gravitational effects on the neutrino oscillation _Phys. Rev. D_ 56 1895 (hep-ph/9611231)
* (22) Cardall C Y and Fuller G M 1997 Neutrino oscillations in curved spacetime: A heuristic treatment _Phys. Rev. D_ 55 7960 (hep-ph/9610494)
* (23) Zhang C M and Beesham A 2001 The General Treatment of the High and Low Energy Particle Interference Phase in a Gravitational Field _Gen. Rel. Grav._ 33 1011 (gr-qc/0004048)
* (24) Zhang C M and Beesham A 2003 On the Mass Neutrino Phase Along the Geodesic Line and the Null Line in Curved and Flat Spacetime _Int. J. Mod. Phys. D_ 12 727
* (25) Pereira J G and Zhang C M 2000 Some Remarks on the Neutrino Oscillation Phase in a Gravitational field _Gen. Rel. Grav._ 32 1633 (gr-qc/0002066)
* (26) Huang X J and Wang Y J 2003 Interference Phase of Mass Neutrinos in Kerr Space-Time _Commun. Theor. Phys._ 40 742; 2004 Mass neutrino oscillations in Reissner Nordstrom space time _Chin. Phys._ 13 1588; 2006 Mass neutrino oscillations in Robertson Walker space time _Chin. Phys._ 15 229
* (27) Mann R B and Sarkar U 1996 Test of the Equivalence Principle from Neutrino Oscillation Experiments _Phys. Rev. Lett._ 76 865 (hep-ph/9505353)
* (28) Bhattacharya T, Habib S and Mottola E 1999 Gravitationally induced neutrino oscillation phases in static spacetimes _Phys. Rev. D_ 59 067301 (1999).
* (29) Lipkin H J 2000 Neutrino oscillations as two-slit experiments in momentum space _Phys. Lett. B_ 477 195
* (30) Lipkin H J 1999 Quantum Mechanics of Neutrino Oscillations - Hand Waving for Pedestrians (hep-ph/9901399)
* (31) Grossman Y and Lipkin H J 1997 Flavor oscillations from a spatially localized source: A simple general treatment _Phys. Rev. D_ 55 2760 (hep-ph/9607201)
* (32) Gasperini M 1988 Testing the principle of equivalence with neutrino oscillations _Phys. Rev. D_ 38 2635
* (33) Mureika J R and Mann R B 1996 Mass or gravitationally-induced neutrino oscillations? a comparison of 8B neutrino flux spectra in a three-generation framework _Phys. Lett. B_ 368 112 (hep-ph/9511220)
* (34) Mureika J R and Mann R B 1996 Three-flavor gravitationally induced neutrino oscillations and the solar neutrino problem _Phys. Rev. D_ 54 2761 (hep-ph/9603335).
* (35) Capozziello S and Lambiase G 2000 Inertial effects on neutrino oscillations _Eur. Phys. J. C_ 12 343 (gr-qc/9910016)
* (36) Lambiase G 2001 Neutrino oscillations in non-inertial frames and the violation of the equivalence principle Neutrino mixing induced by the equivalence principle violation _Europ. J. Phys. C_ 19 553
* (37) Alimohammadi M and Shariati A 1999 Neutrino oscillation in a space-time with torsion _Mod. Phys. Lett. A_ 14 267 (gr-qc/9808066)
* (38) Capozziello S, Iovane G and Lambiase G et al 1999 Fermion helicity flip induced by torsion field _Europhys. Lett._ 46 710 (astro-ph/9904199).
* (39) Capozziello S, Lambiase G and Stornaiolo C 2001 Geometric classification of the torsion tensor in space-time _Annalen Phys._ 10 713 (gr-qc/0101038)
* (40) Lambiase G 2005 Pulsar Kicks Induced by Spin Flavor Oscillations of Neutrinos in Gravitational Fields _Mon. Not. Roy. Astron. Soc._ 362 867-871 (astro-ph/0411242)
* (41) Lambiase G, Papini G, Punzi R and Scarpetta G 2005 Neutrino optics and oscillations in gravitational fields _Phys. Rev. D_ 71 073011 (gr-qc/0503027)
* (42) Lychkovskiy O 2006 Constraints on neutrino mixing angle $theta_{13}$ and Supernova neutrino fluxes from the LSD neutrino signal from SN1987A (hep-ph/0604113)
* (43) R. Adhikari, A. Datta and B. Mukhopadhyaya 2007 Neutrino mass scale and the mixing angle $\theta_{13}$ for quasidegenerate Majorana neutrinos _Phys. Rev. D_ 76 073003 (hep-ph/0703318)
* (44) Eby D A, Frampton P H and Matsuzaki S 2009 Predictions of neutrino mixing angles in a T model _Phys. Lett. B_ 671 386 (hep-ph/0810.4899).
* (45) Klinkhamer F R 2006 Possible new source of T and CP violation in neutrino oscillations _Phys. Rev. D_ 73 057301
* (46) Schwetz T 2007 What is the probability that 13 and CP violation will be discovered in future neutrino oscillation experiments? _Phys. Lett. B_ 648 54-59 (hep-ph/0612223).
* (47) Gava J and Volpe C 2008 Collective neutrino oscillations in matter and CP violation _Phys. Rev. D_ 78 083007 (astro-ph/0807.3418).
* (48) Altarelli G and Meloni D 2009 CP violation in neutrino oscillations and new physics _Nucl. Phys. B_ 809 158-182
* (49) Cuesta H J M and Lambiase G 2008 Neutrino Mass Spectrum from Gravitational Waves Generated by Double Neutrino Spin-Flip in Supernovae _Astrophysical Journal_ 689 371
* (50) Akhmedov E K, Maltoni M and Smirnov A Y 2008 Neutrino oscillograms of the Earth: effects of 1-2 mixing and CP-violation _Journal of High Energy Physics_ 0806, 072(hep-ph/0804.1466)
* (51) Ohanian H C and Ruffini R 1994 Gravitation and Spacetime (New York: Norton & Company)
* (52) Audretsch J 1981 Trajectories and spin motion of massive spin-1/2 particles in gravitational fields _J. Phys. A_ 14 411
* (53) L. Stodolsky 1979 Matter and light wave interferometry in gravitational fields _Gen. Relativ. Gravit._ 11 391
* (54) Boehm F and Vogel P 1992 _Physics of Massive Neutrino_ (Cambridge University Press, Cambridge)
* (55) Zuber K 1998 On the physics of massive neutrinos _Phys. Rep._ 305 295 (hep-ph/9811267)
* (56) Bahcall J N 1994 _Neutrino Astrophysics_ (Cambridge University Press, Cambridge) 1983
* (57) Chandrasekhar S 1983 _Mathematical Theory of Black Holes_ (Clarendon Press, Oxford; Oxford University Press, New York), 326-350.
* (58) Landau L D and Lifshitz E M 1987 _The Classical Theory of Field_ (Forth Revised English Edition, Butterworth Heinemann) 235
* (59) Stodolsky L 1998 When the wavepacket is unnecessary _Phys. Rev. D_ 58 036006
* (60) Giunti C 2001 Energy and Momentum of Oscillating Neutrinos _Mod. Phys. Lett. A_ 16 2363 (hep-ph/0104148)
* (61) De Leo S, Ducati G and Rotelli P 2000 Remarks upon the mass oscillation formulas _Modern Physics Letters A_ 15 2057 (hep-ph/9906460)
* (62) Giunti C and Kim C W 2001 Quantum Mechanics of Neutrino Oscillations _Found. Phys. Lett._ 14, 213 (hep-ph/0011074)
|
arxiv-papers
| 2010-02-03T05:35:22 |
2024-09-04T02:49:08.160370
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Ren and Cheng-Min Zhang",
"submitter": "Jun Ren",
"url": "https://arxiv.org/abs/1002.0648"
}
|
1002.0808
|
hlvtwo nohl end inline
# Patterning of ultrathin YBCO nanowires using a new focused-ion-beam process
N Curtz1,2, E Koller2, H Zbinden1, M Decroux2, L Antognazza2, Ø Fischer2 and N
Gisin1 1 Group of Applied Physics, University of Geneva, 1211, Geneva 4,
Switzerland 2 Department of Condensed Matter Physics, University of Geneva,
1211, Geneva 4, Switzerland Noe.Curtz@unige.ch
###### Abstract
Manufacturing superconducting circuits out of ultrathin films is a challenging
task when it comes to pattern complex compounds, which are likely to be
deteriorated by the patterning process. With the purpose of developing high-Tc
superconducting photon detectors, we designed a novel route to pattern
ultrathin YBCO films down to the nanometric scale. We believe that our method,
based on a specific use of a focused ion beam, consists in locally implanting
Ga${}^{\text{3+}}$ ions and/or defects instead of etching the film. This
protocol could be of interest to engineer high-Tc superconducting devices
(SQUIDS, SIS/SIN junctions, Josephson junctions), as well as to treat other
sensitive compounds.
###### pacs:
85.25.Am, 81.16.Nd, 74.78.Bz, 81.15.Cd, 85.25.Pb
## 1 Introduction
Superconducting Single-Photon Detectors (SSPDs) are superconducting devices
designed with the purpose of detecting light down to single-photon level. They
present a good quantum efficiency (QE > 10%), low dark-count rate (DK < 10
Hz), and high operating frequency (> GHz), outperforming in a number of cases
InGaAs Avalanche PhotoDiodes (APDs) [1]. These characteristics make them a
premium candidate for single-photon telecommunication and applications like
Quantum Key Distribution. Their underlying mechanism is based upon the
formation of a hotspot in a current-biased superconducting stripe [2]. The
creation of the hotspot is triggered by the incoming photon whose energy
locally thermalizes the stripe, confining the bias current hence raising its
density, up to the point of overcoming its critical value, resulting in a
local transition of the stripe.
To achieve that, the circuit geometry has to fulfill drastic geometrical
constraints. First, the stripe has to be narrow enough, otherwise the
variation of the current density isn’t sufficiently significant, preventing
the transition to take place and the voltage pulse to be detectable. The
section of the nanowire should also be extremely homogeneous, since any
constriction locally lowers the critical current, hence affects the whole
device with for aftermath a drop of the QE; the closer to 1 is the ratio
$I_{\text{bias}}/I_{\text{c}}$, the closer to the dissipative state is the
device, and therefore the more important is the section homogeneity. Finally,
the detector’s recovery time is governed by the device thickness,
necessitating devices less than 15 nm thick. Whereas such devices have been
successfully produced with low-Tc superconductors such as NbN operated at 2.6
K [3], their realization with high-Tc compounds remains a challenge. High-Tc
SSPDs would nevertheless allow a higher working temperature, hence a
significant reduction of the associated cryogenic costs.
Among high-T${}_{\text{c}}$ materials, cuprates present the advantage of a low
kinetic inductance, leading to fast response times. From a purely structural
point of view,
Nd${}_{\text{1+}x}$Ba${}_{\text{2-}x}$Cu${}_{\text{3}}$O${}_{\text{7+}\delta}$
presents excellent crystallographic and planeity properties [4], which are
interesting features given the aforementioned geometrical constraints of
SSPDs. It turns out however that the intrinsic loose stoichiometry of Nd
atoms, who interdiffuse with Ba ones, leads to a high unstability of the
oxygen content and therefore to an important loss of T${}_{\text{c}}$ during
the patterning process.
YBa${}_{\text{2}}$Cu${}_{\text{3}}$O${}_{\text{7-}\delta}$ (YBCO) is much more
stable and appears to be a good candidate for high-T${}_{\text{c}}$ detectors.
Several routes have been carried out to create junctions or patterned
structures out of high-T${}_{\text{c}}$ thin films, such as Selective
Epitaxial Growth [5], Electron Beam Lithography / Ion Beam Etching (EBL/IBE)
[6], Ion Irradiation [7], using an Atomic Force Microscope [8] or a Focused
Electron Beam Irradiation [9]. Nanobridges have been fabricated with a Focused
Ion Beam [10]. However those experiments were performed on films with
thickness $d$ > 20 nm. Here we report a new method using such an apparatus to
write an arbitrary pattern upon an ultrathin (< 20 nm) film, allowing to
manufacture YBCO superconducting circuits. The key point of this method is
that to produce the structure the superconducting phase is locally altered
rather than etched.
## 2 Experimental
### 2.1 Overview of the FIB-based protocol
We designed a 2-step protocol, involving a preliminary chemical etching
followed by a focused ion beam (FIB) managed nanostructuration. An overview of
the scheme used to create YBCO circuits embedding paths to characterize their
transport properties with 4-point measurements is given in figure 1.
Figure 1: Overview of the patterning protocol. (a) Photograph of the
preliminary structure created from a thin film by photolithography and
chemical etching. (b) SEM micrograph of a 20 $\mu$m stripe engineered with a
high-current Focused Ion Beam. (c) SEM micrograph of a 12 nm thick, 15 $\mu$m
long, 500 nm wide stripe patterned with a lower current. (d) SEM micrograph of
a 1 $\mu$m wide meandering circuit written with a lower current.
### 2.2 Growth of YBCO films
$c$-axis YBCO films were deposited by RF magnetron sputtering on (100)
SrTiO${}_{\text{3}}$ substrates. Samples are heated to 700°C and exposed to
the sputtering of a YBCO target in a Argon plasma. Along with the Ar flow a
complementary O${}_{\text{2}}$ flow (1:5 ratio) accounts for the growth of a
tetragonal, non-superconducting phase, under a total pressure
P=8.10${}^{\text{-2}}$ mbar. An in-situ 2-hour-long annealing at 580°C in a
600 mbar O${}_{\text{2}}$ atmosphere causes this
YBa${}_{\text{2}}$Cu${}_{\text{3}}$O6 phase to undergo a tetragonal-
orthorhombic transition to optimally-doped superconducting
YBa${}_{\text{2}}$Cu${}_{\text{3}}$O${}_{\text{7-}\delta}$. The critical
temperature of bulk YBCO is 92 K; this critical temperature decreases with the
film’s thickness, down to T${}_{\text{c0}}$(d=12 nm) $\approx$ 80 K. The high
crystallinity of the films was demonstrated with X-ray diffraction
measurements such as depicted in figure 2. Up to 4 degrees the grazing
incidence scan leads to Kiessig fringes contributions from both films [11].
Around the (001) YBCO Bragg peak are located secondary fringes clearly showing
a finite size effect (Laue oscillations), demonstrating the high quality of
the crystallographic layers. The fringes of X-ray spectra allow to determine a
film’s thickness with unit cell accuracy. In the following, all the films
processed are at d=12 nm, with T${}_{\text{c0}}\approx$ 80 K.
Figure 2: Typical X-ray $\theta$-2$\theta$-spectrum of a 12 nm thick YBCO
film passivated with a 8 nm thick amorphous PBCO layer.
The sputtering process also deposits CuO${}_{\text{2}}$ surface particles
located on the top of the YBCO phase, with a typical diameter estimated at 250
nm; those might indeed be the origin of shortcuts after patterning. No
reproducible way of getting particle-free samples or eliminating them has been
found, however most of the time these particles are not a real obstacle to the
patterning process, since the core structure of the devices can most of the
times be chosen to be located in a clear area (figure 1c). Even when it’s not
the case, such as in figure 1d, table 2 shows that ultimately these particles
are not an issue.
In order to improve electrical contacts, we adapted a gold evaporation system
involving a mechanical mask to deposit gold slots in-situ immediatly after the
YBCO deposition. In addition it was observed that the in-situ deposition of a
8 nm thick amorphous PrBaCuO passivation cap layer over the whole sample
attenuates the loss of critical temperature occurring during the chemical
etching process [12]. Therefore, we end up with a YBCO/Au/PBCO topology.
### 2.3 Photolithography and chemical etching
The patterning of a 50 $\mu$m wide stripe is done by optical lithography and
chemical etching with orthophosphoric acid H${}_{\text{3}}$PO${}_{\text{4}}$
(1%). Four independent structures are patterned in a single run (figure 1a).
One of them, top-left on the figure, is specifically designed to electrically
characterize the sample at this point of the protocol.
### 2.4 FIB writing
As the second step of the protocol, a dual beam (FEI Nova 600 Nanolab scanning
electron microscope / 30 kV focused Ga${}^{\text{3+}}$ ion beam) was used to
carry out a finer patterning of the YBCO films into meandering stripes
suitable for optical characterization of SSPDs. In the context of YBCO thin
layers, some precautions to make a relevant use of the FIB are necessary since
the standard manner of working is too destructive for the samples.
Accordingly, we devised a specific modus operandi to satisfy the needed
requirements.
First of all, due to the extreme thickness of the involved films, the
alignment routines cannot be handled with the standard procedure, as the whole
window, containing areas destined to remain superconducting is exposed to the
Ga${}^{\text{3+}}$ beam during the operation, and irreversibly damaged. To
circumvent this problem we have to synchronize the SEM and the FIB on a non-
critical area, then align the pattern to etch with the sample using the SEM.
Secondly, to find which parameters (intensity, number of passes, numerical
aperture…) are optimal to produce an effective pattern, we tried to separate
two YBCO areas by creating FIB-made barriers and injecting current through
them; results for a 50 pA beam are shown in table 1. We determined that at 50
pA, with a beam diameter $\delta\approx$ 20 nm, 50 passes are optimal to
electrically isolate both areas, while at 1 nA, with a beam diameter
$\delta\approx$ 100 nm, 10 passes are optimal. The fact that for N>20 the
resistance exceeds 3 M$\Omega$ rules out an exclusive YBCO${}_{\text{7}}$ to
YBCO${}_{\text{6}}$ transformation due to oxygen loss, for if it were the
case, the barrier transition would have resulted in a 5 k$\Omega$ resistance
(assuming very conservatively $\rho$(underdoped)=10${}^{\text{-1}}$
$\Omega$.cm at room temperature [13]) for a 1 $\mu$m wide affected width. Even
if the whole stripe had been transformed to YBCO${}_{\text{6}}$ its resistance
wouldn’t have exceeded 150 k$\Omega$. We attribute this result to the fact
that some Ga${}^{\text{3+}}$ ions are implanted or give rise to columnar
defects into the YBCO phase during the exposure, turning it locally into an
insulator.
Table 1: Room-temperature resistance of a FIB-made barrier across a 20 $\mu$m
wide stripe for various numbers of passes (N) with a 50 pA beam.
* N | 0 | 1 | 2 | 5 | 20 | 50 | 100
---|---|---|---|---|---|---|---
R ($\Omega$) | 2k | 2k | 3k | 700k | 3M | >10M | >10M
Lateral contamination is a crucial parameter to consider since it directly
addresses both the issues of smallest reachable dimensions and current
homogeneity. To illustrate this point, we obtained different kinds of
superconducting transitions on structures of different widths by varying the
number of FIB passes N. Figure 3 shows a comparison between resistive profiles
for 20 $\mu$m wide stripes patterned with respectively N=500 and N=10 passes.
A significant loss of T${}_{\text{c}}$ occurs for N=500. This T${}_{\text{c}}$
loss may be due to a reduction of the O${}_{\text{2}}$ stoichiometry, but in
this case it can’t be attributed to local heating occurring during the writing
process. Indeed, the elevation of the film temperature has been estimated
within a simple thermal model and found to be lower than 5 K, under the beam
writing conditions (voltage: 30 kV, current: 1 nA, diameter: 100 nm). Besides,
it is unlikely that Ga ions can travel 10 $\mu$m accross the sample resulting
in implantations responsible for the T${}_{\text{c}}$ losses observed in
figure 3. Consequently N=500 was ruled out for our process. For N=10 no such
loss happens, although the current is confined in the stripe.
Figure 3: Resistive curves of 12 nm thick micron-scale circuits patterned
using a Focused Ion Beam with a beam current i=1 nA. N is the number of passes
used to produce the structure.
Finally, to pattern a meandering wire, we first produce a 20 $\mu$m wide
stripe (figure 1b), then we force the current to follow a serpentine path by
adding some crossing barriers (figure 1d). The supporting stripe has to be
manufactured at 1nA because of window size considerations. Thus we choose to
use two different FIB currents to pattern a meander : 1 nA, 10 passes for the
supporting stripe; 50 pA, 50 passes for the meander itself. Supplementary
barriers parallel to the stripe are added to avoid side effects near to the
1nA-exposed areas. Using this protocol superconducting straight stripes down
to 500 nm wide (figure 1c), and meanders down to 1 $\mu$m wide (figure 1d)
were produced. It’s worth mentioning that at no point we were confronted to
some limitation so in principle it should be possible to achieve even better
resolutions with this method.
## 3 Results and discussion
The $\rho$ vs T curves of the samples can be followed during the different
steps of the protocol: after the film deposition the resistivity is measured
with the Van der Pauw method [14]; after chemical etching a four-point
measurement is carried out along a 50 $\mu$m wide stripe using the top-left
structure of figure 1a. Eventually the final circuit is characterized as shown
in figure 1b. Results are plotted in figure 4, and demonstrate that the
obtained circuits are superconducting, although a small loss of
T${}_{\text{c}}$ as well as a broadening of the resistive transition is
observed. This could be explained by the fact that the samples are
intrinsically inhomogeneous, and by restricting the superconducting geometry
to narrow circuits new areas with lower T${}_{\text{c}}$ enter the current
path; but it could also be material damages occuring during the processing. On
another hand, $\rho_{\text{100K}}$ and $\rho_{\text{300K}}$ are slightly
higher at the end of the process.
As mentioned previously the writing routine with the focused ion beam doesn’t
etch the film; instead the beam turns locally the irradiated area into an
insulating phase by implanting columnar defaults inside it. The penetration
depth of Ga${}^{\text{3+}}$ ions is about 70 nm, far superior to the YBCO and
amorphous PBCO combined thickness. From this point of view this method differs
from EBL/IBE processes where parts of the film are physically removed to
create the superconducting pattern, but it also implies caution if one desires
to implement it on with thicker films. This approach prevents the YBCO phase
from being in contact with air, which could be an escape path for oxygen, both
with the PBCO top passivation layer and the lateral insulating phase. Table 2
presents resistance measurements obtained for different structures, the last
column showing the consistency of the results and ensuring that the method
presented in the paper leads to reproducible structures.
Figure 4: Resistivity vs temperature for different samples, with additional
curves at intermediary steps of the process. Table 2: Summary of
T${}_{\text{c}}$ and resistance for different kinds of samples.
R${}_{\text{norm}}$ is the equivalent resistance computed with the structure
normalized to a 1 $\mu$m wide, 15 $\mu$m long line.
* shape | length | width | T${}_{\text{c0}}$ | R(100K) | R${}_{\text{norm}}$ ($k\Omega$)
---|---|---|---|---|---
meander | 180 $\mu$m | 2 $\mu$m | 72 K | 32 k$\Omega$ | 5.3
meander | 430 $\mu$m | 1 $\mu$m | 70 K | 190 k$\Omega$ | 6.6
meander | 430 $\mu$m | 1 $\mu$m | 75 K | 140 k$\Omega$ | 4.9
stripe | 15 $\mu$m | 1 $\mu$m | 70 K | 5 k$\Omega$ | 5
stripe | 15 $\mu$m | .8 $\mu$m | 75 K | 6.6 k$\Omega$ | 5.3
stripe | 15 $\mu$m | .5 $\mu$m | 65 K | 12 k$\Omega$ | 6
Figure 5 present the current-voltage and the $\rho$-$j$ characteristics of one
sample. Both present a negative curvature in the whole range of temperature
covered, yielding the existence, in every case, of a true critical current
density $j_{\text{c}}$. This ensures that the superconducting phase isn’t in a
flux-creep state in spite of the thinness of the sample, which is consistent
with the observation that above $d$ = 10 nm flux line lattices behave like a
3D system [15].
The critical current is generally defined using the standard criterion of an
electric field of 1$\mu$V/cm. In our case the measurement noise due to the
high impedance of the line sets a resolution limit of 2 $\mu$V across the
whole meander, corresponding to an equivalent electric field E = 110 $\mu$V/cm
as shown on figures 5a and 5b. We clearly see on figure 5b, especially at T =
65 K, that at 110 $\mu$V/cm (or 2 $\mu$V) the critical current densities are
overestimated and that such a determination isn’t relevant.
However, it has been reported [16] that a precise measurement of the flux flow
resistivity at high current density allows to extract $j_{\text{c}}$ from the
best power law fit $V\propto(j-j_{\text{c}})^{n}$, $j_{\text{c}}$ and $n$
being the fitting parameters. Figure 6 shows the very good fits obtained with
power laws, giving good confidence for the determination of $j_{\text{c}}$’s
by this procedure. Moreover, $n$ is found to be around 5 with little variation
over the whole temperature range. A significant advantage of this fitting
method is that the knowledge of the current-voltage characteristic at high
current density is sufficient to infer $j_{\text{c}}$.
The temperature-dependence of $j_{\text{c}}$ is described within the Ginzburg-
Landau theory of superconductivity as $j_{\text{c}}$ =
$j_{\text{c0}}(1-(\frac{T}{T_{\text{c}}})^{2})(1-(\frac{T}{T_{\text{c}}})^{4})^{1/2}$
[17]. Figure 7 shows that this model fits extremely well our experimental data
and allows to extrapolate $j_{c}(0K)\approx$ 4.1 MA/cm${}^{\text{2}}$, which
is two orders of magnitude smaller than the depairing limit
$j_{\text{d}}=0.54\frac{B_{\text{c}}}{\mu_{0}\lambda}\approx 3~{}10^{8}$
A/cm${}^{\text{2}}$, demonstrating the good quality of the samples after the
processing. It’s worth noting that the hypothesis of inhomogeneities in the
sample previously mentioned is reinforced by the fact that the best Ginzburg-
Landau fit is obtained for T${}_{\text{c}}$ = 68 K, below the critical
temperature of 72 K found with resistivity measurements (see table 2).
A last point concerns long-term behavior of the nanowires. Although no
systematic study was carried out, some samples were characterized several
times at low temperature, with more than one month gone by between the
measurements. No deterioration of any kind was observed, which is an
indication that the manufactured structures are unaffected by time and thermal
cycling.
Figure 5: (a) Voltage vs current density at various temperatures for a 2
$\mu$m wide meandering circuit. (b) Resistivity vs current density for the
same sample.
Figure 6: Same data as in figure 5a represented as voltage vs
$j/j_{\text{c}}-1$. Power law fits $V=B(j-j_{\text{c}})^{n}$ (continuous
lines) correctly describe the curves and allow to determine $j_{\text{c}}$. In
inset is reported $n$ as temperature drops from 65 K to 30 K. Figure 7:
Critical current density vs reduced temperature $t=T/T_{\text{c}}$ for the
same meandering circuit. $j_{\text{c}}$ is obtained through the power law fits
presented in figure 6. The solid line is the best fit with the theoretical
Ginzburg-Landau model.
## 4 Conclusion
To summarize, a modus operandi to create ultrathin superconducting YBCO
circuits by implanting Ga${}^{\text{3+}}$ ions with a Focused Ion Beam was
devised. We confined current in straight stripes down to 500 nm wide and
produced 1 $\mu$m wide meandering wires using a 50 pA beam current. The
consistency of the resistivity vs temperature profiles measured on the samples
at the different steps of the processing ensures the reproducibility of this
patterning method for superconducting films. For one sample, the critical
current density extrapolated to 0 K has been found to be only two orders of
magnitude times smaller than the depairing limit, demonstrating its quality.
The most natural application of this protocol would be the manufacturing of
high-T${}_{\text{c}}$ superconducting devices. Photoresponse experiments to
characterize the devices as single-photon detectors are left for future work.
The authors would like to thank Michaël Pavius, Kevin Lister, Samuel Clabecq,
and Philippe Flückiger for providing access and training to EPFL’s focused ion
beam, as well as Jean-Claude Villégier for helpful discussions. This work is
supported by the European project Sinphonia (contract No. NMP4-CT-2005-16433),
and the Swiss poles NCCR MaNEP and NCCR Quantum Photonics.
## References
* [1] R T Thew, N Curtz, P Eraerds, W Walenta, J-D Gautier, E Koller, J Zhang, N Gisin, and H Zbinden. Approaches to single photon detection. Nucl. Instr. and Meth. A, 610 (1):16–19, 2009. doi:10.1016/j.nima.2009.05.031.
* [2] G Gol’tsman, O Okunev, G Chulkova, A Lipatov, A Semenov, K Smirnov, B Voronov, A Dzardanov, C Williams, and R Sobolewski. Picosecond superconducting single-photon optical detector. Appl. Phys. Lett., 79:705, 2001.
* [3] F Marsili, D Bitauld, A Gaggero, R Leoni, F Mattioli, S Hold, M Benkahoul, F Lévy, and A Fiore. High efficiency NbN nanowire superconducting single photon detectors fabricated on MgO substrates from a low temperature process. Optics Express, 16:3191, 2008.
* [4] M Badaye, J G Wen, K Fukushima, N Koshizuka, T Morishita, T Nishimura, and Y Kido. Superior properties of NdBa${}_{\text{2}}$Cu${}_{\text{3}}$Oy over YBa${}_{\text{2}}$Cu${}_{\text{3}}$Ox thin films. Supercond. Sci. Technol., 10:825–830, 1997.
* [5] C A J Damen, H-J H Smilde, D H A Blank, and H Rogalla. Selective epitaxial growth for YBCO thin films. Supercond. Sci. Technol., 11:437–448, 1998.
* [6] H Schneidewind, F Schmidl, S Linzen, and P Seidel. The possibilities and limitations of ion-beam etching of YBa${}_{\text{2}}$Cu${}_{\text{3}}$O${}_{\text{7-}x}$ thin films and microbridges. Physica C, 250:191–201, 1995.
* [7] N Bergeal, X Grison, J Lesueur, G Faini, M Aprili, and J-P Contour. High Tc superconducting quantum interferences devices made by ion irradiation. Appl. Phys. Lett., 89:112515, 2006.
* [8] C Delacour, J Claudon, J-Ph Poizat, B Pannetier, V Bouchiat, R Espiau de Lamaëstre, J-C Villégier, M Tarkhov, A Korneev, B Voronov, and G Gol’tsman. Superconducting single photon detectors made by local oxydation with an atomic force microscope. Appl. Phys. Lett., 90:191116, 2007.
* [9] W E Booij, A J Pauza, E J Tarte, D F Moore, and M G Blamire. Proximity coupling in high-Tc Josephson junctions produced by focused electron beam irradiation. Phys. Rev. B, 455:14600, 1997.
* [10] S-G Lee, S Oh, C S Kang, and S-J Kim. Superconducting nanobridge made from YBa${}_{\text{2}}$Cu${}_{\text{3}}$O${}_{\text{7}}$ film by using focused ion beam. Physica C, 460:1468–1469, 2007.
* [11] K N Stoev and K Sakurai. Review on grazing incidence X-ray spectrometry and reflectometry. Spectrochimica Acta Part B, 54:41–82, 1999.
* [12] A Jaeger, J-C Villégier, P Bernstein, J Bok, and L Force. Field effect in multilayered YBaCuO/PrBaCuO devices. Ieee Trans. Appl. Supercond., 3:2933, 1993.
* [13] K Semba, A Matsudaa, and M Mukaida. Carrier-concentration-driven superconductor-to-insulator transition in YBa${}_{\text{2}}$Cu${}_{\text{3}}$O${}_{\text{6+}x}$. Physica B, 281:904–905, 2000.
* [14] L J Van der Pauw. A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape. Philips Technical Review, 20:220–224, 1958.
* [15] J-M Triscone and Ø Fischer. Superlattices of high-temperature superconductors. Rep. Prog. Phys., 60:1673–1721, 1997.
* [16] L Antognazza, M Decroux, S Reymond, E de Chambrier, J-M Triscone, W Paul, M Chen, and Ø Fischer. Simulation of the behavior of superconducting YBCO lines at high current densities. Physica C, 372-376:1684–1687, 2002.
* [17] C P Poole, H A Farach, and R J Creswick. Superconductivity. Academic Press, 1995.
|
arxiv-papers
| 2010-02-03T17:54:17 |
2024-09-04T02:49:08.167637
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "No\\'e Curtz, Edmond Koller, Hugo Zbinden, Michel Decroux, Louis\n Antognazza, {\\O}ystein Fischer and Nicolas Gisin",
"submitter": "No\\'e Curtz",
"url": "https://arxiv.org/abs/1002.0808"
}
|
1002.0849
|
# Vector field models of modified gravity and the dark sector
J. Zuntz1, T.G Zlosnik2, F. Bourliot3, P.G. Ferreira1, G.D. Starkman4
1Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road,
Oxford OX1 3RH, UK
2Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Waterloo, Ontario N2L 2Y5, Canada
3 CPHT, Ecole Polytechnique, 91128 Palaiseau cedex, France
4Department of Physics/CERCA/ISO,Case Western Reserve University,
Cleveland,OH, 44106-7079
###### Abstract
We present a comprehensive investigation of cosmological constraints on the
class of vector field formulations of modified gravity called Generalized
Einstein-Aether models. Using linear perturbation theory we generate cosmic
microwave background and large-scale structure spectra for general parameters
of the theory, and then constrain them in various ways. We investigate two
parameter regimes: a dark-matter candidate where the vector field sources
structure formation, and a dark-energy candidate where it causes late-time
acceleration. We find that the dark matter candidate does not fit the data,
and identify five physical problems that can restrict this and other theories
of dark matter. The dark energy candidate does fit the data, and we constrain
its fundamental parameters; most notably we find that the theory’s kinetic
index parameter $n_{\mathrm{ae}}$ can differ significantly from its
$\Lambda$CDM value.
###### pacs:
PACS Numbers :
††preprint:
## I Introduction
Over the past few years, a new suite of models for the dark sector has been
proposed. They invoke a vector field which is normally constrained to lie
along the time-like direction and may lead to modifications to the
gravitational sector Zlosnik et al. (2007); Jiménez and Maroto (2008);
Dimopoulos et al. (2009); Koivisto and Mota (2008). Sometimes called Einstein-
Aether models, they tend to entangle two of the main paradigms currently being
considered: on the one hand modified gravity and on the other dark matter and
energy.
Vector field models are attractive because they seem to be able to resolve the
problem of the dark sector (i.e. dark matter and energy) in a unified way.
Most of the emphasis has gone into constraining vector field models that lead
to accelerated expansion Jiménez and Maroto (2008); Dimopoulos et al. (2009);
Koivisto and Mota (2008) although there is a fair amount of work for which the
the vector field leads to a relativistic version of Modified Newtonian
Dynamics (MOND) Milgrom (1983) or can even play the role of dark matter
Zlosnik et al. (2007); Zhao (2007). In fact vector field models seem to
incorporate what seems to be a generic feature of relativistic modified
gravity models Ferreira and Starkman (2009): that it is impossible to
construct relativistic models that just modify the gravitational sector
without introducing new degrees of freedom, which can then behave like either
dark matter or dark energy (although for other approaches see, for example,
Milgrom (2009); Blanchet and Tiec (2009, 2008); Soussa and Woodard (2004);
Bruneton et al. (2009); Sagi (2009)).
There has been significant progress in trying to constrain these models. For
example, at a fundamental level it has been shown that a broad class will lead
to instabilities and the formation of caustics, signaling a break down of the
fundamental theory Contaldi et al. (2008). It has also been shown that for a
general choice of kinetic terms, these theories will be plagued by ghosts or
tachyons Carroll et al. (2009); Eling (2006); Lim (2005). These pathologies
are worrying but do not entirely rule out vector field models- it has been
shown that modifications to the kinetic term, for example, can cure them.
Substantial work has been done on understanding how these fields in these
theories behave on macroscopic scales, either through their interaction with
matter to form galaxies and clusters Dai et al. (2008), or on the largest
scales, affecting the growth of structure and its effect on the CMB Zlosnik et
al. (2008). Indeed for a particular, “vanilla” version of the vector-field
model, detailed and definitive constraints have been placed on the various
coupling constants Zuntz et al. (2008); Jacobson (2007).
In Zlosnik et al. (2008), it was found that one of the key effects that
vectors would have would be to modify the growth rate of structure. This is
not surprising- theories that modify gravity tend to have this effect. We also
found that it lead to a mismatch between the two gravitational potentials a
potentially observable effect Zhang et al. (2007). In this paper we wish to
pursue this analysis and quantify how strong these effects are. Although we
focus on a particular (albeit broad) class of theories, we are interested in
extracting general lessons from these models. We believe that much of what we
learn by looking at these models will shed light on other models of modified
gravity (such as, for example, $f(R)$ theories Bean et al. (2007) and bimetric
theories Bekenstein (2004)).
The structure of this paper is as follows. In Section II we lay out the
essential ingredients for a reasonably broad class of vector-like models and
its background evolution, and specialize to the form used in the remainder of
the paper. In section III.1 we lay out the equations of the perturbed theory,
and how we implement them with theoretical constraints. In section IV we
discuss the problems with modelling dark matter with the theory. In section V
we find and constrain the parameters which let the theory act as dark energy.
In section VI we conclude and draw more general lessons about the dark sector.
## II The Theory
### II.1 Theory Definition
A general action for a vector field $A^{a}$ coupled to gravity can be written
in the form:
$\displaystyle S=\int d^{4}x\sqrt{-g}\left[\frac{R}{16\pi G}+{\cal
L}(g^{ab},A^{b})\right]+S_{M}$ (1)
where $g_{ab}$ is the metric, $R$ the Ricci scalar of that metric, $S_{M}$ the
matter action and $\cal{L}$ is constructed to be generally covariant and
local. By construction $S_{M}$ only couples to the metric, $g_{ab}$ and not to
$A^{a}$.
We will restrict ourselves to consider a Lagrangian that only depends on
covariant derivatives of $A$ and we will consider a unit time-like $A^{a}$.
Such a theory can be written in the form:
$\displaystyle{\cal L}(g^{ab},A^{a})$ $\displaystyle=$
$\displaystyle\frac{M^{2}}{16\pi G}F(K)+\frac{1}{16\pi
G}\lambda(A^{a}A_{a}+1)$ (2) $\displaystyle K$ $\displaystyle=$ $\displaystyle
M^{-2}K^{ab}_{\phantom{ab}mn}\nabla_{a}A^{m}\nabla_{b}A^{n}$ (3)
$\displaystyle K^{ab}_{\phantom{ab}mn}$ $\displaystyle=$ $\displaystyle
c_{1}g^{ab}g_{mn}+c_{2}\delta^{a}_{\phantom{a}m}\delta^{b}_{\phantom{b}n}+c_{3}\delta^{a}_{\phantom{a}n}\delta^{b}_{\phantom{b}m}$
(4)
where $c_{i}$ are dimensionless constants and $M$ has the dimension of mass.
We have removed an additional $c_{4}$ ‘acceleration’ term in accordance with
the transformation described in Eling and Jacobson (2006).
As was the case with TeVeS, the constant $G$ may be a different number from
the locally measured value of Newton’s gravitational constant. $\lambda$ is a
non-dynamical Lagrange-multiplier field with dimensions of mass-squared.
The gravitational field equations for this theory, obtained by varying the
action with respect to $g^{ab}$ (see Zlosnik et al. (2007)) are
$G_{ab}=\tilde{T}_{ab}+8\pi GT^{\mathrm{matter}}_{ab}$ (5)
where the stress-energy tensor for the vector field is given by
$\displaystyle\tilde{T}_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nabla_{c}(F_{K}(J_{(a}^{\phantom{a}c}A_{b)}-J^{c}_{\phantom{c}(a}A_{b)}-J_{(ab)}A^{c}))$
(6) $\displaystyle-F_{K}Y_{(ab)}+\frac{1}{2}g_{ab}M^{2}F+\lambda A_{a}A_{b}$
$\displaystyle F_{K}$ $\displaystyle\equiv$ $\displaystyle\frac{dF}{dK}$ (7)
$\displaystyle J^{a}_{\phantom{a}c}$ $\displaystyle=$
$\displaystyle(K^{ab}_{\phantom{ab}cd}+K^{ba}_{\phantom{ba}dc})\nabla_{b}A^{d}$
(8)
Brackets around indices denote symmetrization222we adopt the convention
$X_{(ab)}=\frac{1}{2}(X_{ab}+X_{ba})$, $X_{[ab)}=\frac{1}{2}(X_{ab}-X_{ba})$
and $Y_{ab}$ is the functional derivative
$\displaystyle
Y_{ab}=\nabla_{c}A^{e}\nabla_{d}A^{f}\frac{\delta(K^{cd}_{\phantom{cd}ef})}{\delta
g^{ab}}$ (9)
The equations of motion for the vector field, obtained by varying with respect
to $A^{b}$ are
$\displaystyle\nabla_{a}(F_{K}J^{a}_{\phantom{a}b})+F_{K}y_{b}$
$\displaystyle=$ $\displaystyle 2\lambda A_{b}$ (10)
where once again we define the functional derivative
$\displaystyle
y_{b}=\nabla_{c}A^{e}\nabla_{d}A^{f}\frac{\delta(K^{cd}_{\phantom{cd}ef})}{\delta
A^{b}}$ (11)
Finally, variations of the action with respect to $\lambda$ will fix
$A^{b}A_{b}=-1$. By inspection, contracting both sides of (10) with $A^{b}$
leads to a solution for $\lambda$ in terms of the the vector field and its
covariant derivatives.
These equations allow us to study a general theory of the form presented in
equation 1 with a unit time-like vector field. For our particular, restricted
choice of $K$ we have
$Y_{ab}=-c_{1}\left[(\nabla_{c}A_{a}(\nabla^{c}A_{b})-(\nabla_{a}A_{c})(\nabla_{b}A^{c})\right]$
and $y_{b}=0$.
### II.2 Background Cosmology
In this paper we will restrict ourselves to background cosmologies where the
spacetime is of the spatially flat Friedmann-Robertson-Walker (henceforth FRW)
form:
$g_{ab}dx^{a}dx^{b}=-{dt}^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}.$ (12)
The energy momentum content of each matter fluid $(i)$ in the background is
taken to be of the usual form
$\displaystyle T^{i}_{ab}=(P^{i}+\rho^{i})u^{i}_{a}u^{i}_{b}+P^{i}g_{ab}.$
(13)
Where $\rho^{i}$ and $P^{i}$ are the co-moving density and pressure of the
$i$th fluid respectively. We assume that all fluids have identical four
velocities $u^{a}=(1,0,0,0)$. Furthermore we define
$\rho\equiv\sum_{i}\rho^{(i)}$.
In spacetimes with FRW symmetries, the vector field must align with the
direction $\partial_{t}$ and so the vector field is entirely fixed to have
components $(1,0,0,0)$ in the co-ordinate system (12). Explicitly, the
background value of the scalar $K$ is given by:
$K_{\mathrm{FRW}}=3\frac{\alpha H^{2}}{M^{2}}$ (14)
Where $H\equiv\partial_{t}\ln(a(t))$ and:
$\alpha\equiv c_{1}+3c_{2}+c_{3}.$ (15)
The Friedmann equation then takes the form Zlosnik et al. (2008):
$\displaystyle\left[1-\alpha
K^{1/2}\frac{d}{dK}\left(\frac{F}{K^{1/2}}\right)\right]H^{2}$
$\displaystyle=$ $\displaystyle\frac{8\pi G}{3}\rho$ (16)
where $\rho$ still includes only the matter components.
We note that the combination of (16) and the aether stress energy tensor
allows us to write down an effective energy density $\rho_{\mathrm{ae}}$ and
pressure $P_{\mathrm{ae}}$ of the aether. From this we may define the
fractional energy density $\Omega_{\mathrm{ae}}\equiv 8\pi
G\rho_{\mathrm{ae}}/3H_{0}^{2}$, and the aether equation of state parameter
$w_{\mathrm{ae}}\equiv P_{\mathrm{ae}}/\rho_{\mathrm{ae}}$:
$\displaystyle\Omega_{\mathrm{ae}}$ $\displaystyle=$
$\displaystyle\frac{M^{2}}{6}\left[\frac{\partial}{\partial
H}\left(\frac{F}{H}\right)\right]_{H=H_{0}}$ (17)
$w_{\mathrm{ae}}=-1-\frac{1}{3H^{2}}\frac{\frac{d^{2}}{dtdH}F}{\frac{d}{dH}\left(\frac{F}{H}\right)}$
(18)
The effect of the vector field on the background expansion may be see as an
expansion rate dependent modification to Newton’s constant i.e. schematically
(16) is an equation of the form $3H^{2}=8\pi G(H^{2})\rho$. It was found in
Zlosnik et al. (2008) that various forms of the function $F$ permitted a wide
variety of cosmological dynamics: the presence of the vector field variously
leading to accelerated expansion, slowed expansion, rescaling of $G$, and
recollapse as summarized in Figure 1
Figure 1: A schematic representation of the types of the late-time background
evolution permitted by the functional form $F=\gamma(-K)^{n}$ as a function of
$(n,\gamma)$ for $n<1$.
### II.3 The functional form of $F(K)$ in cosmology
We must now specify the form of the function $F(K)$ in equation (6). There is
an obvious set of candidates here - we could attempt to be directly consistent
with MOND and use the same branch of $F$ as it uses on small scales. In
appendix A we show that doing so would make it impossible to consistently
generate late-time acceleration behavior in the background cosmology. Instead,
we will use a simple and reasonable ansatz that works for the regime $|K|\gg
1$ that we consider here.
Existing functional forms for F(K) in the MOND regime typically are dominated
for a single monomial term for $|K|\gg 1$ (see for instance Famaey and Binney
(2005)) and so it seems reasonable to restrict the function to take this form:
$\displaystyle F=\gamma(-K)^{n_{\mathrm{ae}}}\ \ $ $\displaystyle K$
$\displaystyle<0$ $\displaystyle F=\gamma(+K)^{n_{\mathrm{ae}}}\ \ $
$\displaystyle K$ $\displaystyle>0$ (19)
where ${n_{\mathrm{ae}}}\leq 1$. This form has sufficient power to express a
wide variety of behavior, and the parameters $\gamma$ and $n$ shall be central
to our further analysis.
## III Linear perturbation Theory
### III.1 Formalism & Theory
We have seen that the vector field can have a significant effect on the
quasistatic, weak field limit and the background cosmological geometry.
Significant evidence for non-baryonic mass persists on the largest
cosmological scales Dodelson and Liguori (2006) therefore it is vital that a
relativistic theory of MOND can account for this. As mentioned, it has been
argued Dai et al. (2008) that even in the quasistatic, weak field limit, a
spatial tilt to the vector field may produce significant deviations under some
circumstances from the local MOND force law. Similarly it was shown Zlosnik et
al. (2008) that in the context of linear cosmological perturbations the energy
density associated with the projection of the vector field onto surfaces of
constant conformal time could, to a degree, act as a cosmological ‘dark
matter’ . In this paper we shall comprehensively address the question of
whether the Lagrangian (2) represents a viable model of the dark sector in
light of precision cosmology.
In the main body of the paper we will consider scalar perturbations. In
Appendix B we derive the equations of motion for the vector field’s two
divergenceless vector modes as well as the gravitational wave tensor modes
(the speed of propagation of which is in general modified by the vector
field). Requiring the stability of these modes puts constraints on (2) (see
Section III.2).
We shall work in the synchronous gauge (see for instance Ma and Bertschinger
(1995)) and so the metric takes the following form:
$g_{\mu\nu}dx^{\mu}dx^{\nu}=-a^{2}d\tau^{2}+a^{2}[\gamma_{ij}+h_{ij}]dx^{i}dx^{j}$
(20)
where $\tau$ is conformal time, $\gamma_{ij}$ is a spatially flat spacelike
3-metric perturbed by $h_{ij}$ which is built from two scalar potentials
$\eta$ and $h$:
$h_{ij}(\textbf{x},\tau)=\int
d^{3}ke^{i\textbf{k}\cdot\textbf{x}}[\hat{\textbf{k}}_{i}\hat{\textbf{k}}_{j}h(\textbf{k},\tau)+(\hat{\textbf{k}}_{i}\hat{\textbf{k}}_{j}-\frac{1}{3}\delta_{ij})6\eta(\textbf{k},\tau)]$
(21)
Similarly we will expand the aether field as:
$A^{\mu}=\frac{1}{a}(1,\partial_{i}V)$ (22)
The zeroth component of the aether field is, by virtue of the gauge choice and
the constraint, fixed as equal to $1$ up to second order in perturbations.
A slight complication in the field equations arises because of the presence of
the function $F$ which depends nonlinearly on the scalar $K$. We assume that
for modes of interest one may consistently regard the perturbation to $K$ as
being much less than unity. Thus one can expand $K$ as $K=K^{0}+K^{\epsilon}$
and $F$ as $F=F^{0}+F_{K}^{0}K^{\epsilon}$ where $K^{\epsilon}\ll 1$ where the
superscript $0$ denotes the quantity corresponding to a function’s background
value. Explicitly we have that:
$\displaystyle K^{0}$ $\displaystyle=$ $\displaystyle 3\frac{\alpha{\cal
H}^{2}}{a^{2}M^{2}}$ (23) $\displaystyle{\cal H}K^{\epsilon}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}K^{0}(k^{2}V-\frac{h^{\prime}}{2})$ (24)
Where primes denote derivatives with respect to conformal time and we have
used the conformal Hubble parameter ${\cal H}\equiv H/a$. We also make use of
the following identity:
$\displaystyle(K^{0})^{\prime}{\cal H}=-4K^{0}({\cal
H}^{2}-\frac{1}{2}\frac{a^{\prime\prime}}{a})$ (25)
Henceforth we will drop the superscripts on $K^{0}$ and $F^{0}$ i.e. $K$ and
$F$ shall be assumed to represent _background_ values of the fields.
Towards simplifying the form of the equations, we will will rather use the
field $\xi$ instead of $V$, where $\xi$ is defined as:
$\xi\equiv V-\frac{1}{2k^{2}}(h+6\eta)^{\prime}$ (26)
For further compactness of notation we define the variables:
$\displaystyle\hat{\alpha}$ $\displaystyle\equiv$
$\displaystyle\left(1+2\frac{F_{KK}}{F_{K}}K\right)\alpha$ (27)
$\displaystyle\hat{c}_{1}$ $\displaystyle\equiv$
$\displaystyle\left(1+2\frac{F_{KK}}{F_{K}}K\right)c_{1}$ (28)
The vector field equation of motion (10) becomes :
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
c_{1}(1+c_{13}F_{K})\frac{(F_{K}\xi^{\prime})^{\prime}}{F_{K}}+2{\cal
H}c_{1}(1+c_{13}F_{K})\frac{(F_{K}\xi)^{\prime}}{F_{K}}$ (29)
$\displaystyle+[2c_{1}(1+c_{13}F_{K})(\frac{a^{\prime\prime}}{a}-{\cal
H}^{2})+2(\hat{c}_{1}+\hat{\alpha})({\cal
H}^{2}-\frac{1}{2}\frac{a^{\prime\prime}}{a})$
$\displaystyle+c_{1}c_{13}(F_{KK}K^{\prime})^{\prime}+\frac{1}{3}(\hat{\alpha}+2c_{13})k^{2}]\xi$
$\displaystyle+(c_{1}+\hat{\alpha})\eta^{\prime}+(\hat{c}_{1}+\hat{\alpha})\frac{1}{k^{2}}({\cal
H}^{2}-\frac{1}{2}\frac{a^{\prime\prime}}{a})(h^{\prime}+6\eta^{\prime})$
$\displaystyle-\frac{3}{2}\frac{c_{1}}{k^{2}}\frac{(F_{K}\Sigma_{f})^{\prime}}{F_{K}}$
The relevant Einstein equations are:
$\displaystyle(1-\frac{1}{2}\hat{\alpha}F_{K})k^{2}\eta^{\prime}$
$\displaystyle=$ $\displaystyle 4\pi Ga^{2}ik^{j}\delta T^{0}_{\phantom{0}j}$
(30) $\displaystyle+\frac{1}{6}k^{4}(\hat{\alpha}+2c_{13})F_{K}\xi$
and
$\displaystyle(1+\frac{1}{2}c_{1}F_{K})({\cal H}h^{\prime}-2k^{2}\eta)$
$\displaystyle=$ $\displaystyle-8\pi Ga^{2}\delta T^{0}_{\phantom{0}0}$ (31)
$\displaystyle-\frac{1}{2}F_{K}(c_{1}+\hat{\alpha})6{\cal H}\eta^{\prime}$
$\displaystyle-2\alpha F_{KK}K{\cal H}k^{2}\xi$
$\displaystyle+\frac{F_{K}c_{1}k^{2}}{a^{2}}\left(a^{2}(1+c_{13}F_{K})\xi\right)^{\prime}$
$\displaystyle-\frac{3}{2}c_{1}F_{K}\Sigma_{f}$
Where we have used the fact that:
$\displaystyle(h+6\eta)^{\prime\prime}+2{\cal H}(h+6\eta)^{\prime}-2k^{2}\eta$
$\displaystyle=$ $\displaystyle-3\Sigma_{f}$ (32)
$\displaystyle+2c_{13}k^{2}[F_{K}(2{\cal H}\xi+\xi^{\prime})$
$\displaystyle+F_{KK}K^{\prime}\xi]$
The functions $\delta T^{0}_{\phantom{0}j}$ and $\delta T^{0}_{\phantom{0}0}$
are the first order perturbations to the corresponding components of the
matter fields’ stress energy tensors. Summation over field species is assumed.
The field $\Sigma_{f}$ is the scalar component of the total fluid shear i.e.
$\Sigma_{f}=-8\pi
Ga^{2}(\hat{k}_{i}\hat{k}^{j}-\frac{1}{3}\delta^{j}_{\phantom{j}i})\Sigma^{i}_{\phantom{i}j}$
and $\Sigma^{i}_{\phantom{i}j}\equiv\delta
T^{i}_{\phantom{i}j}-\frac{1}{3}\delta^{i}_{\phantom{i}j}\delta
T^{k}_{\phantom{k}k}$. A gauge invariant formulation of the theory’s equations
may be found in Li et al. (2008a).
### III.2 Parameter Constraints
We can immediately see a number of constraints on the $c_{i}$ and the form of
the function $F$. From (29) it can be shown that for in the limit of
timescales shorter than a Hubble time the quantity:
$C_{S}^{2}=\frac{2}{3}\frac{(\frac{\hat{\alpha}}{2}+c_{13})}{c_{1}(1+c_{13}F_{K})}$
(33)
can be interpreted as the squared sound speed of the field $\xi$. The
avoidance of exponentially growing subhorizon modes dictates that $C_{S}^{2}$
should be positive definite.
Similarly, one may consider the field equations of the two divergenceless
‘vector’ modes of the vector field and the two transverse traceless ‘tensor’
modes of of the metric. Each respectively has a squared sound speed function
(named $C_{V}^{2}$ and $C_{T}^{2}$ respectively) which, as in the scalar case,
should be positive definite. These functions are calculated in Appendix B and
are as follows:
$C_{T}^{2}=\frac{1}{1+c_{13}F_{K}}$ (34)
$C_{V}^{2}=\frac{F_{K}}{2c_{1}}\frac{2c_{1}+F_{K}(c_{1}^{2}-c_{3}^{2})}{1+c_{13}F_{K}}$
(35)
Note that the gravitational wave tensor modes now generically have a time
dependent speed of propagation.
Collectively the three positivity constraints imply the following constraints
on the $c_{i}$ and function $F$ parameter space:
$\displaystyle 1+F_{K}c_{13}$ $\displaystyle>$ $\displaystyle 0$ (36)
$\displaystyle(\hat{\alpha}+2c_{13})/2c_{1}$ $\displaystyle>$ $\displaystyle
0$ (37) $\displaystyle F_{K}(1+F_{K}\frac{c_{1}^{2}-c_{3}^{2}}{2c_{1}})$
$\displaystyle>$ $\displaystyle 0$ (38)
Now we turn to the Einstein equations (30) and (31). Terms in the perturbed
vector field stress energy tensor $\delta\tilde{T}_{ab}$ may contain terms
proportional (up to a time-dependent function of the background fields) to
$\delta G_{ab}$ and indeed this can be seen in (30) and (31) through the
appearance of terms proportional to the $c_{i}$ on the left hand sides of the
equations. If the effect $\xi$ is negligible then the effect of the aether
stress energy terms proportional to components of the Einstein tensor may
always be absorbed into a redefinition of Newton’s constant as a time-
dependent effective gravitational coupling. Thus, even if $\xi$ has
comparatively little effect, there may be a considerable modification to the
link between the matter fields and the gravitational field. The resulting
gravitational couplings should be greater than or equal to zero otherwise the
gravitational field will interpret normal matter as violating energy
conditions, and so risking the appearance of instabilities. This restriction
implies the following constraints:
$\displaystyle(1-\frac{1}{2}\hat{\alpha}F_{K})>0$ (39)
$\displaystyle(1+\frac{1}{2}c_{1}F_{K})>0$ (40)
Throughout our analysis, we will only consider regions of the model’s
parameter space which satisfy these constraints.
### III.3 Computation
To study the effects of the vector field in detail we have modified the
structure formation Boltzmann code CMBEASY Doran (2005). We add a Newton-
Raphson solver for the Hubble parameter, with added aether components. The
perturbation evolution is also modified to include the aether components $\xi$
and $\xi^{\prime}$, and their contribution to the density, pressure and shear
perturbations. We also include the altered metric perturbations in the
calculation of the CMB source function. We use adiabatic initial
conditionsZlosnik (2009). Since Boltzmann codes are very highly optimized for
$\Lambda$CDM models, care must be taken to ensure that modifications are
performed in a consistent manner - for example an unmodified Friedmann
equation is often assumed for computational efficiency.
To explore parameter spaces, CMBEASY is coupled to a Monte-Carlo Markov Chain
(MCMC) engine Doran and Mueller (2004). We extended this engine to include our
new Aether parameters: $\Omega_{\mathrm{ae}}$, $c_{+}=(c_{1}+c_{3})$,
$c_{-}=(c_{1}-c_{3})$, $C_{S}^{2}$, $n_{\mathrm{ae}}$ and $M$. We include the
full ranges of these parameters by allowing the kinetic term to take two
branches, for positive and negative $K$ as in (19).
We constrain models using both CMB data from the WMAP experiment Komatsu et
al. (2009) and large-scale structure from the SDSS survey Adelman-McCarthy et
al. (2008), though not always at the same time. As we shall see, some Aether
models are extremely poor fits to the combined data sets; to illustrate such
problems we want to find models that fit only the large scale structure data.
In regimes where fits are extremely poor, MCMC does not work particularly
well. We ameliorate such situations by running larger numbers of shorter
Markov chains and re-starting from their best-fit positions, and sometimes by
abandoning MCMC altogether and simply performing random searches for good
parameter combinations.
## IV The vector field as Dark Matter
### IV.1 Background evolution & doppler peak positions
An unusual property of the model considered here is that at the level of
cosmological perturbations the field can mimic a perturbed pressureless fluid
in the formation of large scale structure whilst behaving entirely differently
in the cosmological background Zlosnik et al. (2008) . In this paper we would
like to consider not just large scale structure but also other cosmological
probes. The anisotropies in the CMB temperature are sensitive to the
background dynamics and perturbed dynamics of source of cosmic mass
discrepancies in a largely distinguishable manner.
If the aether plays the role of dark matter only in the perturbations, then
the background expansion is dark energy-dominated at an earlier time. For a
given $H_{0}$ this reduces the expansion rate of the universe between
recombination and now and so decreases the angle subtended on the sky by given
distance at last scattering. This moves the CMB doppler peaks to higher
$\ell$. Although the aether can give a suitable time dependence to the
effective gravitational coupling $G$ in the Friedmann equations so as to yield
the same expansion rate as $\Lambda$CDM, the necessary functional forms of $F$
are extremely contrived. For instance these forms essentially contain a new
constant scale roughly equal to the Hubble parameter at matter-radiation
equality. Even in the event of such a construction, it may not be possible for
the squared sound speed of vector field perturbations $C_{S}^{2}$ to remain
sufficiently small as to behave like cold dark matter in perturbations Zlosnik
et al. (2007).
The other input to the peak position is the sound horizon at last scattering.
The physics of this is sufficiently robust that changes compensatory to the
alteration in the distance to last scattering are not feasible without
exceptional fine tuning of the perturbational behavior of the aether. Indeed,
it has been argued that only additional components that behave like non-
relativistic matter in the background might fix this problem Ferreira et al.
(2008).
Could there indeed be such a non-relativistic matter in the background which
allows acoustic peak positions which are consistent with the data whilst
leaving the aether in its role as the seed of the formation of large scale
structure? A appropriate candidate would appear to be massive neutrinos. A
suitable mass of such particles so as to account for the right effective
contribution to the dust component of the background typically implies that
the neutrinos themselves are unable to clump on small enough scales so as to
be a good candidate for all of the dark matter. Such a solution as has been
proposed in Skordis et al. (2006) and Angus (2008).
### IV.2 Perturbation evolution
We now argue that even with the inclusion of massive neutrinos, one
generically expects the aether to have an unacceptable influence on the large
scale CMB anisotropy if it is to also play the dominant role in structure
formation.
The first requirement for successful perturbation evolution is that structure
can form at all. One necessary condition for this is that the sound speed of
the structure seed not be too large, since this would wash out structure. We
require that the sound horizon in the model be less than the smallest scales
where structure can form linearly: $C_{S}k_{\mathrm{max}}\tau\lesssim 1$,
where $k_{\mathrm{max}}\sim 0.2h/M\mathrm{pc}$. For matter power observations
at $\tau\sim 3\times 10^{4}$, the present epoch, this yields $C_{S}\lesssim
10^{-4}$.
There are two underlying physical processes that further constrain the models.
The first is a change to the growth rate of perturbation amplitude. This can
cause discrepancies between the amplitudes we expect in the matter power
spectrum and the CMB, since the evolution between the two is different. It can
also lead to an integrated Sachs-Wolfe (ISW - see below) effect during the
matter era since $\Phi$ will accrue a time dependence.
The second is the increased presence of a $\Phi-\Psi$ metric shear. This also
leads (directly) to a matter era ISW.
### Observable 1: ISW
Under the assumption of adiabaticity we have that the anisotropy in the CMB,
$\Delta T({\hat{n}})/T$ on large scales in a given spatial direction
${\hat{n}}$ is given by
$\frac{\Delta
T({\hat{n}})}{T}\simeq-\frac{1}{3}\Psi(\tau_{*},d_{*}{\hat{n}})-\int_{\tau_{*}}^{\tau_{0}}d\tilde{\tau}({\Psi^{\prime}}+{\Phi^{\prime}})[\tilde{\tau},(\tau_{0}-\tilde{\tau}){\hat{n}}]$
(41)
where $\tau_{*}$ is the conformal time of last scattering, $\tau_{0}$ is the
conformal time today, $d_{*}$ is the comoving radius of the surface of last
scattering, and $\Phi$ and $\Psi$ are the conformal Newtonian gauge
gravitational potentials. The integral in (41) is the integrated Sachs-Wolfe
(ISW) effect. Writing the integrand of (41) as
$\Psi^{\prime}+\Phi^{\prime}=-(\Phi-\Psi)^{\prime}+2\Phi^{\prime}$, we can see
this as time derivatives of a shear part and a growth rate part.
In the standard cosmological model, the field $\Phi$ has negligible time
dependence during matter domination. It gains a time dependence only when the
background starts accelerating, and only then can the resulting growth rate
ISW contribution be considerable. In the model considered here, the situation
may be rather different. It was found in Zlosnik et al. (2008) that
substantial contributions to the ISW may occur even during the matter era. For
the aether field to seed structure formation the field $\xi$ must have a
suitable growing mode solution in the matter era. Typically the corresponding
spatial curvature perturbation $k^{2}\Phi$ will then have a time dependence
via the Poisson equation.
The shear part can also gain a time dependence in the aether model, which in
the $\Lambda$CDM is very small even during acceleration.
Each of these effects depend on the functional form of $F$, the time-
dependence of the $\xi$ growing mode and the choice of the parameters $c_{i}$.
It is extremely challenging to find combinations of the parameters which allow
for a realistic growth of structure whilst making the ISW acceptably low. This
is most easily illustrated by considering the theory TeVeS Bekenstein (2004)
which has many of the same properties as the model considered here. It may be
shown that TeVeS can be written as a single metric theory with a timelike
vector field of unfixed norm Zlosnik et al. (2006),Skordis (2009). As in the
model considered here, the longitudinal component of the vector field can
source the growth of structure Skordis et al. (2006), Dodelson and Liguori
(2006). We will call this field $V_{T}$. We consider a matter dominated era
where $V_{T}$ is responsible for the dominant source in the Poisson and shear
equations. These equations respectively then areSkordis (2006):
$\displaystyle k^{2}\Phi$ $\displaystyle\approx$ $\displaystyle-f_{s}{\cal
H}k^{2}V_{T}-\frac{K_{B}}{2}k^{2}V_{T}^{\prime}$ (42) $\displaystyle
k^{2}(\Psi-\Phi)$ $\displaystyle\approx$ $\displaystyle f_{s}k^{2}(2{\cal
H}V_{T}-V_{T}^{\prime})$ (43) $\displaystyle f_{s}(\tau)$
$\displaystyle\equiv$ $\displaystyle\frac{(1-\bar{A}^{4})}{\bar{A}^{4}}$ (44)
where $K_{B}$ is a positive constant of the action and $\bar{A}^{4}$ is the
norm of the vector squared again (equal to unity in the fixed norm case, but
in TeVeS the deviation of $|\bar{A}^{2}|$ from unity is essentially the
background variation of the ‘scalar field’ degree of freedom). Therefore:
$\displaystyle\Psi+\Phi=-(f_{s}(\tau)+K_{B})V^{\prime}_{T}$ (45)
In this era the vector field equation is:
$V_{T}^{\prime\prime}+b_{1}\frac{V_{T}^{\prime}}{\tau}+b_{2}\frac{V_{T}}{\tau^{2}}=S[\Phi,\Psi]$
(46)
where
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle 2(3-\bar{A}^{4})$ (47)
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle
2(2-\bar{A}^{4})+\frac{12\bar{A}^{4}}{K_{B}}(1-\bar{A}^{4})$ (48)
and $S[\Phi,\Psi]$ is a source term.
For this situation to arise, there must be a growing mode in $V_{T}$ Dodelson
and Liguori (2006). Therefore we require that $b_{2}<0$. The function
$\bar{A}^{4}$ will be rather close to unity Bourliot et al. (2007) so
$b_{2}\sim 2+12f_{s}(\tau)/K_{B}$ (49)
Therefore we require that $f_{s}(\tau)/K_{B}\lesssim-1/6$ (recalling that
$K_{B}$ is a positive number). By (45) we see that the contribution to the ISW
will be proportional to $f_{s}(\tau)+K_{B}$. Meanwhile we see from (42) that
the comparative contribution to Poisson’s equation is also independently
weighted by $f_{s}$ and $K_{B}$. If there is no time dependence in the Poisson
equation (42) due to the vector field then the vector field will make no
contribution to the ISW effect. If there is a time dependence in the Poisson
equation, there may yet be no contribution to the ISW if $f_{s}(\tau)\sim-
K_{B}$ between last scattering and now (though the overdensities of the baryon
field will then not generally grow as $a$, thus contributing to the ISW
effect). This would be consistent with the condition for a growing mode in
$V_{T}$ but it does not guarantee that the resulting growing mode would be
suitable.
Indeed, it was found Skordis et al. (2006) that in seeding the growth of large
scale structure in TeVeS there was a significant danger of incurring
unacceptably high temperature anisotropies in the CMB on large scales.
Although involving a larger number of terms, the same reasoning carries over
to the model considered here - i.e. parameters which allow realistic structure
formation will typically lead to an unacceptable ISW effect. This is vividly
illustrated in Figure 4 which shows best fit models as compared to SDSS large
scale structure data. In every case, the corresponding temperature anisotropy
displays a dramatically poor fit to the data at low $\ell$.
### Observable 2: Amplitudes
The ratio of the observed amplitudes of the CMB anisotropy and the matter
power spectrum is consistent with a growth rate proportional to $a$ (though
see Bean (2009)). Any uncompensated change to this growth rate in the aether
model over this period would lead to a different ratio. The bias parameter
between the galaxy distribution and the underlying density field can be used
to rectify this difference, but only if the change is relatively small and
unphysically large bias parameters (larger than $\sim 10$) are not required.
### IV.3 Summary
We have seen that although the vector field may play a number of the roles
that dark matter plays, it seemingly cannot do all at once 111It is
interesting to compare this with similar results found in Li et al. (2008b)
for a framework for generalizations of dark matter. It is not clear whether
the model considered in this paper fits within this framework.
The position of the acoustic peaks in the CMB temperature anisotropy should be
taken as a strong indication of additional nonbaryonic nonrelativistic mattter
present in the universe during matter domination. Such an effect can be
achieved in this model by a rescaling of the value of Newton’s constant.
However, this rescaling cannot persist into the radiation era Carroll and Lim
(2004). Thus, the functional form would have to be approaching the rescaling
solution only after one would expect matter (including cold dark
matter)-radiation equality to happen.
This implies the presence of a new scale in the theory, roughly corresponding
to the Hubble parameter $H_{eq}$ at this time. It seems fair to say in general
that a model such as that considered here, is more likely to be a
cosmologically viable candidate for dark matter if the scale $M$ in the theory
is closer to $H_{eq}$ and not $H_{0}$. It is tempting to speculate whether a
theory where the scale $M$ itself is dynamical may find more success, but that
will not be explored in this work.
Furthermore we have seen that even if the background is consistent with
observations, the effect on the evolution of perturbations may be
unacceptable, notably either through the ISW effect or comparing the
respective amplitudes of the CMB anisotropy and matter power spectrum today.
### IV.4 Example Problem Spectra
It has previously been shown that the Einstein Aether can produce acceptable
matter power spectra with certain parameter combinations Zlosnik et al.
(2008). Here we show that such combinations do not provide an acceptable fit
to CMB measurements. Despite extensive searches we have been unable to find
any parameter set within the model that does fit the WMAP data well; this is
entirely in line with the problems discussed above.
We use a parameter set which is consistent with BBN limits on
$\Omega_{B}h^{2}$ and the HST key project measurement of $H_{0}$. The standard
cosmological parameters are: $\Omega_{b}h^{2}=0.0193$, $n_{s}=0.83$,
$H_{0}=89.3\,\mathrm{km/s/Mpc}$, $\Omega_{c}h^{2}=0$. The new Aether variables
are $c_{+}=-4.72$, $c_{-}=-6.11$, $n_{\mathrm{ae}}=0.34$,
$\Omega_{\mathrm{ae}}=0.82$, $M_{\mathrm{ae}}=111.3\,\mathrm{km/s/Mpc}$ with
$c_{2}$ set by requiring zero sound speed. This parameter combination is in no
sense optimal, but it does provide an illustration of all the problems that
arise here.
Figure 4 shows power spectra from our modified Boltzmann code for this
parameter set. The matter power is a realistic fit to the SDSS data (this was
the criterion for our choice of parameters). The CMB spectra shows various
problems. In the low-$\ell$ regime a large ISW effect is clearly present,
destroying the fit at large scales, as described in section (IV). The
positions of the peaks are poorly fit by the model, as expected and discussed
in section (IV.1). Finally, in the plot we have rescaled the amplitude of the
matter power spectrum by a factor $0.02$, corresponding to a galaxy bias of
$0.14$ in order to reconcile the relative amplitudes of the two spectra with
the data; such a scaling is unphysically small. This corresponds to the
changed growth rate described in section (IV). All these effects cause severe
problems when attempting to simultaneously fit the CMB and large scale
structure.
Figures 2 and 3 illustrate the sources of the extreme ISW effects shown in
Figure 4; the time derivatives of the metric quantities plotted create an ISW
effect as shown in equation (41). The onset of background acceleration in each
case is marked by a turnover in the curves at late time. The GEA universe
exhibits a dramatically increased $|\Phi-\Psi|$ and time dependence of
$|\Phi|$ during the matter era as compared to the $\Lambda$CDM universe.
Although the $|\Phi-\Psi|$ has a smaller magnitude its time dependence can be
significant for the total ISW effect. Note that values of $\tau$ between the
two universes do not correspond to the same physical time or redshift since
the universes expand at different rates.
Figure 2: Exotic behavior of metric potentials for $k\sim$ $10^{-2}$
$Mpc^{-1}$. The panels show the fields $|\Phi|$ and $|\Phi-\Psi|$ for a
$\Lambda$CDM universe (green solid line) and GEA universe (dashed blue line)
as a function of $k(\tau-\tau_{r})$ where $\tau_{r}$ is the conformal time of
recombination.
Figure 3: Equivalent of Figure 2 for k $\sim$ $10^{-4}$ $Mpc^{-1}$
.
Figure 4: Matter power (top) and CMB (bottom) power spectra for the $\Lambda
CDM$ (dashed green) and typical GEA (solid red) models, with WMAP and SDSS
constraints.
## V The vector field as Dark Energy
### V.1 Dark Energy Regime
As shown in figure 1, our vector field can produce late-time acceleration and
so play the role of dark energy. Indeed, for the form of the vector field used
here, as the index $n_{\mathrm{ae}}\rightarrow 0$ the theory becomes the same
as a cosmological constant for both the background and the perturbations.
Since the model can fit the data well we can use our MCMC engine to find
constraints on the parameters of the vector field, telling us exactly how
close to the $n_{\mathrm{ae}}=0$ cosmological constant case the theory must be
to fit the CMB and LSS data.
If the model can fit the data only extremely close to $n_{\mathrm{ae}}=0$ then
it does not provide a compelling alternative to the cosmological constant. If,
on the other hand, there is significant flexibility in the model and no fine
tuning, or if it can provide a better fit than $\Lambda$CDM, then it is
somewhat more interesting.
In this section we will consider the resulting background evolution, CMB
temperature anisotropy, and matter power spectrum for a universe containing
the vector field, cold dark matter, the conventional matter fields, and no
cosmological constant. The acceleration will arise solely from the vector
field’s modification to the Friedmann equation.
### V.2 Constraints on aether dark energy from data
Our MCMC generated the constraints on the vector field parameters shown in
figures 5 to 8; these curves are the smoothed histograms from our combined
Markov chains.
Figure 5: Constraints on the three coupling terms of the theory. Figure 6:
Constraints on the kinetic term power law index parameter. Figure 7:
Constraints on the vector field sound speed parameter. Figure 8: Constraints
on the parameter $\alpha=c_{1}+3c_{2}+c_{3}$
The most important trend evident in these results is closeness of
$n_{\mathrm{ae}}$ to the $\Lambda$ value of zero. We find the best fit value
$n_{\mathrm{ae}}=4.2\cdot 10^{-3}$, with a 95% upper limit
$n_{\mathrm{ae}}<0.126$.
The best fit value in the MCMC run is very close to the $\Lambda$CDM
likelihood of the same data, at the cost of six extra parameters, meaning that
it is unlikely to be favored by any model comparison exercise. It does,
however, demonstrate the validity of modified gravity-related dark energy
candidates.
Having obtained these constraints we can determine their origin. There are two
ways in which the vector field must behave like $\Lambda$ to provide a good
fit. The first is that the late-time acceleration should be close to that
given by $\Lambda$. The second is that any perturbations in the field (which
are not present in $\Lambda$) should not affect the observable spectra.
### V.3 Constraint origins - acceleration rates
The consistency of the acceleration of the universe with the cosmological
constant equation of state $w=-1$ is being measured with increasing precision
in supernova and baryon acoustic oscillation experiments (which are beyond the
scope of this paper). Here, they will be constrained by the late-time ISW
effect induced by dark energy, and by the perturbation growth rate.
We can assess how closely vector-induced acceleration mimics $\Lambda$-driven
expansion at late times with the equation of state $w_{\mathrm{ae}}$ of the
vector field in the background:
$w_{\mathrm{ae}}=-1-\frac{1}{3H^{2}}\frac{\frac{d^{2}}{dtdH}F}{\frac{d}{dH}\left(\frac{F}{H}\right)}.$
(50)
For the monomial form of $F(K)$ we have that:
$\displaystyle w_{\mathrm{ae}}=-1-\frac{2n}{3(2n-1)}\frac{\dot{H}}{H^{2}}$
(51)
Thus the equation of state will generically deviate from $-1$ whenever $n\neq
0$ and so the acceleration for these values will not be degenerate with a
cosmological constant. We see immediately from equation (51) that
$w_{\mathrm{ae}}(\tau)<-1$ for $0<n<1/2$ and $w_{\mathrm{ae}}(\tau)>-1$ for
$n>1/2$. This is clearly visible in Figure 9.
Figure 9: The vector field’s equation of state as a function of redshift $z$,
for various values of the kinetic term index $n_{\mathrm{ae}}.$
### V.4 Constraint origins - perturbation evolution
Even if the background expansion is rather close to the the $\Lambda$CDM
model, the evolution of perturbations need not be. This is most easily
illustrated by considering the Poisson equation on large scales (see Zlosnik
et al. (2007) for a derivation). On these large scales there is a time-
dependent re-scaling of the the metric perturbation $\Phi$, which we can cast
as a modification of the effective gravitational constant $G$:
$\displaystyle k^{2}\Phi$ $\displaystyle=$ $\displaystyle-4\pi
G^{(1)}_{\mathrm{eff}}a^{2}\sum_{i}\bar{\rho}_{i}\delta_{i}$ (52)
$\displaystyle G^{(1)}_{\mathrm{eff}}$ $\displaystyle\equiv$
$\displaystyle\frac{G}{1+\frac{c_{1}}{2}F_{K}}$ (53)
where we have assumed that terms proportional to the velocity divergence are
ignorable and provisionally considered the effect of the field $\xi$ to be
subdominant.
The Friedmann equation may be used to cast the above equation in a more
familiar form by eliminating he background $\rho_{i}$ in favour of background
expansion rate of the universe and the time-dependent fractional energy
density $\Omega_{i}(\tau)\equiv 8\pi G\rho(\tau)/(3H(\tau)^{2})$. This yields:
$\displaystyle k^{2}\Phi$ $\displaystyle=$ $\displaystyle-\frac{3}{2}{\cal
H}^{2}\frac{G^{(1)}_{\mathrm{eff}}}{G^{(0)}_{\mathrm{eff}}}\sum_{i}\Omega_{i}(\tau)\delta_{i}$
(54)
where
$\displaystyle G^{(0)}_{\mathrm{eff}}$ $\displaystyle\equiv$
$\displaystyle\frac{G}{1-\alpha
K^{\frac{1}{2}}\frac{d}{dK}\left(\frac{F}{K^{\frac{1}{2}}}\right)}$ (55)
The $n=0$ $\Lambda$CDM Poisson equation may be cast in this form by taking
$G^{(1)}_{\mathrm{eff}}=G$ and
$G^{(0)}_{\mathrm{eff}}=G/(1-\Lambda/(3H^{2}))$. For the case where $n$
differs from $0$, the function $G^{(1)}_{\mathrm{eff}}$ will generically
possess a time dependence during the background evolution. Therefore the link
between the time evolution of the functions $G^{(0)}_{\mathrm{eff}}$,
$\delta_{i}$ and $\Phi$ will differ from the case where acceleration is caused
by a cosmological constant. We may thus expect the ISW effect to be of a non-
standard form. This is vividly illustrated in Figure 10 where it can be seen
that for a given set of $(c_{i},\gamma,M)$, variation of $n$ results in a
considerable variation in the large scale CMB temperature anisotropy. Also
evident is a variation in the matter power spectrum amplitude with $n$,
evident on all scales.
Variation of parameters other than $n$ could also have a significant impact on
the success of the models. Given the results of the previous section, it seems
unlikely that any influence of the field $\xi$ would tend to improve the
models. This indeed seems to be the case. Figure 11 depicts various models
where the function is varied $C_{S}^{2}$ for fixed values of the other
parameters. In particular, $n$ takes the value $0.1$. The function $C_{S}^{2}$
is ultimately a measure of the ability of the field $\xi$ to sustain any
homogeneous growing behavior for $(k\tau)>1$; higher values will tend to limit
the effect of the vector field to larger and larger scales. The sets of
parameters were chosen so such a growing solution indeed existed on
superhorizon scales. The figure indicates that a growing $\xi$ field will
indeed have deleterious effects on scales where it is not suppressed. There is
a significant ISW effect evident for the red curve CMB; the corresponding
model must be considered as being at the edge of acceptability. The
corresponding large scale matter power spectrum exhibits exotic oscillations,
entirely unrelated to the baryon-acoustic oscillations which occur on other
scales. Their presence is in this model is thus reflective of dynamics in the
dark energy sector.
Figure 10: Matter power (top) and CMB (bottom) power spectra for the various
GEA dark energy models, with WMAP and SDSS constraints. The power-law
function’s exponent $n$ is varied
.
Figure 11: Matter power (top) and CMB (bottom) power spectra for the various
GEA dark energy models, with WMAP and SDSS constraints. The squared speed of
sound of vector field perturbations is varied.
## VI Conclusions
### VI.1 Being Dark Matter is hard
Generalized Einstein-Aether can, with different parameter choices, resemble
dark matter in some important ways but never all of them at once.
Specifically, the new degrees of freedom introduced by the model may conspire
to identically replicate one or more but not all of the following properties
of cold dark matter.
#### VI.1.1 Background dynamics
To accomplish identical background dynamics to cold dark matter, one must
introduce considerable fine tuning into the function $F$ of the theory.
Specifically the function must change form either side of (dark) matter-
radiation equality. The parameter we tune to make this happen is in the action
itself, unlike the usual case where we simply alter the abundance
$\Omega_{c}$. Changes to fit cosmological observations can therefore have a
larger impact on the small scale behavior of gravity.
#### VI.1.2 The speed of sound
If the speed of sound is too high then structure cannot form on small enough
scales. In this model we may reduce the sound speed to be close to zero, at
the cost of one of our parameters. When designing new gravity theories this is
perhaps the easiest structure formation constraint to investigate, and it
should be examined to see if it conflicts with other constraints needed to
make the theory useful - for an example, see Seifert (2007).
#### VI.1.3 Growth rate of ‘overdensity’
Theories of modified gravity designed to replace dark matter must necessarily
have growing modes of fluctuation in at least one of the new degrees of
freedom they introduce, in order to sufficiently source gravitational collapse
and structure formation on scales within their own sound horizon. There
remains some flexibility in the perturbation growth rate, since the bias on
the galaxy power spectrum measurements is a free factor. As measurements of
weak lensing (which samples gravitation directly) and semi-analytic models
(which predict bias) improve this freedom will be reduced.
#### VI.1.4 Absence of anisotropic stress and contribution the cosmological
Poisson equation
A sufficiently small anisotropic stress associated with the vector field may
be implemented by fine tuning the parameter $c_{13}$ to be very small (see
equation (32)). However, as was discussed in Subsection IV, even this will
tend to come at the expense of other desired behavior of the field. An
appreciable time variation of the anisotropic stress over the time from last
scattering to today can result in a very poor fit to the low-$\ell$ CMB
$C_{\ell}$. This problem is likely to be common in theories of modified
gravity.
As we have seen, it is a combination of time variation of the anisotropic
stress and time variation of the field $\Phi$ via the vector field’s effect on
the cosmological Poisson equation that contribute to the ISW effect. Though
both effects are absent in the cold dark matter case, one may imagine both
effects being present in the vector field model but being of equal and
opposite sign. As with the case of the isolated anisotropic shear
contribution, it seems this is not possible whilst maintaining the other
constraints like the existence of a growing mode.
#### VI.1.5 Effective minimal coupling to the gravitational field
Even if the vector field growing mode gives an appropriate (dark matter-like)
contribution to the Poisson equation, the link between the overdensity and the
corresponding $\Phi$ can differ from the CDM case. This difference can come
from curvature terms in the vector field stress energy tensor, and its main
consequence is a time-dependent rescaling of Newton’s constant $G$. Thus
$\Phi$ may gain a time dependence during the matter era even there is a
completely standard dark matter contribution to the Poisson equation. The
converse may also be possible: the time dependence of an incorrect Poisson
contribution could be counteracted by a time dependence of the effective $G$.
### VI.2 Being Dark Energy is easy
As a model for dark energy the Generalized Einstein-Aether theory is more
successful: we have obtained constraints on its parameters and found that it
generates spectra that fit the data across a reasonable range of its parameter
space. It is clear that modified gravity approaches to explaining late-time
acceleration are viable and can provide motivated explanations for dark energy
(though this theory retains the co-incidence problem in the guise of the
parameter $M$).
#### VI.2.1 Closeness to $\Lambda$
The most interesting constraint on this branch of the theory is on the
parameter $n_{\mathrm{ae}}$ and is shown in figure 6. In some sense this
parameter describes how closely the theory mimics $\Lambda$ (which has
$n_{\mathrm{ae}}=0$). The fact that this parameter is rather free,
$n_{\mathrm{ae}}<0.126$, (95% CL), is consistent with the fact that a wide
variety of other theories can also explain dark energy: present structure
formation data is not very informative about the nature of dark energy, and
deeper require expansion probes like baryon acoustic oscillation and
supernovae.
#### VI.2.2 Sound speed
The other notable constraint on Einstein-Aether dark energy, which may extend
to other modified gravity approaches, is the limits on the sound speed,
illustrated in figure 7. As shown in figure 11, an incorrect sound speed can
lead to large scale oscillations by modifying the other parameters of the
theory and permitting a growing mode excitation at late time.
#### VI.2.3 Other constraints
The other constraints on the theory (which are easily fulfilled by choosing
the $c$ parameters) come from ensuring that no growing mode can disrupt the
power spectra, that the acceleration is close to the $\Lambda$CDM value, and
that the value of effective $G$ remains positive at all times.
### VI.3 Future issues for modified gravity and structure formation
Because $\Lambda$CDM is such a good fit to current cosmological data, modified
gravity will never be favoured in a model comparison exercise using only
current data about linear structure. It is only in combination with physics on
galactic and smaller scales that it can be be persuasive. This work highlights
a few issues for future model-building in this vein.
The Generalized Einstein Aether model is a member of a class of models in
which the scale $M\sim H_{0}$ associated with Dark Energy is visible to dark
matter. Many of the issues raised here will be relevant to any such models
which try to use a dark matter scale consistent with small-scale modifications
to gravity.
A combination of probes sensitive only to the background (like Type 1A
supernovae) and to the behavior of cosmological perturbations is needed to
fully constrain these theories. For example, a value $n_{\mathrm{ae}}=0.3$ has
$w(z)\sim 1$ at low redshift but is ruled out by our constraints. Similarly
there are models with $n_{\mathrm{ae}}\sim 0.75$ that provide reasonable
spectra, but they are ruled out by $w(z)$ constraints.
There are, of course, a number of extensions to the theory and features of it
that could be change; we could, for example, allow $M$ to vary dynamically, or
add more terms to the kinetic component $F$ in equation (19). There are also
myriad possibilities in more changing more general aspects of the primordial
conditions or cosmological parameters: what happens if we add add tensors? Can
we include an isocurvature mode? Would massive neutrinos help? Or curvature?
This leads us to a key caveat that applies to this and all similar work
constraining new physics with linear structure: a simple constraint from the
data alone is worthless, since any of the numerous other parameters we could
change might conspire to counteract whatever problem it solves. We need a
physical explanation of a constraint’s origin to understand whether it is
robust to the cosmologist’s tinkering.
Acknowledgments: We thank Constantinos Skordis and David Jacobs for useful
discussions. GDS was supported by a grant to the CWRU particle/astrophysics
theory group from the US DOE. JZ is supported by an STFC rolling grant. TGZ is
supported by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported by the Government of Canada through Industry
Canada and by the Province of Ontario through the Ministry of Research &
Innovation.
## References
* Zlosnik et al. (2007) T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys. Rev. D75, 044017 (2007), eprint astro-ph/0607411.
* Jiménez and Maroto (2008) J. B. Jiménez and A. L. Maroto, Phys. Rev. D 78, 063005 (2008), eprint 0801.1486.
* Dimopoulos et al. (2009) K. Dimopoulos, M. Karciauskas, and J. M. Wagstaff, ArXiv e-prints (2009), eprint 0907.1838.
* Koivisto and Mota (2008) T. S. Koivisto and D. F. Mota, JCAP 0808, 021 (2008), eprint 0805.4229.
* Milgrom (1983) M. Milgrom, Astrophys. J. 270, 365 (1983).
* Zhao (2007) H. Zhao, Astrophys. J. Lett. 671, L1 (2007), eprint 0710.3616.
* Ferreira and Starkman (2009) P. G. Ferreira and G. D. Starkman, Science 326, 812 (2009), eprint 0911.1212.
* Milgrom (2009) M. Milgrom, Phys. Rev. D80, 123536 (2009), eprint 0912.0790.
* Blanchet and Tiec (2009) L. Blanchet and A. L. Tiec, Phys. Rev. D80, 023524 (2009), eprint 0901.3114.
* Blanchet and Tiec (2008) L. Blanchet and A. L. Tiec, Phys. Rev. D78, 024031 (2008), eprint 0804.3518.
* Soussa and Woodard (2004) M. E. Soussa and R. P. Woodard, Phys. Lett. B578, 253 (2004), eprint astro-ph/0307358.
* Bruneton et al. (2009) J.-P. Bruneton, S. Liberati, L. Sindoni, and B. Famaey, JCAP 0903, 021 (2009), eprint 0811.3143.
* Sagi (2009) E. Sagi, Phys. Rev. D80, 044032 (2009), eprint 0905.4001.
* Contaldi et al. (2008) C. R. Contaldi, T. Wiseman, and B. Withers, Phys. Rev. D 78, 044034 (2008), eprint 0802.1215.
* Carroll et al. (2009) S. M. Carroll, T. R. Dulaney, M. I. Gresham, and H. Tam, Phys. Rev. D 79, 065011 (2009), eprint 0812.1049.
* Eling (2006) C. Eling, Phys. Rev. D73, 084026 (2006), eprint gr-qc/0507059.
* Lim (2005) E. A. Lim, Phys. Rev. D71, 063504 (2005), eprint astro-ph/0407437.
* Dai et al. (2008) D. Dai, R. Matsuo, and G. Starkman, Phys. Rev. D 78, 104004 (2008), eprint 0806.4319.
* Zlosnik et al. (2008) T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys. Rev. D77, 084010 (2008), eprint 0711.0520.
* Zuntz et al. (2008) J. A. Zuntz, P. G. Ferreira, and T. G. Zlosnik (2008), eprint 0808.1824.
* Jacobson (2007) T. Jacobson, PoS QG-PH, 020 (2007), eprint 0801.1547.
* Zhang et al. (2007) P. Zhang, M. Liguori, R. Bean, and S. Dodelson, Physical Review Letters 99, 141302 (2007), eprint 0704.1932.
* Bean et al. (2007) R. Bean, D. Bernat, L. Pogosian, A. Silvestri, and M. Trodden, Phys. Rev. D 75, 064020 (2007), eprint arXiv:astro-ph/0611321.
* Bekenstein (2004) J. D. Bekenstein, Phys. Rev. D70, 083509 (2004), eprint astro-ph/0403694.
* Eling and Jacobson (2006) C. Eling and T. Jacobson, Classical and Quantum Gravity 23, 5625 (2006), eprint arXiv:gr-qc/0603058.
* Famaey and Binney (2005) B. Famaey and J. Binney, Mon. Not. Roy. Astron. Soc. 363, 603 (2005), eprint astro-ph/0506723.
* Dodelson and Liguori (2006) S. Dodelson and M. Liguori, Phys. Rev. Lett. 97, 231301 (2006), eprint astro-ph/0608602.
* Dai et al. (2008) D.-C. Dai, R. Matsuo, and G. Starkman, Phys. Rev. D78, 104004 (2008), eprint 0806.4319.
* Ma and Bertschinger (1995) C.-P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995), eprint astro-ph/9506072.
* Li et al. (2008a) B. Li, D. Fonseca Mota, and J. D. Barrow, Phys. Rev. D77, 024032 (2008a), eprint 0709.4581.
* Doran (2005) M. Doran, JCAP 0510, 011 (2005), eprint astro-ph/0302138.
* Zlosnik (2009) T. Zlosnik, In preparation (2009).
* Doran and Mueller (2004) M. Doran and C. M. Mueller, JCAP 0409, 003 (2004), eprint astro-ph/0311311.
* Komatsu et al. (2009) E. Komatsu et al. (WMAP), Astrophys. J. Suppl. 180, 330 (2009), eprint 0803.0547.
* Adelman-McCarthy et al. (2008) J. K. Adelman-McCarthy et al. (SDSS), Astrophys. J. Suppl. 175, 297 (2008), eprint 0707.3413.
* Ferreira et al. (2008) P. G. Ferreira, C. Skordis, and C. Zunckel, Phys. Rev. D78, 044043 (2008), eprint 0806.0116.
* Skordis et al. (2006) C. Skordis, D. F. Mota, P. G. Ferreira, and C. Boehm, Phys. Rev. Lett. 96, 011301 (2006), eprint astro-ph/0505519.
* Angus (2008) G. W. Angus (2008), eprint 0805.4014.
* Zlosnik et al. (2006) T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys. Rev. D74, 044037 (2006), eprint gr-qc/0606039.
* Skordis (2009) C. Skordis, Class. Quant. Grav. 26, 143001 (2009), eprint 0903.3602.
* Skordis (2006) C. Skordis, Phys. Rev. D74, 103513 (2006), eprint astro-ph/0511591.
* Bourliot et al. (2007) F. Bourliot, P. G. Ferreira, D. F. Mota, and C. Skordis, Phys. Rev. D75, 063508 (2007), eprint astro-ph/0611255.
* Bean (2009) R. Bean, ArXiv e-prints (2009), eprint 0909.3853.
* Carroll and Lim (2004) S. M. Carroll and E. A. Lim, Phys. Rev. D70, 123525 (2004), eprint hep-th/0407149.
* Seifert (2007) M. D. Seifert, Phys. Rev. D76, 064002 (2007), eprint gr-qc/0703060.
* Li et al. (2008b) B. Li, J. D. Barrow, D. F. Mota, and H. Zhao, Phys. Rev. D78, 064021 (2008b), eprint 0805.4400.
* Cardone and Radicella (2009) V. F. Cardone and N. Radicella, Phys. Rev. D80, 063515 (2009), eprint 0908.0095.
## Appendix A The MOND regime F(K) cannot yield late-time acceleration
The MOND branch of $F$ applies in the regime $0\leq K\ll 1$; outside this
range MOND does not prescribe its form. Inside that range the MOND value is:
$\displaystyle c_{1}F$ $\displaystyle=$
$\displaystyle-2K+\frac{2}{3}K^{\frac{3}{2}}$ (56) $\displaystyle(0\leq$
$\displaystyle K$ $\displaystyle\ll 1)$
The weak field limit of the Einstein Aether theory is:
$K^{\mathrm{ae}}_{\mathrm{WF}}=-c_{1}\frac{(\nabla\phi)^{2}}{M^{2}},$ (57)
where $\phi$ is the conformal Newtonian potential. The weak field MOND limit
is:
$K^{\mathrm{MOND}}_{\mathrm{WF}}=\frac{(\nabla\phi)^{2}}{a_{0}^{2}}.$ (58)
Clearly we can equate these by setting M${}^{2}=-c_{1}a_{0}^{2}$. Using this
relation with equation (14) gives us this value for the present-day
cosmological $K$:
$K_{\mathrm{FRW}}(t_{0})=-3\frac{\alpha}{c_{1}}\frac{H_{0}^{2}}{a_{0}^{2}}$
(59)
There are now two cases: $\alpha/c_{1}<0$ and $\alpha/c_{1}>0$. In either case
we need the measured values of $H_{0}$ and $a_{0}$; they are suggestively
similar:
$\displaystyle H_{0}$ $\displaystyle\equiv$ $\displaystyle\kappa a_{0}$
$\displaystyle\kappa$ $\displaystyle\approx$ $\displaystyle 6$ (60)
where
If $\alpha/c_{1}<0$ then $K>0$ and we are directly in the MOND regime. Then we
obtain the modified Friedman equation:
$H^{2}\left(1+\frac{\alpha}{c_{1}}+\frac{3}{2}\kappa^{2}\left(\frac{H}{H_{0}}\right)\left(\frac{-3\alpha}{c_{1}}\right)^{\frac{3}{2}}\right)=8\pi
G\rho$ (61)
A self-accelerating solution to this is only possible if the quantity in
brackets is positive definite, so that $H\rightarrow$ _const_ as
$\rho\rightarrow 0$. This could only happen if $\alpha/c1<-1$, but that would
violate our requirements that $K\ll 1$.
In the other case $\alpha/c_{1}>0$ we must extrapolate the MOND form of $F$ in
equation (A) to $K<0$. To make the extension continuous across the $K=0$, we
should set $F(-K)=-F(K)$, so that the MOND form becomes:
$\displaystyle c_{1}F$ $\displaystyle=$
$\displaystyle-2K-\frac{2}{3}(-K)^{\frac{3}{2}}$ (62) $\displaystyle(0\leq$
$\displaystyle-K$ $\displaystyle\ll 1)$
The Friedman equation then becomes:
$H^{2}\left(1+\frac{\alpha}{c_{1}}-\frac{3}{2}\kappa^{2}\left(\frac{H}{H_{0}}\right)\left(\frac{3\alpha}{c_{1}}\right)^{\frac{3}{2}}\right)=8\pi
G\rho$ (63)
which does have an accelerating solution. Unfortunately, solving this equation
at the present day for reasonable values of $\Omega_{m}$ shows that it
requires $K\sim 2$, which again violates our requirement that $|K|\ll 1$.
Having exhausted our other options we are forced to require $K\geq 1$, outside
the true MOND regime. This is consistent with another separate analysis of the
solutions of (16) Cardone and Radicella (2009). It seems likely that if the
vector field is responsible for the late time acceleration then it is a result
of behavior of the function away from the MONDian limit. There is still,
though a role for the near numerical coincidence of $a_{0}$ and $H_{0}$ \- it
lessens the fine tuning of the other parameters in $F$ in order for the
acceleration to happen at suitably late times.
## Appendix B Vector And Tensor Modes
In this appendix, we provide the perturbed equations of motion for the vector
and tensor modes of the various fields appearing in these models, namely the
metric, the vector field and the Lagrange multiplier. The latter, being
scalar, only have a spin$-0$ mode. The other two fields, $g_{ij}$ and $A^{i}$
are perturbed as
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-a^{2}d\tau^{2}+a^{2}B_{i}d\tau
dx^{i}+a^{2}(\gamma_{ij}+h_{ij})dx^{i}dx^{j}$ $\displaystyle h_{ij}$
$\displaystyle=$ $\displaystyle 2\partial_{(i}E_{j)}+2E_{ij}$ $\displaystyle
A^{\mu}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{a},\frac{A^{i}}{a}\right),$ $\displaystyle
A_{\mu}$ $\displaystyle=$ $\displaystyle\left(-a,aV_{i}\right),$ (64)
where the different fields introduced satisfy
$\partial_{i}B^{i}=0=\partial^{j}E_{ij}=\partial_{i}E^{i}=E^{i}_{i}\text{ and
}\partial_{i}A^{i}=0.$ (65)
We also introduce the useful quantity $V^{i}=A^{i}+B^{i}$ and remind that all
the latin indices on the perturbed fields are raised and lowered thanks to the
Kronecker flat metric $\delta_{ij}$.
In the following we use the unperturbed results
$\displaystyle K$ $\displaystyle=$ $\displaystyle
3M^{-2}\mathcal{H}^{2}\alpha,$ $\displaystyle J^{0}_{\ 0}=6c_{2}\mathcal{H}$ ,
$\displaystyle J^{i}_{\ j}=2\mathcal{H}\alpha\delta^{i}_{j},$ $\displaystyle
I^{\sigma}_{00}=6c_{2}\mathcal{H}\delta^{\sigma}_{0}$ , $\displaystyle
I^{\sigma}_{0i}=0,\
I^{\sigma}_{ij}=-2\mathcal{H}\alpha\delta^{\sigma}_{0}\delta_{ij}.$ (66)
The Einstein Equation (5) without matter introduces the stress tensor
$\displaystyle\tilde{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nabla_{\sigma}\left[I^{\sigma}_{\alpha\beta}\right]+\hat{T}_{\alpha\beta}$
$\displaystyle\hat{T}_{\alpha\beta}$ $\displaystyle=$ $\displaystyle-
F_{K}Y_{(\alpha\beta)}+\frac{1}{2}M^{2}g_{\alpha\beta}\mathcal{F}+\lambda
A_{\alpha}A_{\beta},$
where
$I^{\sigma}_{\alpha\beta}=F_{K}\left[J_{(\alpha}^{\ \
\sigma}A_{\beta)}-J^{\sigma}_{\
(\alpha}A_{\beta)}-J_{(\alpha\beta)}A^{\sigma}\right].$ (67)
As intermediate results we have
$\delta K^{\alpha\beta}_{\ \ \ \gamma\delta}=c_{1}\left(\delta
g^{\alpha\beta}g_{\gamma\delta}+g^{\alpha\beta}\delta
g_{\gamma\delta}\right),$ (68)
$\delta\left(\nabla_{\alpha}A^{\gamma}\right)=\delta^{\gamma}_{i}\partial_{\alpha}\left[\frac{A^{i}}{a}\right]+\Gamma^{\gamma}_{\alpha
k}\frac{A^{k}}{a}+\frac{\delta\Gamma^{\gamma}_{\alpha 0}}{a},$ (69) $\delta
Y_{0i}=\delta Y_{i0}=c_{1}\left(\mathcal{H}V_{i}+V_{i}^{\prime}\right),$ (70)
$\displaystyle\delta\hat{T}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M^{2}Fa^{2}h_{ij},$ $\displaystyle\delta\hat{T}_{0i}$
$\displaystyle=$ $\displaystyle\frac{1}{2}a^{2}B_{i}M^{2}F-\lambda
a^{2}V_{i}-F_{K}\delta Y_{(0i)},$ (71) $\displaystyle\delta J^{0}_{\ i}$
$\displaystyle=$
$\displaystyle\frac{2\mathcal{H}c_{3}V_{i}}{a}-\frac{2c_{1}V_{i}^{\prime}}{a},$
$\displaystyle\delta J^{\ i}_{0}$ $\displaystyle=$ $\displaystyle-\delta
J^{0}_{\ i}+\frac{2\mathcal{H}}{a}(c_{1}+c_{3})B^{i},$ (72)
$\displaystyle\delta J^{\ 0}_{i}$ $\displaystyle=$ $\displaystyle-\delta
J^{j}_{\ 0}\delta_{ij}+\frac{2\mathcal{H}}{a}(c_{1}+c_{3})B_{i},$ (73)
$\displaystyle\delta J^{i}_{\ 0}$ $\displaystyle=$ $\displaystyle
2\frac{c_{13}B^{i}}{a}\mathcal{H}-2\frac{c_{1}\mathcal{H}}{a}V^{i}+\frac{2c_{3}V^{i^{\prime}}}{a};$
(74)
and
$\displaystyle\delta J^{i}_{\ j}$ $\displaystyle=$ $\displaystyle
2c_{1}\left[\frac{\partial^{i}A_{j}}{a}+\frac{h^{i^{\prime}}_{j}}{2a}+\frac{\partial^{[i}B_{j]}}{a}\right],$
(75)
$\displaystyle+2c_{3}\left[\frac{\partial_{j}A^{i}}{a}+\frac{h^{i^{\prime}}_{j}}{2a}+\frac{\partial_{[j}B^{i]}}{a}\right].$
We then obtain,
$\displaystyle\delta I^{\sigma}_{0i}$ $\displaystyle=$ $\displaystyle
F_{K}\left[2\mathcal{H}c_{3}V_{i}-2c_{1}V_{i}^{\prime}-2\mathcal{H}\alpha
B_{i}\right]\delta^{\sigma}_{0}$
$\displaystyle+2F_{K}(c_{1}-c_{3})\partial^{[k}V_{i]}$
with $\alpha=c_{1}+3c_{2}+c_{3}$, and
$\displaystyle\delta I^{i}_{00}$ $\displaystyle=$ $\displaystyle
F_{K}[-2c_{1}V^{i^{\prime}}-2c_{1}\mathcal{H}V^{i}+2\mathcal{H}c_{3}V^{i}$
(77) $\displaystyle+2c_{3}V^{i^{\prime}}+6c_{2}\mathcal{H}A^{i}].$
We also get
$\delta I^{\sigma}_{ij}=-\delta^{\sigma}_{0}F_{K}\delta J_{(ij)}.$ (78)
To obtain the perturbed Einstein equation, we then plug these results into the
relations
$\displaystyle\delta\nabla_{\sigma}I^{\sigma}_{0i}$ $\displaystyle=$
$\displaystyle\partial_{0}\left(\delta
I^{0}_{0i}\right)+\partial_{k}\left(\delta
I^{k}_{0i}\right)+2\mathcal{H}\delta I^{0}_{0i}$
$\displaystyle-(B^{j^{\prime}}+\mathcal{H}B^{j})I^{0}_{ji}-6\mathcal{H}^{2}c_{2}B_{i}F_{K}-\Gamma^{0}_{ji}\delta
I^{j}_{00},$ $\displaystyle\delta\nabla_{\sigma}I^{\sigma}_{ij}$
$\displaystyle=$ $\displaystyle\partial_{0}\left(\delta
I^{0}_{ij}\right)+2\mathcal{H}\delta
I^{0}_{ij}-I^{0}_{kj}\delta\Gamma^{k}_{0i}-\mathcal{H}\delta_{ki}\delta
I^{k}_{0j}$
$\displaystyle-\delta\Gamma^{k}_{0j}I^{0}_{ik}-\mathcal{H}\delta_{kj}\delta
I^{k}_{i0}.$
We decompose these equations into Fourier components:
$\displaystyle X^{i}(t,\overrightarrow{x})$ $\displaystyle=$
$\displaystyle\sum_{\overrightarrow{k}}\sum_{m=0,1}X(t,\overrightarrow{k})Y^{i(\pm
m)}_{\overrightarrow{k}},$ $\displaystyle T^{ij}(t,\overrightarrow{x})$
$\displaystyle=$
$\displaystyle\sum_{\overrightarrow{k}}\sum_{m=0,1,2}T(t,\overrightarrow{k})Y^{ij(\pm
m)}_{\overrightarrow{k}}.$ (79)
where the orthonormal modes
$Y^{(0)},\ Y^{(\pm 1)},\ Y^{(\pm 2)}$ (80)
are eigenmodes of the Laplace-Beltrami operator: $\Delta
Y_{\mathbf{I}}^{(m)}=-k^{2}Y_{\mathbf{I}}^{(m)}$, $\mathbf{I}$ being an
arbitrary set of Lorentz indices. For more information on these functions, see
Lim (2005). The perturbed vector field equation can then be written
$\displaystyle 2\lambda aV$ $\displaystyle=$ $\displaystyle
F_{KK}K^{\prime}\left(2\mathcal{H}c_{3}V-2c_{1}V^{\prime}\right)-2c_{1}F_{K}V^{\prime\prime}$
(81)
$\displaystyle-4F_{K}c_{1}\mathcal{H}V^{\prime}+V\left(2\frac{a^{\prime\prime}}{a}c_{3}F_{K}+\mathcal{H}^{2}(2c_{3}+2c_{1})F_{K}\right)$
$\displaystyle-
F_{K}\left(2c_{1}k^{2}A+(c_{1}-c_{3})k^{2}B\right)+3F_{K}c_{2}k^{2}E^{\prime}.$
The spin$-2$ part of the $ij$ Einstein equations gives
$E^{{}^{\prime\prime}}+2\mathcal{H}E^{{}^{\prime}}+\frac{k^{2}E}{1+F_{K}(c_{1}+c_{3})}+2F_{KK}K^{\prime}(c_{1}+c_{3})E^{{}^{\prime}}=0,$
(82)
while the spin$-1$ part of the ${}^{0}_{i}$ Einstein equations gives the same
equation as (81) and the spin$-1$ component of the $ij$ Einstein equation is,
in space conventions,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{(i}E_{j)}^{{}^{\prime\prime}}+\frac{2(1-2\alpha)\mathcal{H}+2c_{13}\frac{1}{a^{2}}(a^{2}F_{K})^{\prime}+4\alpha\mathcal{H}F_{K}}{1+2F_{K}c_{13}}\partial_{(i}E_{j)}^{{}^{\prime}}$
(83)
$\displaystyle-\frac{\partial_{(i}B_{j)}^{{}^{\prime}}-c_{13}F_{K}\partial_{(i}A_{j)}^{{}^{\prime}}}{1+2F_{K}c_{13}}$
$\displaystyle-\frac{2(2\mathcal{H}^{\prime}+\mathcal{H}^{2})+M^{2}\mathcal{F}a^{2}+4\alpha({\cal
H}F_{K})^{\prime}+8\mathcal{H}^{2}\alpha\mathcal{F}}{1+2F_{K}c_{13}}\partial_{(i}E_{j)}$
$\displaystyle-\frac{2\mathcal{H}\partial_{(i}B_{j)}}{1+2F_{K}c_{13}}$
$\displaystyle+\frac{c_{13}F_{KK}K^{\prime}+2c_{13}\mathcal{H}F_{K}}{1+2F_{K}c_{13}}\partial_{(i}A_{j)}.$
In order to compare with the results of Lim (2005) let us consider the gauge
$E_{j}=0$. The spin$-1$ part of the $ij$ Einstein equations (83) implies
$B_{i}=c_{13}F_{K}A_{i}.$ (84)
After that we can rewrite (81) as
$\displaystyle A^{{}^{\prime\prime}}$ $\displaystyle=$
$\displaystyle-\left(2\mathcal{H}+\frac{F_{KK}}{c_{1}F_{K}}K^{{}^{\prime}}\right)A^{{}^{\prime}}-\frac{F_{K}}{2c_{1}}\frac{2c_{1}+F_{K}(c_{1}^{2}-c_{3}^{2})}{1+F_{K}c_{13}}k^{2}A$
$\displaystyle+\left[\mathcal{H}\frac{c_{3}F_{KK}}{c_{1}F_{K}}K^{{}^{\prime}}+\frac{a^{\prime\prime}}{a}\frac{c_{3}}{c_{1}}+\mathcal{H}^{2}\frac{c_{13}}{c_{1}}-\frac{\lambda
a}{F_{K}c_{1}}\right]A$
and obtaining the exact expression of $\lambda$ in the case $F(K)=K$,
$\lambda=3c_{13}\frac{\mathcal{H}^{2}}{a}-3c_{2}\frac{a^{\prime\prime}}{a^{2}}+6c_{2}\frac{\mathcal{H}^{2}}{a},$
(85)
implies in this latter case
$\displaystyle c_{1}\left(2\mathcal{H}A^{\prime}+A^{\prime\prime}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left[c_{1}-c_{3}+\frac{c_{13}}{1+c_{13}}\right]k^{2}A-$
(86)
$\displaystyle\left[2\alpha\mathcal{H}^{2}-\alpha\frac{a^{\prime\prime}}{a}+c_{1}\frac{a^{\prime\prime}}{a}\right]A.$
It is interesting to find that in the limit where $\mathcal{F}(K)=K$ we
recover results from Lim (2005) for (82,B) and (84).
A quick look at (82) and (B) implies the existence of two different speeds
$\displaystyle C_{T}^{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{1+F_{K}c_{13}},$ $\displaystyle C_{V}^{2}$
$\displaystyle\equiv$
$\displaystyle\frac{F_{K}}{2c_{1}}\frac{2c_{1}+F_{K}(c_{1}^{2}-c_{3}^{2})}{1+F_{K}c_{13}},$
(87)
being the sound speeds respectively of the tensor perturbation $E_{ij}$ and
the vector perturbation $A_{i}$. These speeds must be positive, so that it
implies constraints on the parameters of the theory and the function $F(K)$.
In particular, $F_{K}\neq\text{const.}$ will imply non-trivial constraints
between the fields of the theory and the parameters $c_{i}$:
$\left\\{\begin{array}[]{ccc}1+F_{K}c_{13}&>&0\\\
F_{K}\left[1+F_{K}\frac{c_{1}^{2}-c_{3}^{2}}{2c_{1}}\right]&>&0\end{array}\right.$
(88)
|
arxiv-papers
| 2010-02-03T21:09:09 |
2024-09-04T02:49:08.184871
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Zuntz, T. G. Zlosnik, F. Bourliot, P. G. Ferreira, G. D. Starkman",
"submitter": "Joseph Zuntz",
"url": "https://arxiv.org/abs/1002.0849"
}
|
1002.0879
|
September 2008
# Coherence for Categorified Operadic Theories
Miles Gould
To my father
$1+2+\dots+n={n\over 2}(n+1)$
## Abstract
Given an algebraic theory which can be described by a (possibly symmetric)
operad $P$, we propose a definition of the _weakening_ (or _categorification_)
of the theory, in which equations that hold strictly for $P$-algebras hold
only up to coherent isomorphism. This generalizes the theories of monoidal
categories and symmetric monoidal categories, and several related notions
defined in the literature. Using this definition, we generalize the result
that every monoidal category is monoidally equivalent to a strict monoidal
category, and show that the “strictification” functor has an interesting
universal property, being left adjoint to the forgetful functor from the
category of strict $P$-categories to the category of weak $P$-categories. We
further show that the categorification obtained is independent of our choice
of presentation for $P$, and extend some of our results to many-sorted
theories, using multicategories.
###### Contents
1. Abstract
2. Acknowledgements
3. Declaration
4. 0 Introduction
1. 0.1 Remarks on notation
5. 1 Theories
1. 1.1 Syntactic approach
2. 1.2 Clones
3. 1.3 Lawvere theories
4. 1.4 Finitary monads
5. 1.5 Equivalences
6. 2 Operads
1. 2.1 Plain operads
2. 2.2 Symmetric operads
3. 2.3 Finite product operads
4. 2.4 Adjunctions
5. 2.5 Existence and monadicity
6. 2.6 Explicit construction of ${F_{\rm pl}}$ and ${F_{\Sigma}^{\rm pl}}$
7. 2.7 Syntactic characterization of the forgetful functors
8. 2.8 Operads and syntactic classes of theories
9. 2.9 Enriched operads and multicategories
10. 2.10 Maps of algebras as algebras for a multicategory
7. 3 Factorization Systems
8. 4 Categorification
1. 4.1 Desiderata
2. 4.2 Categorification of strongly regular theories
3. 4.3 Examples
4. 4.4 A more general approach: factorization systems
5. 4.5 Examples
6. 4.6 Symmetric monoidal categories
7. 4.7 Multicategories
8. 4.8 Examples
9. 4.9 Evaluation
9. 5 Coherence
1. 5.1 Strictification
2. 5.2 Universal property of st
3. 5.3 Presentation-independence
10. 6 Other Approaches
1. 6.1 Pseudo-algebras for 2-monads
2. 6.2 Laplaza sets
3. 6.3 Non-algebraic definitions
11. Index
###### List of Figures
1. 2.1 Composition in a multicategory
2. 2.2 Composition in the operad $\mathcal{S}$ of symmetries
3. 2.3 Composition in the little 2-discs operad
4. 2.4 “Combing out” the -action
5. 2.5 Grafting of trees
6. 2.6 Composition in $\mathcal{S}\times P$
7. 4.1 Part of ${{\mbox{Wk($1$)}}}_{3}$
8. 4.2 A multigraph
9. 4.3 A multigraph enriched in directed graphs
## Acknowledgements
First and foremost, I must thank my supervisor, Dr Tom Leinster, for his help
and encouragement with all aspects of this thesis and my research. I could not
have wished for a better supervisor. I would like to thank Steve Lack for
invaluable help with pseudo-algebras: the argument of Theorem 6.1.9 is due to
him (any errors, of course, are mine). I would like to thank Michael Batanin
for suggesting I consider the construction of Definition 4.4.1 in the context
of symmetric operads. I would like to thank Jeff Egger and Colin Wright for
invaluable motivational advice, which in both cases came exactly when it was
most needed. I would like to thank Jon Cohen for suggesting Examples 4.5.2 and
4.5.3, and Hitesh Jasani for helping me to see the benefits of isolating the
concept of labelling functions. I would like to thank the night staff at
Schiphol airport, for providing quite the best environment for doing
mathematics I’ve ever encountered. I would like to thank Wilson Sutherland,
both for his excellent teaching of undergraduate mathematics and for
encouraging me to apply for this PhD, and Samson Abramsky and Bob Coecke, for
first showing me the beauty of category theory. Thanks to all those who
commented on drafts of this thesis: Rami Chowdhury, Malcolm Currie, Susannah
Fleming, Cath Howdle, John Kirk, Avril Korman and Michael Prior-Jones. Thanks
to Hannah Johnson for Sumerological ratification. I am grateful to EPSRC, for
funding this research.
Much of the challenge of this PhD has been retaining some semblance of sanity
throughout, so the people below are those who have provided welcome
distraction (as opposed to the unwelcome kind). Thanks must go to Ruth Elliot,
my co-organizer for the Scottish Juggling Convention 2008, who did far more
than her share despite being in recovery from a serious motorbike accident; to
the rest of Glasgow Juggling Club (and to Alia Sheikh for first teaching me to
juggle); to the hillwalking and rock climbing crowd, namely Katie Edwards,
Martin Goodman, Michael Jenkins, Andy Miller, Elsie Riley, Jo Stewart, Richard
Vale, Bart Vlaar, Dan and Becca Winterstein, Stuart White, and especially
Philipp Reinhard; to my office-mates, James Ferguson, Martin Hamilton, Gareth
Vaughan and John Walker, for the good times; to the members of NO2ID Scotland,
particularly Geraint Bevan, Richard Clay, James Hammerton, Alex Heavens, Bob
Howden, Jaq Maitland, Roddy McLachlan, Charlotte Morgan, and John Welford; to
the students and instructors at Glasgow Capoeira; and to all at the theatre
group Two Shades of Blue. I would like to extend sincere and heartfelt thanks
to the Chinese emperor Shen Nung, the Ethiopian goat herder Kaldi, the
Sumerian goddess Ninkasi, and the unnamed Irish monk who, according to legend,
discovered or invented tea, coffee, beer and whisky respectively.
I would like to thank my parents, Dick and Jackie Gould, for their patience
and support, and particularly my father for showing me that mathematics could
be beautiful in the first place. Finally, I would like to thank my wonderful
girlfriend Ciorstaidh MacGlone, who (when not contending for the computer) has
been an endless source of love, sympathy, support and tea.
## Declaration
I declare that this thesis is my own original work, except where credited to
others. This thesis does not include work forming part of a thesis presented
for another degree.
## Chapter 0 Introduction
Many definitions exist of categories with some kind of “weakened” algebraic
structure, in which the defining equations hold only up to coherent
isomorphism. The paradigmatic example is the theory of weak monoidal
categories, as presented in [ML98], but there are also definitions of
categories with weakened versions of the structure of groups [BL04], Lie
algebras [BC04], crossed monoids [Age02], sets acted on by a monoid [Ost03],
rigs [Lap72], vector spaces [KV94] and others. A general definition of such
categories-with-weakened-structure is obviously desirable, but hard in the
general case. In this thesis, we restrict our attention to the case of
theories that can be described by (possibly symmetric) operads, and present
possible definitions of weak $P$-category and weak $P$-functor for any
symmetric operad $P$. We show that this definition is independent (up to
equivalence) of our choice of presentation for $P$; this generalizes the
equivalence of classical and unbiased monoidal categories. In support of our
definition, we present a generalization of Joyal and Street’s result from
[JS93] that every weak monoidal category is monoidally equivalent to a strict
monoidal category: this holds straightforwardly when $P$ is a plain operad.
This generalization includes the classical theorem that every symmetric
monoidal category is equivalent via symmetric monoidal functors and
transformations to a symmetric monoidal category whose associators and unit
maps are identities.
The idea is to consider the strict models of our theory as algebras for an
operad, then to obtain the weak models as (strict) algebras for a weakened
version of that operad (which will be a Cat-operad). In particular, we do not
make use of the pseudo-algebras of Blackwell, Kelly and Power, for which see
[BKP89]. Their definition is related to ours in the non-symmetric case,
however: we explore the connections in Chapter 6. We weaken the operad using a
similar approach to that used in Penon’s definition of $n$-category: see
[Pen99], or [CL04] for a non-rigorous summary.
In Chapters 1, 2 and 3, we review some essential background material on
theories, operads and factorization systems. Most of this is well-known, and
only one result (in Section 2.8) is new. In Chapter 4, we present our
definitions of weak $P$-category, weak $P$-functor and $P$-transformation. We
start with a naïve, syntactic definition that is only effective for strongly
regular (plain-operadic) theories. We then re-state this definition using the
theory of factorization systems, which allows us to apply it to the more
general symmetric operads. Section 4.6 uses this definition to explicitly
calculate the categorification of the theory of commutative monoids with their
standard signature, and shows that this is exactly the classical theory of
symmetric monoidal categories. In Chapter 5, we treat the problem of different
presentations of a given operad: we use this to prove that the weakening of a
given theory is independent of the choice of presentation. We also prove some
theorems about strictification of weak $P$-categories. In Chapter 6, we
compare our approach to other approaches to categorification which have been
proposed in the literature.
Material in this thesis has appeared in two previous papers: the material on
strictification for strongly regular theories was in my preprint [Gou06], and
the material on signature-independence was in my paper [Gou07], which was
presented at the 85th Peripatetic Seminar on Sheaves and Logic in Nice in
March 2007, and at CT 2007 in Carvoeiro, Portugal.
### 0.1 Remarks on notation
Throughout this thesis, the set of natural numbers is taken to include 0. We
shall occasionally adopt the ∙ notation from chain complexes and write, for
instance, $p_{\bullet}$ for a finite sequence $p_{1},\dots,p_{n}$ and
$p_{\bullet}^{\bullet}$ for a double sequence. We shall use the notation
${\underline{n}}$ to refer to the set $\\{1,\dots,n\\}$ for all
$n\in\natural$: the set ${\underline{0}}$ is the empty set. We shall use the
symbol 1 to refer to terminal objects of categories and identity arrows, as
well as to the first nonzero natural number; it is my hope that no confusion
results.
## Chapter 1 Theories
The first step will be to obtain a mathematical description of the notion of
an algebraic theory, of which the familiar theories of groups, rings, modules
etc. are examples. In this chapter, we present some standard ways of doing
this, and prove that they are equivalent. The most convenient description for
our purposes will be the notion of _clone_ , which appears to have been
introduced by Philip Hall in unpublished lecture notes in the 1960s, and may
be found on [Coh65] page 132, under the name “abstract clone”. The treatment
here follows [Joh94]. The remainder of the material in this chapter is all
well-known, and may be found in e.g. [Bor94] chapters 3 and 4, or [AR94]
chapter 3.
In the next chapter, we shall describe _operads_ , which allow us to capture
certain algebraic theories in an especially simple way, suitable for
categorification, and we shall show how operads relate to the clones described
in this chapter.
### 1.1 Syntactic approach
The most traditional way of formalizing algebraic theories is syntactic. In
this approach, we abstract from the standard “operations plus equations”
description (used to describe e.g. the theory of groups) to create
presentations of algebraic theories, and define a notion of an algebra for a
presentation.
###### Definition 1.1.1.
A signature $\Phi$ is an object of Set.
In other words, a signature is a sequence of sets
$\Phi_{0},\Phi_{1},\Phi_{2},\dots$.
Fix a countably infinite set $X=\\{x_{1},x_{2},\dots,\\}$, whose elements we
call variables. Throughout, let $\Phi$ be a signature.
###### Definition 1.1.2.
Let $n\in\natural$. An $n$-ary $\Phi$-term is defined by the following
inductive clauses:
* •
$x_{1},x_{2},\dots,x_{n}$ are $n$-ary terms.
* •
If $\phi\in\Phi_{m}$ and $t_{1},\dots,t_{m}$ are $n$-ary terms, then
$\phi(t_{1},\dots,t_{m})$ is an $n$-ary term.
A $\Phi$-term is an $n$-ary $\Phi$-term for some $n\in\natural$.
###### Definition 1.1.3.
Let $t$ be an $n$-ary $\Phi$-term. Then $\mathop{\rm{var}}(t)$ is the sequence
of elements of $\\{x_{1},\dots,x_{n}\\}$ given as follows:
* •
$\mathop{\rm{var}}(x_{i})=(x_{i})$,
* •
$\mathop{\rm{var}}(\phi(t_{1},\dots,t_{n}))=\mathop{\rm{var}}(t_{1})\mathrel{++}\mathop{\rm{var}}(t_{2})\mathrel{++}\dots\mathrel{++}\mathop{\rm{var}}(t_{n})$,
where $\mathrel{++}$ is concatenation.
###### Definition 1.1.4.
Let $t$ be an $n$-ary $\Phi$-term. Then $\mathop{\rm{supp}}(t)$, the support
of $t$, is the subset of $\\{x_{1},\dots,x_{n}\\}$ given as follows:
* •
$\mathop{\rm{supp}}(x_{i})=\\{x_{i}\\}$,
* •
$\mathop{\rm{supp}}(\phi(t_{1},\dots,t_{n}))=\mathop{\rm{supp}}(t_{1})\cup\mathop{\rm{supp}}(t_{2})\cup\dots\cup\mathop{\rm{supp}}(t_{n})$,
###### Definition 1.1.5.
Let $t$ be an $n$-ary $\Phi$-term, with
$\mathop{\rm{var}}(t)=(x_{i_{1}},\dots,x_{i_{m}})$. The labelling function
$\mathop{\rm{label}}(t)$ of $t$ is the function
${\underline{m}}\to{\underline{n}}$ sending $j$ to $i_{j}$.
###### Definition 1.1.6.
An $n$-ary $\Phi$-equation is a pair $(s,t)$ of $n$-ary $\Phi$-terms. A
$\Phi$-equation is an $n$-ary $\Phi$-equation for some $n\in\natural$.
###### Definition 1.1.7.
An $n$-ary term $t$ is linear if $\mathop{\rm{label}}(t)$ is a bijection, and
strongly regular if $\mathop{\rm{label}}(t)$ is an identity. An equation
$(s,t)$ is linear if both $s$ and $t$ are linear, and strongly regular if both
$s$ and $t$ are strongly regular.
In other words, a term is linear if every variable is used exactly once, and
strongly regular if every variable is used exactly once in the order
$x_{1},\dots,x_{n}$. Up to trivial relabellings, an equation is linear if
every variable is used exactly once on both sides, though not necessarily in
the same order: an example is the commutative equation
$x_{1}.x_{2}=x_{2}.x_{1}$. An equation is strongly regular if every variable
is used exactly once in the same order on both sides. An example is the
associative equation $x_{1}.(x_{2}.x_{3})=(x_{1}.x_{2}).x_{3}$, though some
care is needed. Strictly, a $\Phi$-equation is a pair $(n,(s,t))$ where
$n\in\natural$ and $s,t$ are $n$-ary $\Phi$-terms. The equation
$(3,((x_{1}.x_{2}).x_{3},x_{1}.(x_{2}.x_{3})))$ is strongly regular, but the
equation $(4,((x_{1}.x_{2}).x_{3},x_{1}.(x_{2}.x_{3})))$ is not.
Classically, an $n$-ary equation $(s,t)$ is regular if
$\mathop{\rm{label}}(t)$ and $\mathop{\rm{label}}(s)$ are surjections:
however, we will not consider regular equations further. The term “linear” is
borrowed from linear logic, and the term “strongly regular” is due to Carboni
and Johnstone (from [CJ95]).
###### Definition 1.1.8.
A presentation of a (one-sorted) algebraic theory is
* •
a signature $\Phi$,
* •
a set $E$ of $\Phi$-equations.
Elements of $\Phi_{n}$ are called ($n$-ary) generating operations.
###### Definition 1.1.9.
Let $P=(\Phi,E)$ be a presentation of an algebraic theory. $P$ is linear if
every equation in $E$ is linear, and strongly regular if every equation in $E$
is strongly regular.
We will return to the consideration of linear and strongly regular
presentations once we have defined operads.
###### Definition 1.1.10.
Let $\Phi$ be a signature. An algebra for $\Phi$ is
* •
a set $A$,
* •
for each $n$-ary operation $\phi$, a map $\phi_{A}:A^{n}\to A$. These are
called the primitive operations of the algebra $A$.
Let $\Phi$ be a signature, and $A$ a $\Phi$-algebra. Each $n$-ary $\Phi$-term
$t$ gives rise to an $n$-ary derived operation $t_{A}:A^{n}\to A$, defined
recursively as follows:
* •
if $t=x_{i}$, then $t_{A}$ is projection onto the $i$th factor,
* •
if $t=\phi(t_{1},\dots,t_{m})$, then $t_{A}$ is the composite
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.43066pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-8.43066pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.7996pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{((t_{1})_{A},\dots,(t_{m})_{A})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.43066pt\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}}{\hbox{\kern
32.43066pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 62.43066pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
85.0329pt\raise 6.11389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{\phi_{A}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 104.84753pt\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}}{\hbox{\kern
104.84753pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{A}$}}}}}}}}}}}\ignorespaces.$
Let $\mathop{\rm{term}}_{n}{\Phi}$ denote the set of $n$-ary derived
operations over $\Phi$. Then $\mathop{\rm{term}}{\Phi}$ is a signature for
every signature $\Phi$. A morphism of signatures $f:\Phi\to\Psi$ induces a map
$\bar{f}:\mathop{\rm{term}}\Phi\to\mathop{\rm{term}}\Psi$. Indeed,
$\mathop{\rm{term}}$ is an endofunctor on Set, and in Section 2.8 we shall
show that it is actually a monad.
###### Definition 1.1.11.
Let $P=(\Phi,E)$ be a presentation of an algebraic theory. A $P$-algebra is a
$\Phi$-algebra $A$ such that, for every equation $(s,t)$ in $E$, the derived
operations $s_{A},t_{A}$ are equal.
An algebra for $\Phi$ is an algebra for $(\Phi,\\{\\})$. Conversely, every
algebra for $(\Phi,E)$ is an algebra for $\Phi$.
###### Definition 1.1.12.
Let $\Phi$ be a signature, and $A$ and $B$ be $\Phi$-algebras. A morphism of
$\Phi$-algebras $f:A\to B$ is a map $f:A\to B$ which commutes with every
primitive operation:
$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{n}}$$\scriptstyle{\phi_{A}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{B}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
for every $n\in\natural$ and every $n$-ary primitive operation $\phi$. If
$P=(\Phi,E)$ is a presentation, then a morphism of $P$-algebras is a morphism
of $\Phi$-algebras.
By an easy induction, a morphism of $\Phi$-algebras will commute with every
derived operation too.
Given a presentation $P$, there is a category Alg$(P)$ whose objects are
$P$-algebras and whose arrows are $P$-algebra morphisms. We shall call a
category $\mathcal{C}$ a variety of algebras (or simply a variety) if
$\mathcal{C}$ is isomorphic to Alg$(P)$ for some presentation $P$.
We will need to consider closures of sets of equations; the idea is that the
closure of $E$ contains the members of $E$ and all of their consequences.
###### Definition 1.1.13.
Let $t$ be an $n$-ary $\Phi$-term, and $t_{1},\dots,t_{n}$ be $\Phi$-terms.
Then the graft $t(t_{1},\dots,t_{n})$ is the $\Phi$-term defined recursively
as follows.
* •
If $t=x_{i}$, then $t(t_{1},\dots,t_{n})=t_{i}$.
* •
If $t=\phi(s_{1},\dots,s_{m})$, where $\phi\in\Phi_{m}$ and
$s_{1},\dots,s_{m}$ are $n$-ary $\Phi$-terms, then
$t(t_{1},\dots,t_{n})=\phi((s_{1}(t_{1},\dots,t_{n})),\dots,s_{m}(t_{1},\dots,t_{n}))$.
###### Definition 1.1.14.
Let $\Phi$ be a signature and $E$ be a set of $\Phi$-equations. The closure
$\bar{E}$ of $E$ is the smallest equivalence relation on
$\mathop{\rm{term}}\Phi$ which contains $E$ and is closed under grafting of
terms:
* •
if $(s,t)\in\bar{E}$, then
$(s(t_{1},\dots,t_{n}),t(t_{1},\dots,t_{n}))\in\bar{E}$ for all
$t_{1},\dots,t_{n}$.
* •
if $(s_{i},t_{i})\in\bar{E}$ for $i=1,\dots,n$, then
$(t(s_{1},\dots,s_{n}),t(t_{1},\dots,t_{n}))\in\bar{E}$ for all $t$.
### 1.2 Clones
Clones attempt to capture theories directly: a clone is to a presentation of
an algebraic theory as a group is to a presentation of that group.
###### Definition 1.2.1.
A clone $K$ is
* •
a sequence of sets $K_{0},K_{1},\dots$,
* •
for all $m,n\in\natural$, a function $\bullet:K_{n}\times(K_{m})^{n}\to
K_{m}$,
* •
for each $n\in\natural$ and each $i\in\\{1,\dots,n\\}$, an element
$\delta^{i}_{n}\in K_{n}$
such that
* •
for each $f\in K_{n}$, $g_{1},\dots,g_{n}\in K_{m}$, $h_{1},\dots,h_{m}\in
K_{p}$,
$f\bullet(g_{1}\bullet(h_{1},\dots,h_{m}),\dots,g_{n}\bullet(h_{1},\dots,h_{m}))=(f\bullet(g_{1},\dots,g_{n}))\bullet(h_{1},\dots,h_{m})$
* •
for all $n$, all $i\in{1,\dots,n}$ and all $f_{1},\dots,f_{n}\in K_{m}$,
$\delta^{i}_{n}\bullet(f_{1},\dots,f_{n})=f_{i}$
* •
for all $n$ and $f\in K_{n}$,
$f\bullet(\delta_{n}^{1},\dots,\delta_{n}^{n})=f$
###### Example 1.2.2.
Let $\mathcal{C}$ be a finite product category, and $A$ be an object of
$\mathcal{C}$. The endomorphism clone of $A$, $\mathop{\rm{End}}(A)$, is
defined as follows:
* •
$\mathop{\rm{End}}(A)_{n}=\mathcal{C}(A^{n},A)$ for each $n\in\natural$,
* •
for all $n\in\natural$ and $i\in\\{1,\dots,n\\}$, the map $\delta^{i}_{n}$ is
the projection of $A^{n}$ onto its $i$th factor,
* •
for all $n,m\in\natural$, all $f\in\mathop{\rm{End}}(A)_{n}$, and all
$g_{1},\dots,g_{n}\in\mathop{\rm{End}}(A)_{m}$, the morphism
$f\bullet(g_{1},\dots,g_{n})$ is the composite $fh$, where $h$ is the unique
arrow $A^{m}\to A^{n}$ induced by the maps $g_{1},\dots,g_{n}$ and the
universal property of $A^{n}$.
###### Definition 1.2.3.
A morphism of clones $f:K\to K^{\prime}$ is a map in Set which commutes with
the composition operations and $\delta$s.
###### Definition 1.2.4.
Let $K$ be a clone, and $\mathcal{C}$ a finite product category with specified
finite powers. An algebra for $K$ in $\mathcal{C}$ is an object
$A\in\mathcal{C}$ and a morphism of clones $K\to\mathop{\rm{End}}(A)$.
Equivalently, an algebra for a clone $K$ in a finite product category
$\mathcal{C}$ with specified powers is
* •
an object $A$ of $\mathcal{C}$,
* •
for each $n\in\natural$ and each $k\in K_{n}$, a morphism $\hat{k}:A^{n}\to A$
such that
* •
for all $n\in\natural$ and all $i\in\\{1,\dots n\\}$, the morphism
$\widehat{\delta}^{i}_{n}$ is the projection of $A^{n}$ onto its $i$th factor;
* •
for all $n,m\in\natural$, all $f\in K_{n}$, and all $g_{1},\dots,g_{n}\in
K_{m}$, the diagram
---
|
$\textstyle{A^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g_{1}}}$$\scriptstyle{h}$$\scriptstyle{\widehat{g_{n}}}$$\scriptstyle{\widehat{f\bullet(g_{1},\dots,g_{n})}}$$\textstyle{A}$$\textstyle{A}$$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\delta}^{1}_{n}}$$\scriptstyle{\widehat{\delta}^{n}_{n}}$$\scriptstyle{f}$$\textstyle{A}$
commutes, where $h$ is the unique arrow induced by the universal property of
$A^{n}$.
###### Definition 1.2.5.
Let $A$ and $B$ be algebras for a clone $K$ in a finite product category
$\mathcal{C}$ with specified finite powers. A morphism of algebras $A\to B$ is
a morphism $F:A\to B$ in $\mathcal{C}$ such that the diagram
$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{n}}$$\scriptstyle{\hat{k}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{k}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{B}$
commutes for all $n\in\natural$ and all $k\in K_{n}$.
Algebras for a clone and their morphisms form a category: we call this
category $\mbox{{{Alg}}}_{\mathcal{C}}(K)$, or Alg$(K)$ in the case where
$\mathcal{C}=\mbox{{{Set}}}$.
Clones can be enriched in any finite product category $\mathcal{V}$ in an
obvious way: the sequence of sets $K_{0},K_{1},\dots$ becomes a sequence of
objects of $\mathcal{V}$, and so on.
### 1.3 Lawvere theories
Lawvere theories are a particularly elegant approach to describing algebraic
theories, introduced by Lawvere in his thesis [Law63]. Like a clone, a Lawvere
theory (sometimes called a finite product theory) is an object that represents
the semantics of the theory directly; in Lawvere theories, the data are
encoded into a category. Algebras for the theory are then certain functors
from the Lawvere theory to Set.
###### Definition 1.3.1.
A Lawvere theory is a category $\mathcal{T}$ whose objects form a denumerable
set $\\{\bf{0},\bf{1},\bf{2},\dots\\}$, such that $\bf{n}$ is the $n$-th power
of $\bf{1}$. A morphism of Lawvere theories $\mathcal{T}\to\mathcal{S}$ is an
identity-on-objects functor $\mathcal{T}\to\mathcal{S}$ which preserves
projection maps. The category of Lawvere theories and their morphisms is
called Law. An algebra for $\mathcal{T}$ is a functor
$F:\mathcal{T}\to\mbox{{{Set}}}$ which preserves finite products. A morphism
of algebras is a natural transformation. The category of
$\mathcal{T}$-algebras is the full subcategory of
$[\mathcal{T},\mbox{{{Set}}}]$ whose objects are finite-product-preserving
functors.
Lawvere theories encode algebraic theories by storing the $n$-ary operations
of the theory as morphisms $\bf{n}\to\bf{1}$.
We can consider algebras for Lawvere theories in categories other than Set: an
algebra for a Lawvere theory $\mathcal{T}$ in a finite product category
$\mathcal{C}$ is just a finite-product-preserving functor
$\mathcal{T}\to\mathcal{C}$. This captures our usual notions of, for instance,
topological groups: a topological group is just an algebra for the Lawvere
theory of groups in the category Top. Much the same could be said for clones
and presentations, of course, but in this case the definition is especially
economical.
We may generalize this definition as follows:
###### Definition 1.3.2.
Let $S$ be a set. An $S$-sorted finite product theory is a small finite
product category whose underlying monoidal category is strict and whose monoid
of objects is the free monoid on $S$. Elements of $S$ will be called sorts.
Algebras and morphisms of algebras are defined as above.
### 1.4 Finitary monads
Recall that a monad on a category $\mathcal{C}$ is a monoid object in the
category $[\mathcal{C},\mathcal{C}]$ of endofunctors on $\mathcal{C}$.
Concretely, a monad is a triple $(T,\mu,\eta)$ where
* •
$T:\mathcal{C}\to\mathcal{C}$ is a functor,
* •
$\mu:T^{2}\to T$ is a natural transformation,
* •
$\eta:1_{\mathcal{C}}\to T$ is a natural transformation,
and $\mu,\eta$ satisfy coherence axioms which are analogues of the usual
associativity and unit laws for monoids, namely
---
$\textstyle{T^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu
T}$$\scriptstyle{T\mu}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T}$
(1.1)
---
$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\eta}$$\scriptstyle{1_{T}}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta
T}$$\scriptstyle{1_{T}}$$\textstyle{T}$ (1.2)
We shall often abuse notation and refer to the monad $(T,\mu,\eta)$ as simply
$T$.
###### Definition 1.4.1.
Let $(T_{1},\mu_{1},\eta_{1}),(T_{2},\mu_{2},\eta_{2})$ be monads on a
category $\mathcal{C}$. A morphism of monads
$(T_{1},\mu_{1},\eta_{1})\to(T_{2},\mu_{2},\eta_{2})$ is a natural
transformation $\alpha:T_{1}\to T_{2}$ such that the diagrams
$\textstyle{T_{1}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\alpha*\alpha}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T_{2}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{T_{2}}$
(1.3)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{2}}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T_{2}}$
(1.4)
commute.
Monads on $\mathcal{C}$ and monad morphisms form a category
$\mbox{{{Mnd}}}(\mathcal{C})$. This notion (or rather, a 2-categorical
version) was introduced and studied by Street in [Str72].
###### Definition 1.4.2.
A category $\mathcal{C}$ is filtered if every finite diagram in $\mathcal{C}$
admits a cocone.
Equivalently, $\mathcal{C}$ is filtered if:
* •
$\mathcal{C}$ is nonempty;
* •
for every pair of parallel arrows
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$
in $\mathcal{C}$, there is an arrow $h:B\to C$ such that $hf=hg$;
* •
for every pair of objects $A,B$, there is an object $C$ and arrows
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$
Filteredness generalizes the notion of directedness for posets (a directed
poset is a poset in which every finite subset has an upper bound). A filtered
category which is also a poset is precisely a directed poset.
###### Definition 1.4.3.
A filtered colimit in a category $\mathcal{C}$ is the colimit of a diagram
$D:\I\to\mathcal{C}$, where is a filtered category.
###### Theorem 1.4.4.
Every object in Set is a filtered colimit of finite sets.
###### Proof.
Let $X\in\mbox{{{Set}}}$, and consider the subcategory of Set whose objects
are finite subsets of $X$ and whose morphisms are inclusions. This is a
directed poset, and thus a filtered category. $X$ is the colimit of the
inclusion of into Set. ∎
###### Theorem 1.4.5.
Let be a small category. Colimits of shape in Set commute with all finite
limits iff is filtered.
###### Proof.
See [MLM92], Corollary VII.6.5. ∎
###### Definition 1.4.6.
A functor $F:\mathcal{C}\to\mathcal{D}$ is finitary if it preserves filtered
colimits.
###### Definition 1.4.7.
A monad $(T,\mu,\eta)$ on $\mathcal{C}$ is finitary if $T$ is finitary.
A finitary monad on Set is determined by its behaviour on finite sets, in the
following sense: since every set $X$ is a filtered colimit of its finite
subsets, then $TX$ has to be the colimit of the images under $T$ of the finite
subsets of $X$.
### 1.5 Equivalences
Let $(\Phi,E)$ be a presentation of an algebraic theory. We define
$K_{(\Phi,E)}$ to be the clone whose operations are elements of the quotient
signature $(\mathop{\rm{term}}\Phi)/\bar{E}$, with composition given by
grafting, and $\delta^{i}_{n}=x_{i}$ for all $i,n\in\natural$. By definition
of $\bar{E}$, grafting gives a well-defined family of composition functions on
$K_{(\Phi,E)}$. Conversely, given a clone $K$, we may define a presentation of
an algebraic theory $(\Phi_{K},E_{K})$, by taking $(\Phi_{K})_{n}=K_{n}$ for
all $n\in\natural$, and for all $n,m\in\natural$, all $k\in K_{n}$ and all
$k_{1},\dots,k_{n}\in K_{m}$, letting $E_{m}$ contain the equation
$k(k_{1}(x_{1},\dots,x_{m}),\dots,k_{n}(x_{1},\dots,x_{m}))=k\bullet(k_{1},\dots,k_{n})(x_{1},\dots,x_{m}).$
###### Lemma 1.5.1.
Let $K$ be a clone. Then $K_{(\Phi_{K},E_{K})}$ is isomorphic to $K$.
###### Proof.
See [Joh94], Lemma 1.7. ∎
###### Lemma 1.5.2.
Let $(\Phi,E)$ be a presentation of an algebraic theory. Let
$(\Phi^{\prime},E^{\prime})$ be the presentation obtained from the clone
$K_{(\Phi,E)}$. Then the category Alg$(\Phi,E)$ is isomorphic to the category
Alg$(\Phi^{\prime},E^{\prime})$
###### Proof.
See [Joh94], Lemma 1.8. ∎
###### Definition 1.5.3.
Let $K$ be a clone. We say that $K$ is strongly regular (resp. linear) if
there exists a strongly regular (resp. linear) presentation $P$ such that
$K=K_{(\Phi,E)}$.
Given a clone $K$, we construct a Lawvere theory $\mathcal{T}_{K}$ for which
$\mathcal{T}_{K}({\bf n},{\bf m})=(K_{n})^{m}$. Suppose
$f=(f_{1},\dots,f_{m})\in\mathcal{T}_{K}(\bf{n},\bf{m})$ and
$g=(g_{1},\dots,g_{p})\in\mathcal{T}_{K}(\bf{m},\bf{p})$, then the composite
$gf$ is
$(g_{1}\bullet(f_{1},\dots,f_{m}),\dots,g_{p}\bullet(f_{1},\dots,f_{m}))$. By
the axioms for a clone, this is a category, with the identity map on $\bf{n}$
being $(\delta^{1}_{n},\dots,\delta^{n}_{n})$. It remains to show that ${\bf
n}$ is the $n$th power of $\bf 1$ for every $n\in\natural$. The $i$th
projection of $\bf n$ onto $1$ is evidently $\delta^{i}_{n}$: we must show
that these have the requisite universal property. Take $m,n\in\natural$, and
$n$ maps $f_{1},\dots,f_{n}:{\bf m}\to{\bf 1}$ in $\mathcal{T}_{K}$. The
diagram
$\textstyle{{\bf
m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{h}$$\scriptstyle{f_{n}}$$\textstyle{{\bf
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{1}_{n}}$$\scriptstyle{\delta^{n}_{n}}$$\textstyle{{\bf
1}}$$\textstyle{\dots}$$\textstyle{{\bf 1}}$
commutes if and only if $h=(f_{1},\dots,f_{n})$, and hence ${\bf n}$ is indeed
the $n$th power of ${\bf 1}$, and so $\mathcal{T}_{K}$ is a Lawvere theory.
The Lawvere theories so constructed evidently respect isomorphisms of clones.
Furthermore, the diagram
---
$\textstyle{\@ensuremath{\mbox{{{Clone}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{T}_{(-)}}$Alg$\textstyle{\@ensuremath{\mbox{{{Law}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Alg$\textstyle{\@ensuremath{\mbox{{{CAT}}}}^{\rm{op}}}$
commutes up to equivalence:
###### Theorem 1.5.4.
Let $K$ be a clone. Then
$\mbox{{{Alg$(K)$}}}\simeq\mbox{{{Alg$(\mathcal{T}_{K})$}}}$.
###### Proof.
Let $A$ be a $K$-algebra. We define a $\mathcal{T}_{K}$-algebra $F_{A}$ as
follows:
* •
$F_{A}{\bf n}=A^{n}$ for all $n\in\natural$;
* •
If $k\in\mathcal{T}_{K}({\bf n},1)=K_{n}$, then $F_{A}k=\hat{k}$;
* •
if $(k_{1},\dots,k_{n}):{\bf m}\to{\bf n}$ in $\mathcal{T}_{K}$, then
$F_{A}(k_{1},\dots,k_{n})$ is the unique arrow $A^{m}\to A^{n}$ such that the
diagram
$\textstyle{A^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{k}_{1}}$$\scriptstyle{\widehat{k}_{n}}$$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\delta^{1}_{n}}}$$\scriptstyle{\widehat{\delta_{n}^{n}}}$$\textstyle{A}$$\textstyle{\dots}$$\textstyle{A}$
commutes.
Let $f:A\to B$ be a morphism of $K$-algebras. Then the diagram
$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{n}}$$\scriptstyle{\widehat{k}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{k}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
commutes for all $n\in\natural$ and all $k\in K_{n}$. By the universal
property of $B^{m}$, the diagram
$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{n}}$$\scriptstyle{F_{A}(k_{1},\dots,k_{m})}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{B}(k_{1},\dots,k_{m})}$$\textstyle{A^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{m}}$$\textstyle{B^{m}}$
commutes for all $n,m\in\natural$ and all $(k_{1},\dots,k_{m}):{\bf n}\to{\bf
m}$ in $\mathcal{T}_{K}$. Hence $F_{f}=(f^{n})_{{\bf n}}$ is a natural
transformation $F_{A}\to F_{B}$, and hence a morphism of
$\mathcal{T}_{K}$-algebras. This defines a functor
$F_{(-)}:\mbox{{{Alg$(K)$}}}\to\mbox{{{Alg$(\mathcal{T}_{K})$}}}$; we wish to
show that it is an equivalence.
For every $\mathcal{T}_{K}$-algebra $G$, we may define a $K$-algebra $A$ by
setting $A=G{\bf 1}$ and $\hat{k}=G({\bf n}\stackrel{{\scriptstyle
k}}{{\longrightarrow}}{\bf 1})$ for all $k\in K_{n}$ and all $n\in\natural$.
Then $F_{A}$ is isomorphic as a $\mathcal{T}_{K}$-algebra to $G$, and hence
the functor $F_{(-)}:\mbox{{{Alg$(K)$}}}\to\mbox{{{Alg$(\mathcal{T}_{K})$}}}$
is essentially surjective on objects. We shall show further that it is full
and faithful. Let $A$ and $B$ be $K$-algebras, and let $\alpha_{n}:F_{A}\to
F_{B}$ be a morphism between their associated $\mathcal{T}_{K}$-algebras.
Since the diagram
$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n}}$$\scriptstyle{\widehat{\delta^{i}_{n}}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\delta^{i}_{n}}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\textstyle{B}$
commutes for all $n\in\natural$ and all $i\in\\{1,\dots,n\\}$, it must be the
case that $\alpha_{n}=\alpha_{1}^{n}$ for all $n$. Hence, the diagram
$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}^{n}}$$\scriptstyle{\widehat{k}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{k}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\textstyle{B}$
must commute for all $n\in\natural$ and all $k\in K_{n}$. So $\alpha_{1}$ is a
$K$-algebra morphism, and $\alpha_{n}=F_{\alpha_{1}}$. Hence $F_{(-)}$ is
full. Suppose $F_{f}=F_{g}$; then $(F_{f})_{\bf 1}=(F_{g})_{\bf 1}$, so $f=g$.
Hence $F_{(-)}$ is faithful; and hence it is an equivalence of categories. ∎
Given a Lawvere theory $\mathcal{T}$, we can construct a clone
$K_{\mathcal{T}}$, as follows:
* •
Let $(K_{\mathcal{T}})_{n}=\mathcal{T}(\bf{n},\bf{1})$ for all $n\in\natural$.
* •
For all $n,m\in\natural$, all $f\in(K_{\mathcal{T}})_{n}$ and all
$g_{1},\dots,g_{n}\in(K_{\mathcal{T}})_{m}$, let
$f\bullet(g_{1},\dots,g_{n})=f\circ(g_{1}+\dots+g_{n})\circ\Delta$, where
$(g_{1}+\dots+g_{n})$ is the unique map ${\bf mn}\to{\bf n}$ in $\mathcal{T}$
such that the diagram
---
$\textstyle{{\bf
mn}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}+\dots+g_{n}}$$\textstyle{{\bf
m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{{\bf
m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{n}}$$\textstyle{{\bf
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
1}}$$\textstyle{\dots}$$\textstyle{{\bf 1}}$
commutes, and $\Delta:{\bf m}\to{\bf mn}$ is the diagonal map (or
equivalently, the image of the codiagonal function ${\bf mn}\to{\bf m}$ under
the contravariant embedding of into $\mathcal{T}$).
* •
For all $n\in\natural$ and all $i\in{1,\dots,n}$, let $\delta_{n}^{i}$ be the
$i$th projection $\bf{n}\to\bf{1}$.
This extends to a functor $K_{(-)}:\mbox{{{Law}}}\to\mbox{{{Clone}}}$, as
follows: given Lawvere theories $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, and a
morphism of Lawvere theories $F:\mathcal{T}_{1}\to\mathcal{T}_{2}$, let
$K_{F}$ be the map of signatures sending
$k\in(K_{\mathcal{T}_{1}})_{n}=\mathcal{T}_{1}(\bf{n},\bf{1})$ to
$Fk\in(K_{\mathcal{T}_{2}})_{n}=\mathcal{T}_{2}(\bf{n},\bf{1})$. Since $F$ is
a functor, and thus commutes with composition in
$\mathcal{T}_{1},\mathcal{T}_{2}$, then $K_{F}$ must commute with composition
in $K_{\mathcal{T}_{1}}$ and $K_{\mathcal{T}_{2}}$. Since $F$ preserves finite
products, it commutes with the projection maps in $K_{\mathcal{T}_{1}}$ and
$K_{\mathcal{T}_{2}}$. Thus, $K_{F}$ is a morphism of clones.
###### Theorem 1.5.5.
The functor $K_{(-)}$ is pseudo-inverse to the functor $\mathcal{T}_{(-)}$.
###### Proof.
Since every object in a Lawvere theory is a copower of ${\bf 1}$, a Lawvere
theory $\mathcal{T}$ is entirely determined (up to isomorphism) by the hom-
sets $\mathcal{T}({\bf n},{\bf 1})$, and thus by $K_{\mathcal{T}}$. The
theorem follows straightforwardly. ∎
Given a Lawvere theory $\mathcal{T}$, we construct a monad $(T,\mu,\eta)$ on
Set as follows:
* •
If $X$ is a set, let $TX=\int^{n\in\bbF}\mathcal{T}({\bf n},{\bf 1})\times
X^{n}$.
* •
If $x\in X$, then $\eta(x)=(1,x)\in TX$.
* •
If $f:{\bf n}\to{\bf 1}$ in $T$ and
$(f_{i},x^{i}_{\bullet})\in\mathcal{T}({\bf k_{i}},{\bf 1})\times X^{k_{i}}$
for $i=1,\dots,n$, then
$\mu(f,((f_{1},x^{1}_{\bullet}),\dots,(f_{n},x^{n}_{\bullet})))=(f\circ(f_{1}+\dots+f_{n}),x_{\bullet}^{\bullet})$
###### Theorem 1.5.6.
The monad so constructed is finitary.
###### Proof.
See [AR94], Theorems 3.18 and 1.5, and Remarks 3.4(4) and 3.6(6). ∎
Given a finitary monad $T$ on Set, we can construct a Lawvere theory
$\mathcal{T}$. Take the full subcategory T of the Kleisli category
$\mbox{{{Set}}}_{T}$ whose objects are finite sets. Now let $\mathcal{T}$ be
the skeleton of the dual of T. The monad induced by this Lawvere theory is
isomorphic to the original monad: see [AR94], Remark 3.17 and Theorem 3.18.
The moral of the above theorems is that presentations, clones, Lawvere
theories and finitary monads on Set all capture the same notion, and may be
used interchangeably. Further, the notion that is captured corresponds to our
usual intuitive understanding of equational algebraic theories.
The equivalence between (finitary monads on $\mathcal{C}$) and (monads on
$\mathcal{C}$ that may be described by a finitary presentation) may actually
be generalized to the case where $\mathcal{C}$ is an arbitrary finitely
presentable category: see [KP93].
## Chapter 2 Operads
Operads arose in the study of homotopy theory with the work of Boardman and
Vogt [BV73], and May [May72]. In that field they are an invaluable tool:
[MSS02] describes a diverse range of applications. Independently,
multicategories (which are to operads as categories are to monoids) had arisen
in categorical logic with the work of Lambek [Lam69]. Multicategories are
sometimes called “coloured operads”.
We will use multicategories and operads as tools to approach universal
algebra: while operads are not as expressive as Lawvere theories, they can be
easily extended to be so, and the theories that _can_ be represented by
operads provide a useful “toy problem” to help us get started.
Informally, categories have objects and arrows, where an arrow has one source
and one target; multicategories have objects and arrows with one target but
multiple sources (see Fig. 2.1); and operads are one-object multicategories.
Multicategories (and thus operads) have a composition operation that is
associative and unital.
Figure 2.1: Composition in a multicategory
### 2.1 Plain operads
###### Definition 2.1.1.
A plain multicategory (or simply “multicategory”) $\mathcal{C}$ consists of
the following:
* •
a collection $\mathcal{C}_{0}$ of _objects_ ,
* •
for all $n\in\natural$ and all $c_{1},\dots,c_{n},d\in\mathcal{C}_{0}$, a set
of _arrows_ $\mathcal{C}(c_{1},\dots,c_{n};d)$,
* •
for all $n,k_{1},\dots k_{n}\in\natural$ and
$c_{1}^{1}\dots,c_{k_{n}}^{n},d_{1},\dots,d_{n},e\in\mathcal{C}_{0}$, a
function called _composition_
$\circ:\mathcal{C}(d_{1},\dots,d_{n};e)\times\mathcal{C}(c^{1}_{1},\dots,c^{1}_{k_{1}};d_{1})\times\dots\times\mathcal{C}(c^{n}_{1},\dots,c^{n}_{k_{n}};d_{n})\to\mathcal{C}(c^{1}_{1},\dots,c^{n}_{k_{n}};e)$
* •
for all $c\in\mathcal{C}$, an _identity arrow_ $1_{c}\in\mathcal{C}(c;c)$
satisfying the following axioms:
* •
_Associativity:_ $f\circ(g_{\bullet}\circ h_{\bullet}^{\bullet})=(f\circ
g_{\bullet})\circ h_{\bullet}^{\bullet}$ wherever this makes sense (we borrow
the ∙ notation for sequences from chain complexes)
* •
_Units:_ $1\circ f=f=f\circ(1,\dots,1)$ for all $f$.
A plain multicategory $\mathcal{C}$ is small if $\mathcal{C}_{0}$ forms a set.
In line with the definition above, we shall take all our multicategories to be
locally small: this restriction is not essential.
We say that an arrow in $\mathcal{C}(c_{1},\dots,c_{n};d)$ is $n$-ary, or has
arity $n$. We remark that taking $n=0$ gives us _nullary_ arrows. This is in
contrast to the definition used by some authors, who do not allow nullary
arrows.
###### Definition 2.1.2.
A morphism of multicategories $F:\mathcal{C}\to\mathcal{D}$ is a map
$F:\mathcal{C}_{0}\to\mathcal{D}_{0}$ together with maps
$F:\mathcal{C}(c_{1},\dots,c_{n};c)\to\mathcal{D}(Fc_{1},\dots,Fc_{n};Fc)$
which commute with $\circ$ and identities. A transformation of multicategory
maps $\alpha:F\to G$ is a family of arrows $\alpha_{c}\in\mathcal{D}(Fc;Gc)$,
one for each $c\in\mathcal{C}$, satisfying the analogue of the usual
naturality squares: for all maps $f:c_{1},\dots,c_{k}\to c$ in $\mathcal{C}$,
we must have
$\alpha_{c}\circ Ff=Gf\circ(\alpha_{c_{1}},\dots,\alpha_{c_{k}})$
One is tempted to write this last condition as
$\textstyle{Fc_{1},\dots,Fc_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ff}$$\scriptstyle{\alpha_{c_{1}},\dots,\alpha_{c_{k}}}$$\textstyle{Fc\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{c}}$$\textstyle{Gc_{1},\dots,Gc_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gf}$$\textstyle{Gc}$
but care must be taken: in a general multicategory,
$\alpha_{c_{1}},\dots,\alpha_{c_{k}}$ does not correspond to any single map,
as it would in a monoidal category.
Small plain multicategories, their morphisms and their transformations form a
2-category: we shall use the notation Multicat for both this 2-category and
its underlying 1-category.
To simplify the presentation of our first example, we recall the notion of
unbiased monoidal category from [Lei03] section 3.1:
###### Definition 2.1.3.
An unbiased weak monoidal category $(\mathcal{C},\otimes,\gamma,\iota)$
consists of
* •
a category $\mathcal{C}$,
* •
for each $n\in\natural$, a functor $\otimes_{n}:\mathcal{C}^{n}\to\mathcal{C}$
called $n$-fold tensor and written
$(a_{1},\dots,a_{n})\mapsto(a_{1}\otimes\dots\otimes a_{n})$
* •
for each $n,k_{1},\dots,k_{n}\in\natural$, a natural isomorphism
$\gamma:\otimes_{n}\circ(\otimes_{k_{1}}\times\dots\times\otimes_{k_{n}})\longrightarrow\otimes_{\sum
k_{i}}$
* •
a natural isomorphism
$\iota:1_{A}\to\otimes_{1}$
satisfying
* •
associativity: for any triple sequence $a^{\bullet}_{\bullet\bullet}$ of
objects in $\mathcal{C}$, the diagram
---
$\textstyle{(((\otimes a_{11}^{\bullet})\otimes\dots\otimes(\otimes
a_{1k_{1}}^{\bullet}))\otimes\dots\otimes((\otimes
a_{n1}^{\bullet})\otimes\dots\otimes(\otimes
a_{nk_{n}}^{\bullet})))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{((\otimes
a_{1\bullet}^{\bullet})\otimes\dots\otimes(\otimes
a_{n\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{((\otimes
a_{11}^{\bullet})\otimes\dots\otimes(\otimes
a_{nk_{n}}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(a_{11}^{1}\otimes\dots\otimes
a_{nk_{n}}^{m_{k_{n}}})}$
commutes.
* •
identity: for any $n\in\natural$ and any sequence $a_{1},\dots,a_{n}$ of
objects in $\mathcal{C}$, the diagrams
$\textstyle{(a_{1}\otimes\dots\otimes
a_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota\otimes\dots\otimes\iota)}$$\scriptstyle{1}$$\textstyle{((a_{1})\otimes\dots\otimes(a_{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{(a_{1}\otimes\dots\otimes
a_{n})}$ $\textstyle{(a_{1}\otimes\dots\otimes
a_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{1}$$\textstyle{((a_{1}\otimes\dots\otimes
a_{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{(a_{1}\otimes\dots\otimes
a_{n})}$
commute.
###### Example 2.1.4.
Let $\mathcal{C}$ be a locally small unbiased weak monoidal category. The
underlying multicategory $\mathcal{C}^{\prime}$ of $\mathcal{C}$ has
* •
objects: objects of $\mathcal{C}$;
* •
arrows:
$\mathcal{C}^{\prime}(a_{1},\dots,a_{n};b)=\mathcal{C}(a_{1}\otimes\dots\otimes
a_{n},b)$;
* •
composition given as follows: if
$f_{i}\in\mathcal{C}^{\prime}(a_{1}^{i},\dots,a_{k_{i}}^{i};b_{i})$ for
$i=1,\dots,n$ and $g\in\mathcal{C}^{\prime}(b_{1},\dots,b_{n};c)$, then we
define $g\circ(f_{1},\dots,f_{n})$ as
|
---|---
$\textstyle{\bigotimes_{i,j}a^{i}_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\circ(g_{1},\dots,g_{n})}$$\scriptstyle{\gamma\otimes\dots\otimes\gamma}$$\textstyle{\bigotimes_{i}(\bigotimes_{j}a^{i}_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigotimes_{i}{f_{i}}}$$\textstyle{c}$$\textstyle{\bigotimes_{i}b_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$
###### Definition 2.1.5.
Let $M$ and $\mathcal{C}$ be plain multicategories. An algebra for $M$ in
$\mathcal{C}$ is a morphism of multicategories $M\to\mathcal{C}$.
###### Definition 2.1.6.
Let $M$ be a plain multicategory, and $\mathcal{C}$ be an unbiased monoidal
category. An algebra for $M$ in $\mathcal{C}$ is a morphism of multicategories
from $M$ to the underlying multicategory of $\mathcal{C}$.
A plain operad (or simply “operad”) is now a one-object multicategory.
Morphisms and transformations of operads are defined as for general
multicategories. As before, we use the notation Operad for both the 2-category
of operads, morphisms and transformations, and its underlying 1-category.
Operads are to multicategories as monoids are to categories: just as with
monoids, this allows us to present the theory of operads in a simplified way.
###### Lemma 2.1.7.
An operad $P$ can be given by the following data:
* •
A sequence $P_{0},P_{1},\dots$ of sets
* •
For all $n,k_{1},\dots,k_{n}\in\natural$, a function $\circ:P_{n}\times
P_{k_{1}}\times\dots\times P_{k_{n}}\to P_{\sum k_{i}}$
* •
An identity element $1\in P_{1}$
satisfying the following axioms:
* •
_Associativity:_ $f\circ(g_{\bullet}\circ h_{\bullet}^{\bullet})=(f\circ
g_{\bullet})\circ h_{\bullet}^{\bullet}$ wherever this makes sense
* •
_Units:_ $1\circ f=f=f\circ(1,\dots,1)$ for all f.
###### Proof.
Using the symbol $*$ for the unique object, let $P_{n}=P(*,\dots,*;*)$, where
the input is repeated $n$ times. The rest of the conditions follow trivially
from the definition of a multicategory. ∎
###### Lemma 2.1.8.
Let $P$ and $Q$ be operads. A morphism $f:P\to Q$ consists of a function
$f_{n}:P_{n}\to Q_{n}$ for each $n\in\natural$ such that, for all
$n,k_{1},\dots,k_{n}$, the diagram
$\textstyle{P_{n}\times P_{k_{1}}\times\dots\times
P_{k_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}\times
f_{k_{1}}\times\dots\times f_{k_{n}}}$$\scriptstyle{\circ}$$\textstyle{P_{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\sum
k_{i}}}$$\textstyle{Q_{n}\times Q_{k_{1}}\times\dots\times
Q_{k_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{Q_{\sum
k_{i}}}$
commutes, and that $f_{1}$ preserves the identity object.
If $f$ and $g$ are morphisms of operads from $P$ to $Q$, then a transformation
from $f$ to $g$ is an element $\alpha\in Q_{1}$ such that $\alpha\circ
Fp=Gp\circ(\alpha,\dots,\alpha)$ for all $n\in\natural$ and all $p\in P_{n}$.
###### Proof.
Trivial. ∎
###### Definition 2.1.9.
If a morphism of operads $f:P\to Q$ is such that $f_{n}$ has some property $X$
for all $n\in\natural$, we say that $f$ is levelwise $X$.
###### Example 2.1.10.
Let $A$ be an object of a multicategory $\mathcal{C}$. The endomorphism operad
of $A$ is the full sub-multicategory $\mathop{\rm{End}}(A)$ of $\mathcal{C}$
whose only object is $A$. In terms of the description in Lemma 2.1.7,
$\mathop{\rm{End}}(A)_{n}$ is the set of $n$-ary arrows from $A,\dots,A$ to
$A$. Composition is as in $\mathcal{C}$.
In particular, if $\mathcal{C}$ is the underlying multicategory of some
monoidal category $\mathcal{C}^{\prime}$, then
$\mathop{\rm{End}}(A)_{n}=\mathcal{C}^{\prime}(A\otimes\dots\otimes A,A)$.
This is the case we shall use most frequently.
###### Example 2.1.11.
There is an operad $\mathcal{S}$ for which each $\mathcal{S}_{n}$ is the
symmetric group $S_{n}$. Operadic composition is given as follows: if
$\sigma\in S_{n}$, and $\tau_{i}\in S_{k_{i}}$ for $i=1,\dots,n$, then
$\sigma\circ(\tau_{1},\dots,\tau_{n}):\sum_{i=1}^{j}k_{i}+m\mapsto\sum_{i:\sigma(i)<\sigma(j+1)}k_{i}+\tau_{j+1}(m)$
for all $j\in\\{1,\dots,n\\}$ and $m\in\\{0,\dots,k_{j+1}-1\\}$. Informally,
the inputs are divided into “blocks” of length $k_{1},k_{2},\dots,k_{n}$,
which are then permuted by $\sigma$: the elements of each block are then
permuted by the appropriate $\tau_{i}$. For an example, see Figure 2.2.
Figure 2.2: Composition in the operad $\mathcal{S}$ of symmetries
###### Example 2.1.12.
There is an operad $\mathcal{B}$ for which each $\mathcal{B}_{n}$ is the Artin
braid group $B_{n}$. Composition is analogous to that for $\mathcal{S}$: the
inputs are divided into blocks, which are braided, and then the elements of
the blocks are braided.
###### Example 2.1.13.
Fix an $m\in\natural$. There is an operad $LD$ for which each $LD_{n}$ is an
embedding of $n$ copies of the closed unit disc $D_{m}$ into $D_{m}$.
Composition is by gluing – see Figure 2.3.
Figure 2.3: Composition in the little 2-discs operad
$LD$ is known as the little $m$-discs operad.
Since we wish to use operads to represent theories, we need to have some way
of describing the models of those theories.
###### Definition 2.1.14.
Let $P$ be an operad. An algebra for $P$ in a multicategory $\mathcal{C}$ is
an object $A\in\mathcal{C}$ and a morphism of operads
$(\hat{\phantom{\alpha}}):P\to\mathop{\rm{End}}(A)$.
Where $\mathcal{C}$ is a monoidal category, this is equivalent to requiring an
object $A\in\mathcal{C}$, and for each $p\in P_{n}$ a morphism
$\hat{p}:A^{\otimes n}\to A$ such that $\hat{1}=1_{A}$ and
$\hat{p}\circ(\hat{q}_{1}\otimes\dots\otimes\hat{q}_{n})=\widehat{p\circ(q_{1}\otimes\dots\otimes
q_{n})}$ for all $p,q_{1},\dots,q_{n}\in P$. A third equivalent definition is,
for each $n\in\natural$, a map $h_{n}:P_{n}\otimes A^{\otimes n}\to A$, such
that $h_{n}(p,h_{n}(q_{\bullet},-))=h_{\sum k_{i}}(p\circ q_{\bullet},-)$ for
all $p\in P_{n}$, $q_{i}\in P_{k_{i}}$, and $h_{1}(1,-)=1_{A}$. We leave the
proofs of these equivalences as an easy exercise for the reader, and will make
use of whichever formulation is most convenient at the time.
###### Definition 2.1.15.
Let $P$ be a plain operad, and $(A,(\hat{\phantom{\alpha}}))$ and
$(B,(\check{\phantom{\alpha}}))$ be algebras for $P$ in a multicategory
$\mathcal{C}$. A morphism of algebras is an arrow $F:A\to B$ in $\mathcal{C}$
such that, for all $n\in\natural$, the diagram
$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\hat{\phantom{\alpha}})}$$\scriptstyle{(\check{\phantom{\alpha}})}$$\textstyle{\mathop{\rm{End}}(B)_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\circ(F,\dots,F)}$$\textstyle{\mathop{\rm{End}}(A)_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\circ-}$$\textstyle{\@ensuremath{\mathcal{C}}(A,\dots,A;B)}$
commutes.
The definition of morphism may be stated equivalently in terms of any of the
three characterizations of algebras given above.
### 2.2 Symmetric operads
###### Definition 2.2.1.
A symmetric multicategory is a multicategory $\mathcal{C}$ and, for every
$n\in\natural$, every $\sigma\in S_{n}$, and every $A_{1},\dots
A_{n},B\in\mathcal{C}$, a map
$\begin{array}[]{rccl}\sigma\cdot-:&\mathcal{C}(A_{1},\dots,A_{n};B)&\longrightarrow&\mathcal{C}(A_{\sigma
1},\dots,A_{\sigma n};B)\\\ &f&\longmapsto&\sigma\cdot f\end{array}$
such that
* •
For each $f\in\mathcal{C}(A_{1},\dots,A_{n};B)$, $1\cdot f=f$.
* •
For each $\sigma,\rho\in S_{n}$, and each
$f\in\mathcal{C}(A_{1},\dots,A_{n};B)$,
$\rho\cdot(\sigma\cdot f)=(\rho\sigma)\cdot f$
* •
For each permutation $\sigma\in S_{n}$, all objects
$A^{1}_{1},\dots,A^{n}_{k_{n}},B_{1},\dots,B_{n},C\in\mathcal{C}$ and all
arrows $f_{i}\in\mathcal{C}(A^{i}_{1},\dots,A^{i}_{k_{i}};B_{i})$ and
$g\in\mathcal{C}(B_{1},\dots,B_{n};C)$,
$(\sigma\cdot g)\circ(f_{\sigma 1},\dots,f_{\sigma
n})=(\sigma\circ(1,\dots,1))\cdot(g\circ(f_{1},\dots f_{n})).$
* •
For each $A^{1}_{1},\dots,A^{n}_{k_{n}},B_{1},\dots,B_{n},C\in\mathcal{C}$,
$\sigma_{i}\in S_{k_{i}}$ for $i=1,\dots,n$, and each
$f_{i}\in\mathcal{C}(A^{i}_{1},\dots,A^{i}_{k_{i}};B_{i}),g\in\mathcal{C}(B_{1},\dots,B_{n};C)$,
$g\circ(\sigma_{1}\cdot f_{1},\dots,\sigma_{n}\cdot
f_{n})=(1\circ(\sigma_{1},\dots,\sigma_{n}))\cdot(g\circ(f_{1},\dots,f_{n})).$
where $\sigma\circ(1,\dots,1)$ and $1\circ(\sigma_{1},\dots,\sigma_{n})$ are
as defined in Example 2.1.11.
This definition is unusual in that the symmetric groups act on the left rather
than on the right as is more common: however, this change is essential for our
later generalization to finite product multicategories in Section 2.3.
###### Definition 2.2.2.
Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be symmetric multicategories. A
morphism (or map) $F$ of symmetric multicategories is a map
$F:\mathcal{C}_{1}\to\mathcal{C}_{2}$ of multicategories such that
$F(\sigma\cdot f)=\sigma\cdot F(f)$ for all $n\in\natural$, all $n$-ary $f$ in
$\mathcal{C}_{1}$, and all $\sigma\in S_{n}$.
###### Definition 2.2.3.
Let $M$ and $\mathcal{C}$ be symmetric multicategories. An algebra for $M$ in
$\mathcal{C}$ is a morphism of symmetric multicategories $M\to\mathcal{C}$.
###### Definition 2.2.4.
A symmetric operad is a symmetric multicategory with only one object.
In this case, the definition is equivalent to the following:
###### Definition 2.2.5.
A symmetric operad is an operad $P$ together with an action of the symmetric
group $S_{n}$ on each $P_{n}$, which is compatible with the operadic
composition:
---
$\textstyle{P_{n}\times\prod
P_{k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\sigma\cdot-)\times
1\times\dots\times
1}$$\scriptstyle{1\times\sigma_{*}}$$\textstyle{P_{n}\times\prod
P_{k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{P_{n}\times\prod
P_{\sigma
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{P_{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\sigma\circ(1,\dots,1))\cdot-}$$\textstyle{P_{\sum
k_{i}}}$
---
$\textstyle{P_{n}\times\prod
P_{k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times(\rho_{1}\cdot-)\times\dots\times(\rho_{n}\cdot-)}$$\scriptstyle{\circ}$$\textstyle{P_{n}\times\prod
P_{k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{P_{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1\circ(\rho_{1},\dots,\rho_{n}))\cdot-}$$\textstyle{P_{\sum
k_{i}}}$
$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\cdot-}$$\scriptstyle{1}$$\textstyle{P_{n}}$
Maps of symmetric operads are just maps of symmetric multicategories.
###### Example 2.2.6.
The operad $\mathcal{S}$ of symmetric groups, as given in Example 2.1.11. The
action of $S_{n}$ on $\mathcal{S}_{n}$ is given by
$\sigma\cdot\tau=\tau\sigma^{-1}$.
###### Example 2.2.7.
Let $\mathcal{C}$ be a symmetric multicategory, and $A\in\mathcal{C}$. The
symmetric endomorphism operad $\mathop{\rm{End}}(A)$ of $A$ is the full
sub-(symmetric multicategory) of $\mathcal{C}$ whose only object is $A$.
If $\mathcal{C}$ is the underlying symmetric multicategory of a symmetric
monoidal category, then $\mathop{\rm{End}}(A)_{n}=\mathcal{C}(A^{\otimes
n};A)$ for each $n\in\natural$, and the actions of the symmetric groups are
given by composition with the symmetry maps.
###### Definition 2.2.8.
Let $P$ be a symmetric operad. An algebra for $P$ in a multicategory
$\mathcal{C}$ is an object $A$ and a map $h:P\to\mathop{\rm{End}}(A)$ of
symmetric operads. A morphism $(A,h)\to(A^{\prime},h^{\prime})$ of
$P$-algebras is an arrow $F:A\to B$ in $\mathcal{C}$ such that
$h^{\prime}F=Fh$.
As with plain operads, the definitions of an algebra for a symmetric operad
$P$ and of morphisms between those algebras may be stated in several
equivalent ways.
### 2.3 Finite product operads
The definition of categorification in Chapter 4 is couched in terms of
operads. To generalize it, therefore, we might generalize the definition of
operad so that it is capable of expressing every (one-sorted) algebraic
theory.
This generalization is not new: our “finite product operads” were presented by
Tronin under the name “FinSet-operads”. Our Theorem 2.3.12 appears in [Tro02],
and Theorem 2.3.13 appears as Theorem 1.2 in [Tro06]. A fuller treatment was
given by T. Fiore (who called them “the functional forms of theories”) in
[Fio06]. Tronin’s paper constructs an isomorphism between the category of
finite product operads and the category of algebraic clones which commutes
with the forgetful functors to Set; Fiore’s constructs an equivalence between
the category of finite product operads and that of Lawvere theories, and also
shows that this equivalence preserves the categories of algebras.
Let be a skeleton of the category of finite sets and functions, with objects
the sets 0, 1, 2, …, where ${\underline{n}}=\\{1,2,\dots,n\\}$.
###### Definition 2.3.1.
A finite product multicategory is:
* •
A plain multicategory $\mathcal{C}$;
* •
for every morphism $f:{\underline{n}}\to{\underline{m}}$ in , and for all
objects $C_{1},\dots,C_{n},D\in\mathcal{C}$, a function
$f\cdot-:\mathcal{C}(C_{1},\dots,C_{n};D)\to\mathcal{C}(C_{f(1)},\dots,C_{f(n)};D)$
satisfying the following axioms:
* •
the -action is functorial: $f\cdot(g\cdot p)=(f\circ g)\cdot p$, and ${\rm
id}_{{\underline{n}}}\cdot p=p$ wherever these equations make sense;
* •
the -action and multicategorical composition interact by “combing out”:
$(f\cdot p)\circ(f_{1}\cdot p_{1},\dots,f_{n}\cdot
p_{n})=(f\circ(f_{1},\dots,f_{n}))\cdot(p\circ(p_{f(1)},\dots,p_{f(n)}))$
where $(f\circ(f_{1},\dots,f_{n}))$ is given as follows:
Let $f:{\underline{n}}\to{\underline{m}}$, and
$f_{i}:{{\underline{k}}}_{i}\to{{\underline{j}}_{i}}$ for $i=1,\dots,n$. Then
$\begin{array}[]{rcccl}f\circ(f_{1},\dots,f_{n})&:&{\underline{\sum
k_{i}}}&\to&{\underline{\sum j_{i}}}\\\
f\circ(f_{1},\dots,f_{n})&:&\left(\sum_{i=1}^{p-1}k_{f(i)}\right)+h&\mapsto&\left(\sum_{i=1}^{f(p)-1}j_{i}\right)+f_{p}(h)\end{array}$
for all $p\in\\{1,\dots,n\\}$ and all $h\in\\{1,\dots,k_{f(p)}\\}$. See Figure
2.4. The small specks represent inputs to the arrow that are ignored.
Figure 2.4: “Combing out” the -action
It is now possible to see why we chose to have our symmetries acting on the
left in Definition 2.2.1: in this more general case, only a left action is
possible.
###### Definition 2.3.2.
A finite product operad is a finite product multicategory with only one
object.
We will see in Section 2.8 that finite product operads are equivalent in
expressive power to Lawvere theories or clones: hence, every finitary
algebraic theory provides an example of a finite product operad. As before,
the sets $P_{n}$ contain the $n$-ary operations in the theory. For
illustrative purposes, we work out two examples now:
###### Example 2.3.3.
Let $R$ be a ring, and $P_{n}=R[x_{1},\dots,x_{n}]$ (the set of polynomials in
$n$ commuting variables over $R$) for all $n\in\natural$. If $p\in P_{n}$ and
$q_{i}\in P_{k_{i}}$ for $i=1,\dots,n$, then
$(p\circ(q_{1},\dots,q_{n}))(x_{1},\dots,x_{\sum_{i=1}^{n}k_{i}})=p(q_{1}(x_{1},\dots,x_{k_{1}}),\dots,q_{n}(x_{(\sum_{i=1}^{n-1}k_{i})+1},\dots,x_{\sum_{i=1}^{n}k_{i}}))$
and if $f:{\underline{n}}\to{\underline{m}}$, then
$(f\cdot p)(x_{1},\dots,x_{m})=p(x_{f(1)},\dots,x_{f(n)})$
###### Example 2.3.4.
Let $P_{n}$ be the set of elements of the free commutative monoid on $n$
variables $x_{1},\dots,x_{n}$. Elements of $P_{n}$ are in one-to-one-
correspondence with elements of n. We call the $n$th component of $p\in P_{n}$
the multiplicity of the $n$th argument. Composition is defined as follows:
$\left[\begin{array}[]{c}{p_{1}}\\\ \vdots\\\
{p_{n}}\end{array}\right]\circ\left(\left[\begin{array}[]{c}{q^{1}_{1}}\\\
\vdots\\\
{q^{1}_{k_{1}}}\end{array}\right],\dots,\left[\begin{array}[]{c}{q^{n}_{1}}\\\
\vdots\\\
{q^{n}_{k_{n}}}\end{array}\right]\right)=\left[\begin{array}[]{c}{p_{1}q_{1}^{1}}\\\
\vdots\\\ {p_{n}q^{n}_{k_{n}}}\end{array}\right]$
and if $f:{\underline{n}}\to{\underline{m}}$,
$f\cdot\left[\begin{array}[]{c}{p_{1}}\\\ \vdots\\\
{p_{n}}\end{array}\right]=\left[\begin{array}[]{c}{\sum_{f(i)=1}p_{i}}\\\
\vdots\\\ {\sum_{f(i)=m}p_{i}}\end{array}\right]$
Or, in more familiar notation:
$\displaystyle(x_{1}^{p_{1}}\dots x_{n}^{p_{n}})\circ(x_{1}^{q^{1}_{1}}\dots
x_{1}^{q^{1}_{k_{1}}},\dots x_{(\sum_{i=1}^{n-1}k_{i})+1}^{q^{n}_{1}}\dots
x_{\sum_{i=1}^{n}k_{i}}^{q^{n}_{k_{n}}})$ $\displaystyle=$ $\displaystyle
x_{1}^{p_{1}q^{1}_{1}}x_{2}^{p_{1}q^{1}_{1}}\dots
x_{\sum_{i=1}^{n}k_{i}}^{p_{n}q^{n}_{k_{n}}}$ $\displaystyle
f\cdot(x_{1}^{p_{1}}\dots x_{n}^{p_{n}})$ $\displaystyle=$ $\displaystyle
x_{f(1)}^{p_{1}}\dots x_{f(n)}^{p_{n}}$ $\displaystyle=$ $\displaystyle
x_{1}^{\sum_{f(i)=1}p_{i}}\dots x_{m}^{\sum_{f(i)=m}p_{i}}$
###### Example 2.3.5.
Let $\mathcal{C}$ be a finite product category, and $A$ be an object of
$\mathcal{C}$. Then there is a finite product operad $\mathop{\rm{End}}(A)$,
the endomorphism operad of $A$, where
$\mathop{\rm{End}}(A)_{n}=\mathcal{C}(A^{n},A)$, and $f\cdot p$ is $p$
composed with the appropriate combination of projections to relabel its
arguments by $f$.
###### Definition 2.3.6.
Let $M$, $N$ be finite product multicategories. A morphism $F:M\to N$ consists
of
* •
for each object $m\in M$, an object $Fm\in N$;
* •
for each $n\in\natural$ and all $m_{1},\dots,m_{n},m\in M$, a map
$F_{m_{1},\dots,m_{n},m}:M(m_{1},\dots m_{n};m)\to N(Fm_{1},\dots,Fm_{n};Fm)$
commuting with the -action, the unit and composition.
###### Definition 2.3.7.
Let $M$ be a finite product multicategory. An algebra for $M$ in a finite
product multicategory $\mathcal{C}$ is a map of finite product multicategories
$M\to\mathcal{C}$. An algebra for $M$ in a finite product category
$\mathcal{C}$ is a map of finite product multicategories from $M$ to the
underlying finite product multicategory of $\mathcal{C}$. Finite product
multicategories and their morphisms form a category called FP-Multicat.
In the special case of finite product operads, these definitions are
equivalent to the following:
###### Definition 2.3.8.
Let $P$, $Q$ be finite product operads. A morphism $F:P\to Q$ is a sequence of
maps $F_{i}:P_{i}\to Q_{i}$ commuting with the action, the unit and
composition.
###### Definition 2.3.9.
Let $P$ be a finite product operad. An algebra for $P$ in a finite product
category $\mathcal{C}$ is an object $A\in\mathcal{C}$ and a map of finite
product operads $P\to\mathop{\rm{End}}(A)$.
Finite product operads and their morphisms form a category called FP-Operad.
###### Example 2.3.10.
The algebras in $\mathcal{C}$ for the operad described in Example 2.3.3 are
associative $R$-algebras in $\mathcal{C}$.
###### Example 2.3.11.
The algebras for the operad described in Example 2.3.4 are commutative monoid
objects in $\mathcal{C}$.
###### Theorem 2.3.12.
${\mbox{{{FP-Operad}}}}\cong\mbox{{{Clone}}}$.
###### Proof.
We shall construct a functor $K_{(-)}:{\mbox{{{FP-
Operad}}}}\to\mbox{{{Clone}}}$, and show that it is bijective on objects, full
and faithful.
If $P$ is a finite product operad, let $K_{P}$ be the following clone:
* •
$(K_{P})_{n}=P_{n}$ for all $n\in\natural$,
* •
composition is given by composition in $P$: if $p\in P_{n}$ and
$p_{1},\dots,p_{n}\in P_{m}$, then $p\bullet(p_{1},\dots,p_{n})\in(K_{P})_{m}$
is $f\cdot(p\circ(p_{1},\dots,p_{n}))\in P_{m}$, where
$\begin{array}[]{lccl}f:&{\underline{nm}}&\to&{\underline{m}}\\\
&x&\mapsto&((x-1)\mod m)+1,\end{array}$ (2.1)
* •
for all $n\in\natural$ and all $i\in{\underline{n}}$, the projection
$\delta^{i}_{n}$ is $f^{i}_{n}\cdot 1$, where
$\begin{array}[]{lccl}f^{i}_{n}:&{\underline{1}}&\to&{\underline{n}}\\\
&1&\mapsto&i.\end{array}$ (2.2)
It is easily checked that $K_{P}$ satisfies the axioms for a clone given in
Definition 1.2.1.
On morphisms, $K_{(-)}$ acts trivially: morphisms of clones and of finite
product operads are simply maps of signatures commuting with the extra
structure, and $K_{(-)}$ preserves the underlying map of signatures.
Let $K$ be a clone. Let $P_{K}$ be the finite product operad for which
* •
$(P_{K})_{n}=K_{n}$ for all $n\in\natural$,
* •
$1=\delta^{1}_{1}$,
* •
$p\circ(p_{1},\dots,p_{n})=p\bullet(p_{1}\bullet(\delta^{1}_{m},\dots,\delta^{k_{1}}_{\sum
k_{i}}),\dots,p_{n}\bullet(\delta^{k_{1}+\dots+k_{n-1}+1}_{\sum
k_{i}},\dots,\delta^{\sum k_{i}}_{\sum k_{i}}))$ for all $n$ and
$k_{1},\dots,k_{n}\in\natural$, all $p\in K_{n}$, and all $p_{1}\in
K_{k_{1}},\dots,p_{n}\in K_{k_{n}}$,
* •
$f\cdot p=p\bullet(\delta^{f(1)}_{m},\dots,\delta^{f(n)}_{m})$ for all
$n,m\in\natural$, all $f:{\underline{n}}\to{\underline{m}}$ and all $p\in
K_{n}$.
We will show that $K_{P_{K}}=K$ for all $K\in\mbox{{{Clone}}}$, and that
$P_{K_{P}}=P$ for all $P\in{\mbox{{{FP-Operad}}}}$. Let $K$ be a clone. Then
$(K_{P_{K}})_{n}=(P_{K})_{n}=K_{n}$ for all $n\in\natural$. If
$n,m\in\natural$, $k\in K_{n}$ and $k_{1},\dots,k_{n}\in K_{m}$, then the
composite $k\bullet(k_{1},\dots,k_{n})$ in $K_{P_{K}}$ is given by the
composite $f\cdot(k\circ(k_{1},\dots,k_{n}))$ in $P_{K}$, where $f$ is given
by (2.1) above. This in turn is given by the composite
$\displaystyle(k$ $\displaystyle\bullet$
$\displaystyle(k_{1}\bullet(\delta^{1}_{nm},\dots,\delta^{m}_{nm}),\dots,k_{n}\bullet(\delta^{(n-1)m+1}_{nm},\dots,\delta^{nm}_{nm})))$
$\displaystyle\bullet$
$\displaystyle(\delta^{1}_{m},\dots,\delta^{m}_{m},\dots,\delta^{1}_{m},\dots,\delta^{m}_{m})$
in $K$. By the associativity law for clones, this is equal to
$\begin{array}[]{rl}k\bullet(&k_{1}\bullet(\delta^{1}_{nm},\dots,\delta^{m}_{nm})\bullet(\delta^{1}_{m},\dots,\delta^{m}_{m},\dots,\delta^{1}_{m},\dots,\delta^{m}_{m}),\\\
&\dots,\\\
&k_{n}\bullet(\delta^{(n-1)m+1}_{nm},\dots,\delta^{nm}_{nm})\bullet(\delta^{1}_{m},\dots,\delta^{m}_{m},\dots,\delta^{1}_{m},\dots,\delta^{m}_{m}))\end{array}$
which in turn may be simplified to $k\bullet(k_{1},\dots,k_{n})$ as required.
For every $n\in\natural$ and every $i\in{\underline{n}}$, the projection
$\delta^{i}_{n}$ in $K_{P_{K}}$ is given by $f^{i}_{n}\cdot 1$, where
$f^{i}_{n}$ is defined in (2.2): this in turn is given by
$1\circ(\delta^{i}_{n})=\delta^{1}_{1}\bullet(\delta^{i}_{n})=\delta^{i}_{n}$.
Hence $K_{P_{K}}=K$.
Conversely, let $P$ be a finite product operad we shall show that
$P_{K_{P}}=P$. For every $n\in\natural$, the set $(P_{K_{P}})_{n}$ is equal to
$P_{n}$. The unit element is given by $1=\delta^{1}_{1}=f^{1}_{1}\cdot 1=1$.
If $p\in P_{n}$ and $p_{i}\in P_{k_{i}}$ for $i=1,\dots,n$, then the composite
$p\circ(p_{1},\dots,p_{n})$ is given by
$p\bullet(p_{1}\bullet(\delta^{1}_{m},\dots,\delta^{k_{1}}_{\sum
k_{i}}),\dots,p_{n}\bullet(\delta^{k_{1}+\dots+k_{n-1}+1}_{\sum
k_{i}},\dots,\delta^{\sum k_{i}}_{\sum k_{i}}))$ in $K_{P}$, which in turn is
given by (after simplification) $p\circ(p_{1},\dots,p_{n})$ in $P$. Hence
$P=P_{K_{P}}$, and $K_{(-)}$ is bijective on objects. The reasoning above also
suffices to show that $K_{(-)}$ is well-defined on morphisms and full (since
preserving a finite product operad structure amounts exactly to preserving the
associated clone structure). Since the morphisms of both categories are simply
maps of signatures with extra properties and $K_{(-)}$ commutes with the
forgetful functors to Set, then $K_{(-)}$ is faithful. Hence $K_{(-)}$ is an
isomorphism of categories, and ${\mbox{{{FP-Operad}}}}\cong\mbox{{{Clone}}}$.
∎
###### Theorem 2.3.13.
Let $P$ be a finite product operad. Then
$\mbox{{{Alg$(P)$}}}\cong\mbox{{{Alg$(K_{P})$}}}$.
###### Proof.
Let $(A,(\hat{\phantom{\alpha}}))$ be a $P$-algebra. Since the elements of the
finite product endomorphism operad $\mathop{\rm{End}}(A)$ are endomorphisms of
$A$, and composition is given by composition of morphisms, then
$K_{\mathop{\rm{End}}(A)}=\mathop{\rm{End}}(A)$, the endomorphism clone of
$A$. Since the functor $K_{(-)}:{\mbox{{{FP-Operad}}}}\to\mbox{{{Clone}}}$ is
an isomorphism, a morphism of finite product operads
$P\to\mathop{\rm{End}}(A)$ is exactly a map of clones
$K_{P}\to\mathop{\rm{End}}(A)$. Hence an algebra for $P$ is exactly an algebra
for $K_{P}$. A morphism between $P$-algebras is a morphism between their
underlying objects that commutes with $\hat{p}$ for every $p\in P_{n}$ and
every $n\in\natural$; this is true iff it commutes with $\hat{k}$ for every
$k\in(K_{P})_{n}$ and every $n\in\natural$. ∎
### 2.4 Adjunctions
In the next few sections, we shall show that there is a chain of monadic
adjunctions
FP-Operad$\scriptstyle{{U^{\rm fp}_{\Sigma}}}$$\scriptstyle{{U^{\rm fp}_{\rm
pl}}}$$\scriptstyle{{U^{\rm
fp}}}$$\textstyle{\@ensuremath{\mbox{{{$\Sigma$-Operad}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\Sigma}_{\rm
pl}}}$$\scriptstyle{{F_{\rm
fp}^{\Sigma}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\Sigma}}}$$\textstyle{\@ensuremath{\mbox{{{Operad}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\Sigma}^{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\rm pl}}}$$\scriptstyle{{F_{\rm
fp}^{\rm
pl}}}$$\textstyle{\@ensuremath{\mbox{{{Set}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{F_{\Sigma}}}$$\scriptstyle{{F_{\rm
fp}}}$ (2.3)
The notation is chosen such that $F^{x}_{y}\dashv U_{x}^{y}$, and
$U_{x}^{y}U_{y}^{z}=U_{x}^{z}$. The notation is inspired by the exponential
notation used for hom-objects: the source category of one of these functors is
determined by its superscript, and the target category is determined by its
subscript. The “pl” stands for “plain”. A similar chain of adjunctions (for
PROPs rather than operads) was discussed in [Bae], pages 51–59.
We refer to the monad $U_{x}^{y}F^{x}_{y}$ as $T^{x}_{y}$. The right adjoints
${U^{\rm pl}},{U^{\Sigma}_{\rm pl}}$ and ${U^{\rm fp}_{\Sigma}}$ are found by
forgetting respectively the compositional structure, the symmetric structure,
and the actions of all non-bijective functions, and will not be described
further. By standard properties of adjunctions, the composite functors are
adjoint: ${F_{\Sigma}}\dashv{U^{\Sigma}}$ etc.
### 2.5 Existence and monadicity
All the left adjoints in (2.3) are examples of a more general construction. We
shall now investigate this general case, and show that the adjunction which
arises is always monadic.
But first, we have so far only asserted that ${U^{\rm fp}},{U^{\rm
fp}_{\Sigma}}$ and ${U^{\rm fp}_{\rm pl}}$ have left adjoints. We shall show
that these left adjoints must exist for general reasons.
Let FP be the category of small categories with finite products and product-
preserving functors.
###### Lemma 2.5.1.
Let and be small finite-product categories, let $\mathcal{C}$ be cartesian
closed and have all small colimits, and let $Q:\Csmall\to\Dsmall$ preserve
finite products. Then the adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.77779pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-10.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{[\Dsmall,\mathcal{C}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.45529pt\raise 10.39165pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.69722pt\hbox{$\scriptstyle{Q_{!}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.05556pt\raise-1.125pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\bot}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.77779pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{[\Csmall,\mathcal{C}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.01086pt\raise-11.2736pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.17639pt\hbox{$\scriptstyle{Q^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
10.7778pt\raise-4.73611pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}}}}}\ignorespaces,$
where $Q^{*}$ is composition with $Q$ and $Q_{!}=\mathop{\rm{Lan}}_{Q}$,
restricts to an adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
45.53311pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-45.53311pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\@ensuremath{\mbox{{{FP}}}}(\Dsmall,\mathcal{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.72186pt\raise-1.78056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.69722pt\hbox{$\scriptstyle{Q_{!}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.32213pt\raise 9.7361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\bot}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.55559pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
15.57806pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\@ensuremath{\mbox{{{FP}}}}(\Csmall,\mathcal{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.2999pt\raise 1.80138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.17639pt\hbox{$\scriptstyle{Q^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
18.5556pt\raise-4.73611pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}}}}}\ignorespaces,$
###### Proof.
Certainly $Q^{*}$ restricts in this way, since $Q$ preserves finite products.
$\mbox{{{FP}}}(\Csmall,\mathcal{C})$ and $\mbox{{{FP}}}(\Dsmall,\mathcal{C})$
are full subcategories of $[\Csmall,\mathcal{C}]$ and $[\Dsmall,\mathcal{C}]$,
so if we can show that $Q_{!}$ restricts to a functor
$\mbox{{{FP}}}(\Csmall,\mathcal{C})\to\mbox{{{FP}}}(\Dsmall,\mathcal{C})$,
then it is automatically left adjoint to the restriction of $Q^{*}$.
Let $X:\Csmall\to\mathcal{C}$ preserve finite products. We must show that
$Q_{!}X:\Dsmall\to\mathcal{C}$ preserves finite products. We shall proceed by
showing that $Q_{!}X$ preserves terminal objects and binary products.
Recall that
$(Q_{!}X)(b)\cong\int^{a}\Dsmall(Qa,b)\times Xa$
for all $b\in\Dsmall$. Hence, using $1$ for the terminal objects in and
$\mathcal{C}$,
$\displaystyle(Q_{!}X)(1)$ $\displaystyle\cong$
$\displaystyle\int^{a}\Dsmall(Qa,1)\times Xa$ $\displaystyle\cong$
$\displaystyle\int^{a}1\times Xa$ $\displaystyle\cong$
$\displaystyle\int^{a}Xa$ $\displaystyle\cong$ $\displaystyle X1$
$\displaystyle\cong$ $\displaystyle 1$
since $X$ preserves finite products and the colimit of a diagram $D$ over a
category with a terminal object $1$ is simply $D1$.
Now, let $b_{1},b_{2}\in\Dsmall$. Then
$\displaystyle(Q_{!}X)(b_{1}\times b_{2})$ (2.4) $\displaystyle\cong$
$\displaystyle\int^{a}\Dsmall(Qa,b_{1}\times b_{2})\times Xa$
$\displaystyle\cong$
$\displaystyle\int^{a}\Dsmall(Qa,b_{1})\times\Dsmall(Qa,b_{2})\times Xa$ (2.5)
$\displaystyle\cong$
$\displaystyle\int^{a}\left(\int^{c_{1}}\Dsmall(Qc_{1},b_{1})\times\Csmall(a,c_{1})\right)\times\left(\int^{c_{2}}\Dsmall(Qc_{2},b_{2})\times\Csmall(a,c_{2})\right)\times
Xc$ (2.6) $\displaystyle\cong$
$\displaystyle\int^{a,c_{1},c_{2}}\Dsmall(Qc_{1},b_{1})\times\Dsmall(Qc_{2},b_{2})\times\Csmall(a,c_{1})\times\Csmall(a,c_{2})\times
Xa$ (2.7) $\displaystyle\cong$
$\displaystyle\int^{a,c_{1},c_{2}}\Dsmall(Qc_{1},b_{1})\times\Dsmall(Qc_{2},b_{2})\times\Csmall(a,c_{1}\times
c_{2})\times Xa$ (2.8) $\displaystyle\cong$
$\displaystyle\int^{c_{1},c_{2}}\Dsmall(Qc_{1},b_{1})\times\Dsmall(Qc_{2},b_{2})\times\left(\int^{a}\Csmall(a,c_{1}\times
c_{2})\times Xa\right)$ (2.9) $\displaystyle\cong$
$\displaystyle\int^{c_{1},c_{2}}\Dsmall(Qc_{1},b_{1})\times\Dsmall(Qc_{2},b_{2})\times
X(c_{1}\times c_{2})$ (2.10) $\displaystyle\cong$
$\displaystyle\int^{c_{1},c_{2}}\Dsmall(Qc_{1},b_{1})\times\Dsmall(Qc_{2},b_{2})\times
Xc_{1}\times Xc_{2}$ (2.11) $\displaystyle\cong$
$\displaystyle\int^{c_{1},c_{2}}\Dsmall(Qc_{1},b_{1})\times
Xc_{1}\times\Dsmall(Qc_{2},b_{2})\times Xc_{2}$ (2.12) $\displaystyle\cong$
$\displaystyle\left(\int^{c_{1}}\Dsmall(Qc_{1},b_{1})\times
Xc_{1}\right)\times\left(\int^{c_{2}}\Dsmall(Qc_{2},b_{2})\times
Xc_{2}\right)$ (2.13) $\displaystyle\cong$
$\displaystyle(Q_{!}X)(b_{1})\times(Q_{!}X)(b_{2})$ (2.14)
(2.4) is the definition of $Q_{!}$; (2.5), (2.8) and (2.11) are from the
definition of products; (2.6) and (2.10) are applications of the Density
Formula; (2.7), (2.9) and (2.14) use the distributivity of products over
colimits in $\mathcal{C}$ (since $\mathcal{C}$ is cartesian closed), and
(2.12) uses the fact that $X$ preserves finite products.
So $Q_{!}$ preserves terminal objects and binary products, and hence all
finite products. ∎
###### Corollary 2.5.2.
The functors ${U^{\rm fp}_{\Sigma}},{U^{\rm fp}_{\rm pl}}$ and ${U^{\rm fp}}$
all have left adjoints.
###### Lemma 2.5.3.
Let $S$ be a set, whose elements we will call sorts. Let $T$ and $T^{\prime}$
be $S$-sorted finite product theories, such that $T^{\prime}$ is a subcategory
of $T$ and the inclusion of $T^{\prime}$ into $T$ preserves finite products.
Let Alg$(T)$ be the category of $T$-algebras and morphisms in some finite
product category $\mathcal{C}$, and Alg$(T^{\prime})$ be the category of
$T^{\prime}$-algebras and morphisms in $\mathcal{C}$. Then the free/forgetful
adjunction
$\textstyle{\@ensuremath{\mbox{{{Alg$(T^{\prime})$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\bot}$$\textstyle{\@ensuremath{\mbox{{{Alg$(T)$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$
is monadic, provided the left adjoint $F$ exists.
###### Proof.
We will make use of Beck’s theorem to prove monadicity: precisely, we shall
make use of the version in [ML98] VI.7.1, which states that $U$ is monadic if
it has a left adjoint and it strictly creates coequalizers for $U$-absolute
coequalizer pairs. Recall that a functor $G:\mathcal{C}\to\mathcal{D}$
strictly creates coequalizers for a diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$
in $\mathcal{C}$ if, for every coequalizer $e:GB\to E$ of $Gf$ and $Gg$ in
$\mathcal{D}$, there are a unique object $E^{\prime}$ in $\mathcal{C}$ and a
unique arrow $e^{\prime}:B\to E^{\prime}$ such that $GE^{\prime}=E$ and
$Ge^{\prime}=e$, and moreover that $e^{\prime}$ is a coequalizer of
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$.
Let
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$
be a $U$-absolute coequalizer pair in Alg$(T)$, and $e:UB\to E$ be the
coequalizer of
$\textstyle{UA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Uf}$$\scriptstyle{Ug}$$\textstyle{UB}$.
We wish to extend $E$ to a functor $E^{\prime}:T\to\mathcal{C}$. Define
$E^{\prime}$ to be equal to $E$ on objects. On arrows, we shall define
$E^{\prime}$ using the universal property of $E$ and the $U$-absolute property
of
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$.
For each arrow $\phi:s_{1}\times\dots\times s_{n}\to r_{1}\times\dots\times
r_{m}$ in $T$ (where $s_{i},r_{j}\in S$), consider the diagram
$\textstyle{\prod
UAs_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
Uf_{s_{i}}}$$\scriptstyle{\prod
Ug_{s_{i}}}$$\scriptstyle{A\phi}$$\textstyle{\prod
UBs_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
e_{s_{i}}}$$\scriptstyle{B\phi}$$\textstyle{\prod
Es_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E^{\prime}\phi}$$\textstyle{\prod
UAr_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
f_{r_{j}}}$$\scriptstyle{\prod g_{r_{j}}}$$\textstyle{\prod
UBr_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
e_{r_{j}}}$$\textstyle{\prod Er_{j}}$ (2.15)
in $\mathcal{C}$, where $e:UB\to E$ is a coequalizer for
$\textstyle{UA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Uf}$$\scriptstyle{Ug}$$\textstyle{UB}$.
Since
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$
is a $U$-absolute coequalizer pair, $\prod e_{s_{i}}:\prod UBs_{i}\to\prod
Es_{i}$ is a coequalizer. Since $f$ and $g$ are $T$-homomorphisms, (2.15)
serially commutes, so $(\prod e_{r_{j}})\phi$ factors uniquely through $\prod
e_{s_{i}}$. Define $E^{\prime}\phi$ to be this map, as shown (and note that
$E^{\prime}\phi=E\phi$ if $\phi$ is in $T^{\prime}$). This definition
straightforwardly makes $E^{\prime}$ into a functor $T\to\mathcal{C}$. Since
$E$ is a $T^{\prime}$-algebra, and products in $T$ are the same as products in
$T^{\prime}$, we may deduce that $E^{\prime}:T\to C$ preserves finite
products, and thus is a $T$-algebra. Clearly, $E^{\prime}$ is the unique
extension of $E$ to a $T$-algebra such that $e$ is a $T$-algebra morphism. It
remains to show that $e$ is a coequalizer map for
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B}$
in Alg$(T)$.
Suppose
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{D}$
is a fork in Alg$(T)$. Then
$\textstyle{UA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Uf}$$\scriptstyle{Ug}$$\textstyle{UB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ud}$$\textstyle{UD}$
is a fork in Alg$(T^{\prime})$, so $Ud$ factors through $e$; say $Ud=he$. We
must show that $h$ is a $T$-homomorphism. As before, take
$\phi:s_{1}\times\dots\times s_{n}\to r_{1}\times\dots\times r_{m}$ in $T$,
and consider the diagram
$\textstyle{\prod
UAs_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
Uf_{s_{i}}}$$\scriptstyle{\prod
Ug_{s_{i}}}$$\scriptstyle{A\phi}$$\textstyle{\prod
UBs_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
e_{s_{i}}}$$\scriptstyle{B\phi}$$\scriptstyle{\prod
Ud_{s_{i}}}$$\textstyle{\prod
E^{\prime}s_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E^{\prime}\phi}$$\scriptstyle{\prod
h_{s_{i}}}$$\textstyle{\prod
UDs_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D\phi}$$\textstyle{\prod
UAr_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
f_{r_{j}}}$$\scriptstyle{\prod g_{r_{j}}}$$\textstyle{\prod
UBr_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
e_{r_{j}}}$$\scriptstyle{\prod Ud_{r_{j}}}$$\textstyle{\prod
E^{\prime}r_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod
h_{r_{j}}}$$\textstyle{\prod Dr_{j}}$ (2.16)
We must show that the curved square on the far right commutes. Now
$(D\phi)\circ(\prod d_{s_{i}})=(D\phi)\circ(\prod h_{s_{i}})\circ(\prod
e_{s_{i}})$, and $(\prod Ud_{r_{j}})\circ(UB\phi)=h\circ
e\circ(UB\phi)=h\circ(E\phi)\circ(\prod e_{s_{i}})$, since $e$ is a
$T$-algebra homomorphism. But $(\prod Ud_{r_{j}})\circ\phi=\phi\circ(\prod
d_{i})$, so $D\phi\circ(\prod h_{s_{i}})\circ(\prod
e_{s_{i}})=h\circ(E\phi)\circ(\prod e_{s_{i}})$. And $\prod e_{s_{i}}$ is
(regular) epic, so $h\circ(E\phi)=(D\phi)\circ(\prod h_{s_{i}})$.
So $h$ is a $T$-algebra homomorphism. Hence $U$ strictly creates coequalizers
for $U$-absolute coequalizer pairs, and hence is monadic. ∎
This result could also have been deduced from the Sandwich Theorem of Manes:
see [Man76] Theorem 3.1.29 (page 182).
###### Theorem 2.5.4.
All the adjunctions in diagram 2.3, namely ${F_{\rm fp}^{\Sigma}}\dashv{U^{\rm
fp}_{\Sigma}},{F_{\rm fp}^{\rm pl}}\dashv{U^{\rm fp}_{\rm pl}},{F_{\rm
fp}}\dashv{U^{\rm fp}},{F_{\Sigma}^{\rm pl}}\dashv{U^{\Sigma}_{\rm
pl}},{F_{\Sigma}}\dashv{U^{\Sigma}}$ and ${F_{\rm pl}}\dashv{U^{\rm pl}}$, are
monadic.
###### Proof.
Each category mentioned is a category of algebras for some -sorted theory, and
the monadicity of each adjunction mentioned is obtained by a simple
application of Lemma 2.5.3. For instance, symmetric operads are algebras for
the theory presented by
* •
_operations_ : one of the appropriate arity for each composition operation in
Definition 2.2.5, and an operation $\sigma\cdot-$ for each $n\in\natural$ and
each $\sigma$ in $S_{n}$.
* •
_equations_ : one for each instance of the axioms in Definition 2.2.5, and an
equation $(\sigma\cdot-)\circ(\rho\cdot-)=\sigma\rho\cdot-$ for each
$\sigma,\rho\in S_{n}$ and every $n\in\natural$.
∎
### 2.6 Explicit construction of ${F_{\rm pl}}$ and ${F_{\Sigma}^{\rm pl}}$
The previous section showed that ${F_{\rm pl}}$ and ${F_{\Sigma}^{\rm pl}}$
exist for general reasons, but it will be useful later to have an explicit
construction of these functors. For this reason, we shall now explicitly
construct functors $\mbox{{{Set}}}\to\mbox{{{Operad}}}$ and
$\mbox{{{Operad}}}\to\mbox{{{$\Sigma$-Operad}}}$, and prove that they are left
adjoint to ${U^{\rm pl}}$ and ${U^{\Sigma}_{\rm pl}}$.
###### Definition 2.6.1.
Let $\Phi$ be a signature. An $n$-ary strongly regular tree labelled by $\Phi$
is an element of the set $\mathop{\mbox{tr}}_{n}\Phi$, which is recursively
defined as follows:
* •
$|$ is an element of $\mathop{\mbox{tr}}_{1}\Phi$.
* •
If $\phi\in\Phi_{n}$, and
$\tau_{1}\in\mathop{\mbox{tr}}_{k_{1}}\Phi,\dots,\tau_{n}\in\mathop{\mbox{tr}}_{k_{n}}\Phi$,
then $\phi\circ(\tau_{1},\dots,\tau_{n})\in\mathop{\mbox{tr}}_{\sum
k_{i}}\Phi$.
In graph-theoretic terms, all our trees are planar and rooted. They need not
be level.
We shall abuse notation and write $\phi$ instead of $\phi\circ(|,\dots,|)$,
for $\phi\in\Phi_{n}$.
Given a signature $\Phi$, the objects of the plain operad $({F_{\rm
pl}}\Phi)_{n}$ are the elements of $\mathop{\mbox{tr}}_{n}\Phi$, and
composition is given by grafting of trees:
* •
$|\circ(\tau)$ = $\tau$
* •
If
$\tau_{1}\in\mathop{\mbox{tr}}_{k_{1}}\Phi,\dots,\tau_{n}\in\mathop{\mbox{tr}}_{k_{n}}\Phi$,
then
$(\phi\circ(\tau_{1},\dots,\tau_{n}))\circ(\sigma_{1},\dots,\sigma_{\sum
k_{i}})=\phi\circ(\tau_{1}\circ(\sigma_{1},\dots,\sigma_{k_{1}}),\dots,\tau_{n}\circ(\sigma_{(\sum
k_{i})-k_{n}+1},\dots,\sigma_{\sum k_{i}}))$
See Figure 2.5.
Figure 2.5: Grafting of trees
The unary tree $|$ is thus the identity in $({F_{\rm pl}}\Phi)$.
${F_{\rm pl}}$ acts on arrows as follows. Let $f:\Phi\to\Psi$ be a map of
signatures. Then:
* •
$({F_{\rm pl}}f)|=|$
* •
$({F_{\rm pl}}f)(\phi\circ(\tau_{1},\dots,\tau_{n}))=(f\phi)\circ(({F_{\rm
pl}}f)\tau_{1},\dots,({F_{\rm pl}}f)\tau_{n})$
It is readily verified that with this definition ${F_{\rm pl}}$ is a functor
$\mbox{{{Set}}}\to\mbox{{{Operad}}}$.
We define natural transformations $\eta:1_{\mbox{{{Set}}}}\to{U^{\rm
pl}}{F_{\rm pl}}$ and $\epsilon:{F_{\rm pl}}{U^{\rm pl}}\to
1_{\mbox{{{Operad}}}}$ as follows:
$\displaystyle\eta_{\Phi}(\phi)$ $\displaystyle=$
$\displaystyle\phi\circ(|,\dots,|)$ (2.17) $\displaystyle\epsilon_{P}(|)$
$\displaystyle=$ $\displaystyle 1_{P}$ (2.18)
$\displaystyle\epsilon_{P}(\phi\circ(\tau_{1},\dots,\tau_{n}))$
$\displaystyle=$
$\displaystyle\phi\circ(\epsilon_{P}(\tau_{1}),\dots,\epsilon_{P}(\tau_{n}))$
(2.19)
where $P\in\mbox{{{Operad}}},\Phi\in\mbox{{{Set}}},\phi\in\Phi$, and
$\tau_{1},\dots,\tau_{n}$ are arrows of $P$.
In other words, $\epsilon_{P}$ is given by applying composition in $P$ to the
formal tree ${F_{\rm pl}}{U^{\rm pl}}P$.
###### Lemma 2.6.2.
$({F_{\rm pl}},{U^{\rm pl}},\eta,\epsilon)$ is an adjunction.
###### Proof.
We proceed by checking the triangle identities. We require to show that
(2.24) (2.29)
commute. For (2.24), we proceed by induction on trees. We shall suppress all
subscripts on natural transformations in the interest of legibility. For the
base case:
$\displaystyle\epsilon{F_{\rm pl}}({F_{\rm pl}}\eta(|))$ $\displaystyle=$
$\displaystyle\epsilon{F_{\rm pl}}(|\circ(|))$ $\displaystyle=$
$\displaystyle|\circ(\epsilon(|))$ $\displaystyle=$ $\displaystyle|$
$\displaystyle=$ $\displaystyle 1_{F_{\rm pl}}(|).$
For the inductive step, let $\Phi$ be a signature, $\phi$ be an $n$-ary
element of $\Phi$, and $\tau_{1},\dots,\tau_{n}$ be trees labelled by $\Phi$.
Then:
$\displaystyle(\epsilon{F_{\rm pl}})(({F_{\rm
pl}}\eta)(\phi\circ(\tau_{1},\dots,\tau_{n})))$ $\displaystyle=$
$\displaystyle\epsilon{F_{\rm
pl}}(\phi\circ(\tau_{1},\dots,\tau_{n})\circ(|,\dots,|))$ $\displaystyle=$
$\displaystyle\phi\circ(\tau_{1},\dots,\tau_{n})\circ((\epsilon{F_{\rm
pl}})(|),\dots,(\epsilon{F_{\rm pl}})(|))$ $\displaystyle=$
$\displaystyle\phi\circ(\tau_{1},\dots,\tau_{n})$ $\displaystyle=$
$\displaystyle 1_{F_{\rm pl}}(\phi\circ(\tau_{1},\dots,\tau_{n}))$
Hence $\epsilon{F_{\rm pl}}\circ{F_{\rm pl}}\eta=1{F_{\rm pl}}$, as required.
For (2.29), let $P$ be a plain operad, and let $p$ be an $n$-ary arrow in $P$.
$\displaystyle({U^{\rm pl}}\epsilon)((\eta{U^{\rm pl}})(p))$ $\displaystyle=$
$\displaystyle{U^{\rm pl}}\epsilon(p\circ(|,\dots,|))$ $\displaystyle=$
$\displaystyle p\circ(({U^{\rm pl}}\epsilon)(|),\dots,({U^{\rm
pl}}\epsilon)(|))$ $\displaystyle=$ $\displaystyle p\circ(1,\dots,1)$
$\displaystyle=$ $\displaystyle p$ $\displaystyle=$ $\displaystyle 1_{U^{\rm
pl}}(p)$
So ${U^{\rm pl}}\epsilon\circ\eta{U^{\rm pl}}=1_{U^{\rm pl}}$, as required. ∎
We now consider the “free symmetric operad” functor ${F_{\Sigma}^{\rm pl}}$.
We shall explicitly define a functor
$\mathcal{S}\times-:\mbox{{{Operad}}}\to\mbox{{{$\Sigma$-Operad}}}$ and show
that it is left adjoint to ${U^{\Sigma}_{\rm pl}}$, and hence isomorphic to
${F_{\Sigma}^{\rm pl}}$.
If $P$ is a plain operad, an element of $(\mathcal{S}\times P)_{n}$ is a pair
$(\sigma,p)$, where $p\in P_{n}$ and $\sigma\in S_{n}$; i.e.,
$(\mathcal{S}\times P)_{n}=S_{n}\times P_{n}$. Composition is given as
follows:
$(\sigma,p)\circ((\tau_{1},q_{1}),\dots,(\tau_{n},q_{n}))=(\sigma\circ(\tau_{1},\dots,\tau_{n}),p\circ(q_{\sigma(1)},\dots,q_{\sigma(n)}))$
The symmetric group action is given by $\rho\cdot(\sigma,p)=(\rho\sigma,p)$.
Figure 2.6: Composition in $\mathcal{S}\times P$
###### Lemma 2.6.3.
$(\mathcal{S}\times-)$ is left adjoint to ${U^{\Sigma}_{\rm pl}}$. The unit of
the adjunction is given by
$\displaystyle\eta:1$ $\displaystyle\to$ $\displaystyle{U^{\Sigma}_{\rm
pl}}(\mathcal{S}\times-)$ $\displaystyle\eta_{P}:p$ $\displaystyle\mapsto$
$\displaystyle(1,p),$
and the counit is given by
$\displaystyle\epsilon:(\mathcal{S}\times-){U^{\Sigma}_{\rm pl}}$
$\displaystyle\to$ $\displaystyle 1$ $\displaystyle\epsilon_{P}:(\sigma,p)$
$\displaystyle\mapsto$ $\displaystyle\sigma\cdot p.$
###### Proof.
As before, we proceed by checking the triangle identities. First, let $P$ be a
plain operad, $p$ be an $n$-ary arrow in $P$, and $\sigma\in S_{n}$. Then
$(\sigma,p)$ is an element of $\mathcal{S}\times P$.
$\displaystyle(\epsilon(\mathcal{S}\times-))(\mathcal{S}\times\eta)(\sigma,p))$
$\displaystyle=$ $\displaystyle(\epsilon(\mathcal{S}\times-))(\sigma,(1,p))$
$\displaystyle=$ $\displaystyle\sigma\cdot(1,p)$ $\displaystyle=$
$\displaystyle(\sigma\circ 1,p)$ $\displaystyle=$ $\displaystyle(\sigma,p)$
Now let $P^{\prime}$ be a symmetric operad, and $p^{\prime}$ be an $n$-ary
arrow in $P^{\prime}$.
$\displaystyle({U^{\Sigma}_{\rm pl}}\epsilon)(\eta{U^{\Sigma}_{\rm
pl}}(p^{\prime}))$ $\displaystyle=$ $\displaystyle({U^{\Sigma}_{\rm
pl}}\epsilon)((1,p^{\prime}))$ $\displaystyle=$ $\displaystyle 1\cdot
p^{\prime}$ $\displaystyle=$ $\displaystyle p^{\prime}$
So the triangle identities are indeed satisfied, and
$(\mathcal{S}\times-)\dashv{U^{\Sigma}_{\rm pl}}$. ∎
Hence, ${F_{\Sigma}^{\rm pl}}=\mathcal{S}\times-$.
###### Definition 2.6.4.
Let $\Phi$ be a signature. An $n$-ary permuted tree labelled by $\Phi$ is an
element of $({F_{\Sigma}^{\rm pl}}{F_{\rm
pl}}\Phi)_{n}=({F_{\Sigma}}\Phi)_{n}$. An $n$-ary finite product tree labelled
by $\Phi$ is an element of $({F_{\rm fp}}\Phi)_{n}$.
By Lemma 2.6.3, a permuted tree is a pair $(\sigma,t)$, where
$t\in\mathop{\mbox{tr}}_{n}\Phi$ and $\sigma\in S_{n}$, and (by analogous
reasoning) an $n$-ary finite product tree is a pair $(f,t)$, where
$t\in\mathop{\mbox{tr}}_{m}\Phi$ and $f:{\underline{m}}\to{\underline{n}}$.
### 2.7 Syntactic characterization of the forgetful functors
There is also a syntactic characterization of the forgetful functor
${U^{\Sigma}_{\rm pl}}$. Given a symmetric operad $P$, we take the signature
given by all operations in $P$ (in other words, the signature
${U^{\Sigma}}P$). We then impose all the plain-operadic equations that are
true in $P$, and take the plain operad corresponding to this strongly regular
theory. This operad is ${U^{\Sigma}_{\rm pl}}P$.
We start by making this precise.
###### Definition 2.7.1.
Let $\Phi$ be a signature. A plain-operadic equation over $\Phi$ in $n$
variables is an element of $(({U^{\rm pl}}{F_{\rm pl}}\Phi)_{n})^{2}$ (that
is, a pair of $n$-ary strongly regular trees over $\Phi$), and a plain-
operadic equation over $\Phi$ is an element of $\sum_{n}(({U^{\rm pl}}{F_{\rm
pl}}\Phi)_{n})^{2}$.
We shall show that a plain-operadic equation over $\Phi$ is the same thing as
a strongly regular equation over $\Phi$.
###### Definition 2.7.2.
Let $P$ be a plain operad. A presentation for $P$ is a signature $\Phi$, a
signature $E$, and maps $e_{1},e_{2}:{F_{\rm pl}}E\to{F_{\rm pl}}\Phi$ such
that, for some $\phi$,
$\textstyle{{F_{\rm
pl}}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{2}}$$\textstyle{{F_{\rm
pl}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{P}$
is a coequalizer. We say that a regular epi $\phi:{F_{\rm pl}}\Phi\to P$
generates $P$, or that $\phi$ (or, where the choice of $\phi$ is clear,
$\Phi$) is a generator of $P$.
Presentations and generators for symmetric and finite product operads are
defined analogously.
We will see how these “presentations” are related to presentations of
algebraic theories in Section 2.8. We now wish to describe the family of all
strongly regular equations that are true in a given symmetric operad $P$: we
will then show that this, together with the signature given by
${U^{\Sigma}}P$, is a presentation for ${U^{\Sigma}_{\rm pl}}P$ as claimed.
###### Definition 2.7.3.
Let $P$ be a plain operad, and $\phi:{F_{\rm pl}}\Phi\to P$ be a generator for
$P$. Let $E$ be a subsignature of $({U^{\rm pl}}{F_{\rm pl}}\Phi)^{2}$, so
that each $E_{n}$ is a set of $n$-ary $\Phi$-equations. Let $i$ be the
inclusion map
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({U^{\rm
pl}}{F_{\rm pl}}\Phi)^{2}}$, and $\pi_{1},\pi_{2}$ be the projection maps
$({U^{\rm pl}}{F_{\rm pl}}\Phi)^{2}\to{U^{\rm pl}}{F_{\rm pl}}\Phi$. Then $P$
satisfies all equations in $E$ if the diagram
$\textstyle{{F_{\rm
pl}}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\pi_{1}i}}$$\scriptstyle{\overline{\pi_{2}i}}$$\textstyle{{F_{\rm
pl}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{P}$
is a fork.
We say that a symmetric or finite product operad satisfies a signature of
equations if the analogous condition holds in $\Sigma$-Operad or FP-Operad.
Recall the notion of the “kernel pair” of a morphism:
###### Definition 2.7.4.
Let $f:A\to B$ in some category $\mathcal{C}$. The kernel pair of $f$ is the
pair
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{A}$
of maps in the pullback square
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textstyle{\lrcorner}}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
if this pullback exists.
###### Lemma 2.7.5.
Let $\epsilon$ be the counit of the adjunction ${F_{\rm pl}}\dashv{U^{\rm
pl}}$. Let
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{{F_{\rm
pl}}{({U^{\Sigma}}P)}}$ be the kernel pair of the component
$\epsilon_{{U^{\Sigma}_{\rm pl}}P}:{F_{\rm pl}}{U^{\Sigma}}P={F_{\rm
pl}}{U^{\rm pl}}{U^{\Sigma}_{\rm pl}}P\to{U^{\Sigma}_{\rm pl}}P,$
of $\epsilon$. Let $h$ be the unique map $Q\to({F_{\rm pl}}{U^{\Sigma}}P)^{2}$
induced by $\pi_{1},\pi_{2}$. Then the image of ${U^{\rm pl}}h$ is the largest
signature of plain-operadic ${U^{\Sigma}}P$-equations satisfied by $P$.
###### Proof.
$Q,\pi_{1},\pi_{2}$ are given by the diagram
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textstyle{\lrcorner}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{{F_{\rm
pl}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{{F_{\rm
pl}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{{U^{\Sigma}_{\rm
pl}}P}$
As a right adjoint, ${U^{\rm pl}}$ preserves pullbacks; we take the standard
construction of pullbacks in Set as subobjects of products, in which case $h$
is an inclusion map. An element of $({U^{\rm pl}}Q)_{n}$ is then a pair
$(e_{1},e_{2})$ of $n$-ary strongly regular ${U^{\Sigma}}P$ trees such that
$\epsilon(e_{1})=\epsilon(e_{2})$. Hence, $Q$ is a signature of plain-operadic
${U^{\Sigma}}P$-equations satisfied by $P$. Conversely, let $E$ be a signature
of plain-operadic ${U^{\Sigma}}P$-equations satisfied by $P$, and let
$(e_{1},e_{2})$ be an element of $E_{n}$: then
$\epsilon(e_{1})=\epsilon(e_{2})$ and so $(e_{1},e_{2})$ is an element of
$({U^{\rm pl}}Q)_{n}$. ∎
###### Corollary 2.7.6.
Let $R$ be the plain operad generated by ${U^{\Sigma}}P$, satisfying exactly
those plain-operadic equations satisfied by $P$. Then
$\textstyle{{F_{\rm pl}}{U^{\rm
pl}}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{{U^{\rm
pl}}\pi_{1}}}$$\scriptstyle{\overline{{U^{\rm
pl}}\pi_{2}}}$$\textstyle{{F_{\rm pl}}{U^{\Sigma}}P}$
is a presentation for $R$, where the overbars refer to transposition with
respect to the adjunction ${F_{\rm pl}}\dashv{U^{\rm pl}}$.
We recall some standard results.
###### Lemma 2.7.7.
The counit of a monadic adjunction is componentwise regular epi.
###### Proof.
See [AHS04] 20.15. ∎
###### Lemma 2.7.8.
If
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Z}$
is a coequalizer in some category, and if $e:W\to X$ is epi, then
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{fe}$$\scriptstyle{ge}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Z}$
is a coequalizer.
###### Proof.
Suppose
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{fe}$$\scriptstyle{ge}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{A}$
is a fork. Then $ife=ige$, so $if=ig$ since $e$ is epi. So
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{A}$
is a fork, and hence $i$ factors uniquely through $h$. ∎
###### Lemma 2.7.9.
In categories with all kernel pairs, every regular epi is the coequalizer of
its kernel pair.
###### Proof.
Let $\mathcal{C}$ have all kernel pairs, and
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{C}$
be a coequalizer diagram in $\mathcal{C}$. Let
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{B}$
be the kernel pair of $e$. We will show that
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{C}$
is a coequalizer diagram. Since $ef=eg$, we may uniquely factor $(f,g)$
through W:
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{i}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textstyle{\lrcorner}}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{C}$
Suppose
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{D}$
is a fork. $hp=hq$, so $hpi=hqi$, so $hf=hg$. By the universal property of
$e$, we may factor $h$ uniquely through $e$. So
$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{C}$
is a coequalizer diagram, as required. ∎
###### Lemma 2.7.10.
Let $P$ be a symmetric operad, and let $Q,\pi_{1},\pi_{2}$ be as in Lemma
2.7.5. Then the coequalizer of the diagram
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{1}}$$\textstyle{{F_{\rm
pl}}{U^{\Sigma}}P}$
is ${U^{\Sigma}_{\rm pl}}P$.
###### Proof.
Let $\epsilon^{\prime}$ be the unit of the adjunction ${F_{\Sigma}^{\rm
pl}}\dashv{U^{\Sigma}_{\rm pl}}$. This adjunction is monadic, so
$\epsilon_{{U^{\Sigma}_{\rm pl}}P}$ is regular epi by Lemma 2.7.7. By Lemma
2.7.9, $\epsilon_{{U^{\Sigma}_{\rm pl}}P}$ is the coequalizer of its kernel
pair, i.e.
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{\sum_{n}({U^{\rm
pl}}{F_{\rm
pl}}{U^{\Sigma}}P)_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{{U^{\Sigma}_{\rm
pl}}P}}$$\textstyle{{U^{\Sigma}_{\rm pl}}P}$
is a coequalizer diagram. ∎
###### Theorem 2.7.11.
Let $P$ be a symmetric operad. Then ${U^{\Sigma}_{\rm pl}}P$ is the plain
operad whose operations are those in $P$, satisfying exactly those plain-
operadic equations which are true in $P$.
###### Proof.
The adjunction ${F_{\rm pl}}\dashv{U^{\rm pl}}$ is monadic, so if
$\epsilon^{\prime}$ is its counit, then $\epsilon^{\prime}_{Q}:{F_{\rm
pl}}{U^{\rm pl}}Q\to Q$ is (regular) epi by Lemma 2.7.7. Hence, by Lemma
2.7.8,
$\textstyle{{F_{\rm pl}}{U^{\rm
pl}}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}\epsilon^{\prime}_{Q}}$$\scriptstyle{\pi_{2}\epsilon^{\prime}_{Q}}$$\textstyle{{F_{\rm
pl}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{{U^{\Sigma}_{\rm
pl}}P}}$$\textstyle{{U^{\Sigma}_{\rm pl}}P}$
is a coequalizer. But $\pi_{1}\epsilon^{\prime}_{Q}=\overline{{U^{\rm
pl}}\pi_{1}}$, and similarly $\pi_{2}\epsilon^{\prime}_{Q}=\overline{{U^{\rm
pl}}\pi_{2}}$. Hence, by Corollary 2.7.6, ${U^{\Sigma}_{\rm pl}}P$ is the
plain operad generated by ${U^{\Sigma}}P$, satisfying all plain-operadic
equations true in $P$. Since ${U^{\Sigma}}P={U^{\rm pl}}{U^{\Sigma}_{\rm
pl}}P$, the $n$-ary operations of ${U^{\Sigma}_{\rm pl}}P$ are exactly the
$n$-ary operations of $P$. ∎
We may generalize this as follows:
###### Theorem 2.7.12.
Let $\mathcal{C}$ be a category with pullbacks, and $T$ be a monad on
$\mathcal{C}$. Let $(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.36633pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-10.36633pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{T}{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.70811pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.36633pt\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}}{\hbox{\kern
34.36633pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A}$}}}}}}}}}}}\ignorespaces)\in\mathcal{C}^{T}$. Let
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\phi_{2}}$$\textstyle{TA}$
be the kernel pair of $a$ in $\mathcal{C}$. Then
$\textstyle{F_{T}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\phi}_{1}}$$\scriptstyle{\bar{\phi}_{2}}$$\textstyle{F_{T}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{(A,a)}}$$\textstyle{(A,a)}$
is a coequalizer in $\mathcal{C}^{T}$, where
$F_{T}:\mathcal{C}\to\mathcal{C}^{T}$ is the free functor, and $\epsilon$ is
the counit of the adjunction $F_{T}\dashv U_{T}$.
###### Proof.
As above. ∎
###### Corollary 2.7.13.
Let $P$ be a finite product operad. Then ${U^{\rm fp}_{\Sigma}}P$ is the
symmetric operad whose operations are given by those of $P$, satisfying all
linear equations that are true in $P$, and ${U^{\rm fp}_{\rm pl}}P$ is the
plain operad whose operations are given by those in $P$, satisfying all
strongly regular equations that are true in $P$.
### 2.8 Operads and syntactic classes of theories
We have defined notions of algebras for plain, symmetric and finite product
operads. We might ask how these are related to the algebraic theories of
Chapter 1: are the algebras for an operad $P$ algebras for some algebraic
theory $\mathcal{T}_{P}$? If so, what can we say about the theories that so
arise?
We will show the following:
* •
Plain operads are equivalent in expressive power to _strongly regular_
theories.
* •
Symmetric operads are equivalent in expressive power to _linear_ theories.
* •
Finite product operads are equivalent in expressive power to general algebraic
theories.
The first equivalence is proved in [Lei03]. The second has long been folklore
(see, for instance, [Bae] page 50), but as far as I know no proof has appeared
before. An (independently found) proof does appear in an unpublished paper of
Adámek and Velebil, who also consider the enriched case. The third equivalence
was proved in two stages by Tronin, in [Tro02] and [Tro06].
Recall the definitions of strongly regular and linear terms from Definition
1.1.7, and the definitions of strongly regular, permuted and finite product
trees (Definitions 2.6.1 and 2.6.4).
Let $\Phi$ be a signature. We will show that there is an isomorphism between
the set $({T_{\rm fp}}\Phi)_{n}$ and the set of $n$-ary words in $\Phi$, and
that this isomorphism restricts to further isomorphisms as follows:
(2.30)
The maps in the left-hand column can be viewed as inclusions between different
sets of finite product trees, or equivalently as maps arising from the units
of the adjunctions ${F_{\rm fp}^{\Sigma}}\dashv{U^{\rm fp}_{\Sigma}}$ and
${F_{\Sigma}^{\rm pl}}\dashv{U^{\Sigma}_{\rm pl}}$.
Let $\Phi$ be a signature. Observe that trees in $\Phi$ give rise to terms
according to the following recursive algorithm:
* •
Let $\tau$ be an $n$-ary strongly regular tree, and
$Y=(y_{1},y_{2},\dots,y_{n})$ a finite sequence of variables. The term
$\mathop{\rm{term}}(\tau,Y)$ arising from $\tau$ with working alphabet $Y$ is
given as follows:
* –
If $\tau=|$, then $\mathop{\rm{term}}(\tau,Y)=y_{1}$.
* –
If $\tau=\phi\circ(\tau_{1},\dots,\tau_{n})$, then
$\mathop{\rm{term}}(\tau,Y)=\phi(\mathop{\rm{term}}(\tau_{1},(y_{1},\dots,y_{i_{1}})),\dots,\mathop{\rm{term}}(\tau_{n},(y_{1+i_{n-1}},\dots,y_{i_{n}}))),$
where $i_{1}$ is the arity of $\tau_{1}$, and $i_{j}-i_{j-1}$ is the arity of
$\tau_{j}$ for $j>1$.
* •
The term $\mathop{\rm{term}}(\tau)$ arising from $\tau$ is
$\mathop{\rm{term}}(\tau,(x_{1},x_{2},\dots,x_{n}))$.
* •
Let $\sigma\cdot\tau$ be a permuted tree. Then
$\mathop{\rm{term}}(\sigma\cdot\tau)=\mathop{\rm{term}}(\tau,(x_{\sigma
1},x_{\sigma 2},\dots,x_{\sigma n}))$.
* •
Let $f\cdot\tau$ be a finite product tree. Then
$\mathop{\rm{term}}(f\cdot\tau)=\mathop{\rm{term}}(\tau,(x_{f(1)},x_{f(2)},\dots,x_{f(n)})$
###### Definition 2.8.1.
Let $t$ be a $\Phi$-term. We define a plain-operadic tree
$\mathop{\rm{tree}}(t)$ recursively:
* •
if $t$ is a variable, let $\mathop{\rm{tree}}(t)=|$.
* •
if $t=\phi(t_{1},\dots,t_{n})$, let
$\mathop{\rm{tree}}(t)=\phi\circ(\mathop{\rm{tree}}(t_{1}),\dots,\mathop{\rm{tree}}(t_{n}))$.
###### Lemma 2.8.2.
Every $\Phi$-term $t$ is equal to $\mathop{\rm{term}}(f\cdot\tau)$ for a
unique finite product tree $(f\cdot\tau)$.
###### Proof.
We will show
1. 1.
if $t$ is a $\Phi$-term, then
$\mathop{\rm{term}}(\mathop{\rm{label}}(t)\cdot\mathop{\rm{tree}}(t))=t$;
2. 2.
if $(f\cdot\tau)$ is a finite product tree, then
$f=\mathop{\rm{label}}(\mathop{\rm{term}}(f\cdot\tau))$ and
$\tau=\mathop{\rm{tree}}(\mathop{\rm{term}}(f\cdot\tau))$.
(1) Let $t$ be a $\Phi$-term. Let $f=\mathop{\rm{label}}(t)$, and
$\tau=\mathop{\rm{tree}}(t)$. Then $\mathop{\rm{term}}(f\cdot\tau)$ is
$\mathop{\rm{term}}(\tau,(x_{f(1)},\dots,x_{f(n)}))$. We proceed by induction.
* •
if $t=x_{i}$, then $\mathop{\rm{term}}(f\cdot\tau)$ is
$\mathop{\rm{term}}(|,(x_{i}))$, which is $x_{i}$.
* •
if $t=\phi(t_{1},\dots,t_{n})$, where each $t_{i}$ has arity $k_{i}$, then
$\displaystyle\mathop{\rm{term}}(f\cdot\tau)$ $\displaystyle=$
$\displaystyle\mathop{\rm{term}}(\phi\circ(\mathop{\rm{tree}}(t_{1}),\dots,\mathop{\rm{tree}}(t_{n})),(x_{f(1)},\dots,x_{f(n)}))$
$\displaystyle=$
$\displaystyle\phi(\mathop{\rm{term}}(\mathop{\rm{tree}}(t_{1}),(x_{f(1)},\dots,x_{f(k_{1})})),\dots,$
$\displaystyle\phantom{\phi(}\mathop{\rm{term}}(\mathop{\rm{tree}}(t_{n}),(x_{f((\sum_{i=1}^{n-1}k_{i})+1)},\dots,x_{f(\sum_{i=1}^{n}k_{i})})))$
$\displaystyle=$ $\displaystyle\phi(t_{1},\dots,t_{n})$ $\displaystyle=$
$\displaystyle t.$
(2) Let $\tau$ be an $n$-ary plain-operadic tree in $\Phi$, and $f$ a function
of finite sets with codomain ${\underline{m}}$. Let
$t=\mathop{\rm{term}}(f\cdot\tau)$. We proceed as usual by induction on
$\tau$.
* •
If $\tau=|$, then $t=x_{f(1)}$; then $\mathop{\rm{tree}}(t)=|=\tau$ and
$\mathop{\rm{label}}(t)$ is the function ${\underline{1}}\to{\underline{m}}$
sending 1 to $f(1)$, i.e. $\mathop{\rm{label}}(t)=f$.
* •
If $\tau=\phi\circ(\tau_{1},\dots,\tau_{n})$, then
$\displaystyle t$ $\displaystyle=$
$\displaystyle\mathop{\rm{term}}(\phi\circ(\tau_{1},\dots,\tau_{n}),(x_{f(1)},\dots,x_{f(\sum
k_{i})}))$ $\displaystyle=$
$\displaystyle\phi(\mathop{\rm{term}}(\tau_{1},(x_{f(1)},\dots,x_{f(k_{1})})),\dots,\mathop{\rm{term}}(\tau_{n},(x_{f((\sum_{i=1}^{n-1}k_{i})+1)})\dots,x_{f(\sum_{i=1}^{n}k_{i})})))$
By induction, $\mathop{\rm{var}}(t)=(x_{f(1)},\dots,x_{f(\sum k_{i})})$, so
$\mathop{\rm{label}}(t)=f$, and
$\mathop{\rm{tree}}(t)=\phi\circ(\tau_{1},\dots,\tau_{n})=\tau$ as required.
∎
We have now established the isomorphism in the top line of 2.30. If we use
this isomorphism to identify finite product operads with finitary monads on
Set, we may view the functor ${F_{\rm fp}^{\rm pl}}$ as the well-known functor
sending a plain operad to its associated monad on Set.
###### Lemma 2.8.3.
Let $t$ be a $\Phi$-term. Then $t$ is linear iff $t=\mathop{\rm{term}}(\tau)$
for some permuted tree $\tau$, and strongly regular iff
$t=\mathop{\rm{term}}(\tau)$ for some strongly regular tree $\tau$.
###### Proof.
In Lemma 2.8.2, we factored every $\Phi$-term $t$ into a strongly regular tree
$\mathop{\rm{tree}}(t)$ and a labelling function $\mathop{\rm{label}}(t)$. By
definition, $t$ is linear iff $\mathop{\rm{label}}(t)$ is a bijection, which
occurs iff $t=\mathop{\rm{term}}(\sigma\cdot\tau)$ for some plain-operadic
tree $\tau$ and some bijection $\sigma$. Hence, the linear terms and permuted
trees are in one-to-one correspondence. Similarly, strongly regular terms and
plain-operadic trees are in one-to-one correspondence. ∎
The commutativity of 2.30 now follows from our explicit construction of
${F_{\Sigma}}$ and ${F_{\rm pl}}$ in Section 2.4.
###### Lemma 2.8.4.
Let $(\Phi,E)$ be a presentation of an algebraic theory. Then $(\Phi,E)$ is
linear if and only if the projection maps $\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
6.97916pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-6.97916pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.98407pt\raise 7.30415pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.97916pt\raise
2.15277pt\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
12.98407pt\raise-7.30415pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
30.97916pt\raise-2.15277pt\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
30.97916pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm{term}}}$}}}}}}}}}}}\ignorespaces(\Phi)$ may
be factored through the map
$\textstyle{{T_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{{T_{\rm
fp}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathop{\rm{term}}{\Phi}}$:
---
###### Proof.
By definition, the presentation is linear iff $\pi_{1},\pi_{2}$ factor through
the signature of linear $\Phi$-terms. By Lemma 2.8.3, this signature is
isomorphic to ${T_{\Sigma}}\Phi$, so we are done. ∎
###### Theorem 2.8.5.
Let $Q\in{\mbox{{{FP-Operad}}}}$. Then
1. 1.
$Q$ is strongly regular iff there exists a $P\in\mbox{{{Operad}}}$ such that
$Q\cong{F_{\rm fp}^{\rm pl}}P$;
2. 2.
$Q$ is linear iff there exists a $P\in\mbox{{{$\Sigma$-Operad}}}$ such that
$Q\cong{F_{\rm fp}^{\Sigma}}P$;
###### Proof.
We will consider the linear case; the strongly regular case is proved
analogously.
If $Q$ is linear, then there exists a linear presentation
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{F_{\rm
fp}}\Phi}$ for $Q$. We may regard $E$ as a subobject of the signature of
$\Phi$-equations. By assumption, $E$ consists only of linear equations; by
Lemma 2.8.3, every $(s,t)\in E$ is
$(\mathop{\rm{term}}(\sigma_{1}\cdot\tau_{1}),\mathop{\rm{term}}(\sigma_{2}\cdot\tau_{2}))$
for some pair $(\sigma_{1}\cdot\tau_{1},\sigma_{2}\cdot\tau_{2})$ of permuted
trees. So the diagram
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{F_{\rm
fp}}\Phi}$ in FP-Operad is the image under ${F_{\rm fp}^{\Sigma}}$ of a
diagram
$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{F_{\Sigma}}\Phi}$
in $\Sigma$-Operad. This diagram has a coequalizer: call it $P$. The functor
${F_{\rm fp}^{\Sigma}}$ is a left adjoint, and thus preserves coequalizers:
hence, $Q$ is the image under ${F_{\rm fp}^{\Sigma}}$ of $P$.
Now suppose $Q={F_{\rm fp}^{\Sigma}}P$ for some symmetric operad $P$. We may
take the canonical presentation of $P$:
$\textstyle{{F_{\Sigma}}{U^{\Sigma}}{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon{F_{\Sigma}}{U^{\Sigma}}}$$\scriptstyle{{F_{\Sigma}}{U^{\Sigma}}\epsilon}$$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P}$
and apply ${F_{\rm fp}^{\Sigma}}$ to it:
$\textstyle{{F_{\rm
fp}}({U^{\Sigma}}{F_{\Sigma}}{U^{\Sigma}}P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\rm
fp}^{\Sigma}}\epsilon{F_{\Sigma}}{U^{\Sigma}}}$$\scriptstyle{{F_{\rm
fp}}{U^{\Sigma}}\epsilon}$$\textstyle{{F_{\rm
fp}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\rm
fp}^{\Sigma}}\epsilon}$$\textstyle{{F_{\rm fp}^{\Sigma}}P=Q}$
Since ${F_{\rm fp}^{\Sigma}}$ is a left adjoint, it preserves coequalizers, so
the transpose of this parallel pair is a presentation for $Q$. Take this
transpose:
$\textstyle{{U^{\Sigma}}{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\epsilon{F_{\Sigma}}{U^{\Sigma}}}}$$\scriptstyle{\overline{{F_{\Sigma}}{U^{\Sigma}}\epsilon}}$$\textstyle{{U^{\Sigma}}{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}}$$\textstyle{{{U^{\rm
fp}}{F_{\rm
fp}}{U^{\Sigma}}P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\rm
fp}}{F_{\rm fp}^{\Sigma}}\epsilon}$$\textstyle{{U^{\rm fp}}Q}$
where $\eta^{\prime}$ is the unit of the adjunction ${F_{\rm fp}}\dashv{U^{\rm
fp}}$, and the bars refer to transposition with respect to the adjunction
${F_{\Sigma}}\dashv{U^{\Sigma}}$.
This is in precisely the form required for Lemma 2.8.4. ∎
###### Example 2.8.6.
The theories of monoids and pointed sets are strongly regular, because the
finite product operads corresponding to these theories are in the image of
${F_{\rm fp}^{\rm pl}}$; the theory of commutative monoids is linear but not
strongly regular, because the finite product operad whose algebras are
commutative monoids is in the image of ${F_{\rm fp}^{\Sigma}}$ but not in the
image of ${F_{\rm fp}^{\rm pl}}$.
There is a little more to be said about these classes of theories.
###### Definition 2.8.7.
A wide pullback is a limit of a (possibly infinite) diagram of the form
---
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots}$
###### Definition 2.8.8.
A natural transformation $\alpha:F\to G$ is cartesian if every naturality
square
$\textstyle{FA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ff}$$\scriptstyle{\alpha_{A}}$$\textstyle{FB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{B}}$$\textstyle{GA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gf}$$\textstyle{GB}$
is a pullback square.
###### Definition 2.8.9.
A monad $(T,\mu,\eta)$ is cartesian if $T$ preserves pullbacks and $\mu,\eta$
are cartesian natural transformations.
###### Theorem 2.8.10.
A plain operad is equivalent to a cartesian monad on Set equipped with a
cartesian map of monads to the free monoid monad.
###### Proof.
See [Lei03] 6.2.4. Let 1 be the terminal plain operad; algebras for 1 are
monoids. Since 1 is terminal, every plain operad $P$ comes equipped with a map
$!:P\to 1$. This induces a cartesian map of monads $T_{!}:T_{P}\to T_{1}$, and
$T_{1}$ is the free monoid monad. ∎
###### Lemma 2.8.11.
Let $T,S$ be endofunctors on a category , let $\alpha:T\to S$ be a cartesian
natural transformation, and let $S$ preserve wide pullbacks. Then $T$
preserves wide pullbacks.
###### Proof.
This follows from the facts that wide pullbacks are products in slice
categories and that the functor $f^{*}:\Acat/B\to\Acat/A$ induced by a map
$f:A\to B$ is product-preserving. ∎
###### Corollary 2.8.12.
Let $P$ be a plain operad. Then the functor part of the monad $T_{P}$ arising
from $P$ preserves wide pullbacks.
###### Definition 2.8.13.
A functor $F:\mathcal{C}\to\mbox{{{Set}}}$ is familially representable if $F$
is a coproduct of representable functors. A monad $(T,\mu,\eta)$ on Set is
familially representable if $T$ is familially representable.
###### Theorem 2.8.14.
(Carboni-Johnstone) Let $\mathcal{C}$ be a complete, locally small, well-
powered category with a small cogenerating set, and let
$F:\mathcal{C}\to\mbox{{{Set}}}$ be a functor. The following are equivalent:
1. 1.
$F$ is familially representable;
2. 2.
$F$ preserves wide pullbacks.
###### Proof.
See [CJ95], Theorem 2.6. ∎
###### Corollary 2.8.15.
The monad associated to a strongly regular theory is familially representable.
However, the inclusion is only one-way: there exist cartesian monads
$(T,\mu,\eta)$ such that $T$ is familially representable but the induced
theory is not strongly regular. For instance, take the theory of involutive
monoids:
###### Definition 2.8.16.
An involutive monoid (or monoid with involution) is a monoid $(M,.,1)$
equipped with an involution $i:M\to M$, satisfying $i(a.b)=i(b).i(a)$.
The theory of involutive monoids is familially representable, but not strongly
regular — see [CJ04].
### 2.9 Enriched operads and multicategories
In the previous sections we considered operads $P$ where
$P_{0},P_{1},\dots\in\mbox{{{Set}}}$, and composition was given by functions.
It is possible to consider operads where $P_{0},P_{1},\dots$ lie in some other
category; the resulting objects are called _enriched operads_. Enriched
operads have many applications and a rich theory: for instance, topologists
often consider operads enriched in Top or in some category of vector spaces.
Our treatment here will be brief, sufficient only to set up the definitions of
Chapter 4: for more on enriched operads, see [MSS02].
Throughout this section, let
$(\mathcal{V},\otimes,I,\alpha,\lambda,\rho,\tau)$ be a symmetric monoidal
category.
###### Definition 2.9.1.
A $\mathcal{V}$-multicategory $\mathcal{C}$ consists of the following:
* •
a collection $\mathcal{C}_{0}$ of _objects_ ,
* •
for all $n\in\natural$ and all $c_{1},\dots,c_{n},d\in\mathcal{C}_{0}$, an
object $\mathcal{C}(c_{1},\dots,c_{n};d)\in\mathcal{V}$ called the _arrows_
from $c_{1},\dots,c_{n}$ to $d$,
* •
for all $n,k_{1},\dots k_{n}\in\natural$ and
$c_{1}^{1}\dots,c_{k_{n}}^{n},d_{1},\dots,d_{n},e\in\mathcal{C}_{0}$, an arrow
in $\mathcal{V}$ called _composition_
$\circ:\mathcal{C}(d_{1},\dots,d_{n};e)\otimes\mathcal{C}(c^{1}_{1},\dots,c^{1}_{k_{1}};d_{1})\otimes\dots\otimes\mathcal{C}(c^{n}_{1},\dots,c^{n}_{k_{n}};d_{n})\to\mathcal{C}(c^{1}_{1},\dots,c^{n}_{k_{n}};e)$
* •
for all $c\in\mathcal{C}$, a _unit_ $u_{c}:I\to\mathcal{C}(c;c)$
satisfying the following axioms:
* •
_Associativity:_ For all
$b^{\bullet}_{\bullet\bullet},c_{\bullet}^{\bullet},d_{\bullet},e\in\mathcal{C}$,
the following diagram commutes:
---
$\mathcal{C}(d_{\bullet};e)\otimes\mathcal{C}(c^{1}_{\bullet};d_{1})\otimes\dots\otimes\mathcal{C}(c^{n}_{\bullet};d_{n})$
---
$\otimes\mathcal{C}(b^{1}_{1\bullet};c^{1}_{1})\otimes\dots\otimes\mathcal{C}(b^{n}_{k_{n}\bullet};c^{n}_{k_{n}})$
$\scriptstyle{\circ}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(d_{\bullet};e)\otimes\mathcal{C}(b^{1}_{\bullet\bullet};d_{1})\otimes\dots\otimes\mathcal{C}(b^{n}_{\bullet\bullet};d_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(c_{\bullet}^{\bullet};e)\otimes\mathcal{C}(b^{1}_{1\bullet};c^{1}_{1})\otimes\dots\otimes\mathcal{C}(b^{n}_{k_{n}\bullet};c^{n}_{k_{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(b^{\bullet}_{\bullet\bullet};e)}$
* •
_Units:_ For all $c_{\bullet},d\in\mathcal{C}$, the following diagram
commutes:
$\textstyle{\@ensuremath{\mathcal{C}}(c_{\bullet};d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\scriptstyle{1}$$\scriptstyle{\rho^{n}}$$\textstyle{I\otimes\mathcal{C}(c_{\bullet};d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u\otimes
1}$$\textstyle{\@ensuremath{\mathcal{C}}(c_{\bullet};d)\otimes
I\otimes\dots\otimes
I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes
u^{\otimes
n}}$$\textstyle{\@ensuremath{\mathcal{C}}(d;d)\otimes\mathcal{C}(c_{\bullet};d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(c_{\bullet};d)\otimes\mathcal{C}(c_{1};c_{1})\otimes\dots\otimes\mathcal{C}(c_{n};c_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(c_{\bullet};d)}$
(We suppress the symmetry maps in $\mathcal{V}$ for clarity).
###### Definition 2.9.2.
A symmetric $\mathcal{V}$-multicategory is a $\mathcal{V}$-multicategory
$\mathcal{C}$ and, for every $n\in\natural$, every $\sigma\in S_{n}$, and
every $a_{1},\dots a_{n},b\in\mathcal{C}$, an arrow
$\sigma\cdot-:\mathcal{C}(a_{1},\dots,a_{n};b)\longrightarrow\mathcal{C}(a_{\sigma
1},\dots,a_{\sigma n};b)$
in $\mathcal{V}$ such that
* •
for each $n\in\natural$ and each $a_{1},\dots,a_{n},b\in\mathcal{C}$, the
arrow
$1_{n}\cdot-:\mathcal{C}(a_{1},\dots,a_{n};b)\to\mathcal{C}(a_{1},\dots,a_{n};b)$
is the identity arrow on $\mathcal{C}(a_{1},\dots,a_{n};b)$,
* •
for each $\sigma,\rho\in S_{n}$,
$(\rho\cdot-)(\sigma\cdot-)=(\rho\sigma)\cdot-$
* •
for each $n,k_{1},\dots,k_{n}\in n$, each $\sigma\in S_{n}$ and $\rho_{i}\in
S_{k_{i}}$ for $i=1,\dots,n$, and for all
$a^{1}_{1},\dots,a^{n}_{k_{n}},b_{1},\dots,b_{n},c\in\mathcal{C}$, the diagram
---
|
$\textstyle{\@ensuremath{\mathcal{C}}(b_{1},\dots,b_{n};c)\otimes\bigotimes_{i=1}^{n}{\mathcal{C}(a^{i}_{1},\dots,a^{i}_{n};b_{i})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\sigma\cdot-)\otimes\bigotimes_{i=1}^{n}{(\rho_{i}\cdot-)}}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(b_{\sigma
1},\dots,b_{\sigma
n};c)\otimes\bigotimes_{i=1}^{n}{\mathcal{C}(a^{i}_{\rho_{i}1},\dots,a^{i}_{\rho_{i}n};b_{i})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\sigma_{*}}$$\textstyle{\@ensuremath{\mathcal{C}}(a^{1}_{1},\dots,a^{n}_{k_{n}};c)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\circ(\rho_{1},\dots,\rho_{n})}$$\textstyle{\@ensuremath{\mathcal{C}}(b_{\sigma
1},\dots,b_{\sigma n};c)\otimes\bigotimes_{i=1}^{n}\mathcal{C}(a^{\sigma
i}_{\rho_{\sigma i}1},\dots,a^{\sigma i}_{\rho_{\sigma i}n};b_{\sigma
i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{C}}(a^{\sigma
1}_{\rho_{\sigma 1}1},\dots,a^{\sigma n}_{\rho_{\sigma n}k_{n}};c)}$
commutes, where $\sigma\circ({\rho_{1},\dots,\rho_{n}})$ is as defined in
Example 2.1.11.
In the case $\mathcal{V}=\mbox{{{Set}}}$ (with the cartesian monoidal
structure), this is equivalent to Definition 2.2.1.
Let be a skeleton of the category of finite sets and functions, with objects
the sets 0, 1, 2, …, where ${\underline{n}}=\\{1,2,\dots,n\\}$.
###### Definition 2.9.3.
A finite product $\mathcal{V}$-multicategory is
* •
A plain $\mathcal{V}$-multicategory $\mathcal{C}$;
* •
for every function $f:{\underline{n}}\to{\underline{m}}$ in , and for all
objects $C_{1},\dots,C_{n},D\in\mathcal{C}$, a morphism
$f\cdot-:\mathcal{C}(C_{1},\dots,C_{n};D)\to\mathcal{C}(C_{f(1)},\dots,C_{f(n)};D)$
in $\mathcal{V}$
satisfying the conditions given in Definition 2.9.2, where
$(f\circ(f_{1},\dots,f_{n}))$ is as given in Definition 2.3.1.
In the case $\mathcal{V}=\mbox{{{Set}}}$, this is equivalent to Definition
2.3.1.
###### Definition 2.9.4.
A (plain, symmetric, finite product) $\mathcal{V}$-operad is a (plain,
symmetric, finite product) $\mathcal{V}$-multicategory with only one object.
###### Definition 2.9.5.
Let $\mathcal{C}$, $\mathcal{D}$ be plain $\mathcal{V}$-multicategories. A
morphism $F:\mathcal{C}\to\mathcal{D}$ is
* •
for each object $C\in\mathcal{C}$, a choice of object $FC\in\mathcal{D}$,
* •
for each $n\in\natural$ and all collections of objects
$A_{1},\dots,A_{n},B\in\mathcal{C}$, an arrow
$\mathcal{C}(A_{1},\dots,A_{n};B)\to\mathcal{D}(FA_{1},\dots,FA_{n};FB)$ in
$\mathcal{V}$
such that
* •
for all $A\in\mathcal{C}$, the diagram
---
$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{u}$$\textstyle{\@ensuremath{\mathcal{C}}(A;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\@ensuremath{\mathcal{D}}(FA;FA)}$
commutes;
* •
for all $n,k_{1},\dots,k_{n}\in\natural$ and all
$C,B_{1},\dots,B_{n},A^{1}_{1},\dots,A^{n}_{k_{n}}\in\mathcal{C}$, the diagram
$\textstyle{\@ensuremath{\mathcal{C}}(B_{\bullet};C)\otimes\bigotimes_{i=1}^{n}\mathcal{C}(A^{i}_{\bullet};B_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\scriptstyle{F\otimes\dots\otimes
F}$$\textstyle{\@ensuremath{\mathcal{C}}(A_{\bullet}^{\bullet};C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\@ensuremath{\mathcal{D}}(FB_{\bullet};FC)\otimes\bigotimes_{i=1}^{n}\mathcal{D}(FA^{i}_{\bullet};FB_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\@ensuremath{\mathcal{D}}(FA_{\bullet}^{\bullet};FC)}$
commutes.
Suppose that $\mathcal{V}$ is cocomplete. Let $Q$ be a (plain, symmetric,
finite product) $\mathcal{V}$-operad, and $A$ an object of $\mathcal{V}$. Let
$Q\circ A$ denote the coend
$\int^{n\in\mathcal{C}}Q_{n}\times A^{n}$
where $\mathcal{C}$ is
* •
the discrete category on if $Q$ is a plain operad;
* •
a skeleton of the category of finite sets and bijections if $Q$ is a symmetric
operad;
* •
a skeleton of the category of finite sets and all functions if $Q$ is a finite
product operad.
This notation is taken from Kelly’s papers [Kel72a] and [Kel72b] on clubs.
The various endomorphism operads defined in Examples 2.1.10, 2.2.7 and 2.3.5
transfer straightforwardly to the $\mathcal{V}$-enriched setting. An algebra
for a (plain, symmetric, finite product) $\mathcal{V}$-operad $P$ in a (plain,
symmetric, finite product) $\mathcal{V}$-multicategory $\mathcal{C}$ is an
object $A\in\mathcal{C}$ and a morphism
$(\hat{\phantom{\alpha}}):P\to\mathop{\rm{End}}(A)$ of the appropriate type.
Equivalently, an algebra for $P$ in $\mathcal{C}$ is an object
$A\in\mathcal{C}$ and a morphism $h:P\circ A\to A$ such that the diagram
$\textstyle{P\circ P\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\circ
h}$$\scriptstyle{\circ}$$\textstyle{P\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{P\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{A}$
commutes, and $h(1_{P},-)$ is the identity on $A$.
###### Remark 2.9.6.
There is another possibility, that of considering _internal_ multicategories
in the category $\mathcal{V}$, which gives a different notion: now
$\mathcal{C}_{0}$ is an object in $\mathcal{V}$ rather than a collection. An
internal operad in $\mathcal{V}$ is an internal multicategory $\mathcal{C}$
such that $\mathcal{C}_{0}$ is terminal in $\mathcal{V}$. We shall not
consider internal multicategories or operads further.
We shall in particular consider the case $\mathcal{V}=\mbox{{{Cat}}}$, and
Cat-operads again have a simple concrete description:
###### Lemma 2.9.7.
A (plain) Cat-operad $Q$ is a sequence of categories $Q_{0},Q_{1},\dots$, a
family of composition functors $\circ:Q_{n}\times Q_{k_{1}}\times\ldots\times
Q_{k_{n}}\to Q_{\sum k_{i}}$ and an identity $1_{Q}\in Q_{1}$, satisfying
(strict) functorial versions of the axioms given in 2.1.7.
###### Lemma 2.9.8.
A symmetric Cat-operad is a plain Cat-operad $Q$ with a left group action of
each symmetric group $S_{n}$ on the corresponding category $Q_{n}$, strictly
satisfying equations as in Definition 2.2.5.
###### Lemma 2.9.9.
A finite product Cat-operad is a plain Cat-operad $Q$ equipped with functors
$f\cdot-:Q_{n}\to Q_{m}$ for each function
$f:{\underline{n}}\to{\underline{m}}$ of finite sets, strictly satisfying
equations as in Definition 2.3.1.
All of these lemmas can be established by a straightforward check of the
definitions.
Just as 2-category theory has a special flavour distinct from the theory of
$\mathcal{V}$-categories in the case $\mathcal{V}=\mbox{{{Cat}}}$, so the
theories of Cat-operads and Cat-multicategories have unique features:
###### Definition 2.9.10.
Let $Q$ be a finite product Cat-operad, and let $Q\circ
A\stackrel{{\scriptstyle\alpha}}{{\to}}A,Q\circ
B\stackrel{{\scriptstyle\beta}}{{\to}}B$ be algebras for $Q$ in Cat. A lax
morphism of $Q$-algebras $A\to B$ consists of a 1-cell $F:A\to B$ and a 2-cell
$\phi:\beta F\to F\alpha$ satisfying the following conditions:
| | |
---|---|---|---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\eta}$$\textstyle{\scriptstyle
1}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\scriptstyle
1}$$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\circ
F}$$\scriptstyle{\alpha}$$\textstyle{\scriptstyle\phi}$$\textstyle{Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{=}$$\textstyle{A\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle
F}$$\textstyle{\scriptstyle
F}$$\textstyle{B}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{B}$
(2.31)
---
$\textstyle{Q\circ Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{1\circ
1\circ F}$$\textstyle{Q\circ Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{Q\circ
Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{1\circ
1\circ
F}$$\scriptstyle{1\circ\alpha}$$\textstyle{\scriptstyle\phi}$$\textstyle{Q\circ
Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{1\circ\beta}$$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\circ
F}$$\scriptstyle{\alpha}$$\textstyle{\scriptstyle\phi}$$\textstyle{Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{=}$$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\circ
F}$$\scriptstyle{\alpha}$$\textstyle{\scriptstyle\phi}$$\textstyle{Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{B}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{B}$
(2.32)
---
$\textstyle{Q_{m}\times
A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
f^{*}}$$\scriptstyle{1\times F^{n}}$$\textstyle{Q_{m}\times
A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
f^{*}}$$\scriptstyle{1\times F^{n}}$$\scriptstyle{f_{*}\times
1}$$\textstyle{Q_{m}\times
A^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{1\times
F^{m}}$$\textstyle{Q_{m}\times
B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
f^{*}}$$\scriptstyle{f_{*}\times 1}$$\textstyle{Q_{m}\times
A^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{Q_{m}\times
B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}\times
1}$$\textstyle{\scriptstyle\phi}$$\textstyle{Q_{n}\times
B^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{=}$$\textstyle{Q_{n}\times
A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{1\times
F^{n}}$$\textstyle{\scriptstyle\phi}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{Q_{n}\times
B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{Q_{n}\times
B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{B}$$\textstyle{B}$
(2.33)
for all functions $f:{\underline{m}}\to{\underline{n}}$.
A morphism $(F,\phi)$ is weak if $\phi$ is invertible, and strict if $\phi$ is
an identity.
Lax morphisms for algebras of plain Cat-operads are required to satisfy 2.31
and 2.32, and lax morphisms for algebras of symmetric Cat-operads are required
to satisfy 2.31, 2.32 and the restriction of 2.33 to the case where $f$ is a
bijection.
We shall make use of a more explicit formulation in the plain case.
###### Lemma 2.9.11.
Let $Q$ be a plain Cat-operad, and let $(A,h)$ and $(B,h^{\prime})$ be
$Q$-algebras. A lax map of $Q$-algebras $(A,h)\to(B,h^{\prime})$ is a pair
$(G,\psi)$, where $G:A\to B$ is a functor and $\psi$ is a sequence of natural
transformations $\psi_{i}:h^{\prime}_{i}(1\times G^{i})\to Gh_{i}$, called the
coherence maps, such that the following equation holds, for all
$n,k_{1},\dots,k_{n}\in\natural$:
$\textstyle{Q_{n}\times Q_{k_{1}}\times\dots\times Q_{k_{n}}\times A^{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
1^{n}\times G^{\sum k_{i}}}$$\scriptstyle{h_{k_{1}}\times\dots\times
h_{k_{n}}}$$\scriptstyle{\psi_{k_{1}}\times\dots\times\psi_{k_{n}}}$$\textstyle{Q_{n}\times
A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
G^{n}}$$\scriptstyle{h_{n}}$$\scriptstyle{\psi_{n}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{Q_{n}\times
Q_{k_{1}}\times\dots\times Q_{k_{n}}\times A^{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}_{k_{1}}\times\dots\times
h^{\prime}_{k_{n}}}$$\textstyle{Q_{n}\times
B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\textstyle{B}$
$\scriptstyle{=}$$\textstyle{Q_{n}\times Q_{k_{1}}\times\dots\times
Q_{k_{n}}\times A^{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
1^{n}\times G^{\sum k_{i}}}$$\scriptstyle{h_{\sum
k_{i}}}$$\scriptstyle{\psi_{\sum
k_{i}}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{Q_{n}\times
Q_{k_{1}}\times\dots\times Q_{k_{n}}\times B^{\sum
k_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}_{\sum
k_{i}}}$$\textstyle{B}$ (2.34)
and the diagram
$\textstyle{Ga\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\prime}_{1}}$$\scriptstyle{1}$$\textstyle{h^{\prime}(1_{P},Ga)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{1}}$$\textstyle{Ga\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\delta_{1}}$$\textstyle{Gh(1_{P},a)}$
(2.35)
commutes. The morphism is weak if every $\psi$ is invertible, and strict if
every $\psi$ is an identity.
###### Proof.
This can be established by a straightforward check of the definition. ∎
###### Definition 2.9.12.
Let $Q,A,B$ etc. be as above, and let $(F,\phi),(G,\gamma)$ be lax morphisms
of $Q$-algebras $A\to B$. A $Q$-transformation $F\to G$ is a natural
transformation $\sigma:F\to G$ such that
|
---|---
$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle
1\circ F}$$\textstyle{\scriptstyle 1\circ
G}$$\textstyle{\scriptstyle\sigma}$$\scriptstyle{\alpha}$$\textstyle{\scriptstyle\gamma}$$\textstyle{Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\circ
F}$$\scriptstyle{\alpha}$$\textstyle{\scriptstyle\phi}$$\textstyle{Q\circ
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{=}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{B}$$\textstyle{A\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle
F}$$\textstyle{\scriptstyle G}$$\textstyle{\scriptstyle\sigma}$$\textstyle{B}$
(2.36)
###### Lemma 2.9.13.
A $Q$-transformation $\sigma:(F,\phi)\to(G,\psi)$ is invertible as a natural
transformation if and only if it is invertible as a $Q$-transformation.
###### Proof.
“If” is obvious: we concentrate on “only if”. It is enough to show that
$\sigma^{-1}$ is a $Q$-transformation, which is to say that
$\textstyle{h(q,Ga_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{h(q,\sigma^{-1}_{a_{\bullet}})}$$\textstyle{Gh(q,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{-1}_{h(q,a_{\bullet})}}$$\textstyle{h(q,Fa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Fh(q,a_{\bullet})}$
(2.37)
commutes for all $(q,a_{\bullet})\in Q\circ A$, and this follows from the fact
that
${\sigma_{h(q,a_{\bullet})}}\circ\phi=\psi\circ{h(q,\sigma_{a_{\bullet}})}$.
∎
Finite product Cat-operads, their morphisms and transformations form a
2-category called Cat-FP-Operad. Similarly, there is a 2-category Cat-Operad
of plain Cat-operads, their morphisms and transformations, and a 2-category
Cat-$\Sigma$-Operad, of symmetric operads, their morphisms and
transformations.
###### Theorem 2.9.14.
There is a chain of monadic adjunctions
Cat-FP-Operad$\scriptstyle{{U^{\rm fp}_{\Sigma}}}$$\scriptstyle{{U^{\rm
fp}_{\rm pl}}}$$\scriptstyle{{U^{\rm
fp}}}$$\textstyle{\@ensuremath{\mbox{{{Cat-$\Sigma$-Operad}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\Sigma}_{\rm
pl}}}$$\scriptstyle{{F_{\rm
fp}^{\Sigma}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\Sigma}}}$$\textstyle{\@ensuremath{\mbox{{{Cat-
Operad}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\Sigma}^{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\rm pl}}}$$\scriptstyle{{F_{\rm
fp}^{\rm
pl}}}$$\textstyle{\@ensuremath{\mbox{{{Cat}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{F_{\Sigma}}}$$\scriptstyle{{F_{\rm
fp}}}$ (2.38)
###### Proof.
This follows from Lemmas 2.5.1 and 2.5.3, via an application of the argument
of Theorem 2.5.4. ∎
Since operads can be considered as one-object multicategories, a Cat-operad
$P$ (of whatever type) is really a 2-dimensional structure. We will therefore
refer to the objects and morphisms of the categories $P_{i}$ as 1-cells and
2-cells of $P$, respectively.
### 2.10 Maps of algebras as algebras for a multicategory
Let $P$ be a plain operad. We form a multicategory $\bar{P}=\mbox{{{2}}}\times
P$, where 2 is the category $(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
4.38889pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-4.38889pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
28.38889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
28.38889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\cdot}$}}}}}}}}}}}\ignorespaces)$. We may
describe $\bar{P}$ as follows: there are two objects, labelled 0 and 1; the
hom-sets $\bar{P}(0,\dots,0;0)$ and $\bar{P}(x_{1},\dots,x_{n};1)$ are copies
of $P_{n}$, for $x_{i}\in\\{0,1\\}$, and
$\bar{P}(x_{1},\dots,x_{n};0)=\emptyset$ if any of the $x_{i}$s are 1.
Composition is given by composition in $P$. An algebra for $\bar{P}$ is a pair
$A_{0},A_{1}$ of $P$-algebras, and a morphism of $P$-algebras $A_{0}\to
A_{2}$. See [Mar], Example 2.4 for more details.
We can extend this construction by defining a multicategory
$\bar{\bar{P}}=\mbox{{{3}}}\times P$, whose algebras are composable pairs of
maps of $P$-algebras, a multicategory $\bar{\bar{\bar{P}}}=\mbox{{{4}}}\times
P$ whose algebras are composable triples of maps of $P$-algebras, and so on.
With the obvious face and degeneracy maps, these multicategories form a
cosimplicial object in the category of plain multicategories.
The same construction can be performed for symmetric and enriched operads, and
the result continues to hold.
## Chapter 3 Factorization Systems
The theory of factorization systems was introduced by Freyd and Kelly in
[FK72] (though it was implicit in work of Isbell in the 1950s). We shall use
it in subsequent chapters to define the weakening of an algebraic theory.
Here, we recall the basic definitions and some relevant theorems.
The material in this chapter is standard, and may be found in (for instance)
[Bor94] or [AHS04]; for an alternative perspective and some more historical
background (as well as the interesting generalization to _weak_ factorization
systems), see [KT93].
###### Definition 3.0.1.
Let $e:a\to b$ and $m:c\to d$ be arrows in a category $\mathcal{C}$. We say
that $e$ is left orthogonal to $m$, written $e\mathop{\bot}m$, if, for all
arrows $f:a\to c$ and $g:b\to d$ such that $mf=ge$, there exists a unique map
$t:b\to c$ such that the following diagram commutes:
$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\forall
f}$$\scriptstyle{e}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!t}$$\scriptstyle{\forall
g}$$\textstyle{d}$
###### Definition 3.0.2.
Let $\mathcal{C}$ be a category. A factorization system on $\mathcal{C}$ is a
pair $(\mathcal{E},\mathcal{M})$ of classes of maps in $\mathcal{C}$ such that
1. 1.
for all maps $f$ in $\mathcal{C}$, there exist $e\in\mathcal{E}$ and
$m\in\mathcal{M}$ such that $f=me$;
2. 2.
$\mathcal{E}$ and $\mathcal{M}$ contain all identities, and are closed under
composition with isomorphisms on both sides;
3. 3.
$\mathcal{E}\mathop{\bot}\mathcal{M}$, i.e. $e\mathop{\bot}m$ for all
$e\in\mathcal{E}$ and $m\in\mathcal{M}$.
###### Example 3.0.3.
Let $\mathcal{C}=\mbox{{{Set}}}$, $\mathcal{E}$ be the epimorphisms, and
$\mathcal{M}$ be the monomorphisms. Then $(\mathcal{E},\mathcal{M})$ is a
factorization system.
###### Example 3.0.4.
More generally, let $\mathcal{C}$ be some variety of algebras, $\mathcal{E}$
be the regular epimorphisms (i.e., the surjections), and $\mathcal{M}$ be the
monomorphisms. Then $(\mathcal{E},\mathcal{M})$ is a factorization system.
###### Example 3.0.5.
Let $\mathcal{C}=\mbox{{{Digraph}}}$, the category of directed graphs and
graph morphisms. Let $\mathcal{E}$ be the maps bijective on objects, and
$\mathcal{M}$ be the full and faithful maps. Then $(\mathcal{E},\mathcal{M})$
is a factorization system.
In deference to Example 3.0.3, we shall use arrows like to denote members of
$\mathcal{E}$ in commutative diagrams, and arrows like to denote members of
$\mathcal{M}$, for whatever values of $\mathcal{E}$ and $\mathcal{M}$ happen
to be in force at the time.
We will use without proof the following standard properties of factorization
systems:
###### Lemma 3.0.6.
Let $\mathcal{C}$ be a category, and $(\mathcal{E},\mathcal{M})$ be a
factorization system on $\mathcal{C}$.
1. 1.
$\mathcal{E}\cap\mathcal{M}$ is the class of isomorphisms in $\mathcal{C}$.
2. 2.
The factorization in 3.0.2 (1) is unique up to unique isomorphism.
3. 3.
The factorization in 3.0.2 (1) is functorial, in the following sense: if the
square
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{D}$
commutes, and $f=me,f^{\prime}=m^{\prime}e^{\prime}$, then there is a unique
morphism $i$ making
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{g}$$\scriptstyle{m}$$\scriptstyle{i}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{\prime}}$$\scriptstyle{m^{\prime}}$$\textstyle{D}$
commute. Thus, given a choice of $e\in\mathcal{E}$ and $m\in\mathcal{M}$ for
each $f$ in $\mathcal{C}$(such that $f=me$), we may construct functors
$\mathcal{E}_{*},\mathcal{M}_{*}:[\mbox{{{2}}},\mathcal{C}]\to[\mbox{{{2}}},\mathcal{C}]$:
$\displaystyle\mathcal{E}_{*}:f\mapsto e$
$\displaystyle\mathcal{E}_{*}:(g,h)\mapsto(g,i)$
$\displaystyle\mathcal{M}_{*}:f\mapsto m$
$\displaystyle\mathcal{M}_{*}:(g,h)\mapsto(i,h).$
These functors are determined by $\mathcal{E}$ and $\mathcal{M}$ uniquely up
to unique isomorphism.
4. 4.
$\mathcal{E}$ and $\mathcal{M}$ are closed under composition.
5. 5.
$\mathcal{E}^{\bot}=\mathcal{M}$ and ${}^{\bot}\mathcal{M}=\mathcal{E}$, where
$\mathcal{E}^{\bot}=\\{f\mbox{ in }\mathcal{C}:e\mathop{\bot}f\mbox{ for all
}e\in\mathcal{E}\\}$ and ${}^{\bot}\mathcal{M}=\\{f\mbox{ in
}\mathcal{C}:f\mathop{\bot}m\mbox{ for all }m\in\mathcal{M}\\}$.
Proofs of these statements may be found in [AHS04] section 14.
We will also use the following fact:
###### Lemma 3.0.7.
Let $\mathcal{C}$ be a category with a factorization system
$(\mathcal{E},\mathcal{M})$. Let $T$ be a monad on $\mathcal{C}$ and let
$\overline{}\mathcal{E}=\\{f\mathop{\mbox{in}}\
\mathcal{C}:Uf\in\mathcal{E}\\}$ and
$\overline{}\mathcal{M}=\\{f\mathop{\mbox{in}}\
\mathcal{C}:Uf\in\mathcal{M}\\}$, where $U$ is the forgetful functor
$\mathcal{C}^{T}\to\mathcal{C}$. Then
$(\overline{}\mathcal{E},\overline{}\mathcal{M})$ is a factorization system on
$\mathcal{C}^{T}$ if $T$ preserves $\mathcal{E}$-arrows.
###### Proof.
This is established in [AHS04], Proposition 20.24: however, we shall provide a
proof for the reader’s convenience. We shall establish the axioms listed in
Definition 3.0.2.
1. Take an algebra map
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tf}$$\scriptstyle{a}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
Applying axiom 1 to the factorization system $(\mathcal{E},\mathcal{M})$, we
obtain a decomposition $f=me$, where $e:A\to I$ and $m:I\to B$. We wish to
lift this to a decomposition of $f$ as an algebra map. In other words, we need
a map $i:TI\to I$ making the diagram
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Te}$$\scriptstyle{a}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tm}$$\scriptstyle{i}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{B}$
commute, such that $(I,i)$ is a $T$-algebra. Since $T$ preserves
$\mathcal{E}$-arrows, $Te\mathop{\bot}m$, and we may obtain $i$ by applying
this orthogonality to the diagram
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{Te}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tm}$$\scriptstyle{\exists!i}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{B.}$
It remains to show that $(I,i)$ is a $T$-algebra. For the unit axiom, consider
the diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{\eta_{A}}$$\scriptstyle{1}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\scriptstyle{\eta_{I}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{B}}$$\scriptstyle{1}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Te}$$\scriptstyle{a}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tm}$$\scriptstyle{i}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{B}$
The top squares commute by naturality, and the outside triangles commute since
$(A,a)$ and $(B,b)$ are $T$-algebras. Hence the diagram
---
$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{e}$$\textstyle{B}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$
commutes if the dotted arrow is either $1_{I}$ or $i\eta_{I}$. By
orthogonality, $i\eta_{I}=1_{I}$.
For the multiplication axiom, observe that the diagrams
$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}e}$$\scriptstyle{\mu_{A}}$$\textstyle{T^{2}I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}m}$$\scriptstyle{\mu_{I}}$$\textstyle{T^{2}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{B}}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Te}$$\scriptstyle{a}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tm}$$\scriptstyle{i}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{B}$
$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}e}$$\scriptstyle{Ta}$$\scriptstyle{a\mu_{A}}$$\textstyle{T^{2}I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}m}$$\scriptstyle{Ti}$$\textstyle{T^{2}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tb}$$\scriptstyle{b\mu_{B}}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Te}$$\scriptstyle{a}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tm}$$\scriptstyle{i}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{B}$
both commute. So the diagram
$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{A}}$$\scriptstyle{T^{2}e}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{T^{2}I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}m}$$\textstyle{T^{2}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{B}}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{B}$
commutes if we take the dotted arrow to be either $i\mu_{I}$ or $i(Ti)$. By
orthogonality, $i\mu_{I}=i(Ti)$.
2. The image under $U$ of an isomorphism in $\mathcal{C}^{T}$ is an isomorphism in $\mathcal{C}$. The class $\mathcal{E}$ contains all isomorphisms in $\mathcal{C}$, so $\overline{}\mathcal{E}=U^{-1}(\mathcal{E})$ contains all isomorphisms in $\mathcal{C}^{T}$. By similar reasoning, $\overline{}\mathcal{E}$ is closed under composition with isomorphisms, and $\overline{}\mathcal{M}$ also satisfies these conditions.
3. We wish to show that $\overline{}\mathcal{E}\mathop{\bot}\overline{}\mathcal{M}$. Take $T$-algebras
$\mbox{$\left(\begin{matrix}{T}{A}\cr\phantom{a}\Big{\downarrow}{a}\cr{A}\end{matrix}\right)$},\mbox{$\left(\begin{matrix}{T}{B}\cr\phantom{b}\Big{\downarrow}{b}\cr{B}\end{matrix}\right)$},\mbox{$\left(\begin{matrix}{T}{I}\cr\phantom{i}\Big{\downarrow}{i}\cr{I}\end{matrix}\right)$},\mbox{$\left(\begin{matrix}{T}{J}\cr\phantom{j}\Big{\downarrow}{j}\cr{J}\end{matrix}\right)$}$
and algebra maps
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.36633pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\\\\{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-10.36633pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.20522pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{Te}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.36633pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-9.70012pt\raise-15.0pt\hbox{{}\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.36633pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.57292pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.30833pt\hbox{$\scriptstyle{i}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.57292pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-6.75pt\raise-36.83331pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.73665pt\raise-32.32637pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.98265pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.98265pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{I}$}}}}}}}}}}}\ignorespaces,\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
9.8698pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\\\\{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-9.8698pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{TJ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.26534pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{Tm}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.8698pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-9.28334pt\raise-15.0pt\hbox{{}\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.62779pt\hbox{$\scriptstyle{j}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.8698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.52953pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 44.52953pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-6.25346pt\raise-36.83331pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.79675pt\raise-32.32637pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{m}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.48613pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.48613pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B}$}}}}}}}}}}}\ignorespaces,\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
10.36633pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\\\\{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-10.36633pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.74463pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{Tf}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.36633pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-9.70012pt\raise-15.0pt\hbox{{}\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.36633pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{TJ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.23613pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.62779pt\hbox{$\scriptstyle{j}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 44.23613pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-6.75pt\raise-36.83331pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.27605pt\raise-30.72221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.98267pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.98267pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J}$}}}}}}}}}}}\ignorespaces,\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
9.20659pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\\\\{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-9.20659pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.8802pt\raise 6.07222pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{Tg}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.20659pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-8.41159pt\raise-15.0pt\hbox{{}\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30833pt\hbox{$\scriptstyle{i}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.20659pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.86632pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.86632pt\raise-27.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-5.59027pt\raise-36.83331pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.41164pt\raise-31.64581pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
36.82292pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.82292pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B}$}}}}}}}}}}}\ignorespaces$
where the first two maps are in $\overline{}\mathcal{E}$ and
$\overline{}\mathcal{M}$ respectively. Suppose that $ge=mf$. Now,
$e\in\mathcal{E}$ and $m\in\mathcal{M}$, so $e\mathop{\bot}m$, and there is a
unique map $t$ in $\mathcal{C}$ such that
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{e}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!t}$$\scriptstyle{g}$$\textstyle{B}$
commutes. We wish to show that $t$ is a map of $T$-algebras.
Consider the diagram
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Te}$$\scriptstyle{a}$$\scriptstyle{Tf}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tt}$$\scriptstyle{i}$$\scriptstyle{Tg}$$\textstyle{TJ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tm}$$\scriptstyle{j}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{f}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{g}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{B}$
We wish to show that the middle square commutes: the assumptions tell us that
all other squares commute. Recall that $Te\mathop{\bot}m$, and apply
orthogonality to the square
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{jTf}$$\scriptstyle{Te}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{TI\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!u}$$\scriptstyle{gi}$$\textstyle{B}$
Now,
$\begin{array}[]{rcll}j(Tf)&=&fa&\mbox{($f$ is a map of $T$-algebras)}\\\
&=&tea&\mbox{(Definition of $t$)}\\\ &=&ti(Te)&\mbox{($e$ is a map of
$T$-algebras)}\\\ \end{array}$
and $mti=gi$ by definition of $t$, so $ti=u$ by uniqueness. Similarly,
$\begin{array}[]{rcll}gi&=&b(Tg)&\mbox{($g$ is a map of $T$-algebras)}\\\
&=&b(Tm)(Tt)&\mbox{(Definition of $t$)}\\\ &=&mj(Tt)&\mbox{($m$ is a map of
$T$-algebras)}\\\ \end{array}$
and $j(Tf)=j(Tt)(Te)$ by definition of $t$, so $j(Tt)=u$ by uniqueness. Hence
$j(Tt)=ti$, and $t$ is a map of $T$-algebras.
By construction, $t$ is unique. So $e\mathop{\bot}m$ in $\mathcal{C}^{T}$, so
$\overline{}\mathcal{E}\mathop{\bot}\overline{}\mathcal{M}$. All the axioms
are satisfied, and so $(\overline{}\mathcal{E},\overline{}\mathcal{M})$ is a
factorization system on $\mathcal{C}^{T}$. ∎
###### Example 3.0.8.
Let $(\mathcal{E},\mathcal{M})$ be the factorization system on Digraph
described in Example 3.0.5 above, and let $T$ be the free category monad. Cat
is monadic over Digraph, and $T$ preserves the property of being bijective on
objects. Hence, this gives a factorization system
$(\overline{}\mathcal{E},\overline{}\mathcal{M})$ on Cat where
$\overline{}\mathcal{E}$ is the collection of bijective-on-objects functors,
and $\overline{}\mathcal{M}$ is the collection of full and faithful functors.
###### Example 3.0.9.
Similarly, there is a factorization system on Digraph, where $\mathcal{E}$ is
the class of maps that are pointwise bijective on objects, and $\mathcal{M}$
is the class of maps that are pointwise full and faithful. This lifts to a
factorization system $(\overline{}\mathcal{E},\overline{}\mathcal{M})$ on Cat,
in which $\overline{}\mathcal{E}$ is the class of pointwise bijective-on-
objects arrows, and $\overline{}\mathcal{M}$ is the class of pointwise full-
and-faithful arrows.
###### Example 3.0.10.
Let $\mathcal{C}=\mbox{{{Cat}}}$, $\mathcal{E}$ be the pointwise bijective-on-
objects maps, and $\mathcal{M}$ be those that are pointwise full and faithful.
Since Cat-Operad is monadic over Cat and the monad preserves bijective-on-
objects maps, this gives a factorization system
$(\overline{}\mathcal{E},\overline{}\mathcal{M})$ on Cat-Operad where
$\overline{}\mathcal{E}$ is the class of levelwise bijective-on-objects maps,
and $\overline{}\mathcal{M}$ is the class of levelwise full and faithful ones.
Similarly, there is a factorization system
$(\overline{}\mathcal{E}^{\prime},\overline{}\mathcal{M}^{\prime})$ on
Cat-$\Sigma$-Operad where $\overline{}\mathcal{E}^{\prime}$ is the class of
bijective-on-objects maps, and $\overline{}\mathcal{M}^{\prime}$ is the class
of levelwise full and faithful ones.
We shall need one final piece of background:
###### Theorem 3.0.11.
If $X$ is a set and $T$ is a monad on $\mbox{{{Set}}}^{X}$ then the regular
epis in $(\mbox{{{Set}}}^{X})^{T}$ are the pointwise surjections. In other
words, the forgetful functor $U:(\mbox{{{Set}}}^{X})^{T}\to\mbox{{{Set}}}^{X}$
preserves and reflects regular epis.
###### Proof.
See again [AHS04] section 20, in particular Definition 20.21 and Proposition
20.30. ∎
## Chapter 4 Categorification
### 4.1 Desiderata
Many categorifications of individual theories have been proposed in the
literature. We aim to replace these with a general definition, which should
satisfy the following criteria insofar as possible:
* •
Broad: it should cover as large a class of theories as possible.
* •
Consistent with earlier work: where a categorification of a given theory is
known, ours should agree with this categorification or be demonstrably better
in some way.
* •
Canonical: it should be free of arbitrary tunable parameters (and if possible
should be given by some universal property).
We shall return to these criteria in Section 4.9 and evaluate how close we
have come to achieving them.
Our strategy is as follows: we start with a naïve version of categorification
for strongly regular theories, which closely parallels Mac Lane and Benabou’s
categorification of the theory of monoids. This will be an _unbiased_
categorification, which treats all operations equally, without regarding any
as “primitive”: for instance, if $P$ is the terminal operad (whose strict
algebras are monoids), then the weak $P$-categories will have tensor products
of all arities, not just 0 and 2. We then re-express our definition of
categorification in terms of factorization systems, which allows us to
generalize our definition in two directions simultaneously: to symmetric
operads, and to operads with presentations. We then use this new definition to
recover the classical theory of symmetric monoidal categories (at which
several other proposed general definitions of categorification fail), and
investigate what it yields in the case of some other linear theories.
### 4.2 Categorification of strongly regular theories
The idea is to consider the strict models of our theory as algebras for an
operad, then to obtain the weak models as (strict) algebras for a weakened
version of that operad (which will be a Cat-operad). We weaken the operad
using a similar approach to that used in Penon’s definition of $n$-category,
as described in [Pen99]. A non-rigorous summary of Penon’s construction can be
found in [CL04].
Throughout this section, let $P$ be a plain (Set-)operad.
Let $D_{*}:\mbox{{{Operad}}}\to\mbox{{{Cat-Operad}}}$ be the functor which
takes discrete categories levelwise; i.e., $(D_{*}P)_{n}$ is the discrete
category on the set $P_{n}$. In terms of the “$n$-cell” terminology introduced
in Chapter 3, the 1-cells of $(D_{*}P)_{n}$ are $n$-ary arrows in $P$, and the
only 2-cells are identities.
###### Definition 4.2.1.
The unbiased weakening of $P$, Wk($P$), is the following Cat-operad:
* •
_1-cells:_ 1-cells of $D_{*}{F_{\rm pl}}{U^{\rm pl}}P$;
* •
_2-cells:_ if $A,B\in({F_{\rm pl}}{U^{\rm pl}}P)_{n}$, there is a single
2-cell $A\to B$ if $\epsilon(A)=\epsilon(B)$ (where $\epsilon$ is the counit
of the adjunction ${F_{\rm pl}}\dashv{U^{\rm pl}}$), and no 2-cells $A\to B$
otherwise;
* •
_Composition of 2-cells:_ the composite of two arrows $A\rightarrow
B\rightarrow C$ is the unique arrow $A\rightarrow C$, and in particular, the
arrows $A\rightarrow B$ and $B\rightarrow A$ are inverses;
* •
_Operadic composition:_ on 1-cells, as in ${F_{\rm pl}}{U^{\rm pl}}P$, and on
2-cells, determined by the uniqueness property.
See Fig. 4.1, which illustrates a fragment of the unbiased weakening of the
terminal operad 1. Since $1_{n}$ is a singleton set for every $n\in\natural$,
then ${\mbox{Wk($1$)}}_{n}$ is the indiscrete category whose objects are
unlabelled $n$-ary strongly regular trees for all $n\in\natural$. We may embed
the discrete category on each $P_{n}$ in ${\mbox{Wk($P$)}}_{n}$, via the map
$p\mapsto p\circ(|,\dots,|)$. We shall occasionally abuse notation and
consider some $p\in P_{n}$ as a 1-cell of ${\mbox{Wk($P$)}}_{n}$.
Figure 4.1: Part of ${{\mbox{Wk($1$)}}}_{3}$
###### Theorem 4.2.2.
Wk($P$) is the unique Cat-operad with the following properties:
* •
Wk($P$) has the same 1-cells as $D_{*}{F_{\rm pl}}{U^{\rm pl}}P$;
* •
we may extend the counit $\epsilon_{P}:{F_{\rm pl}}{U^{\rm pl}}P\to P$ to a
map of Cat-operads ${\mbox{Wk($P$)}}\to D_{*}P$, which is full and faithful
levelwise.
###### Proof.
Immediate. ∎
We may now make the following definition:
###### Definition 4.2.3.
A weak $P$-category is an algebra for Wk($P$).
In the case $P=1$, this reduces exactly to Leinster’s definition of unbiased
monoidal category in [Lei03] section 3.1. There, two 1-cells $\phi$ and $\psi$
have the same image under $\epsilon$ iff they have the same arity, so the
categories ${\mbox{Wk($1$)}}_{i}$ are indiscrete. If $h:{\mbox{Wk($P$)}}\circ
A\to A$ is a weak $P$-category, we refer to the image under $h$ of a 2-cell
$q\to q^{\prime}$ in Wk($P$) as $\delta_{q,q^{\prime}}$. This is clearly a
natural transformation $h(q,-)\to h(q^{\prime},-)$. As a special case, we
write $\delta_{q}$ for $\delta_{q,\epsilon(q)}$ (where we consider
$\epsilon(q)$ as a 1-cell of Wk($P$) as described above).
###### Definition 4.2.4.
A strict $P$-category is an algebra for $D_{*}P$.
Equivalently, a strict $P$-category is a weak $P$-category in which every
component of $\delta$ is an identity arrow.
###### Definition 4.2.5.
Let $(A,h)$ and $(B,h^{\prime})$ be weak $P$-categories. A weak $P$-functor
from $(A,h)$ to $(B,h^{\prime})$ is a weak map of Wk($P$)-algebras. A strict
$P$-functor from $(A,h)$ to $(B,h^{\prime})$ is a strict map of
Wk($P$)-algebras.
Equivalently, a strict $P$-functor is a weak $P$-functor for which all the
coherence maps are identities. These definition are natural generalizations of
the definition of weak and strict unbiased monoidal functors given in [Lei03]
section 3.1.
###### Definition 4.2.6.
Let $(F,\phi)$ and $(G,\psi)$ be weak $P$-functors $(A,h)\to(B,h^{\prime})$. A
$P$-transformation $\sigma:(F,\phi)\to(G,\psi)$ is a Wk($P$)-transformation
$(F,\phi)\to(G,\psi)$, in the sense of Definition 2.9.12.
Note that there is only one possible level of strictness here.
There is a 2-category, Wk-$P$-Cat, whose objects are weak $P$-categories,
whose 1-cells are weak $P$-functors, and whose 2-cells are
$P$-transformations. Similarly, there is a 2-category Str-$P$-Cat of strict
$P$-categories, strict $P$-functors, and $P$-transformations, which can be
considered a sub-2-category of Wk-$P$-Cat.
###### Definition 4.2.7.
A $P$-equivalence is an equivalence in the 2-category Wk-$P$-Cat.
###### Lemma 4.2.8.
Let $P$ be a plain operad, $(A,h)$ and $(B,h^{\prime})$ be weak
$P$-categories, and $(F,\phi),(G,\psi):(A,h)\to(B,h^{\prime})$ be weak
$P$-functors. A $P$-transformation $\sigma:(F,\phi)\to(G,\psi)$ is invertible
as a $P$-transformation if and only if it is invertible as a natural
transformation.
###### Proof.
This is a straightforward application of Lemma 2.9.13. ∎
### 4.3 Examples
Unfortunately, few well-studied theories are strongly regular. We will
consider the following examples:
1. 1.
the trivial theory (in other words, the theory of sets);
2. 2.
the theory of pointed sets;
3. 3.
the theory of monoids;
4. 4.
the theory of $M$-sets, for a monoid $M$.
While we could easily invent a new strongly regular theory to categorify, this
would not help us to see how well our definition of weakening accords with our
intuitions. Further examples will be considered later, when the machinery to
categorify theories-with-generators and linear theories has been developed.
We will first need to introduce an auxiliary definition:
###### Definition 4.3.1.
Let $\mathcal{C}$ be a category, and $(T,\mu,\eta)$ be a monad on
$\mathcal{C}$. We say that $(T,\mu,\eta)$ is trivial if $\eta$ is a natural
isomorphism.
###### Lemma 4.3.2.
The identity monad on $\mathcal{C}$ is initial in the category
$\mbox{{{Mnd}}}(\mathcal{C})$ of monads on $\mathcal{C}$, with the unique
morphism of monads $(1_{\mathcal{C}},1,1)\to(T,\mu,\eta)$ being $\eta$.
###### Proof.
First we show that $\eta$ is a morphism of monads in the sense of Street
(Definition 1.4.1). One axiom corresponds to the outside of the diagram
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{1}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta
T}$$\scriptstyle{1}$$\scriptstyle{1}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{T}$
commuting; all the inner segments commute (the top right triangle by the unit
axiom for monads), so the outside must commute. The other axiom corresponds to
the diagram
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\eta}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{T}$
and this commutes trivially. Hence, $\eta$ is a morphism of monads $1\to T$.
Now suppose that $\alpha:1\to T$ is a morphism of monads. From the unit axiom
for monad morphisms, the diagram
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\eta}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T}$
must commute, so $\eta=\alpha$. ∎
###### Corollary 4.3.3.
A monad $(T,\mu,\eta)$ on $\mathcal{C}$ is trivial if and only if it is
isomorphic to the identity monad on $\mathcal{C}$.
###### Proof.
If $T$ is isomorphic to the identity monad, then by Lemma 4.3.2 the
isomorphism concerned must be $\eta$, so $\eta$ must be invertible. It is
readily checked that if $\eta$ is invertible, then $\eta^{-1}$ must be a
morphism of monads, so if $T$ is trivial then it is isomorphic to the identity
monad. ∎
###### Example 4.3.4.
_The trivial theory:_ Let $0$ be the initial operad, whose algebras are sets.
An unbiased weak $0$-category is a category equipped with a specified trivial
monad, for the following reason. $0$ has only one operator (call it $I$), of
arity one. Hence $({F_{\rm pl}}{U^{\rm pl}}0)_{1}\cong\natural$, and all other
$({F_{\rm pl}}{U^{\rm pl}}0)_{n}$’s are empty. All derived operations in the
theory of sets are composites of identities, and thus equivalent to the
identity. So all objects of ${\mbox{Wk($0$)}}_{1}$ are isomorphic. Hence,
$I\cong\mbox{id}$. All diagrams commute: in particular, those giving the monad
and monad morphism axioms commute, so in any weak $0$-category
$(\mathcal{C},(\hat{\phantom{\alpha}}))$, the functor $\hat{I}$ is a monad,
and the isomorphism $\hat{I}\to 1_{\mathcal{C}}$ is an isomorphism of monads.
By Corollary 4.3.3, $\hat{I}$ must be trivial.
Conversely, suppose $T$ is a trivial monad on a category $\mathcal{C}$. We
wish to show that $\mu$ is also invertible, and thus that $\mathcal{C}$ is an
unbiased weak $0$-category. From the monad axioms, we have that
$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta
T}$id$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T}$
commutes. But $\eta T$ is invertible, so $\mu$ must be its inverse. So
$\mbox{id}\cong T\cong T^{2}\cong T^{3}\cong\ldots$, and all diagrams commute.
Hence $\mathcal{C}$ is an unbiased weak $0$-category.
If $(\mathcal{C},S)$ and $(\mathcal{D},T)$ are weak $0$-categories, then a
weak $0$-functor $(\mathcal{C},S)\to(\mathcal{D},T)$ is a functor
$F:\mathcal{C}\to\mathcal{D}$ and a natural isomorphism $\phi:TF\to FS$, such
that the equations
| |
---|---|---
$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{S}$$\scriptstyle{F}$$\textstyle{\scriptstyle\phi^{-1}}$$\textstyle{\@ensuremath{\mathcal{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\scriptstyle\mu}$$\textstyle{D}$$\textstyle{=}$
---
$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{F}$$\textstyle{\scriptstyle\phi^{-1}}$$\textstyle{\@ensuremath{\mathcal{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{T}$$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{F}$$\textstyle{\scriptstyle\phi^{-1}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{D\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\mu}$
and
|
---|---
$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle 1_{\mathcal{C}}}$$\textstyle{\scriptstyle\eta}$$\scriptstyle{S}$$\scriptstyle{F}$$\textstyle{\scriptstyle\phi^{-1}}$$\textstyle{\@ensuremath{\mathcal{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\@ensuremath{\mathcal{D}}}$$\textstyle{=}$ |
---|---
$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\phi^{-1}}$$\scriptstyle{F}$$\scriptstyle{1_{\mathcal{C}}}$$\textstyle{\@ensuremath{\mathcal{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle
T}$$\textstyle{\scriptstyle\eta}$$\scriptstyle{1_{\mathcal{D}}}$$\textstyle{\@ensuremath{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\@ensuremath{\mathcal{D}}}$
are satisfied.
###### Example 4.3.5.
_Pointed sets:_ Let $P$ be the operad with a single element of arity 0 (call
it $*$) and a single element of arity 1 (the identity). Strict algebras for
$P$ in Set are pointed sets. The set $({F_{\rm pl}}{U^{\rm pl}}P)_{0}$ is
countable (it has elements $*,I*,I^{2}*,I^{3}*,\ldots$, and so is $({F_{\rm
pl}}{U^{\rm pl}}P)_{1}$ (it has elements $\mbox{id},I,I^{2},\ldots$). So an
unbiased weak $P$-category is a category $\mathcal{C}$ equipped with a
distinguished object $\hat{*}$ and a trivial monad $\hat{I}$.
If $(\mathcal{C},(\hat{\phantom{\alpha}}))$ and
$(\mathcal{D},(\bar{\phantom{\alpha}})$ are unbiased weak $P$-categories, then
a weak $P$-functor
$(\mathcal{C},(\hat{\phantom{\alpha}}))\to(\mathcal{D},(\bar{\phantom{\alpha}}))$
is a triple $(F,\phi,\psi)$, where $F$ and $\phi$ are as in Example 4.3.4,
$\psi:\bar{*}\to F\hat{*}$ is an isomorphism, and there is exactly one natural
isomorphism $\bar{I}^{n}\bar{*}\to F\hat{I}^{m}\hat{*}$ composed from $\phi$s
and $\psi$s for each $m$ and each $n\in\natural$.
###### Example 4.3.6.
_Monoids:_ An unbiased weak $1$-category is precisely an unbiased weak
monoidal category in the sense of Definition 2.1.3. An unbiased weak
$1$-functor is an unbiased weak monoidal functor. For a proof, see [Lei03]
Theorem 3.2.2.
###### Example 4.3.7.
_$M$ -sets:_ Let $M$ be a monoid, and $N$ be the operad such that
$\displaystyle N_{1}$ $\displaystyle=$ $\displaystyle M$ $\displaystyle N_{i}$
$\displaystyle=$ $\displaystyle\emptyset\mbox{ whenever $i\neq 1$}$
with composition of arrows of arity 1 given by the multiplication in $M$. An
algebra for $N$ in Set is an $M$-set. An unbiased weak $N$-category is a
category $\mathcal{C}$ with a functor $\hat{m}:\mathcal{C}\to\mathcal{C}$ for
each $m\in\mathcal{M}$. For every equation $m_{1}m_{2}\dots
m_{i}=n_{1}n_{2}\dots n_{j}$ that is true in $M$, there is a natural
isomorphism $\delta_{m_{1}\dots m_{i}}^{n_{1}\dots
n_{j}}:\hat{m}_{1}\hat{m}_{2}\dots\hat{m}_{i}\to\hat{n}_{1}\hat{n}_{2}\dots\hat{n}_{j}$.
If $e$ is the identity element in $M$, then $\hat{e}$ is a trivial monad. All
diagrams involving these natural isomorphisms commute. Hence, an unbiased weak
$N$-category is a category $\mathcal{C}$ together with a weak monoidal functor
$M\to\mathop{\rm{End}}(\mathcal{C}$). If
$(\mathcal{C},(\hat{\phantom{\alpha}}))$ and
$(\mathcal{D},(\bar{\phantom{\alpha}}))$ are unbiased weak $N$-categories, an
unbiased weak $N$-functor is a functor $F:\mathcal{C}\to\mathcal{D}$ together
with natural transformations $\phi_{m}:\bar{m}F\to F\hat{m}$ for all $m\in M$,
such that if $m_{1}m_{2}\dots m_{i}=n_{1}n_{2}\dots n_{j}$ in $M$, there is
precisely one natural isomorphism $\bar{m}_{1}\dots\bar{m}_{i}F\to
F\hat{n}_{1}\dots\hat{n}_{j}$ that can be formed by composing $\delta$s and
$\phi$s.
### 4.4 A more general approach: factorization systems
Recall from Definition 2.7.2 the definition of a presentation and a generator
for an operad. We will define a categorification of any symmetric operad
equipped with a generator, generalizing the unbiased categorification defined
in Section 4.2. In particular, we shall consider categorification with respect
to the component of the counit $\epsilon_{P}:{F_{\Sigma}}{U^{\Sigma}}P\to P$
at a symmetric operad $P$; this is a generator for $P$ since both
$\textstyle{{F_{\Sigma}}{U^{\Sigma}}{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon{F_{\Sigma}}{U^{\Sigma}}}$$\scriptstyle{{F_{\Sigma}}{U^{\Sigma}}\epsilon}$$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P}$
and
$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\times_{P}{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P}$
are coequalizer diagrams (the latter by Lemma 2.7.9). We will then show that
the categorification is independent of our choice of generator, in the sense
that the symmetric Cat-operads which arise are equivalent (and thus have
equivalent categories of algebras).
###### Definition 4.4.1.
Let $\Phi$ be a signature, $P$ be a symmetric operad, and
$\phi:{F_{\Sigma}}\Phi\to P$ be a regular epi in $\Sigma$-Operad. Then the
weakening (or categorification) Wk${}_{\phi}(P)$ of $P$ with respect to $\phi$
is the (unique-up-to-isomorphism) symmetric Cat-operad such that the following
diagram commutes:
---
$\textstyle{D_{*}{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\phi}$$\scriptstyle{b}$$\textstyle{D_{*}P}$Wk${}_{\phi}(P)$$\scriptstyle{f}$
where $f$ is full and faithful levelwise, $b$ is levelwise bijective on
objects, and $D_{*}$ is the levelwise discrete category functor
$\mbox{{{$\Sigma$-Operad}}}\to\mbox{{{Cat-$\Sigma$-Operad}}}$. The existence
and uniqueness of Wk${}_{\phi}(P)$ follow from Lemma 3.0.6 applied to the
factorization system on Cat-$\Sigma$-Operad described in 3.0.10 above.
###### Definition 4.4.2.
Let $\phi,\Phi$ and $P$ be as above. A $\phi$-weak $P$-category is an algebra
for Wk${}_{\phi}(P)$.
Note that any strict algebra for $P$ can be considered as a $\phi$-weak
$P$-category (for any $\phi$), via the map
Wk${}_{\phi}(P)$$\textstyle{D_{*}P}$.
###### Definition 4.4.3.
Let $\phi,\Phi$ and $P$ be as above. A $\phi$-weak $P$-functor is a weak map
of Wk${}_{\phi}(P)$-algebras.
###### Definition 4.4.4.
Let $P$ be a symmetric operad, and
$\textstyle{{F_{\Sigma}}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{2}}$$\textstyle{{F_{\Sigma}}\Phi}$
be a presentation for $P$, with $\phi:{F_{\Sigma}}\Phi\to P$ being the regular
epi in Definition 2.7.2. The weakening of $P$ with respect to $(\Phi,E)$ is
the weakening of $P$ with respect to $\phi$.
###### Definition 4.4.5.
The unbiased weakening of $P$ is the weakening arising from the counit
$\epsilon:{F_{\Sigma}}{U^{\Sigma}}P\to P$ of the adjunction
${F_{\Sigma}}\dashv{U^{\Sigma}}$. Call this symmetric Cat-operad Wk($P$).
###### Lemma 4.4.6.
Let $\phi,\Phi$ and $P$ be as above. Then, for every $n\in\natural$, the
category ${\mbox{Wk${}_{\phi}(P)$}}_{n}$ is the equivalence relation $\sim$ on
the elements of $({F_{\Sigma}}\Phi)_{n}$, where $t_{1}\sim t_{2}$ if
$\phi(t_{1})=\phi(t_{2})$.
###### Proof.
Let $n\in\natural$, and $t_{1},t_{2}\in{\mbox{Wk${}_{\phi}(P)$}}_{n}$. The
objects of ${\mbox{Wk${}_{\phi}(P)$}}_{n}$ are the elements of
$({F_{\Sigma}}\Phi)_{n}$, by construction. Since $\phi_{n}$ factors through a
full functor
$\textstyle{{\mbox{Wk${}_{\phi}(P)$}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(D_{*}P)_{n}}$
and $(D_{*}P)_{n}$ is the discrete category on $P_{n}$, there is an arrow
$t_{1}\to t_{2}$ in ${\mbox{Wk${}_{\phi}(P)$}}_{n}$ iff
$\phi(t_{1})=\phi(t_{2})$. Since this functor is also faithful, such an arrow
must be unique. Hence ${\mbox{Wk${}_{\phi}(P)$}}_{n}$ is a poset; it is
readily checked that it is also an equivalence relation. ∎
An obvious question is how this notion of weakening is related to the version
defined for plain operads in Section 4.2. In light of Theorem 4.2.2, it is
clear that the plain-operadic version can be re-phrased as in Definition 4.4.5
above, but with the factorization occurring in Cat-Operad rather than
Cat-$\Sigma$-Operad. We may generalize it to give a definition of the
weakening of a plain operad $P$ with respect to a generator $\phi$:
###### Definition 4.4.7.
Let $P$ be a plain operad, $\Phi$ be a signature, and $\phi:{F_{\rm
pl}}\Phi\to P$ be a regular epi. The weakening Wk${}_{\phi}(P)$ of $P$ with
respect to $\phi$ is the plain Cat-operad given by the bijective on
objects/levelwise full and faithful factorization
---
$\textstyle{D_{*}{F_{\rm
pl}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\phi}$$\textstyle{D_{*}P}$Wk${}_{\phi}(P)$
in Cat-Operad. A $\phi$-weak $P$-category is an algebra for Wk${}_{\phi}(P)$.
But do the weak algebras for a strongly regular theory $T$ change if we
consider $T$ as a linear theory instead? We now answer that question in the
negative.
###### Theorem 4.4.8.
Let $P$ be a plain operad, let $\Phi$ be a signature, and let $\phi:{F_{\rm
pl}}\Phi\to P$ be a regular epi. Then ${\mbox{Wk${}_{{F_{\Sigma}^{\rm
pl}}\phi}({F_{\Sigma}^{\rm pl}}P)$}}\cong{F_{\Sigma}^{\rm
pl}}({\mbox{Wk${}_{\phi}(P)$}})$ in the category Cat-$\Sigma$-Operad.
###### Proof.
First note that Wk${}_{{F_{\Sigma}^{\rm pl}}\phi}({F_{\Sigma}^{\rm pl}}P)$ is
well-defined: ${F_{\Sigma}^{\rm pl}}$ is a left adjoint, and hence preserves
colimits, so ${F_{\Sigma}^{\rm pl}}\phi$ is a regular epi in $\Sigma$-Operad.
Wk${}_{{F_{\Sigma}^{\rm pl}}\phi}({F_{\Sigma}^{\rm pl}}P)$ is defined by its
universal property, so it is enough to show that the Cat-operad
${F_{\Sigma}^{\rm pl}}({\mbox{Wk${}_{\phi}(P)$}})$ also has this property.
Specifically, it is enough to show that if
---
$\textstyle{D_{*}{F_{\rm
pl}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\phi}$$\scriptstyle{b}$$\textstyle{D_{*}P}$Wk${}_{\phi}(P)$$\scriptstyle{f}$
is the bijective-on-objects/full-and-faithful factorization of $\phi$, then in
the diagram
---
$\textstyle{D_{*}{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}{F_{\Sigma}^{\rm
pl}}\phi}$$\scriptstyle{{F_{\Sigma}^{\rm
pl}}b}$$\textstyle{D_{*}{F_{\Sigma}^{\rm pl}}P}$$\textstyle{{F_{\Sigma}^{\rm
pl}}({\mbox{Wk${}_{\phi}(P)$}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\Sigma}^{\rm
pl}}f}$
the arrow ${F_{\Sigma}^{\rm pl}}b$ is bijective on objects and the arrow
${F_{\Sigma}^{\rm pl}}f$ is levelwise full and faithful (note that
$D_{*}{F_{\Sigma}^{\rm pl}}={F_{\Sigma}^{\rm pl}}D_{*}$). But this follows
straightforwardly from the explicit construction of ${F_{\Sigma}^{\rm pl}}$ in
Section 2.6. ∎
###### Corollary 4.4.9.
Let $P$ be a plain operad, $\phi:{F_{\rm pl}}\Phi\to P$ generate $P$, and $A$
be a $\phi$-weak $P$-category in the sense of Definition 4.4.7. Then $A$ is an
${F_{\Sigma}^{\rm pl}}\phi$-weak ${F_{\Sigma}^{\rm pl}}P$-category in the
sense of Definition 4.4.2. Conversely, every ${F_{\Sigma}^{\rm pl}}\phi$-weak
${F_{\Sigma}^{\rm pl}}P$-category is a weak $P$-category.
###### Proof.
A ${F_{\Sigma}^{\rm pl}}\phi$-weak ${F_{\Sigma}^{\rm pl}}P$-category is a
category $A$ and a morphism ${\mbox{Wk${}_{{F_{\Sigma}^{\rm
pl}}\phi}({F_{\Sigma}^{\rm pl}}P)$}}\to\mathop{\rm{End}}(A)$ of symmetric Cat-
operads. By Theorem 4.4.8, this is equivalent to a morphism ${F_{\Sigma}^{\rm
pl}}({\mbox{Wk${}_{\phi}(P)$}})\to\mathop{\rm{End}}(A)$ in
Cat-$\Sigma$-Operad, which is equivalent by the adjunction ${F_{\Sigma}^{\rm
pl}}\dashv{U^{\Sigma}_{\rm pl}}$ to a morphism of plain Cat-operads
${\mbox{Wk${}_{\phi}(P)$}}\to{U^{\Sigma}_{\rm pl}}\mathop{\rm{End}}(A)$. This
is exactly a $\phi$-weak $P$-category. ∎
Note that we had to apply ${F_{\Sigma}^{\rm pl}}$ to $\phi$ to obtain a
generator for ${F_{\Sigma}^{\rm pl}}P$. This means that the theorem does not
tell us that the unbiased categorification is unaffected by whether we
consider our theory as a linear or a strongly regular one. In fact, it is not
the case that ${\mbox{Wk(${F_{\Sigma}^{\rm pl}}P$)}}\cong{F_{\Sigma}^{\rm
pl}}({\mbox{Wk($P$)}})$ in general.
###### Example 4.4.10.
Consider the terminal plain operad $1$ whose algebras are monoids.
${F_{\Sigma}^{\rm pl}}1$ is the operad $\mathcal{S}$ of Example 2.1.11, for
which each $\mathcal{S}_{n}$ is the symmetric group $S_{n}$. Then the objects
of ${\mbox{Wk($\mathcal{S}$)}}_{n}$ are $n$-leafed permuted trees with each
node labelled by a permutation, whereas the 1-cells of $({F_{\Sigma}^{\rm
pl}}{\mbox{Wk($1$)}})_{n}$ are unlabelled permuted trees. These two sets are
not canonically isomorphic. Hence, there is no canonical isomorphism between
Wk($\mathcal{S}$) and ${F_{\Sigma}^{\rm pl}}{\mbox{Wk($1$)}}$.
However, we can make a weaker statement: the two candidate unbiased weakenings
are _equivalent_ in the 2-category Cat-$\Sigma$-Operad. We shall return to
this point in Corollary 5.3.3.
### 4.5 Examples
###### Example 4.5.1.
Consider the trivial theory (given by the initial operad 0), with the empty
generating set. A weak algebra for this theory (with respect to this
generating set) is simply a category. ${F_{\Sigma}}$ is a left adjoint, and
hence preserves colimits, so ${F_{\Sigma}}\emptyset$ is the initial operad,
and the coequalizer $\phi:{F_{\Sigma}}\emptyset\to 0$ is therefore the
identity. Hence Wk${}_{\phi}(0)$ is also the initial operad, and so a
$\phi$-weak 0-category is just a category. A $\phi$-weak $0$-functor is just a
functor.
###### Example 4.5.2.
Consider the operad $P$ of Example 4.3.5, generated by one nullary operation
$*$. Let $\phi$ be the associated regular epi. Then Wk${}_{\phi}(P)$ has one
nullary object and no objects of any other arity; the only arrow is the
identity on the unique nullary object. In fact,
${\mbox{Wk${}_{\phi}(P)$}}=D_{*}P$. So a weak algebra for this theory and this
generating set is a category $\mathcal{C}$ with a distinguished object
$\hat{*}\in\mathcal{C}$. A $\phi$-weak $P$-functor from
$(\mathcal{C},(\hat{\phantom{\alpha}}))$ to
$(\mathcal{D},(\bar{\phantom{\alpha}}))$ is a functor
$F:\mathcal{C}\to\mathcal{D}$ and an isomorphism
$\bar{*}\stackrel{{\scriptstyle\sim}}{{\to}}\hat{*}$.
###### Example 4.5.3.
Consider again the operad $P$ of Example 4.3.5, this time generated by four
nullary operations $A,B,C,D$ (which are all set equal to each other). Let
$\phi$ be the associated regular epi. Then ${\mbox{Wk${}_{\phi}(P)$}}_{0}$ is
the indiscrete category on the four objects $A,B,C,D$, and
${\mbox{Wk${}_{\phi}(P)$}}_{i}$ is empty for all other $i\in\natural$. Hence a
$\phi$-weak $P$-category is a category $\mathcal{C}$ containing four specified
objects $\hat{A},\hat{B},\hat{C}$ and $\hat{D}$. These four objects are
isomorphic via specified isomorphisms $\delta_{AB},\delta_{AC},\delta_{AD}$
etc, and all diagrams involving these isomorphisms commute:
$\textstyle{\hat{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{AB}}$$\scriptstyle{\delta_{AD}}$$\scriptstyle{\delta_{AC}}$$\textstyle{\hat{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{BD}}$$\scriptstyle{\delta_{BC}}$$\textstyle{\hat{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{CD}}$$\textstyle{\hat{D}}$
and $\delta_{XY}\delta_{YX}=1_{\hat{X}}$ for all $X,Y\in\\{A,B,C,D\\}$.
Let $(\mathcal{C},(\hat{\phantom{\alpha}}))$ and
$(\mathcal{D},(\bar{\phantom{\alpha}}))$ be $\phi$-weak $P$-categories. A
$\phi$-weak $P$-functor
$(\mathcal{C},(\hat{\phantom{\alpha}}))\to(\mathcal{D},(\bar{\phantom{\alpha}}))$
consists of
* •
a functor $F:\mathcal{C}\to\mathcal{D}$,
* •
an isomorphism $\phi_{XY}:\bar{X}\stackrel{{\scriptstyle\sim}}{{\to}}F\hat{X}$
for all $X\in\\{A,B,C,D\\}$,
such that, for all $X,Y\in\\{A,B,C,D\\}$, there is precisely one isomorphism
$\bar{X}\to F\hat{Y}$ formed by compositions of $\delta$s and $\phi$s.
### 4.6 Symmetric monoidal categories
Consider the terminal symmetric operad $P$, whose algebras in Set are
commutative monoids, and the following linear presentation $(\Phi,E)$ for $P$:
* •
$\Phi_{0}=\\{e\\},\Phi_{2}=\\{.\\}$, all other $\Phi_{i}$s are empty;
* •
$E$ contains the equations
1. 1.
$x_{1}.(x_{2}.x_{3})=(x_{1}.x_{2}).x_{3}$
2. 2.
$e.x_{1}=x_{1}$
3. 3.
$x_{1}.e=x_{1}$
4. 4.
$x_{1}.x_{2}=x_{2}.x_{1}$
This linear presentation gives rise to a symmetric-operadic presentation
$(\Phi,E)$, as described in Lemma 2.8.3. Let $\phi:{F_{\Sigma}}\Phi\to P$ be
the coequalizer in the diagram
$\textstyle{{F_{\Sigma}}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{P}$
We shall now prove that the algebras for Wk${}_{\phi}(P)$ are classical
symmetric monoidal categories. More precisely, we shall show the following:
1. 1.
for a given category $\mathcal{C}$, the Wk${}_{\phi}(P)$-algebra structures on
$\mathcal{C}$ are in one-to-one correspondence with the symmetric monoidal
category structures on $\mathcal{C}$;
2. 2.
there exists an isomorphism (which we construct) between the category
Wk-$P$-Cat and the category of symmetric monoidal categories and weak
functors;
3. 3.
the isomorphism in (2) respects the correspondence in (1).
To fix notation, we recall the classical notions of symmetric monoidal
category and symmetric monoidal functor:
###### Definition 4.6.1.
A symmetric monoidal category is a 7-tuple
$(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$, where
* •
$\mathcal{C}$ is a category;
* •
$\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ is a functor;
* •
$I$ is an object of $\mathcal{C}$,
* •
$\alpha:A\otimes(B\otimes C)\to(A\otimes B)\otimes C$ is natural in
$A,B,C\in\mathcal{C}$;
* •
$\lambda:I\otimes A\to A$ and $\rho:A\otimes I\to A$ are natural in
$A\in\mathcal{C}$;
* •
$\tau:A\otimes B\to B\otimes A$ is natural in $A,B\in\mathcal{C}$,
$\alpha,\lambda,\rho,\tau$ are all invertible, and the following diagrams
commute:
---
|
$\textstyle{(A\otimes B)\otimes(C\otimes D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{A\otimes(B\otimes(C\otimes D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\alpha}$$\scriptstyle{\alpha}$$\textstyle{((A\otimes B)\otimes C)\otimes D}$$\textstyle{A\otimes((B\otimes C)\otimes D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{(A\otimes(B\otimes C))\otimes D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\otimes 1}$ (4.1) |
---|---
$\textstyle{A\otimes(I\otimes C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{1\otimes\lambda}$$\textstyle{(A\otimes I)\otimes C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho\otimes 1}$$\textstyle{A\otimes C}$ (4.2) |
---|---
$\textstyle{A\otimes
I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\rho}$$\textstyle{I\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{A}$
(4.3) $\textstyle{(A\otimes B)\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\alpha^{-1}}$$\textstyle{C\otimes(A\otimes
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{A\otimes(B\otimes
C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\tau}$$\textstyle{(C\otimes
A)\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau\otimes
1}$$\textstyle{A\otimes(C\otimes
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{(A\otimes
C)\otimes B}$ $\textstyle{A\otimes(B\otimes
C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\alpha}$$\textstyle{(B\otimes
C)\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{-1}}$$\textstyle{(A\otimes
B)\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau\otimes
1}$$\textstyle{B\otimes(C\otimes
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\tau}$$\textstyle{(B\otimes
A)\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{-1}}$$\textstyle{B\otimes(A\otimes
C)}$ (4.4) $\textstyle{A\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\tau_{A,B}}$$\textstyle{B\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{B,A}}$$\textstyle{A\otimes
B.}$ (4.5)
###### Definition 4.6.2.
Let $M=(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$ and
$N=(\mathcal{C}^{\prime},\otimes^{\prime},I^{\prime},\alpha^{\prime},\lambda^{\prime},\rho^{\prime},\tau^{\prime})$
be symmetric monoidal categories. A lax symmetric monoidal functor $F:M\to N$
consists of
* •
a functor $F:\mathcal{C}\to\mathcal{C}^{\prime}$,
* •
morphisms $F_{2}:(FA)\otimes^{\prime}(FB)\to F(A\otimes B)$, natural in
$A,B\in\mathcal{C}$,
* •
a morphism $F_{0}:I^{\prime}\to FI$ in $\mathcal{C}^{\prime}$,
such that the following diagrams commute:
$\textstyle{FA\otimes^{\prime}(FB\otimes^{\prime}FC)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{1\otimes^{\prime}F_{2}}$$\textstyle{(FA\otimes^{\prime}FB)\otimes^{\prime}FC\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}\otimes^{\prime}1}$$\textstyle{FA\otimes^{\prime}(F(B\otimes
C))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}}$$\textstyle{F(A\otimes
B)\otimes^{\prime}FC\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}}$$\textstyle{F(A\otimes(B\otimes
C))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\alpha}$$\textstyle{F((A\otimes
B)\otimes C)}$ (4.6)
$\textstyle{(FB)\otimes^{\prime}I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
FB}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{1\otimes^{\prime}F_{0}}$$\textstyle{FB}$$\textstyle{(FB)\otimes^{\prime}(FI)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}}$$\textstyle{F(B\otimes
I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\rho}$
$\textstyle{I^{\prime}\otimes^{\prime}(FB)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{\prime}}$$\textstyle{FB}$$\textstyle{(FI)\otimes^{\prime}(FB)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}}$$\scriptstyle{F_{0}\otimes^{\prime}1}$$\textstyle{F(I\otimes
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\lambda}$
(4.7)
$\textstyle{(FA)\otimes^{\prime}(FB)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\prime}}$$\scriptstyle{F_{2}}$$\textstyle{(FB)\otimes^{\prime}(FA)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}}$$\textstyle{F(A\otimes
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\tau}$$\textstyle{F(B\otimes
A).}$ (4.8)
$F$ is said to be weak when $F_{0},F_{2}$ are isomorphisms, and strict when
$F_{0},F_{2}$ are identities.
Recall also the coherence theorem for classical symmetric monoidal categories.
For any $n$-ary permuted $\Phi$-tree $(\sigma\cdot t)$, let $(\sigma\cdot
t)_{M}$ be the functor $M^{n}\to M$ obtained by replacing every . in $t$ by
$\otimes$ and every $e$ by $I$, and permuting the arguments according to
$\sigma$, so $(\sigma\cdot t)_{M}(A_{1},\dots,A_{n})=t_{M}(A_{\sigma
1},\dots,A_{\sigma n})$ for all $A_{1},\dots,A_{n}\in M$. In particular, we do
not make use of the symmetry maps on $M$ in constructing these functors. Then:
###### Theorem 4.6.3.
(Mac Lane) In each weak symmetric monoidal category $M$ there is a function
which assigns to each pair $(\sigma\cdot t_{1},\rho\cdot t_{2})$ of permuted
$\Phi$-trees of the same arity $n$ a unique natural isomorphism
${\mbox{{can}}}_{M}(\sigma\cdot t_{1},\rho\cdot t_{2}):(\sigma\cdot
t_{1})_{M}\to(\rho\cdot t_{2})_{M}:M^{n}\to M$
called the canonical map from $\sigma\cdot t_{1}$ to $\rho\cdot t_{2}$, in
such a way that the identity of $M$ and all instances of $\alpha,\lambda,\rho$
and $\tau$ are canonical, and the composite as well as the $\otimes$-product
of two canonical maps is canonical.
###### Proof.
See [ML98] XI.1. ∎
Finally, recall the coherence theorem for weak monoidal functors:
###### Lemma 4.6.4.
Let $M,N$ be monoidal categories, and $F:M\to N$ be a weak monoidal functor.
For every $n\in\natural$ and every strongly regular $\Phi$-tree $v$ of arity
$n$, there is a unique map $F_{v}:v_{N}(FA_{1},\dots,FA_{n})\to
Fv_{M}(A_{1},\dots,A_{n})$ natural in $A_{1},\dots,A_{n}\in M$ and formed by
taking composites and tensors of $F_{0}$ and $F_{2}$, such that the diagram
$\textstyle{v_{N}(FA_{1},\dots
FA_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{v}}$$\scriptstyle{{\mbox{{can}}}_{N}}$$\textstyle{Fv_{M}(A_{1},\dots,A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F{\mbox{{can}}}_{M}}$$\textstyle{w_{N}(FA_{1},\dots
FA_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{w}}$$\textstyle{F(w)_{M}(A_{1},\dots,A_{n})}$
commutes for all $n\in\natural$, all $v,w\in({F_{\rm pl}}\Phi)_{n}$, and all
$A_{1},\dots,A_{n}\in\mathcal{M}$.
###### Proof.
See [ML98], p. 257. ∎
We may use this result to sketch a proof of a coherence theorem for weak
symmetric monoidal functors:
###### Theorem 4.6.5.
Let $M,N$ be symmetric monoidal categories, and $F:M\to N$ be a weak symmetric
monoidal functor. Let $\sigma\cdot v$ be an $n$-ary permuted $\Phi$-tree. Then
there is a unique natural transformation
$\textstyle{M^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{n}}$$\scriptstyle{(\sigma\cdot
v)_{M}}$$\scriptstyle{F_{\sigma\cdot
v}}$$\textstyle{N^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\sigma\cdot
v)_{N}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{N}$
formed by composing tensor products of $F_{2}$ and $F_{0}$, possibly with
their arguments permuted. Furthermore, if $\rho\cdot w$ is another permuted
$\Phi$-tree, then the diagram
$\textstyle{(\sigma\cdot v)_{N}(FA_{1},\dots
FA_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\sigma\cdot
v}}$$\scriptstyle{{\mbox{{can}}}_{N}}$$\textstyle{F(\sigma\cdot
v)_{M}(A_{1},\dots,A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F{\mbox{{can}}}_{M}}$$\textstyle{(\rho\cdot
w)_{N}(FA_{1},\dots
FA_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\rho\cdot
w}}$$\textstyle{F(\rho\cdot w)_{M}(A_{1},\dots,A_{n})}$
commutes.
###### Proof.
Let $F_{\sigma\cdot
v}(A_{1},\dots,A_{n})=F_{v}(A_{\sigma(1)},\dots,A_{\sigma(n)})$, and similarly
on morphisms. Then $F_{\sigma\cdot v}$ has the required type. We may decompose
${\mbox{{can}}}_{M}(\sigma\cdot v,\rho\cdot w)$ as
$\mbox{{perm}}_{M}(\sigma,\rho)\ {\mbox{{can}}}_{M}(v,w)$, where
$\mbox{{perm}}_{M}(\sigma,\rho):F_{\sigma\cdot v}\to F_{\rho\cdot v}$ is a
composite of $\tau$s.
Equation 4.8 and Lemma 4.6.4 together imply that the diagram
$\textstyle{(\sigma\cdot v)_{N}(FA_{1},\dots
FA_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\sigma\cdot
v}}$$\scriptstyle{\mbox{{perm}}_{N}}$$\textstyle{{F((\sigma\cdot
v)_{M}(A_{1},\dots,A_{n}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\mbox{{perm}}_{M}}$$\textstyle{(\rho\cdot
v)_{N}(FA_{1},\dots
FA_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\rho\cdot
v}}$$\scriptstyle{{\mbox{{can}}}_{N}}$$\textstyle{{F((\rho\cdot
v)_{M}(A_{1},\dots,A_{n}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F{\mbox{{can}}}_{M}}$$\textstyle{{(\rho\cdot
w)_{N}(FA_{1},\dots
FA_{n})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\rho\cdot
w}}$$\textstyle{{F((\rho\cdot w)_{M}(A_{1},\dots,A_{n}))}}$
commutes. It remains to show that $F_{\sigma\cdot v}$ is unique with this
property.
Suppose that $F_{\sigma\cdot v}$ is not unique for some $\sigma\cdot v$, and
that there exists some natural transformation $G:(\sigma\cdot
v)_{N}(FA_{1},\dots,FA_{n})\to F((\sigma\cdot v)_{M}(A_{1},\dots,A_{n}))$,
composed of tensor products of components of $F_{0}$ and $F_{2}$, such that
$G\neq F_{\sigma\cdot v}$. Suppose further that $\sigma\cdot v$ and $G$ have
been chosen to be a minimal counterexample, in the sense that of all such
counterexamples, $\sigma$ may be written as a product of the smallest number
of transpositions. If no transpositions are used, then we have a
contradiction, because then $\sigma=1_{n}$, and Lemma 4.6.4 tells us that
$G=F_{v}$. But suppose $\sigma=t_{1}t_{2}\dots t_{m}$ where each $t_{i}$ is a
transposition: then $t_{1}\cdot G$ is a natural transformation $(\sigma\cdot
v)_{N}(FA_{t_{1}1},\dots,FA_{t_{1}n})\to F((\sigma\cdot
v)_{M}(A_{t_{1}1},\dots,A_{t_{1}n}))$, and thus a transformation
$(t_{1}\sigma\cdot v)_{N}(FA_{\bullet})\to F((t_{1}\sigma\cdot
v)_{M}(A_{\bullet}))$. But $t_{1}\sigma=t_{2}t_{3}\dots t_{m}$, and thus (by
minimality of $\sigma$), it must be the case that $t_{1}\cdot
G=F_{t_{1}\sigma\cdot v}=t_{1}\cdot F_{\sigma\cdot v}$. Hence
$G=F_{\sigma\cdot v}$. ∎
We now proceed to relate the classical theory of symmetric monoidal categories
to the more general notion of categorification we developed in previous
sections.
By Lemma 4.4.6, if $\tau_{1},\tau_{2}$ are $n$-ary 1-cells in Wk${}_{\phi}(P)$
(in other words $n$-ary permuted $\Phi$-trees), there is a (unique) 2-cell
$\tau_{1}\to\tau_{2}$ in Wk${}_{\phi}(P)$ iff $\tau_{1}\sim\tau_{2}$ under the
congruence generated by $E$. By standard properties of commutative monoids,
this relation holds iff $\tau_{1}$ and $\tau_{2}$ take the same number of
arguments, so there is exactly one 2-cell $\tau_{1}\to\tau_{2}$ for every
$n\in\natural$ and every pair $(\tau_{1},\tau_{2})$ of $n$-ary 1-cells in
Wk${}_{\phi}(P)$.
Let SMC denote the category of symmetric monoidal categories and weak maps
between them. We shall define functors
$S:\mbox{{{SMC}}}\to\mbox{{{Wk-$P$-Cat}}}$ and
$R:\mbox{{{Wk-$P$-Cat}}}\to\mbox{{{SMC}}}$, and show that they are inverses of
each other.
Let $M=(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$ be a symmetric
monoidal category. Let $SM$ be the weak $P$-category
$(\hat{\phantom{\alpha}}):{{\mbox{Wk${}_{\phi}(P)$}}}\to\mathop{\rm{End}}(\mathcal{C})$
defined as follows:
* •
On 1-cells of Wk${}_{\phi}(P)$, $(\hat{\phantom{\alpha}})$ is determined by
$\hat{.}=\otimes$ and $\hat{e}=I$.
* •
If $\delta:\tau_{1}\to\tau_{2}$ is an $n$-ary 2-cell in Wk${}_{\phi}(P)$ (i.e.
a morphism in the category ${{\mbox{Wk${}_{\phi}(P)$}}}_{n}$), let
$\hat{\delta}$ be the canonical map $\hat{\tau}_{1}\to\hat{\tau}_{2}$.
###### Lemma 4.6.6.
$SM$ is a well-defined Wk${}_{\phi}(P)$-algebra for all $M\in\mbox{{{SMC}}}$.
###### Proof.
The 1-cells of Wk${}_{\phi}(P)$ are the same as those of ${F_{\Sigma}}\Phi$;
hence, $(\hat{\phantom{\alpha}})$ is entirely determined on 1-cells by a map
of signatures $\Phi\to{U^{\Sigma}}\mathop{\rm{End}}(\mathcal{C})$, which we
have given. On 2-cells, Theorem 4.6.3 and the uniqueness property of 2-cells
in Wk${}_{\phi}(P)$ tell us that if $\delta_{1},\delta_{2}$ are 2-cells in
Wk${}_{\phi}(P)$, then
$\widehat{\delta_{1}.\delta_{2}}=\hat{\delta}_{1}\otimes\hat{\delta}_{2}=\hat{\delta}_{1}\hat{.}\hat{\delta}_{2}$,
and $\widehat{\delta_{1}\delta_{2}}=\hat{\delta_{1}}\hat{\delta_{2}}$ wherever
$\delta_{1},\delta_{2}$ are composable. Hence,
$(\mathcal{C},(\hat{\phantom{\alpha}}))$ is a well-defined
Wk${}_{\phi}(P)$-algebra. ∎
Given symmetric monoidal categories $M$ and $N$, and a weak symmetric monoidal
functor $F:M\to N$, we would like to define a weak $P$-functor
$SF=(F,\psi):SM\to SN$. Let $\psi_{\sigma\cdot v,A_{\bullet}}=F_{\sigma\cdot
v}$ for all $n\in\natural$, all $\sigma\cdot v\in({F_{\Sigma}}\Phi)_{n}$, and
all $A_{1},\dots,A_{n}\in M$. By Theorem 4.6.5, this is natural in
$\sigma\cdot v$ and in $A_{1},\dots,A_{n}$. The other axioms for a weak
$P$-functor are all implied by the coherence theorem (Theorem 4.6.5). This can
be generalized: a lax symmetric monoidal functor $F$ determines a lax
$P$-functor $SF$, and a strict symmetric monoidal functor $F$ determines a
strict $P$-functor $SF$.
Now, let $\mathcal{C}$ be a Wk${}_{\phi}(P)$-algebra, with map
$(\hat{\phantom{\alpha}}):{{\mbox{Wk${}_{\phi}(P)$}}}\to\mathop{\rm{End}}(\mathcal{C})$.
We shall construct a symmetric monoidal category
$R(\mathcal{C},(\hat{\phantom{\alpha}}))=(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$.
Take
* •
$\otimes=\hat{.}$
* •
$I=\hat{e}$
* •
$\alpha=\hat{\delta}_{1}$, where $\delta_{1}:-.(-.-)\to(-.-).-$ in
${{\mbox{Wk${}_{\phi}(P)$}}}_{3}$,
* •
$\lambda=\hat{\delta}_{2}$, where $\delta_{2}:e.-\to-$ in
${{\mbox{Wk${}_{\phi}(P)$}}}_{1}$,
* •
$\rho=\hat{\delta}_{3}$, where $\delta_{3}:-.e\to e$ in
${{\mbox{Wk${}_{\phi}(P)$}}}_{1}$,
* •
$\tau=\hat{\delta}_{4}$, where $\delta_{4}:(-.-)\to(12)\cdot(-.-)$ in
${{\mbox{Wk${}_{\phi}(P)$}}}_{2}$.
###### Lemma 4.6.7.
$R(\mathcal{C},(\hat{\phantom{\alpha}}))$ is a symmetric monoidal category.
###### Proof.
Because there is at most one 2-cell $\tau_{1}\to\tau_{2}$ for any pair of
1-cells $\tau_{1},\tau_{2}$ in Wk${}_{\phi}(P)$, all diagrams involving these
commute. In particular, the axioms for a symmetric monoidal category are
satisfied. The 2-cells in $\mathop{\rm{End}}(\mathcal{C})$ are natural
transformations, so $\alpha,\lambda,\rho$ and $\tau$ (as images of 2-cells in
Wk${}_{\phi}(P)$ under the map
$(\hat{\phantom{\alpha}}):{{\mbox{Wk${}_{\phi}(P)$}}}\to\mathop{\rm{End}}(\mathcal{C})$)
are natural transformations. All 2-cells in Wk${}_{\phi}(P)$ are invertible,
so $\alpha,\lambda,\rho$ and $\tau$ are all natural isomorphisms. Hence
$(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$ is a symmetric monoidal
category. ∎
Let
$(F,\psi):(\mathcal{C},(\hat{\phantom{\alpha}}))\to(\mathcal{C}^{\prime},(\check{\phantom{\alpha}}))$
be a weak morphism of Wk${}_{\phi}(P)$-algebras. Then let
$R(F,\psi):R(\mathcal{C},(\hat{\phantom{\alpha}}))\to
R(\mathcal{C}^{\prime},(\check{\phantom{\alpha}}))$ be the following symmetric
monoidal functor:
* •
the underlying functor is $F$,
* •
$F_{0}$ is $\psi_{e\circ 1}:\check{e}\to F\hat{e}$,
* •
$F_{2}$ is $\psi_{.\circ 1}:(\check{.})F^{2}\to F(\hat{.})$.
The coherence diagrams (4.6), (4.7) and (4.8) all commute by virtue of the
coherence axioms for a weak morphism of Wk${}_{\phi}(P)$-algebras and the
naturality of $\psi$. Hence $(F,F_{0},F_{2})$ is a symmetric monoidal functor.
###### Lemma 4.6.8.
Let $(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$ be a symmetric monoidal
category. Then
$RS(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)=(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau).$
###### Proof.
Let
$RS(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)=(\mathcal{C},\otimes^{\prime},I^{\prime},\alpha^{\prime},\lambda^{\prime},\rho^{\prime},\tau^{\prime})$.
Their underlying categories are equal, both being $\mathcal{C}$.
$\begin{array}[]{rcccl}\otimes^{\prime}&=&\hat{.}&=&\otimes\\\
I^{\prime}&=&\hat{e}&=&I\\\ \alpha^{\prime}&=&\hat{\delta}_{1}&=&\alpha\mbox{,
the unique canonical map of the correct type}\\\
\lambda^{\prime}&=&\hat{\delta}_{2}&=&\lambda\\\
\rho^{\prime}&=&\hat{\delta}_{3}&=&\rho\\\
\tau^{\prime}&=&\hat{\delta}_{4}&=&\tau\\\ \end{array}$
∎
###### Lemma 4.6.9.
Let $(\mathcal{C},(\hat{\phantom{\alpha}}))$ be a Wk${}_{\phi}(P)$-algebra,
and let
$(\mathcal{C}^{\prime},(\check{\phantom{\alpha}}))=SR(\mathcal{C},(\hat{\phantom{\alpha}}))$.
Then
$(\mathcal{C},(\hat{\phantom{\alpha}}))=(\mathcal{C}^{\prime},(\check{\phantom{\alpha}}))$.
###### Proof.
Their underlying categories are the same. As above,
$(\check{\phantom{\alpha}})$ is determined on objects by the values it takes
on . and $e$: these are $\otimes=\hat{.}$ and $I=\hat{e}$ respectively. So
$(\check{\phantom{\alpha}})=(\hat{\phantom{\alpha}})$ on objects. If
$\delta:\tau_{1}\to\tau_{2}$, then $\check{\delta}$ is the unique canonical
map from $\check{\tau}_{1}\to\check{\tau}_{2}$, which, by an easy induction,
must be $\hat{\delta}$. So
$(\check{\phantom{\alpha}})=(\hat{\phantom{\alpha}})$, and hence
$(\mathcal{C},(\hat{\phantom{\alpha}}))=SR(\mathcal{C},(\hat{\phantom{\alpha}}))$.
∎
###### Lemma 4.6.10.
Let $M=(\mathcal{C},\otimes,I,\alpha,\lambda,\rho,\tau)$ and
$N=(\mathcal{C}^{\prime},\otimes^{\prime},I^{\prime},\alpha^{\prime},\lambda^{\prime},\rho^{\prime},\tau^{\prime})$
be symmetric monoidal categories, and let $(F,F_{0},F_{2})$ be a weak
symmetric monoidal functor $M\to N$. Then $RS(F,F_{0},F_{2})=(F,F_{0},F_{2})$.
###### Proof.
Let $(G,G_{0},G_{2})=RS(F,F_{0},F_{2})$. Then $G$ is the underlying functor of
$S(F,F_{0},F_{2})$ which is $F$, and $G_{0},G_{2}$ are both the canonical maps
with the correct types given by Theorem 4.6.5: that is to say, they are
$F_{0}$ and $F_{2}$ respectively. ∎
###### Lemma 4.6.11.
Let $(\mathcal{C},(\hat{\phantom{\alpha}}))$ and
$(\mathcal{C}^{\prime},(\check{\phantom{\alpha}}))$ be
Wk${}_{\phi}(P)$-algebras, and let
$(F,\phi):(\mathcal{C},(\hat{\phantom{\alpha}}))\to(\mathcal{C}^{\prime},(\check{\phantom{\alpha}}))$
be a weak morphism of Wk${}_{\phi}(P)$-algebras. Then $SR(F,\phi)=(F,\phi)$.
###### Proof.
Let $(G,\gamma)=SR(F,\phi)$. Then $G$ is the underlying functor of
$R(F,\phi)$, which is $F$. Let $(F,F_{0},F_{2})=R(F,\phi)$. Each component of
$\gamma$ is then by definition the correct component of the canonical map
arising from $F_{0},F_{2}$ in the process described in Theorem 4.6.5. By the
“uniqueness” part of the Theorem, this must be the corresponding component of
$\phi$. Hence $\gamma=\phi$. ∎
###### Theorem 4.6.12.
$S$ and $R$ form an isomorphism of categories
$\mbox{{{SMC}}}\cong\mbox{{{Wk-$P$-Cat}}}$.
###### Proof.
Lemmas 4.6.8 and 4.6.9 show that $R$ and $S$ are bijective on objects; Lemmas
4.6.10 and 4.6.11 show that $R$ and $S$ are locally bijective on morphisms.
Hence, $R$ and $S$ are a pair of mutually inverse isomorphisms of categories.
∎
### 4.7 Multicategories
We can tell this whole story for (symmetric) multicategories as well as just
operads. We sketch this development briefly here, although the remainder of
the thesis will continue to focus on the special case of operads.
###### Definition 4.7.1.
A (directed) multigraph consists of
1. 1.
a set of vertices $V$,
2. 2.
for each $n\in\natural$ and each sequence $v_{1},v_{2},\dots,v_{n},w$ of
vertices, a set $E(v_{1},\dots,v_{n};w)$ of funnels from $v_{1},\dots,v_{n}$
to $w$.
###### Definition 4.7.2.
Let $M_{1}=(V_{1},E_{1})$ and $M_{2}=(V_{2},E_{2})$ be multigraphs. A morphism
of multigraphs $f:M_{1}\to M_{2}$ is
1. 1.
a function $f_{V}:V_{1}\to V_{2}$,
2. 2.
for each finite sequence $v_{1},v_{2}\dots v_{n},w$ of vertices in $M_{1}$, a
function
$f^{v_{1},\dots,v_{n}}_{w}:E_{1}(v_{1},\dots,v_{n};w)\to
E_{2}(f_{V}(v_{1}),\dots,f_{V}(v_{n});f_{V}(w)).$
We say that a funnel $f\in E(v_{1},\dots,v_{n};w)$ has source
$v_{1},\dots,v_{n}$ and target $w$; we say that two funnels are parallel if
they have the same source and target. The reason for the “funnel” terminology
should be clear from Figure 4.2. We shall say that a multigraph has some
property $P$ locally if every $E(v_{1},\dots,v_{n};w)$ is $P$, and similarly a
morphism $f$ of multicategories is locally $P$ if every
$f^{v_{1},\dots,v_{n}}_{w}$ is $P$.
Multigraphs and their morphisms form a category which we shall call
Multigraph.
Figure 4.2: A multigraph
In order to proceed with the rest of the construction, we will need to
consider subcategories of Multicat, Multigraph etc.
###### Definition 4.7.3.
Let $X$ be a set. Then $\mbox{{{Multigraph}}}_{X}$ is the subcategory of
Multigraph whose objects are multigraphs with vertex set $X$, and whose
morphisms are identity-on-vertices maps of multigraphs. We define
$\mbox{{{Multicat}}}_{X}$ and $\mbox{{{$\Sigma$-Multicat}}}_{X}$ similarly.
For each $X\in\mbox{{{Set}}}$, there is a chain of adjunctions similar to that
given in Section 2.4:
$\textstyle{{\mbox{{{FP-
Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\rm
fp}_{\Sigma}}}$$\scriptstyle{{U^{\rm fp}_{\rm pl}}}$$\scriptstyle{{U^{\rm
fp}}}$$\textstyle{\@ensuremath{\mbox{{{$\Sigma$-Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\Sigma}_{\rm
pl}}}$$\scriptstyle{{F_{\rm
fp}^{\Sigma}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\Sigma}}}$$\textstyle{\@ensuremath{\mbox{{{Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\Sigma}^{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\rm pl}}}$$\scriptstyle{{F_{\rm
fp}^{\rm
pl}}}$$\textstyle{\@ensuremath{\mbox{{{Multigraph}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{F_{\Sigma}}}$$\scriptstyle{{F_{\rm
fp}}}$
These adjunctions are monadic, by Lemma 2.5.1 and Lemma 2.5.3. Note that Set
can be regarded as $\mbox{{{Multigraph}}}_{1}$: thus, the adjunctions of
Section 2.4 are just the restrictions of the adjunctions above to the one-
vertex case.
We can consider multigraphs enriched in some category $\mathcal{V}$:
###### Definition 4.7.4.
Let $\mathcal{V}$ be a category. A $\mathcal{V}$-multigraph $M=(V,E)$ consists
of
1. 1.
a set $V$ of vertices,
2. 2.
for each $n\in\natural$ and each finite sequence $v_{1},v_{2}\dots v_{n},w$ of
vertices, an object of $\mathcal{V}$ called $E(v_{1},\dots,v_{n};w)$ of
funnels from $v_{1},\dots,v_{n}$ to $w$.
###### Definition 4.7.5.
Let $M_{1}=(V_{1},E_{1})$ and $M_{2}=(V_{2},E_{2})$ be
$\mathcal{V}$-multigraphs. A morphism of $\mathcal{V}$-multigraphs $f:M_{1}\to
M_{2}$ is
1. 1.
a function $f_{V}:V_{1}\to V_{2}$,
2. 2.
for each $n\in\natural$ and each finite sequence $v_{1},v_{2}\dots v_{n},w$ of
vertices in $M_{1}$, an arrow $E_{1}(v_{1},\dots,v_{n};w)\to
E_{2}(f_{V}(v_{1}),\dots,f_{V}(v_{n});f_{V}(w))$ in $\mathcal{V}$.
The category of $\mathcal{V}$-multigraphs and their morphisms is called
$\mathcal{V}$-Multigraph. The category whose objects are
$\mathcal{V}$-multigraphs with vertex-set $X$ and whose morphisms are
identity-on-vertices maps is called $\mbox{{{$\mathcal{V}$-Multigraph}}}_{X}$.
In particular, we shall consider multigraphs enriched in the category Digraph
of directed graphs. An object of the category Digraph-Multigraph consists of
1. 1.
vertices (or objects);
2. 2.
funnels, each of which has one object as its target, and a sequence of objects
as its source;
3. 3.
edges, which each have one funnel as a source and one as a target: the source
and target of a given edge must be parallel.
Figure 4.3: A multigraph enriched in directed graphs
The factorization system construction of Example 3.0.10 works in this broader
setting too. Let $X$ be a set. The factorization system
$(\mathcal{E},\mathcal{M})$ on Digraph of Example 3.0.5 gives rise to a
factorization system $(\mathcal{E}^{\prime},\mathcal{M}^{\prime})$ on
$\mbox{{{Digraph-Multigraph}}}_{X}$, where $\mathcal{E}$ consists of maps
which are bijective on objects and funnels, and $\mathcal{M}$ consists of maps
which are locally full-and-faithful. This lifts to a factorization system
$(\mathcal{E}^{\prime\prime},\mathcal{M}^{\prime\prime})$ on $\mbox{{{Cat-
Multigraph}}}_{X}$ via Lemma 3.0.7. By the usual argument, there is a chain of
monadic adjunctions:
$\textstyle{{\mbox{{{$\mbox{{{Cat}}}$-FP-
Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\rm
fp}_{\Sigma}}}$$\scriptstyle{{U^{\rm fp}_{\rm pl}}}$$\scriptstyle{{U^{\rm
fp}}}$$\textstyle{\@ensuremath{\mbox{{{Cat-$\Sigma$-Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\Sigma}_{\rm
pl}}}$$\scriptstyle{{F_{\rm
fp}^{\Sigma}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\Sigma}}}$$\textstyle{\@ensuremath{\mbox{{{Cat-
Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\Sigma}^{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{U^{\rm pl}}}$$\scriptstyle{{F_{\rm
fp}^{\rm pl}}}$$\textstyle{\@ensuremath{\mbox{{{Cat-
Multigraph}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F_{\rm
pl}}}$$\scriptstyle{\dashv}$$\scriptstyle{{F_{\Sigma}}}$$\scriptstyle{{F_{\rm
fp}}}$
Since $\mbox{{{Cat-Multicat}}}_{X}$ is monadic over $\mbox{{{Cat-
Multigraph}}}_{X}$, this in turn lifts to a factorization system on
$\mbox{{{Cat-Multicat}}}_{X}$. Similarly, we obtain a factorization system on
$\mbox{{{Cat-$\Sigma$-Multicat}}}_{X}$.
A generator for a plain multicategory $M$ with object-set $X$ is a multigraph
$\Phi=(X,E)$ together with a regular epi ${F_{\rm pl}}\Phi\to M$ in
$\mbox{{{Multicat}}}_{X}$. Similarly, a generator for a symmetric
multicategory $M$ with object-set $X$ is a multigraph $\Phi=(X,E)$ together
with a regular epi ${F_{\Sigma}}\Phi\to M$ in
$\mbox{{{$\Sigma$-Multicat}}}_{X}$.
We can therefore extend Definition 4.4.7 above, in the obvious way. Let
$D_{*}$ be the embedding of $\mbox{{{Multicat}}}_{X}$ into $\mbox{{{Cat-
Multicat}}}_{X}$ via the (full and faithful) discrete category functor applied
locally.
###### Definition 4.7.6.
Let $M$ be a plain multicategory with object-set $X$, and let $\phi:{F_{\rm
pl}}\Phi\to M$ be a regular epi in $\mbox{{{Multicat}}}_{X}$. Then the
weakening of $M$ with respect to $\phi$ is the unique-up-to-isomorphism Cat-
multicategory Wk${}_{\phi}(M)$ such that the following diagram commutes:
---
$\textstyle{D_{*}F\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\phi}$$\scriptstyle{b}$$\textstyle{D_{*}M}$Wk${}_{\phi}(M)$$\scriptstyle{f}$
where $f$ is locally full and faithful, and $b$ is locally bijective on
objects (i.e., each map of sets of funnels in $b$ is a bijection). The
uniqueness of Wk${}_{\phi}(M)$ follows from properties of the factorization
system on $\mbox{{{Cat-Multicat}}}_{X}$ given above.
###### Definition 4.7.7.
Let $M$ be a (symmetric) multicategory. The unbiased weakening of $M$ is the
weakening of $M$ with respect to the counit $\epsilon$ of the adjunction
${F_{\rm pl}}\dashv{U^{\rm pl}}$ (respectively, the adjunction
${F_{\Sigma}}\dashv{U^{\Sigma}}$).
###### Definition 4.7.8.
Let $M$ be a multicategory, and let $\phi:{F_{\rm pl}}\Phi\to M$ be a regular
epi in $\mbox{{{Multicat}}}_{X}$. A $\phi$-weak $M$-algebra is an algebra for
Wk${}_{\phi}(M)$. An unbiased weak $M$-algebra is an algebra for Wk($M$).
We define weakenings of symmetric multicategories analogously.
### 4.8 Examples
###### Example 4.8.1.
Let $M$ be the multicategory generated by
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{2}$
Then a weak algebra for $M$ in Cat with respect to this generating set
consists of a diagram
$\textstyle{\hat{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{f}}$$\textstyle{\hat{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{g}}$$\textstyle{\hat{2}}$
in Cat, whereas an unbiased weak $M$-algebra is a diagram
---
$\textstyle{\hat{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{f}}$$\scriptstyle{\widehat{gf}}$$\textstyle{\scriptstyle\sim}$$\scriptstyle{\widehat{I_{0}}}$$\textstyle{\hat{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{I_{2}}}$$\textstyle{\hat{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{g}}$$\scriptstyle{\widehat{I_{1}}}$
where $\widehat{I_{0}},\widehat{I_{1}}$ and $\widehat{I_{2}}$ are trivial
monads.
###### Example 4.8.2.
Consider the theory $T$ whose algebras are a monoid $M$ together with an
$M$-set $A$. Then a weak $T$-algebra with respect to the standard presentation
(a binary and nullary operation on $M$, and an operation $M\times A\to A$) is
a classical monoidal category $\hat{M}$, a category $\hat{A}$, and a weak
monoidal functor $\hat{M}\to\mathop{\rm{End}}(\hat{A})$. An unbiased weak
$T$-algebra is an unbiased monoidal category $\hat{M}$, a category $\hat{A}$
equipped with a trivial monad $\hat{I}_{A}$, and an unbiased monoidal functor
$\hat{M}\to\mathop{\rm{End}}(\hat{A})$ which commutes up to coherent
isomorphism with $\hat{I}_{A}$.
###### Example 4.8.3.
Let $P$ be an operad, and let $\bar{P}$ be the multicategory from Section 2.10
whose algebras are pairs of $P$-algebras with a morphism between them. It
seems clear that an unbiased weak $\bar{P}$-category is a pair of unbiased
weak $P$-categories and an unbiased weak $P$-functor between them; a rigorous
proof would first require a coherence theorem to be proven for weak maps of
Cat-operad algebras, and currently no such theorem is known.
### 4.9 Evaluation
At the beginning of this chapter, we proposed three criteria that a successful
definition of categorification should satisfy: namely, it should be broad,
consistent with earlier work, and canonical. The examples considered
throughout the chapter show that our theory agrees with the standard
categorifications that are within its scope. It is determined by the universal
property given by the factorization system on Cat-$\Sigma$-Operad: the only
tunable parameter is the choice of generator of a given theory, and in Chapter
5 we shall see that the weakening of a given theory is independent (up to
equivalence) of the generator used. The main problem is the breadth of our
theory: as presented, it is restricted to linear theories, preventing us from
categorifying the theories of groups, rings, Lie algebras, and many other
interesting nonlinear theories. We shall now show what happens when we try to
extend our theory to general algebraic theories.
###### Lemma 4.9.1.
There is a factorization system $(\mathcal{E},\mathcal{M})$ on Cat-FP-Operad
where $\mathcal{E}$ is the collection of maps which are bijective on objects,
and $\mathcal{M}$ is the collection of maps which are levelwise full and
faithful.
###### Proof.
The proof is exactly as for the proof of existence of the factorization
systems on Cat-Operad and Cat-$\Sigma$-Operad given in Example 3.0.10. ∎
###### Theorem 4.9.2.
Let $P$ be the finite product operad whose algebras are commutative monoids,
and $D_{*}:{\mbox{{{FP-Operad}}}}\to{\mbox{{{$\mbox{{{Cat}}}$-FP-Operad}}}}$
be the levelwise “discrete category” functor. Let $Q$ be the finite product
Cat-operad given by the factorization
---
$\textstyle{D_{*}{F_{\rm fp}}{U^{\rm
fp}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\epsilon_{P}}$$\textstyle{D_{*}P}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Then an algebra for $Q$ is an unbiased symmetric monoidal category
$\mathcal{C}$ such that, for all $A\in\mathcal{C}$, the component $\tau_{AA}$
of the symmetry map $\tau$ is the identity on $A\otimes A$.
###### Proof.
We adopt the notation for elements of $P$ introduced in Example 2.3.4. Let $f$
be the unique function ${\underline{2}}\to{\underline{1}}$, and let
$t:{\underline{2}}\to{\underline{2}}$ be the permutation transposing 1 and 2.
Then $\epsilon(f\cdot\left[1\atop 1\right])=[2]\in P_{1}$. Let
$(\mathcal{C},(\hat{\phantom{\alpha}}))$ be a $Q$-algebra. We shall write
$\hat{\left[}1\atop 1\right](A,B)$ as $A\otimes B$. We may impose a symmetric
monoidal category structure on $\mathcal{C}$, where the symmetry map is the
image under $(\hat{\phantom{\alpha}})$ of the unique map $\delta:\left[1\atop
1\right]\to t\cdot\left[1\atop 1\right]$ in $Q_{1}$. All diagrams in $Q_{1}$
commute, so in particular, the following diagram commutes:
---
$\textstyle{[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f\cdot\left[1\atop
1\right]}$$\textstyle{f\cdot\left[1\atop
1\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\cdot\delta}$
The two unlabelled arrows are mutually inverse. Applying
$(\hat{\phantom{\alpha}})$ to the entire diagram, and evaluating the resulting
functors at $A\in\mathcal{C}$, we see that the following diagram commutes:
---
$\textstyle{\hat{[2]}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes
A}$$\textstyle{A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{AA}}$
and hence $\tau_{AA}=1_{A\otimes A}$. ∎
This is not the case for most interesting symmetric monoidal categories. Hence
this definition of categorification would fail to be consistent with earlier
work.
## Chapter 5 Coherence
There are many “coherence theorems” in category theory, but in practice they
usually fall into one of two classes:
1. 1.
“All diagrams commute”, or more precisely, that diagrams in a given class
commute if and only if some quantity is invariant.
2. 2.
Every “weak” object is equivalent to an appropriate “strict” object.
Since the diagrams of interest in theorems of type 1 will usually commute
trivially in a strict object, a coherence theorem of type 2 usually implies
one of type 1. However, establishing the converse is usually harder. In the
previous chapter, our “weak $P$-categories” were defined explicitly in terms
of an infinite class of commuting diagrams (namely, those diagrams which
become identities under the application of the counit of the adjunctions
${F_{\Sigma}}\dashv{U^{\Sigma}}$ or ${F_{\rm pl}}\dashv{U^{\rm pl}}$): it is
therefore interesting to see if we can prove a theorem of type 2 about them.
We do this in Section 5.1, and investigate an interesting property of the
strictification functor in Section 5.2.
In Section 5.3, we investigate how the operad defining weak $P$-categories is
affected by our choice of presentation for $P$. While the independence result
obtained is not a coherence theorem of the usual form, it can be seen as a
coherence theorem in a higher-dimensional sense: that the process of
categorification is itself coherent.
For other related work, see Power’s paper [Pow89].
### 5.1 Strictification
Let $P$ be a plain operad, and $Q=\mbox{Wk}(P)$, with $\pi:Q\to D_{*}P$ the
levelwise full-and-faithful map in Theorem 4.2.2. We again adopt the ∙
notation from chain complexes and write, for instance, $p_{\bullet}$ for a
finite sequence of objects in $P$, and $p_{\bullet}^{\bullet}$ for a double
sequence. Let $Q\circ
A\mathop{\stackrel{{\scriptstyle\scriptstyle{h}}}{{\longrightarrow}}}A$ be a
weak $P$-category. We shall construct a strict $P$-category st($A$) and a weak
$P$-functor $(F,\phi):\mbox{{{st}}}(A)\to A$, and show that it is an
equivalence of weak $P$-categories. This “strictification” construction is
closely related to that given in [JS93] for monoidal categories; however, it
is more general, and since we work for the moment with unbiased weak
$P$-categories, our construction has some additional pleasant properties.
In fact, st is functorial, and is left adjoint to the forgetful functor
$\mbox{{{Str-$P$-Cat}}}\linebreak[0]\to\mbox{{{Wk-$P$-Cat}}}$ (see Section
5.2). The theorem then says that the unit of this adjunction is pseudo-
invertible, and that the strict $P$-categories and strict $P$-functors form a
weakly coreflective sub-2-category of Wk-$P$-Cat.
If $P$ is a plain operad, let $\iota$ be the embedding
$\displaystyle\iota:{U^{\rm pl}}D_{*}P$ $\displaystyle\to$
$\displaystyle{U^{\rm pl}}{\mbox{Wk($P$)}}$ $\displaystyle\iota(p)$
$\displaystyle=$ $\displaystyle p\circ(|,\dots,|)$
Note that this is a morphism in Cat, not in Cat-Operad.
###### Definition 5.1.1.
Let $P$, $Q$, $h$, $A$, $\iota$ be as above. The strictification of $A$,
written st($A$), is given by the bijective-on-objects/full and faithful
factorization of $h(\iota\circ 1)$ in Cat:
---
$\textstyle{P\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota\circ
1}$$\textstyle{Q\circ
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{A.}$$\textstyle{\@ensuremath{\mbox{{{st}}}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
We shall show that st($A$) is a strict $P$-category. We may describe st($A$)
explicitly as follows:
* •
An object of st($A$) is an object of $P\circ A$.
* •
If $p\in P_{n}$ and $a_{1},\dots,a_{n}\in A$, an arrow
$(p,a_{\bullet})\to(p^{\prime},a_{\bullet}^{\prime})$ in st($A$) is an arrow
$h(p,a_{\bullet})\to h(p^{\prime},a_{\bullet}^{\prime})$ in $A$. We say that
such an arrow is a lifting of $h(p,a_{\bullet})$. Composition and identities
are as in $A$.
We define an action $h^{\prime}$ of $P$ on st($A$) as follows:
* •
On objects, $h^{\prime}$ acts by
$h^{\prime}(q,(p,a_{\bullet})^{\bullet})=(\pi(p\circ(p^{\bullet})),a_{\bullet}^{\bullet})$
where $p\in P_{n}$ and $(p^{i},a_{\bullet}^{i})\in\mbox{{{st}}}(A)$ for
$n\in\natural$ and $i=1,\dots n$.
* •
Let $f_{i}:(p_{i},a_{i})\to(p_{i}^{\prime},a_{i}^{\prime})$ for $i=1,\dots,n$.
Then $h^{\prime}(p,f_{\bullet})$ is the composite
$\begin{array}[]{rcl}h(p\circ(p_{\bullet}),a_{\bullet})&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{p\circ(p_{\bullet})}^{-1}}}}{{\longrightarrow}}}&h(p\circ(p_{\bullet}),a_{\bullet})=h(p,h(p_{1},a_{1}),\dots,h(p_{n},a_{n}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{h(p,f_{\bullet})}}}{{\longrightarrow}}}&h(p,h(p_{1}^{\prime},a_{1}^{\prime}),\dots,h(p_{n}^{\prime},a_{n}^{\prime}))=h(p\circ(p_{\bullet}^{\prime}),a_{\bullet}^{\prime})\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{p\circ(p_{\bullet}^{\prime})}}}}{{\longrightarrow}}}&h(p\circ(p_{\bullet}^{\prime}),a_{\bullet}^{\prime}).\end{array}$
###### Lemma 5.1.2.
st($A$) is a strict $P$-category.
###### Proof.
It is clear that the action we have defined is strict and associative on
objects and that $1_{P}$ acts as a unit: we must show that the action on
arrows is associative. Let
$f_{i}^{j}:(p_{i}^{j},a_{i\bullet}^{j})\to(q_{i}^{j},b_{i\bullet}^{j})$,
$\sigma\in Q_{n}$, and $\tau_{i}\in Q_{k_{i}}$ for $j=1,\dots,k_{i}$ and
$i=1,\dots,n$. We wish to show that
$h^{\prime}(\sigma\circ(\tau_{\bullet}),f_{\bullet}^{\bullet})=h^{\prime}(\sigma,h^{\prime}(\tau_{1},f_{1}^{\bullet}),\dots,h^{\prime}(\tau_{n},f_{n}^{\bullet}))$.
The LHS is
$\begin{array}[]{rcl}h(\sigma\circ(\tau_{\bullet})\circ(p_{\bullet}^{\bullet}),a_{\bullet}^{\bullet})&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{\sigma\circ(\tau_{i})\circ(p_{\bullet}^{\bullet})}^{-1}}}}{{\longrightarrow}}}&h(\sigma\circ(\tau_{\bullet}),h(p_{1}^{1},a_{1\bullet}^{1}),\dots,h(p_{n}^{k_{n}},a_{n\bullet}^{k_{n}}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{h(\sigma\circ(\tau_{\bullet}),f_{\bullet}^{\bullet})}}}{{\longrightarrow}}}&h(\sigma\circ(\tau_{\bullet}),h(q_{1}^{1},b_{1\bullet}^{1}),\dots,h(q_{n}^{k^{\prime}_{n}},b_{n\bullet}^{k^{\prime}_{n}}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet})\circ(q_{\bullet}^{\bullet})}}}}{{\longrightarrow}}}&h(\sigma\circ(\tau_{\bullet})\circ(q_{\bullet}^{\bullet}),b_{\bullet}^{\bullet}).\end{array}$
The RHS is
$\begin{array}[]{rcl}h(\sigma\circ(\tau_{\bullet})\circ(p_{\bullet}^{\bullet}),a_{\bullet}^{\bullet})&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{\sigma\circ(\tau_{i}\circ(p_{\bullet}^{\bullet}))}^{-1}}}}{{\longrightarrow}}}&h(\sigma,h(\tau_{1}\circ(p_{1}^{\bullet}),a_{1\bullet}^{\bullet}),\dots,h(\tau_{n}\circ(p_{n}^{\bullet}),a_{n\bullet}^{\bullet}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{h(\sigma,h^{\prime}(\tau_{\bullet},f_{\bullet}^{\bullet}))}}}{{\longrightarrow}}}&h(\sigma,h(\tau_{1}\circ(q_{1}^{\bullet}),b_{1\bullet}^{\bullet}),\dots,h(\tau_{n}\circ(q_{n}^{\bullet}),b_{n\bullet}^{\bullet}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{\sigma\circ(\tau_{i}\circ(p_{\bullet}^{\bullet}))}}}}{{\longrightarrow}}}&h(\sigma\circ(\tau_{\bullet})\circ(q_{\bullet}^{\bullet}),b_{\bullet}^{\bullet}),\end{array}$
where each $h^{\prime}(\tau_{i},f_{i}^{\bullet})$ is
$\begin{array}[]{rcl}h(\tau_{i}\circ(p_{i}^{\bullet}),a_{i}^{\bullet}))&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{\tau_{i}\circ(p_{i}^{\bullet})}^{-1}}}}{{\longrightarrow}}}&h(\tau_{i},h(p_{i}^{1},a_{i\bullet}^{1}),\dots,h(p_{i}^{k_{i}},a_{i\bullet}^{k_{i}}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{h(\tau_{i},f_{i}^{\bullet})}}}{{\longrightarrow}}}&h(\tau_{i},h(q_{i}^{1},b_{i\bullet}^{1}),\dots,h(q_{i}^{k_{i}},b_{i\bullet}^{k_{i}}))\\\
&\mathop{\stackrel{{\scriptstyle\scriptstyle{\delta_{\tau_{i}\circ(p_{i}^{\bullet})}}}}{{\longrightarrow}}}&h(\tau_{i}\circ(q_{i}^{\bullet}),b_{i}^{\bullet}).\end{array}$
So the equation holds if the following diagram commutes:
---
$\textstyle{h(\sigma\circ(\tau_{\bullet})\circ(p_{\bullet}^{\bullet}),a_{\bullet}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\sigma\circ(\tau_{i})\circ(p_{\bullet}^{\bullet})}^{-1}}$$\scriptstyle{\delta_{\sigma\circ(\tau_{i}\circ(p_{\bullet}^{\bullet}))}^{-1}}$$\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet})\circ(p_{\bullet}^{\bullet})}^{-1}}$$\textstyle{h(\sigma\circ(\tau_{\bullet}),h(p_{\bullet}^{\bullet},a_{\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h(\sigma\circ(\tau_{\bullet}),f_{\bullet}^{\bullet})}$$\textstyle{h(\sigma,h(\tau_{\bullet}\circ(p_{\bullet}^{\bullet}),a_{\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet})}}$$\scriptstyle{h(\sigma,h(\tau_{i},f_{i}^{\bullet}))}$$\textstyle{1}$$\textstyle{2}$$\textstyle{h(\sigma\circ(\tau_{\bullet}),h(p_{\bullet}^{\bullet},a_{\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h(\sigma,\delta_{\tau_{\bullet}\circ(p_{\bullet}^{\bullet})}^{-1})}$$\scriptstyle{h(\sigma,h^{\prime}(\tau_{\bullet},f_{\bullet}^{\bullet}))}$$\textstyle{h(\sigma\circ(\tau_{\bullet}),h(q_{\bullet}^{\bullet},b_{\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet})\circ(q_{\bullet}^{\bullet})}}$$\textstyle{h(\sigma,h(\tau_{\bullet}\circ(q_{\bullet}^{\bullet}),b_{\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet}\circ(q_{\bullet}^{\bullet}))}}$$\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet})}}$$\scriptstyle{h(\sigma,\delta_{\tau_{i}\circ(p_{i}^{\bullet})})}$$\textstyle{h(\sigma,h(\tau_{\bullet}\circ(q_{\bullet}^{\bullet}),b_{\bullet}^{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\sigma\circ(\tau_{\bullet}\circ(q_{\bullet}^{\bullet}))}}$$\textstyle{h(\sigma\circ(\tau_{\bullet})\circ(q_{\bullet}^{\bullet}),b_{\bullet}^{\bullet})}$
The triangles all commute because all $\delta$s are images of arrows in $Q$,
and there is at most one 2-cell between any two 1-cells in $Q$.
$\textstyle{2}$ commutes by the definition of
$h^{\prime}(\tau_{i},f_{i}^{\bullet})$, and $\textstyle{1}$ commutes by
naturality of $\delta$. ∎
###### Lemma 5.1.3.
Let $Q\circ
A\mathop{\stackrel{{\scriptstyle\scriptstyle{h}}}{{\longrightarrow}}}A$ and
$Q\circ
B\mathop{\stackrel{{\scriptstyle\scriptstyle{h^{\prime}}}}{{\longrightarrow}}}B$
be weak $P$-categories, $(F,\pi):A\to B$ be a weak $P$-functor, and
$(F,G,\eta,\epsilon)$ be an adjoint equivalence in Cat. Then $G$ naturally
carries the structure of a weak $P$-functor, and $(F,G,\eta,\epsilon)$ is an
adjoint equivalence in Wk-$P$-Cat.
###### Proof.
We want a sequence $(\psi_{\bullet})$ of natural transformations:
$\textstyle{Q_{i}\times
B^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
G^{i}}$$\scriptstyle{h^{\prime}_{i}}$$\scriptstyle{\psi_{i}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{Q_{i}\times
A^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{i}}$$\textstyle{A}$
Let $\psi_{i}$ be given by
$\textstyle{Q_{i}\times
B^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
G^{i}}$$\scriptstyle{h^{\prime}_{i}}$$\scriptstyle{\psi_{i}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{Q_{i}\times
A^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{i}}$$\textstyle{A}$$\scriptstyle{=}$
---
$\textstyle{Q_{i}\times
B^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{1\times
G^{i}}$$\textstyle{Q_{i}\times
B^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}_{i}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\scriptstyle{1\times\epsilon^{i}}$$\scriptstyle{{\pi^{-1}_{i}}}$$\textstyle{Q_{i}\times
A^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times
F^{i}}$$\scriptstyle{h_{i}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{1}$$\textstyle{A\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\eta}$
We must check that $\psi$ satisfies (2.34) and (2.35) from Lemma 2.9.11. For
(2.34):
LHS $\scriptstyle{=}$ $\scriptstyle{1\times G^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{k_{1}}\times\dots\times
h^{\prime}_{k_{n}}}$$\scriptstyle{\psi_{k_{1}}\times\dots\times\psi_{k_{n}}}$$\scriptstyle{1\times
G^{n}}$$\scriptstyle{h^{\prime}_{n}}$$\scriptstyle{\psi_{n}}$$\scriptstyle{G}$$\scriptstyle{h_{k_{1}}\times\dots\times
h_{k_{n}}}$$\scriptstyle{h}$ $\scriptstyle{=}$
---
$\scriptstyle{1}$$\scriptstyle{1\times G^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{k_{1}}\times\dots\times
h^{\prime}_{k_{n}}}$$\scriptstyle{1}$$\scriptstyle{1\times
G^{n}}$$\scriptstyle{h^{\prime}_{n}}$$\scriptstyle{G}$$\scriptstyle{1\times\epsilon^{\sum
k_{i}}}$$\scriptstyle{\pi^{-1}_{k_{1}}\times\dots\times\pi^{-1}_{k_{n}}}$$\scriptstyle{1\times\epsilon^{n}}$$\scriptstyle{\pi_{n}^{-1}}$$\scriptstyle{1\times
F^{\sum k_{i}}}$$\scriptstyle{h_{k_{1}}\times\dots\times
h_{k_{n}}}$$\scriptstyle{1\times F^{n}}$$\scriptstyle{1}$$\scriptstyle{1\times
F^{n}}$$\scriptstyle{h}$$\scriptstyle{1\times\eta^{n}}$$\scriptstyle{F}$$\scriptstyle{1}$$\textstyle{\scriptstyle\eta}$
$\scriptstyle{=}$ $\scriptstyle{1}$$\scriptstyle{1\times G^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{k_{1}}\times\dots\times
h^{\prime}_{k_{n}}}$$\scriptstyle{1}$$\scriptstyle{h^{\prime}_{n}}$$\scriptstyle{G}$$\scriptstyle{1\times\epsilon^{\sum
k_{i}}}$$\scriptstyle{\pi^{-1}_{k_{1}}\times\dots\times\pi^{-1}_{k_{n}}}$$\scriptstyle{\pi_{n}^{-1}}$$\scriptstyle{1\times
F^{\sum k_{i}}}$$\scriptstyle{h_{k_{1}}\times\dots\times
h_{k_{n}}}$$\scriptstyle{1\times F^{n}}$$\scriptstyle{1}$$\scriptstyle{1\times
F^{n}}$$\scriptstyle{h}$$\scriptstyle{F}$$\scriptstyle{1}$$\textstyle{\scriptstyle\eta}$
$\scriptstyle{=}$
---
$\scriptstyle{1}$$\scriptstyle{1\times G^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{k_{1}}\times\dots\times
h^{\prime}_{k_{n}}}$$\scriptstyle{h^{\prime}_{n}}$$\scriptstyle{G}$$\scriptstyle{1\times\epsilon^{\sum
k_{i}}}$$\scriptstyle{\pi^{-1}_{k_{1}}\times\dots\times\pi^{-1}_{k_{n}}}$$\scriptstyle{\pi_{n}^{-1}}$$\scriptstyle{1\times
F^{\sum k_{i}}}$$\scriptstyle{h_{k_{1}}\times\dots\times
h_{k_{n}}}$$\scriptstyle{1\times
F^{n}}$$\scriptstyle{h}$$\scriptstyle{F}$$\scriptstyle{1}$$\textstyle{\scriptstyle\eta}$
$\scriptstyle{=}$ $\scriptstyle{1}$$\scriptstyle{1\times G^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{\sum
k_{i}}}$$\scriptstyle{G}$$\scriptstyle{1\times\epsilon^{\sum
k_{i}}}$$\scriptstyle{\pi_{\sum k_{i}}^{-1}}$$\scriptstyle{1\times F^{\sum
k_{i}}}$$\scriptstyle{h_{\sum
k_{i}}}$$\scriptstyle{F}$$\scriptstyle{1}$$\textstyle{\scriptstyle\eta}$
$\scriptstyle{=}$ $\scriptstyle{1\times G^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{\sum k_{i}}}$$\scriptstyle{\psi_{\sum
k_{i}}}$$\scriptstyle{G}$$\scriptstyle{h_{\sum k_{i}}}$ $\displaystyle=$ RHS.
For (2.35), consider the following diagram:
|
---|---
$\textstyle{Gb\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1_{Q}}}$$\scriptstyle{1}$$\scriptstyle{\eta
G}$$\textstyle{2}$$\textstyle{h(1_{P},Gb)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\eta}$$\textstyle{1}$$\textstyle{GFGb\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{GF\delta_{1_{Q}}}$$\scriptstyle{1}$$\textstyle{3}$$\textstyle{GFh(1_{P},Gb)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{-1}_{1}}$$\textstyle{5}$$\textstyle{GFGb\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\delta^{\prime}_{1_{Q}}}$$\scriptstyle{G\epsilon}$$\textstyle{4}$$\textstyle{Gh^{\prime}(1_{P},FGb)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gh^{\prime}(1_{P},\epsilon)}$$\textstyle{Gb\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\delta^{\prime}_{1_{Q}}}$$\textstyle{Gh^{\prime}(1_{P},b)}$
(2.35) is the outside of the diagram. $\textstyle{1}$ commutes by the triangle
identities. $\textstyle{2}$ commutes by naturality of $\eta$. $\textstyle{3}$
commutes since $(F,\pi)$ is a $P$-functor. $\textstyle{4}$ commutes by
naturality of $\delta$. $\textstyle{5}$ is the definition of $\psi$. Hence the
whole diagram commutes, and $(G,\psi)$ is a $P$-functor.
To see that $(F,G,\eta,\epsilon)$ is a $P$-equivalence, it is now enough to
show that $\eta$ and $\epsilon$ are $P$-transformations, since they satisfy
the triangle identities by hypothesis.
Write $(GF,\chi)=(G,\psi)\circ(F,\pi)$. We wish to show that $\eta$ is a
$P$-transformation $(1,1)\to(GF,\chi)$. Each $\chi_{q,a_{\bullet}}$ is the
composite
$\textstyle{h(q,GFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{q,Fa_{\bullet}}}$$\textstyle{Gh(q,Fa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\pi_{q,a_{\bullet}}}$$\textstyle{GFh(q,a_{\bullet})}$
Applying the definition of $\psi$, this is
$\textstyle{h(q,GFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{GFh(q,GFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\pi^{-1}}$$\textstyle{Gh(q,FGFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gh_{q}\epsilon
F}$$\textstyle{Gh(q,Fa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\pi}$$\textstyle{GFh(q,a_{\bullet})}$
The axiom on $\eta$ is the outside of the diagram
$\textstyle{h(q,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{h(q,\eta)}$$\scriptstyle{\eta}$$\textstyle{h(q,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{GFh(q,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\pi^{-1}}$$\scriptstyle{GFh(q,\eta)}$$\textstyle{1}$$\textstyle{2}$$\textstyle{Gh(q,Fa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gh(q,F\eta)}$$\scriptstyle{1}$$\textstyle{3}$$\textstyle{h(q,GFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{GFh(q,GFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\pi^{-1}}$$\textstyle{Gh(q,FGFa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gh(q,\epsilon
F)}$$\textstyle{Gh(q,Fa_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G\pi}$$\textstyle{GFh(q,a_{\bullet})}$
$\textstyle{1}$ commutes by naturality of $\eta$, $\textstyle{2}$ commutes by
naturality of $\pi^{-1}$, and $\textstyle{3}$ commutes since $G\pi\circ
G\pi^{-1}=G(\pi\circ\pi^{-1})=G1=1G$. The triangle commutes by the triangle
identities. So the whole diagram commutes, and $\eta$ is a $P$-transformation.
By Lemma 4.2.8, $\eta^{-1}$ is also a $P$-transformation. Similarly,
$\epsilon$ and $\epsilon^{-1}$ are $P$-transformations.
∎
The statement of the lemma is a fragment of the statement that Wk-$P$-Cat is
2-monadic over Cat. Compare the fact that monadic functors reflect isos.
###### Theorem 5.1.4.
Let $Q\circ
A\mathop{\stackrel{{\scriptstyle\scriptstyle{h}}}{{\longrightarrow}}}A$ be a
weak $P$-category. Then $A$ is equivalent to $\mbox{{{st}}}(A)$ in the
2-category Wk-$P$-Cat.
###### Proof.
Let $F:\mbox{{{st}}}(A)\to A$ be given by $F(p,a_{\bullet})=h(p,a_{\bullet})$
and identification of maps. This is certainly full and faithful, and it is
essentially surjective on objects because $\delta^{-1}_{1_{Q}}:h(1_{P},a)\to
a$ is an isomorphism. By Lemma 5.1.3, it remains only to show that $F$ is a
weak $P$-functor.
We must find a sequence $(\phi_{i}:h_{i}(1\times F^{i})\to Fh^{\prime})$ of
natural transformations satisfying equations (2.34) and (2.35) from Lemma
2.9.11. We can take
$(\phi_{i})_{q,(p_{\bullet},a_{\bullet}^{\bullet})}=(\delta_{q\circ(p_{\bullet})})_{a_{\bullet}^{\bullet}}$
for $q\in Q_{n}$ and
$(p_{1},a^{1}_{\bullet}),\dots,(p_{n},a^{n}_{\bullet})\in\mbox{{{st}}}(A)$.
For (2.34), we must show that
$\scriptstyle{1\times F^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{k_{1}}\times\dots\times
h^{\prime}_{k_{n}}}$$\scriptstyle{\phi_{k_{1}}\times\dots\times\phi_{k_{n}}}$$\scriptstyle{1\times
F^{n}}$$\scriptstyle{h^{\prime}_{n}}$$\scriptstyle{\phi_{n}}$$\scriptstyle{F}$$\scriptstyle{h_{k_{1}}\times\dots\times
h_{k_{n}}}$$\scriptstyle{h}$$\scriptstyle{=}$$\scriptstyle{1\times F^{\sum
k_{i}}}$$\scriptstyle{h^{\prime}_{\sum k_{i}}}$$\scriptstyle{\phi_{\sum
k_{i}}}$$\scriptstyle{F}$$\scriptstyle{h_{\sum k_{i}}}$
All 2-cells in this equation are instances of $\delta$. Since there is at most
one 2-cell between two 1-cells in $Q$, the equation holds.
For (2.35) to hold, we must have
$\textstyle{F(p,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\delta_{1_{Q}}}$$\textstyle{h(1_{P},F(p,a_{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1_{P}}}$$\textstyle{F(p,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\delta^{\prime}_{1_{Q}}}$$\textstyle{Fh^{\prime}(1_{P},(p,a_{\bullet}))}$
(5.7)
Since st($A$) is a strict monoidal category, $\delta^{\prime}=1$. Apply this
observation, and the definitions of $F$, $\phi$ and $h^{\prime}$; then (5.7)
becomes
$\textstyle{h(p,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\delta_{1_{Q}}}$$\textstyle{h(1_{P},h(p,a_{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1_{P}\circ(p)}}$$\textstyle{h(p,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{h(p,a_{\bullet})}$
Since there is at most one arrow between two 1-cells in $Q$, this diagram
commutes. So $(F,\phi)$ is a weak $P$-functor, and hence (by Lemma 5.1.3) an
equivalence in Wk-$P$-Cat. ∎
###### Example 5.1.5.
Consider the initial operad $0$, whose algebras are sets. We saw in Example
4.3.4 that unbiased weak $0$-categories are categories equipped with a trivial
monad. By Theorem 5.1.4, every unbiased weak $0$-category is equivalent via
weak $0$-functors to a category equipped with a monad which _is_ the identity:
in other words, a category.
###### Example 5.1.6.
Consider the terminal operad $1$, whose algebras are monoids. Theorem 5.1.4
tells us that every unbiased weak monoidal category is monoidally equivalent
to a strict monoidal category.
### 5.2 Universal property of st
Let $P$ be a plain operad.
###### Theorem 5.2.1.
Let $U^{\prime}$ be the forgetful functor
$\mbox{{{Str-$P$-Cat}}}\to\mbox{{{Wk-$P$-Cat}}}$ (considering both of these as
1-categories). Then st is left adjoint to $U^{\prime}$.
###### Proof.
For each $(A,h)\in\mbox{{{Wk-$P$-Cat}}}$, we construct an initial object
$A\mathop{\stackrel{{\scriptstyle\scriptstyle{(F^{\prime},\psi)}}}{{\longrightarrow}}}\mbox{{{st}}}(A)$
of the comma category $(A\downarrow U^{\prime})$, thus showing that st is
functorial and that ${\rm\textbf{st}}\dashv U^{\prime}$ (and that
$(F^{\prime},\psi)$ is the component of the unit at $A$). Let
$(B,h^{\prime\prime})$ be a strict $P$-category, and $(G,\gamma):A\to
U^{\prime}B$ be a weak $P$-functor. We must show that there is a unique strict
$P$-functor $H$ making the following diagram commute:
---
(5.8)
$(F^{\prime},\psi)$ is given as follows:
* •
If $a\in A$, then ${F^{\prime}}(a)=(1,a)$.
* •
If $f:a\to a^{\prime}$ in $A$ then ${F^{\prime}}f$ is the lifting of $h(1,f)$
with source $(1,a)$ and target $(1,a^{\prime})$.
* •
Each $\psi_{(p,a_{\bullet})}$ is the lifting of
$(\delta_{1_{Q}})_{h(p,a_{\bullet})}:h(p,a_{\bullet})\to
h(1,h(p,a_{\bullet}))$ to a morphism
$h^{\prime}(p,F^{\prime}(a)_{\bullet})=(p,a_{\bullet})\to(1,h(p,a_{\bullet}))=F^{\prime}(h(p,a_{\bullet}))$.
For commutativity of (5.8), we must have $H(1,a)=G(a)$, and for strictness of
$H$, we must have $H(p,a_{\bullet})=h^{\prime\prime}(p,H(1,a)_{\bullet})$.
These two conditions completely determine $H$ on objects.
Now, take a morphism $f:(p,a_{\bullet})\to(p^{\prime},a_{\bullet}^{\prime})$,
which is a lifting of a morphism $g:h(p,a_{\bullet})\to
h(p^{\prime},a_{\bullet}^{\prime})$ in $A$. Then $Hf$ is a morphism
$h^{\prime\prime}(p,Ga_{\bullet})\to
h^{\prime\prime}(p^{\prime},Ga_{\bullet}^{\prime})$: the obvious thing for it
to be is the composite
$\textstyle{h^{\prime\prime}(p,Ga_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{Gh^{\prime\prime}(p,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Gg}$$\textstyle{Gh^{\prime\prime}(p^{\prime},a_{\bullet}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{-1}}$$\textstyle{h^{\prime\prime}(p^{\prime},Ga_{\bullet}^{\prime})}$
and we shall show that this is in fact the only possibility. Consider the
composite
$\textstyle{(1,h(p,a_{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{-1}}$$\textstyle{(p,a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(p^{\prime},a_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{(1,h(p^{\prime},a_{\bullet}^{\prime}))}$
in st($A$). Composition in st($A$) is given by composition in $A$, so this is
equal to the lifting of $\delta_{1_{Q}}\circ g\circ\delta_{1_{Q}}^{-1}=h(1,g)$
to a morphism $(1,h(p,a_{\bullet}))\to(1,h(p^{\prime},a_{\bullet}^{\prime}))$,
namely $F^{\prime}g$. So $f=\psi^{-1}\circ F^{\prime}g\circ\psi$, and
$Hf=H\psi^{-1}\circ HF^{\prime}g\circ H\psi$. By commutativity of (5.8),
$HF^{\prime}=G$ and $H\psi=\gamma$, so $Hf=\gamma^{-1}\circ Gg\circ\gamma$ as
required.
This completely defines $H$. So we have constructed a unique $H$ which makes
(5.8) commute and which is strict. Hence $(F^{\prime},\psi):A\to
U^{\prime}\,\mbox{{{st}}}(A)$ is initial in $(A\downarrow U^{\prime})$, and so
${\rm\textbf{st}}\dashv U^{\prime}$. ∎
The $P$-functor $(F,\phi):\mbox{{{st}}}(A)\to A$ constructed in Theorem 5.1.4
is pseudo-inverse to $(F^{\prime},\psi)$, which we have just shown to be the
$A$-component of the unit of the adjunction st $\dashv U^{\prime}$. We can
therefore say that Str-$P$-Cat is a weakly coreflective sub-2-category of
Wk-$P$-Cat. Note that the counit is _not_ pseudo-invertible, so this is not a
2-equivalence.
###### Example 5.2.2.
Consider again the initial operad $0$, whose algebras are sets. We saw in
Example 4.3.4 that unbiased weak $0$-categories are categories equipped with a
specified trivial monad. Let Triv denote the category of such categories, with
morphisms being functors that preserve the trivial monad up to coherent
isomorphism. A strict unbiased $0$-category is a category equipped with a
monad equal to the identity monad, which is simply a category. So Cat is a
weakly coreflective sub-2-category of Triv.
### 5.3 Presentation-independence
We will now show that the weakening of a symmetric operad $P$ is essentially
independent of the generators chosen. This generalizes Leinster’s result (in
[Lei03] section 3.2) that the theory of weak monoidal categories is
essentially unaffected by the choice of a different presentation for the
theory of monoids.
We will need the following lemma:
###### Lemma 5.3.1.
In Cat-$\Sigma$-Operad, if
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{R}$
is a fork, and $\gamma$ is levelwise full and faithful, then
$\alpha\cong\beta$.
###### Proof.
We shall construct an invertible Cat-$\Sigma$-operad transformation
$\eta:\alpha\to\beta$. We form the $\eta_{n}$s as follows: for all $p\in
P_{n}$, let $\gamma\alpha(p)=\gamma\beta(p)$. Since $\gamma$ is levelwise
full, there exists an arrow $(\eta_{n})_{p}:\alpha(p)\to\beta(p)$ such that
$\gamma_{n}((\eta_{n})_{p})=1_{\gamma\alpha(p)}$. Since $\gamma$ is levelwise
full and faithful, this arrow is an isomorphism. Each $\eta_{n}$ is natural
because, for all $n\in\natural$ and $f:p\to q$ in $P_{n}$, the image under
$\gamma$ of the naturality square
$\textstyle{\alpha(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta_{n})_{p}}$$\scriptstyle{\alpha(f)}$$\textstyle{\beta(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta(f)}$$\textstyle{\alpha(q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta_{n})_{q}}$$\textstyle{\beta(q)}$
is
$\textstyle{\gamma\alpha(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\gamma\alpha(f)}$$\textstyle{\gamma\beta(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\beta(f)}$$\textstyle{\gamma\alpha(q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{\gamma\beta(q)}$
which commutes since $\gamma\alpha=\gamma\beta$. Since $\gamma$ is faithful,
the naturality square commutes, and $\eta_{n}$ is natural. It remains to show
that the collection $(\eta_{n})_{n\in\natural}$ forms a Cat-$\Sigma$-operad
transformation, in other words that the equations
(5.21) $\displaystyle(\eta_{1})_{1}$ $\displaystyle=$ $\displaystyle 1$ (5.22)
(5.35)
hold, for all $n,k_{1}\dots k_{n}\in\natural$ and every $\sigma\in S_{n}$. As
above, it is enough to show that the images of both sides under $\gamma$ are
equal, and this is trivially true by definition of $\eta$. ∎
Let $P$ be a symmetric operad.
###### Theorem 5.3.2.
Let $\Phi\in\mbox{{{Set}}}$ and let $\phi:{F_{\Sigma}}\Phi\to P$ be a regular
epi. Then Wk${}_{\phi}(P)$ is equivalent as a symmetric Cat-operad to Wk($P$).
###### Proof.
Let $Q$ be the weakening of $P$ with respect to $\phi:{F_{\Sigma}}\Phi\to P$.
By the triangle identities, we have a commutative square
$\textstyle{{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{{F_{\Sigma}}\overline{\phi}}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{P}}$$\textstyle{P}$
By functoriality of the factorization system, this gives rise to a unique map
$\chi:Q\to{\mbox{Wk($P$)}}$ such that
$\textstyle{{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{{F_{\Sigma}}\overline{\phi}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{P}}$Wk($P$)$\textstyle{P}$
commutes. We wish to find a pseudo-inverse to $\chi$.
Since $\Sigma$-Operad is monadic over Set, a regular epi in $\Sigma$-Operad is
a levelwise surjection by Theorem 3.0.11. So we may choose a section
$\psi_{n}$ of $\phi_{n}:({F_{\Sigma}}\Phi)_{n}\to P_{n}$ for all
$n\in\natural$. So we have a morphism
$\psi:{U^{\Sigma}}P\to{U^{\Sigma}}{F_{\Sigma}}\Phi$ in Set. We wish to show
that
$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{P}}$$\scriptstyle{\overline{\psi}}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{P}$
commutes. This follows from a simple transpose argument:
---
$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{P}$$\textstyle{{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{{U^{\Sigma}}{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{U^{\Sigma}}\phi}$$\textstyle{{U^{\Sigma}}P}$$\textstyle{=}$$\textstyle{{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{{U^{\Sigma}}P}$$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P.}$
This induces a map
$\textstyle{{F_{\Sigma}}{U^{\Sigma}}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{P}}$$\scriptstyle{\overline{\psi}}$Wk($P$)$\scriptstyle{\omega}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{{F_{\Sigma}}\Phi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$
We will show that $\omega$ is pseudo-inverse to $\chi$. Now,
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$Wk($P$)$\scriptstyle{\chi}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$
commutes. So
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{Q}}$$\scriptstyle{\chi\omega}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$
is a fork. By Lemma 5.3.1, $\chi\omega\cong 1_{Q}$, and similarly
$\omega\chi\cong 1_{{\mbox{Wk($P$)}}}$. So $Q\simeq{\mbox{Wk($P$)}}$ as a
symmetric Cat-operad, as required. ∎
###### Corollary 5.3.3.
Let $P$ be a plain operad. Then ${F_{\Sigma}^{\rm
pl}}({\mbox{Wk($P$)}})\simeq{\mbox{Wk(${F_{\Sigma}^{\rm pl}}P$)}}$.
###### Proof.
Let $\phi:{F_{\rm pl}}{U^{\rm pl}}P\to P$ be the component at $P$ of the
counit of the adjunction ${F_{\rm pl}}\dashv{U^{\rm pl}}$. Let $\epsilon$ be
the counit of the adjunction ${F_{\Sigma}^{\rm pl}}\dashv{U^{\Sigma}_{\rm
pl}}$.
By Theorem 4.4.8, there is an isomorphism ${F_{\Sigma}^{\rm
pl}}({\mbox{Wk${}_{\phi}(P)$}})\cong{\mbox{Wk${}_{{F_{\Sigma}^{\rm
pl}}\phi}({F_{\Sigma}^{\rm pl}}P)$}}$, and by Theorem 5.3.2, there is an
equivalence ${\mbox{Wk${}_{{F_{\Sigma}^{\rm pl}}\phi}({F_{\Sigma}^{\rm
pl}}P)$}}\simeq{\mbox{Wk(${F_{\Sigma}^{\rm pl}}P$)}}$. Hence ${F_{\Sigma}^{\rm
pl}}({\mbox{Wk($P$)}})\simeq{\mbox{Wk(${F_{\Sigma}^{\rm pl}}P$)}}$. ∎
###### Corollary 5.3.4.
Let $P$ be a plain operad. Then the category Wk-$P$-Cat is equivalent to the
category Wk-${F_{\Sigma}^{\rm pl}}P$-Cat.
This tells us that the unbiased categorification of a strongly regular theory
is essentially unaffected by our treating it as a linear theory instead.
###### Example 5.3.5.
Considering again the trivial theory $0$, we see that
$\mbox{{{Triv}}}\simeq\mbox{{{Cat}}}$.
This can be generalised to the multi-sorted situation:
###### Lemma 5.3.6.
Let $X$ be a set, and $f$ be a regular epi in the category $\mbox{{{Cat-
Multicat}}}_{X}$ or in the category $\mbox{{{Cat-$\Sigma$-Multicat}}}_{X}$.
Then $f$ is locally surjective on objects.
###### Proof.
$\mbox{{{Multicat}}}_{X}$ is monadic over $\mbox{{{Multigraph}}}_{X}$ by Lemma
2.5.1 and Theorem 2.5.3, and an object of $\mbox{{{Multigraph}}}_{X}$ can be
considered as an object of $\mbox{{{Set}}}^{Y}$, where $Y=X\times X^{*}$, and
$X^{*}$ is the free monoid on $X$: for each $x\in X$, and each sequence
$x_{1},\dots,x_{n}\in X^{*}$, there is a set of funnels $x_{1},\dots,x_{n}\to
x$. Hence, by 3.0.11, every regular epi in $\mbox{{{Multicat}}}_{X}$ is
locally surjective.
The objects functor $O:\mbox{{{Cat}}}\to\mbox{{{Set}}}$ has both a left
adjoint $D$ and a right adjoint $I$. Hence $O$ and $I$ preserve products, and
hence by Lemma 2.5.1 they induce an adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
60.98924pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-60.98924pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\@ensuremath{\mbox{{{Cat-
Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.73355pt\raise-1.55139pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.92639pt\hbox{$\scriptstyle{O_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.77826pt\raise 9.7361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\bot}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 58.01172pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.0342pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\@ensuremath{\mbox{{{Multicat}}}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.70978pt\raise 1.12083pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.92639pt\hbox{$\scriptstyle{I_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
34.01173pt\raise-4.73611pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}}}}}\ignorespaces.$
Since $O_{*}$ is a left adjoint, it preserves colimits, and in particular
regular epis: hence, every regular epi in $\mbox{{{Cat-Multicat}}}_{X}$ must
be locally surjective on objects.
The symmetric case is proved analogously. ∎
###### Theorem 5.3.7.
Let $M$ be a (symmetric) multicategory, and $\phi:{F_{\rm pl}}\Phi\to M$ (or
in the symmetric case, $\phi:{F_{\Sigma}}\Phi\to M$) be a regular epi. Then
the weakening of $M$ with respect to $\phi$ is equivalent as a Cat-
multicategory to Wk($M$).
###### Proof.
The proof is exactly as for Theorem 5.3.2. ∎
## Chapter 6 Other Approaches
### 6.1 Pseudo-algebras for 2-monads
We begin by recalling some standard notions of 2-monad theory.
###### Definition 6.1.1.
A 2-monad is a monad object in the 2-category of 2-categories, in the sense of
[Str72]; that is to say, a 2-category $\mathcal{C}$, a strict 2-functor
$T:\mathcal{C}\to\mathcal{C}$, and 2-transformations $\mu:T^{2}\to T$ and
$\eta:1_{\mathcal{C}}\to T$ satisfying the usual monad laws:
---
$\textstyle{T^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu
T}$$\scriptstyle{T\mu}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T}$
(6.1)
---
$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\eta}$$\scriptstyle{1_{T}}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta
T}$$\scriptstyle{1_{T}}$$\textstyle{T}$ (6.2)
As is common for ordinary 1-monads, we will usually refer to a 2-monad
$(\mathcal{C},T,\mu,\eta)$ as simply $T$.
The usual notion of an algebra for a monad carries over simply to this case:
###### Definition 6.1.2.
Let $(\mathcal{C},T,\mu,\eta)$ be a 2-monad. A strict algebra for $T$ is an
object $A\in\mathcal{C}$ and a 1-cell $a:TA\to A$ satisfying the following
axioms:
---
$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{Ta}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
(6.3)
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{1_{A}}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
(6.4)
For our purposes, it is more interesting to consider the well-known pseudo-
algebras for a 2-monad. These are algebras “up to isomorphism”:
###### Definition 6.1.3.
Let $(\mathcal{C},T,\mu,\eta)$ be a 2-monad. A pseudo-algebra for $T$ is an
object $A\in\mathcal{C}$, a 1-cell $a:TA\to A$, and invertible 2-cells
---
$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{Ta}$$\scriptstyle{\alpha}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{1_{A}}$$\scriptstyle{\beta}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
satisfying the equations
---
$\textstyle{T^{3}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}a}$$\scriptstyle{\mu
T}$$\scriptstyle{T\mu}$$\textstyle{T^{3}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{2}a}$$\scriptstyle{\mu
T}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ta}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ta}$$\scriptstyle{\mu}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{Ta}$$\scriptstyle{T\alpha}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ta}$$\scriptstyle{\mu}$$\textstyle{=}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\scriptstyle\alpha}$$\textstyle{\scriptstyle\alpha}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\alpha}$$\textstyle{A}$
(6.5)
---
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{T\eta}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{1}$$\scriptstyle{\eta
T}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ta}$$\scriptstyle{\mu}$$\textstyle{=}$$\textstyle{=}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{1}$$\scriptstyle{\beta}$$\textstyle{T^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{Ta}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\alpha}$$\textstyle{A\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\alpha}$
(6.6)
###### Definition 6.1.4.
Let $(\mathcal{C},T,\mu,\eta)$ be a 2-monad, and let
$(A,a,\alpha_{1},\alpha_{2})$ and $(B,b,\beta_{1},\beta_{2})$ be pseudo-
algebras for $T$. A pseudo-morphism of pseudo-algebras
$(A,a,\alpha_{1},\alpha_{2})$ to $(B,b,\beta_{1},\beta_{2})$ is a pair
$(f,\phi)$, where $f:A\to B$ is a 1-cell in $\mathcal{C}$ and $\phi$ is an
invertible 2-cell:
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tf}$$\scriptstyle{a}$$\textstyle{\scriptstyle\phi}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
satisfying the axioms
(6.23) (6.38)
This gives rise to a category Ps-Alg$(T)$ for any 2-monad $T$.
Every cartesian monad $T$ on Set gives rise to a 2-monad $\bar{T}$ on Cat in
an obvious way, and (as we saw in Theorem 2.8.10) every plain operad $P$ gives
rise to a cartesian monad $T_{P}$ on Set. So an alternative definition of
“weak $P$-category” might be “pseudo-algebra for $\bar{T}_{P}$”.
In order to explore the connections between this idea and the notion of weak
$P$-category given in previous chapters, we shall need some theorems from
[BKP89] and related papers.
###### Theorem 6.1.5.
(Blackwell, Kelly, Power) Let $T$ be a 2-monad with rank on a cocomplete
2-category $K$, let $\mbox{{{Alg$(T)$}}}_{\rm str}$ be the 2-category of
strict $T$-algebras and strict morphisms, and $\mbox{{{Alg$(T)$}}}_{\rm wk}$
be the 2-category of strict $T$-algebras and weak morphisms. Then the
inclusion $J:\mbox{{{Alg$(T)$}}}_{\rm str}\to\mbox{{{Alg$(T)$}}}_{\rm wk}$ has
a left adjoint $L$. Thus every strict $T$-algebra $A$ has a pseudo-morphism
classifier $p:A\to A^{\prime}$ (where $A^{\prime}=JLA$), such that for all
$B\in K$, and every pseudo-morphism $f:A\to B$, we may express $f$ uniquely as
the composite of $p$ and a strict morphism:
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{f}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J\bar{f}}$$\textstyle{B}$
###### Proof.
See [BKP89], Theorem 3.13. ∎
###### Theorem 6.1.6.
(Blackwell, Kelly, Power) Let $f:S\to T$ be a strict map between 2-monads with
rank on a cocomplete 2-category $K$. Then the induced map
$f^{*}:\mbox{{{Alg$(T)$}}}_{\rm str}\to\mbox{{{Alg$(S)$}}}_{\rm str}$ has a
left adjoint, and the induced map $f^{*}:\mbox{{{Alg$(T)$}}}_{\rm
wk}\to\mbox{{{Alg$(S)$}}}_{\rm wk}$ has a left biadjoint.
###### Proof.
See [BKP89], Theorem 5.12. ∎
###### Corollary 6.1.7.
Composing this left adjoint with the left adjoint of Theorem 6.1.5 gives us an
adjunction
$\textstyle{\mbox{{{Alg$(S)$}}}_{\rm
wk}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\bot}$$\textstyle{\mbox{{{Alg$(T)$}}}_{\rm
str}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$
###### Theorem 6.1.8.
(Power, Lack) Let $T$ be a 2-monad with rank on a cocomplete 2-category $K$ of
the form $\mbox{{{Cat}}}^{X}$ for some set $X$, and let $T$ preserve pointwise
bijectivity-on-objects. Let $(A,a)$ be a strict $T$-algebra. Then the pseudo-
morphism classifier $A^{\prime}$ for $A$ may be found by factorizing the
structure map $a:TA\to A$ as a pointwise bijective-on-objects map followed by
a locally full-and-faithful map:
---
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
###### Proof.
The construction is given in Power’s paper [Pow89], and the universal property
of the algebra constructed is proved in Lack’s paper [Lac02]. ∎
This argument is due to Steve Lack (private communication).
###### Theorem 6.1.9.
Let $P$ be a plain operad. Let $T_{P}$ be the monad induced by $P$ on Set.
Then a pseudo-algebra for $\bar{T}_{P}$ is a weak $P$-category in the sense of
Definition 4.2.1. Furthermore, there is an isomorphism of categories
$\mbox{{{Ps-Alg$(\bar{T}_{P})$}}}\cong\mbox{{{Alg${}_{\rm wk}(P)$}}}$.
###### Proof.
Cat-Operad is monadic over Cat via one of the special monads of Theorem 6.1.8,
and hence, for every plain Cat-operad $P$, the pseudo-morphism classifier of
$P$ is none other than Wk($P$). Hence, if $A$ is a category, then a strict map
of Cat-operads ${\mbox{Wk($P$)}}\to\mathop{\rm{End}}(A)$ is precisely a weak
map $P\to\mathop{\rm{End}}(A)$, or equivalently a $\bar{T}_{P}$-pseudo-algebra
structure on $A$. ∎
We may also use these ideas to provide a simple proof of the strictification
result in Theorem 5.1.4. The map $P\to{\mbox{Wk($P$)}}$ given by Theorem 6.1.5
is pseudo, but it has a strict retraction $q:{\mbox{Wk($P$)}}\to P$. This is
equivalent to a strict map of monads $T_{\mbox{Wk($P$)}}\to T_{P}$. By
Corollary 6.1.7, this induces a 2-functor $\mbox{{{Alg$(P)$}}}_{\rm
str}\to\mbox{{{Alg$({\mbox{Wk($P$)}})$}}}_{\rm wk}$ with a left adjoint. This
functor is simply the inclusion of the 2-category of strict $P$-categories,
strict $P$-functors and $P$-transformations into the 2-category of weak
$P$-(categories, functors, transformations), and its left adjoint is the
functor st constructed in Section 5.1. The fact that any weak $P$-category $A$
is equivalent to $\mbox{{{st}}}(A)$ is a consequence of the fact that any
pseudo $P$-algebra is equivalent to a strict one, and this holds by the
General Coherence Result of Power.
However, pseudo-algebras are less useful in the case of linear theories. Since
the monads arising from symmetric operads are not in general cartesian, we may
not perform the construction given above. We may, however, use the existence
of colimits in Cat, and consider the 2-monad
$A\mapsto\int^{n\in\bbB}P_{n}\times A^{n}$
for any symmetric operad $P$. If $P$ is the free symmetric operad on a plain
operad $P^{\prime}$, this 2-monad is equal to $\bar{T}_{P^{\prime}}$. Yet this
coend construction also leads to problems.
Let $T$ be the “free commutative monoid” monad on Set, and $S$ be the “free
monoid” monad on Set. Since these both arise from symmetric operads, we may
lift them to 2-monads $T^{\prime},S^{\prime}$ on Cat as described above.
$T^{\prime}$ is the free commutative monoid monad on Cat, which is to say the
free strict symmetric monoidal category 2-monad; similarly, $S^{\prime}$ is
the free strict monoidal category 2-monad. For each category $A$, there is a
functor $\pi_{A}:S^{\prime}A\to T^{\prime}A$ which is full and surjective-on-
objects; hence, if $(A,a,\alpha_{1},\alpha_{2})$ is a pseudo-algebra for
$T^{\prime}$, we obtain an $S^{\prime}$-pseudo-algebra structure by
precomposing with $\pi_{A}$:
$\textstyle{S^{\prime}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\textstyle{T^{\prime}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
---
$\textstyle{S^{\prime
2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi*\pi)_{A}}$$\textstyle{T^{\prime
2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T^{\prime}a}$$\scriptstyle{\mu}$$\textstyle{\scriptstyle\alpha_{1}}$$\textstyle{T^{\prime}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{T^{\prime}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{A}}$$\scriptstyle{\eta}$$\textstyle{S^{\prime}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{1_{A}}$$\scriptstyle{\alpha_{2}}$$\textstyle{T^{\prime}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A}$
The $S^{\prime}$-pseudo-algebra structure so obtained is uniquely determined.
Since $\pi_{A}$ is full and surjective-on-objects, it is epic, and so every
pseudo-algebra for $S^{\prime}$ is a pseudo-algebra for $T^{\prime}$ in at
most one way. Hence we may view all pseudo-algebras for $T^{\prime}$ as
pseudo-algebras for $S^{\prime}$ (that is, as monoidal categories) with extra
properties. But there exist monoidal categories with several choices of
symmetric structure on them. For instance, consider the category of graded
Abelian groups, with tensor product $(A\otimes
B)_{n}=\bigoplus_{i+j=n}A_{i}\otimes B_{j}$. As well as the obvious symmetry,
there is another given by $\tau_{AB}(a\otimes b)=(-1)^{ij}b\otimes a$, where
$a\in A_{i},b\in B_{j}$.
We can say at least something about the extra properties that pseudo-algebras
for $T^{\prime}$ must have:
###### Theorem 6.1.10.
A pseudo-algebra for $T^{\prime}$ is a symmetric monoidal category $A$ in
which $x\otimes y=y\otimes x$ for all $x,y\in A$.
###### Proof.
Recall our construction of the finite product operad whose algebras are
commutative monoids in Example 2.3.4. From this, we may deduce that if $A$ is
a set, then an element of $T_{P}A$ is a function $A\to\natural$ assigning each
element of $A$ its multiplicity: in other words, a multiset of elements of
$A$. Let $(A,a,\alpha,\beta)$ be a pseudo-algebra for $T^{\prime}$ in Cat.
Then we have a binary tensor product:
$x\otimes y:=a(x^{1}y^{1})$
where $x^{1}y^{1}$ is the function $A\to\natural$ sending $x$ and $y$ to 1 and
all other objects of $A$ to 0. The tensor is defined analogously on morphisms.
The components of $\alpha$ and $\beta$ give us associator, symmetry and unit
maps, and it can be shown that they satisfy the axioms for a monoidal
category. However, since the function $x^{1}y^{1}$ is equal to the function
$y^{1}x^{1}$ for all $x,y\in A$, it must be the case that $x\otimes y=y\otimes
x$. ∎
Since not all symmetric monoidal categories satisfy this condition, it is
apparent that a naïve approach to categorification based on pseudo-algebras is
doomed to fail, and that more sophistication is required. In fact, I
conjecture that a stronger condition holds: that the symmetry maps are all
identities.
In the specific case of symmetric monoidal categories, we may remedy the
situation as follows. Let $T$ be the “free symmetric strict monoidal category”
2-monad. Then pseudo-algebras for $T$ are precisely symmetric monoidal
categories.
### 6.2 Laplaza sets
This notion was introduced by T. Fiore, P. Hu and I. Kriz in [FHK], as a
generalization of Laplaza’s categorification of rigs in [Lap72]. It was
introduced as an attempt to correct an error in the earlier definition of
categorification proposed in [Fio06]; the error in question is essentially
that discussed in Section 4.9 above.
###### Definition 6.2.1.
Let $T$ be a finite product operad. A Laplaza set for $T$ is a subsignature of
${U^{\rm fp}}T$.
Concretely, a Laplaza set $S$ for $T$ is a sequence $S_{0}\subset
T_{0},S_{1}\subset T_{1},\dots$ of subsets of $T_{0},T_{1},\dots$.
###### Definition 6.2.2.
Let $T$ be a finite product operad, and $S$ be a Laplaza set for $T$. A
$(T,S)$-pseudo algebra is
* •
a category $\mathcal{C}$
* •
for each $\phi\in T_{n}$, a functor
$\hat{\phi}:\mathcal{C}^{n}\to\mathcal{C}$,
* •
coherence morphisms witnessing all equations that are true in $T$,
such that, if
* •
$s_{1},s_{2},t_{1}$ and $t_{2}$ are elements of $({F_{\rm fp}}{U^{\rm
fp}}T)_{n}$,
* •
$\delta_{1}:\hat{s}_{1}\to\hat{t}_{1}$ and
$\delta_{2}:\hat{s}_{2}\to\hat{t}_{2}$ are coherence morphisms,
* •
$\epsilon(s_{1})=\epsilon(s_{2})\in S$ and $\epsilon(t_{1})=\epsilon(t_{2})\in
S$,
then $\delta_{1}=\delta_{2}$.
This definition can be recast in terms of strict algebras for a finite product
Cat-operad.
By judicious choice of Laplaza set, one can recover the classical notion of
symmetric monoidal category and Laplaza’s categorification of the theory of
rigs.
### 6.3 Non-algebraic definitions
Various definitions have appeared that are inspired by the notions of homotopy
monoids etc. in topology. In [Lei00], Leinster proposes a definition of a
“homotopy $P$-algebra in $M$” for any plain operad $P$ and any monoidal
category $M$; his shorter paper [Lei99] explores this definition in the case
$P=1$. Related (but more general) is Rosický’s work described in [Ros].
These definitions stand roughly in relation to ours as do the “non-algebraic”
definitions of $n$-category in relation to the “algebraic” ones: see [CL04].
## References
* [Age02] Pierre Ageron. Les catégories monoïdales croisées. 78th PSSL, Strasbourg, Feb 2002. Available at http://www.math.unicaen.fr/$\sim$ageron/Strasbourg.dvi.
* [AHS04] Jiří Adámek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Second edition, 2004. Available from http://katmat.math.uni-bremen.de/acc.
* [AR94] Jiří Adámek and Jiří Rosický. Locally Presentable and Accessible Categories. Number 189 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1994.
* [Bae] John Baez. Universal algebra and diagrammatic reasoning. Lectures at Geometry of Computation 2006. Available from http://math.ucr.edu/home/baez/universal/.
* [BC04] John C. Baez and Alissa S. Crans. Higher-dimensional algebra VI: Lie 2-algebras. Theory and Applications of Categories, 12:492–538, 2004, math.QA/0307263.
* [BKP89] R. Blackwell, G.M. Kelly, and A.J. Power. Two-dimensional monad theory. Journal of Pure and Applied Algebra, 59:1–41, 1989.
* [BL04] John C. Baez and Aaron D. Lauda. Higher-dimensional algebra V: 2-groups. Theory and Applications of Categories, 12:423–491, 2004, math.QA/0307200.
* [Bor94] Francis Borceux. Handbook of Categorical Algebra 2: Categories and Structures. Number 51 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.
* [BV73] J.M. Boardman and R.M. Vogt. Homotopy Invariant Algebraic Structures on Topological Spaces. Number 347 in Lecture Notes in Mathematics. Springer, 1973.
* [CJ95] Aurelio Carboni and Peter T. Johnstone. Connected limits, familial representability and Artin glueing. Mathematical Structures in Computer Science, 5:441–459, 1995.
* [CJ04] Aurelio Carboni and Peter T. Johnstone. Corrigenda for ‘Connected limits, familial representability and Artin glueing’. Mathematical Structures in Computer Science, 14:185–187, 2004.
* [CL04] Eugenia Cheng and Aaron Lauda. Higher-dimensional categories: an illustrated guide book, Jun 2004.
* [Coh65] P. M. Cohn. Universal Algebra. Harper’s Series in Modern Mathematics. Harper & Row, 1965.
* [FHK] Thomas M. Fiore, Po Hu, and Igor Kriz. Laplaza sets, or how to select coherence diagrams for pseudo algebras. To appear in Advances of Mathematics, arXiv:0803.1408v2.
* [Fio06] Thomas M. Fiore. Pseudo limits, biadjoints, and pseudo algebras: Categorical foundations of conformal field theory. Memoirs of the American Mathematical Society, 182(860), 2006, math.CT/04028298.
* [FK72] P. Freyd and G.M. Kelly. Categories of continuous functors. Journal of Pure and Applied Algebra, 2:169–191, 1972.
* [Gou06] Miles Gould. Coherence for categorified operadic theories. July 2006, math.CT/0607423.
* [Gou07] Miles Gould. The categorification of a symmetric operad is independent of signature. November 2007, math.CT/0711.4904.
* [Joh94] P.T. Johnstone. Universal algebra. Unpublished lecture notes from Cambridge Part III course, 1994.
* [JS93] André Joyal and Ross Street. Braided tensor categories. Advances in Mathematics, 102:20–78, 1993.
* [Kel72a] G. M. Kelly. An abstract approach to coherence. In Coherence in Categories, number 281 in Lecture Notes in Mathematics, pages 106–174. Springer-Verlag, May 1972.
* [Kel72b] G. M. Kelly. Many-variable functorial calculus I. In Coherence in Categories, number 281 in Lecture Notes in Mathematics, pages 66–105. Springer-Verlag, May 1972.
* [KP93] G. M. Kelly and A. J. Power. Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. Journal of Pure and Applied Algebra, 89:163–179, 1993.
* [KT93] M. Korostenski and W. Tholen. Factorization systems as Eilenberg-Moore algebras. Journal of Pure and Applied Algebra, 85(1):57–72, 1993.
* [KV94] M.M. Kapranov and V.A. Voevodsky. 2-categories and Zamolodchikov tetrahedra equations. In Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods, number 56 in Proceedings of Symposia in Pure Mathematics. AMS, 1994.
* [Lac02] S. Lack. Codescent objects and coherence. Journal of Pure and Applied Algebra, 175:223–241, 2002.
* [Lam69] Joachim Lambek. Deductive systems and categories II: standard constructions and closed categories. In P. Hilton, editor, Category Theory, Homology Theory and their Applications, I (Battelle Institute Conference, Seattle, 1968, Vol. One), number 86 in Lecture Notes in Mathematics. Springer, 1969.
* [Lap72] Miguel L. Laplaza. Coherence for distributivity. In Coherence in Categories, number 281 in Lecture Notes in Mathematics, pages 29–65. Springer-Verlag, May 1972.
* [Law63] F. William Lawvere. Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories. PhD thesis, Columbia University, 1963.
* [Lei99] T. Leinster. Up-to-homotopy monoids. 1999, math.QA/9912084.
* [Lei00] T. Leinster. Homotopy algebras for operads. 2000, math.QA/0002180.
* [Lei03] Tom Leinster. Higher Operads, Higher Categories. Cambridge University Press, 2003, math.CT/0305049.
* [Man76] Ernest G. Manes. Algebraic Theories. Number 26 in Graduate Texts in Mathematics. Springer-Verlag, 1976.
* [Mar] M. Markl. Homotopy algebras are homotopy algebras. math.AT/9907138.
* [May72] J.P. May. The Geometry of Iterated Loop Spaces. Number 271 in Lecture Notes in Mathematics. Springer, 1972.
* [ML98] Saunders Mac Lane. Categories for the Working Mathematician. Number 5 in Graduate Texts in Mathematics. Springer-Verlag, second edition, 1998.
* [MLM92] Saunders Mac Lane and Ieke Moerdijk. Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Universitext. Springer-Verlag, 1992.
* [MSS02] Martin Markl, Steve Shnider, and Jim Stasheff. Operads in Algebra, Topology and Physics, volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, 2002.
* [Ost03] Viktor Ostrik. Module categories, weak Hopf algebras and modular invariants. Transform. Groups, 8(2):159–176, 2003, math.QA/0111139.
* [Pen99] J. Penon. Approche polygraphique des $\infty$-catégories non strictes. Cahiers de Topologie et Géometrie Différentielle Catégoriques, XL(1):31–80, 1999.
* [Pow89] A.J. Power. A general coherence result. Journal of Pure and Applied Algebra, 57:165–173, 1989.
* [Ros] J. Rosický. Homotopy varieties. Talk at CT 2006; slides available from http://www.mscs.dal.ca/$\sim$selinger/ct2006/slides/CT06-Rosicky.pdf.
* [Str72] R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149–168, 1972.
* [Tro02] S.N. Tronin. Abstract clones and operads. Siberian Mathematical Journal, 43(4):746–755, Jul 2002.
* [Tro06] S. N. Tronin. Operads and varieties of algebras defined by polylinear identities. Siberian Mathematical Journal, 47(3):555–573, May 2006.
## Index
* §2.3, 2nd item
* 1- and 2-cells §2.9
* 2-monad Definition 6.1.1
* Adámek, Jiří §2.8
* adjunctions
* for Cat-multicategories §4.7
* for Cat-operads Theorem 2.9.14
* for multicategories §4.7
* for operads §2.4
* Alg §1.2
* Alg$(P)$ §1.1
* algebra
* for a 2-monad Definition 6.1.2
* for a clone Definition 1.2.4
* for a finite product multicategory Definition 2.3.7
* for a finite product operad Definition 2.3.9
* for a Lawvere theory Definition 1.3.1
* for a plain multicategory Definition 2.1.5
* for a plain operad Definition 2.1.14
* for a presentation Definition 1.1.11
* for a ring Example 2.3.10, Example 2.3.3
* for a signature Definition 1.1.10
* for a symmetric multicategory Definition 2.2.3
* for a symmetric operad Definition 2.2.8
* for an enriched operad §2.9
* $\mathcal{B}$ Example 2.1.12
* cartesian
* monad Definition 2.8.9
* natural transformation Definition 2.8.8
* clone Definition 1.2.1
* closure Definition 1.1.14
* coherence
* for monoidal functors Lemma 4.6.4
* for symmetric monoidal categories Theorem 4.6.3
* for symmetric monoidal functors Theorem 4.6.5
* for weak $P$-categories Theorem 5.1.4
* general case Chapter 5
* commutative monoids Example 2.3.11, Example 2.3.4, Example 2.8.6, §4.6, Theorem 6.1.10
* derived operation §1.1
* directed §1.4
* endomorphism clone Example 1.2.2
* endomorphism operad
* finite product Example 2.3.5
* plain Example 2.1.10
* symmetric Example 2.2.7
* equation Definition 1.1.6
* plain-operadic Definition 2.7.1
* factorization system Definition 3.0.2
* bijective on objects/full and faithful
* on Cat Example 3.0.8
* on Cat-Operad Example 3.0.10
* on Digraph Example 3.0.5
* on $\mbox{{{Digraph-Multigraph}}}_{X}$ §4.7
* epi-mono Example 3.0.3
* familial representability Definition 2.8.13
* filtered
* category Definition 1.4.2
* colimit Definition 1.4.3
* finitary
* functor Definition 1.4.6
* monad Definition 1.4.7
* finite product theory, multi-sorted Definition 1.3.2
* Fiore, Tom §6.2
* FP-Operad §2.3, Definition 2.3.7
* funnel item 2
* generating operation Definition 1.1.8
* grafting
* of terms Definition 1.1.13
* of trees §2.6
* Hu, Po §6.2
* involutive monoid Definition 2.8.16
* Kelly, Max §2.9, Chapter 5
* kernel pair Definition 2.7.4
* Kriz, Igor §6.2
* labelling function Definition 1.1.5
* Laplaza set Definition 6.2.1
* Laplaza, Miguel §6.2
* Law Definition 1.3.1
* Lawvere theory Definition 1.3.1
* Lawvere, F. William §1.3
* lax
* morphism of algebras for a symmetric Cat-operad Definition 2.9.10
* symmetric monoidal functor Definition 4.6.2
* Leinster, Tom §6.3
* linear
* clone Definition 1.5.3
* equation Definition 1.1.7
* presentation Definition 1.1.9
* term Definition 1.1.7
* little $m$-discs operad Example 2.1.13
* $M$-sets Example 4.3.7, Example 4.8.2
* Mac Lane, Saunders Theorem 4.6.3
* $\mbox{{{Mnd}}}(\mathcal{C})$ §1.4
* monad §1.4
* monoid with involution, see involutive monoid
* monoids Example 2.8.6, Example 4.3.6, Example 4.4.10, Example 5.1.6
* morphism
* of algebras for a clone Definition 1.2.5
* of algebras for a finite product Cat-operad Definition 2.9.10
* of algebras for a Lawvere theory Definition 1.3.1
* of algebras for a plain operad Definition 2.1.15
* of algebras for a signature Definition 1.1.12
* of algebras for a symmetric Cat-operad Definition 2.9.10
* of algebras for a symmetric operad Definition 2.2.8
* of clones Definition 1.2.3
* of enriched multicategories Definition 2.9.5
* of enriched multigraphs Definition 4.7.5
* of finite product multicategories Definition 2.3.6
* of finite product operads Definition 2.3.8
* of Lawvere theories Definition 1.3.1
* of monads Definition 1.4.1
* of multigraphs Definition 4.7.2
* of plain multicategories Definition 2.1.2
* of plain operads §2.1
* of symmetric multicategories Definition 2.2.2
* Multicat §2.1
* multicategory
* finite product Definition 2.3.1
* enriched Definition 2.9.3
* internal Remark 2.9.6
* plain Definition 2.1.1
* enriched Definition 2.9.1
* symmetric Definition 2.2.1
* enriched Definition 2.9.2
* underlying multicategory of a monoidal category Example 2.1.4
* Multigraph §4.7
* multigraph Definition 4.7.1
* enriched Definition 4.7.4
* $\mbox{{{Multigraph}}}_{X}$ Definition 4.7.3
* Operad §2.1
* monadicity over Set Theorem 2.5.4
* operad
* enriched Definition 2.9.4
* in Cat Lemma 2.9.7
* finite product Definition 2.3.2
* internal Remark 2.9.6
* little $m$-discs, see little $m$-discs operad
* plain §2.1
* concrete description Lemma 2.1.7
* symmetric Definition 2.2.4
* operad of braids, see $\mathcal{B}$
* operad of symmetries, see $\mathcal{S}$
* orthogonal Definition 3.0.1
* pointed sets Example 2.8.6, Example 4.3.5, Example 4.5.2, Example 4.5.3
* presentation
* for a plain operad Definition 2.7.2
* independence of weakenings Theorem 5.3.2
* of an algebraic theory Definition 1.1.8
* primitive operation 2nd item
* Ps-Alg$(T)$ §6.1
* pseudo-algebra
* for a 2-monad Definition 6.1.3
* pseudo-morphisms Definition 6.1.4
* for a finite product operad with Laplaza set Definition 6.2.2
* regular §1.1
* Rosický, Jiří §6.3
* $\mathcal{S}$
* as a plain operad Example 2.1.11
* as a symmetric operad Example 2.2.6
* as ${F_{\Sigma}^{\rm pl}}1$ Example 4.4.10
* relation to ${F_{\Sigma}^{\rm pl}}$ §2.6
* sets Example 4.3.4, Example 4.5.1, Example 5.1.5, Example 5.2.2, Example 5.3.5
* signature Definition 1.1.1
* Street, Ross §1.4
* strict
* morphism of algebras for a finite product Cat-operad Definition 2.9.10
* morphism of algebras for a symmetric Cat-operad Definition 2.9.10
* $P$-category Definition 4.2.4
* $P$-functor Definition 4.2.5
* symmetric monoidal functor Definition 4.6.2
* strictification Definition 5.1.1
* universal property §5.2
* strongly regular
* clone Definition 1.5.3
* equation Definition 1.1.7
* presentation Definition 1.1.9
* term Definition 1.1.7
* tree, see tree, strongly regular
* supp Definition 1.1.4
* symmetric monoidal category Definition 4.6.1, §6.1
* symmetric monoidal functor Definition 4.6.2
* term Definition 1.1.2
* transformation
* between morphisms of algebras for a Cat-operad Definition 2.9.12
* between morphisms of plain multicategories Definition 2.1.2
* between morphisms of plain operads §2.1
* between $P$-functors Definition 4.2.6
* tree
* finite product Definition 2.6.4
* equivalence to terms Lemma 2.8.2
* permuted Definition 2.6.4
* equivalence to linear terms Lemma 2.8.3
* plain-operadic
* equivalence to strongly regular terms Lemma 2.8.3
* strongly regular Definition 2.6.1
* trivial monad Definition 4.3.1
* $U^{\rm fp}_{\Sigma}$
* explicit construction Corollary 2.7.13
* unbiased
* monoidal category Definition 2.1.3
* underlying multicategory Example 2.1.4
* weakening of a multicategory Definition 4.7.7
* weakening of a plain operad Definition 4.2.1
* comparison to symmetric weakening Corollary 5.3.3
* weakening of a symmetric operad Definition 4.4.5
* $U^{\Sigma}_{\rm pl}$
* explicit construction Theorem 2.7.11
* var Definition 1.1.3
* variable §1.1
* Velebil, Jiří §2.8
* weak
* morphism of algebras for a finite product Cat-operad Definition 2.9.10
* morphism of algebras for a symmetric Cat-operad Definition 2.9.10
* $P$-category Definition 4.2.3, Definition 4.4.2
* $P$-functor Definition 4.2.5, Definition 4.4.3, Example 4.8.3
* symmetric monoidal functor Definition 4.6.2
* weakening
* of a multicategory Definition 4.7.6
* of a plain operad Definition 4.2.1
* considered as a symmetric operad Theorem 4.4.8
* of a symmetric operad Definition 4.4.1, Definition 4.4.4
* wide pullback Definition 2.8.7
|
arxiv-papers
| 2010-02-04T02:15:28 |
2024-09-04T02:49:08.241777
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. R. Gould",
"submitter": "Miles Gould",
"url": "https://arxiv.org/abs/1002.0879"
}
|
1002.0985
|
# A New Global Embedding Approach to Study Hawking and Unruh Effects
Rabin Banerjee, Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: rabin@bose.res.inE-mail: bibhas@bose.res.in
###### Abstract
A new type of global embedding of curved space-times in higher dimensional
flat ones is introduced to present a unified description of Hawking and Unruh
effects. Our analysis simplifies as well as generalises the conventional
embedding approach.
## 1 Introduction
After Hawking’s famous work [1] \- the black holes radiate - known as Hawking
effect, it is now well understood that it is related to the event horizon of a
black hole. A closely related effect is the Unruh effect [2], where a similar
type of horizon is experienced by a uniformly accelerated observer on the
Minkowski space-time. A unified description of them was first put forwarded by
Deser and Levin [3, 4] which was a sequel to an earlier attempt [5]. This is
called the global embedding Minkowskian space (GEMS) approach. In this
approach, the relevant detector in curved space-time (namely Hawking detector)
and its event horizon map to the Rindler detector in the corresponding flat
higher dimensional embedding space [6, 7] and its event horizon. Then
identifying the acceleration of the Unruh detector, the Unruh temperature was
calculated. Finally, use of the Tolman relation [8] yields the Hawking
temperature. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [9, 10, 11, 12]. However the results were confined to four dimensions
and the calculations were done case by case, taking specific black hole
metrics. It was not clear whether the technique was applicable to complicated
examples like the Kerr-Newman metric which lacks spherical symmetry.
The motivation of this paper is to give a modified presentation of the GEMS
approach that naturally admits generalization. Higher dimensional black holes
with different metrics, including Kerr-Newman, are considered. Using this new
embedding, the local Hawking temperature (Unruh temperature) will be derived.
Then the Tolman formula leads to the Hawking temperature.
We shall first introduce a new global embedding which embeds only the
($t-r$)-sector of the curved metric into a flat space. It will be shown that
this embedding is enough to derive the Hawking result using the Deser-Levin
approach [3, 4], instead of the full embedding of the curved space-time. Hence
we might as well call this the reduced global embedding. This is actually
motivated from the fact that an $N$-dimensional black hole metric effectively
reduces to a $2$ -dimensional metric (only the ($t-r$)-sector) near the event
horizon by the dimensional reduction technique [13, 14, 15, 16] (for examples
see Appendix 1). Furthermore, this $2$-dimensional metric is enough to find
the Hawking quantities if the back scattering effect is ignored. Several
spherically symmetric static metrics will be exemplified. Also, to show the
utility of this reduced global embedding, we shall discuss the most general
solution of the Einstein gravity - Kerr-Newman space-time, whose full global
embedding is difficult to find. Since the reduced embedding involves just the
two dimensional ($t-r$)-sector, black holes in arbitrary dimensions can be
treated. In this sense our approach is valid for any higher dimensional black
hole.
The organization of the paper is as follows. In section 2 we shall find the
reduced global embedding of several black hole space-times which are
spherically symmetric. In the next section the power of this approach will be
exploited to find the Unruh/Hawking temperature for the Kerr-Newman black
hole. Finally, we shall give our concluding remarks.
## 2 Reduced global embedding
A unified picture of Hawking effect [1] and Unruh effect [2] was established
by the global embedding of a curved space-time into a higher dimensional flat
space [4]. Subsequently, this unified approach to determine the Hawking
temperature using the Unruh effect was applied for several black hole space-
times [9, 10], but usually these are spherically symmetric. For instance, no
discussion on the Kerr-Newman black hole has been given, because it is
difficult to find the full global embedding.
Since the Hawking effect is governed solely by properties of the event
horizon, it is enough to consider the near horizon theory. As already stated,
this is a two dimensional theory obtained by dimensional reduction of the full
theory. Its metric is just the ($t-r$)-sector of the original metric.
In the following sub-sections we shall find the global embedding of the near
horizon effective $2$-dimensional theory. Then the usual local Hawking
temperature will be calculated. Technicalities are considerably simplified and
our method is general enough to include different black hole metrics.
### 2.1 Schwarzschild metric
Near the event horizon the physics is given by just the two dimensional
($t-r$) -sector of the full Schwarzschild metric [13]:
$\displaystyle
ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}=\Big{(}1-\frac{2m}{r}\Big{)}dt^{2}-\frac{dr^{2}}{1-\frac{2m}{r}}.$
(1)
It is interesting to see that this can be globally embedded in a flat $D=3$
space as,
$\displaystyle ds^{2}=(dz^{0})^{2}-(dz^{1})^{2}-(dz^{2})^{2}$ (2)
by the following relations among the flat and curved coordinates:
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{(}1+\frac{r_{H}r^{2}+r_{H}^{2}r+r_{H}^{3}}{r^{3}}\Big{)}^{1/2},$ (3)
where the surface gravity $\kappa=\frac{1}{4m}$ and the event horizon is
located at $r_{H}=2m$. The suffix “$in$” (“$out$”) refer to the inside
(outside) of the event horizon while variables without any suffix imply that
these are valid on both sides of the horizon. We shall follow these notations
throughout the paper. Now if a detector moves according to constant $r$
(Hawking detector) outside the horizon in the curved space, then the
corresponding Unruh detector moves on the constant $z^{2}$ plane and it will
follow the hyperbolic trajectory
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=16m^{2}\Big{(}1-\frac{2m}{r}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(4)
This shows that the Unruh detector is moving in the
($z^{0}_{out},z^{1}_{out}$) flat plane with a uniform acceleration
${\tilde{a}}=\frac{1}{4m}\Big{(}1-\frac{2m}{r}\Big{)}^{-1/2}$. Then, according
to Unruh [2], the accelerated detector will see a thermal spectrum in the
Minkowski vacuum with the local Hawking temperature given by,
$\displaystyle T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar}{8\pi
m}\Big{(}1-\frac{2m}{r}\Big{)}^{-1/2}.$ (5)
So we see that with the help of the reduced global embedding the local Hawking
temperature near the horizon can easily be obtained.
Now the temperature measured by any observer away from the horizon can be
obtained by using the Tolman formula [8] which ensures constancy between the
product of temperatures and corresponding Tolman factors measured at two
different points in space-time. This formula is given by [8]:
$\displaystyle\sqrt{g_{tt}}~{}T=\sqrt{g_{0_{tt}}}~{}T_{0}$ (6)
where, in this case, the quantities on the left hand side are measured near
the horizon whereas those on the right hand side are measured away from the
horizon (say at $r_{0}$). Since away from the horizon the space-time is given
by the full metric, $g_{0_{tt}}$ must correspond to the $dt^{2}$ coefficient
of the full (four dimensional) metric.
For the case of Schwarzschild metric $g_{tt}=1-2m/r$, $g_{0_{tt}}=1-2m/r_{0}$.
Now the Hawking effect is observed at infinity ($r_{0}=\infty$), where
$g_{0_{tt}}=1$. Hence, use of the Tolman formula (6) immediately yields the
Hawking temperature:
$\displaystyle T_{0}={\sqrt{g_{tt}}}~{}T=\frac{\hbar}{8\pi m}.$ (7)
Thus, use of the reduced embedding instead of the embedding of the full metric
is sufficient to get the answer.
### 2.2 Reissner-Nordstr$\ddot{\textrm{o}}$m metric
In this case, the effective metric near the event horizon is given by [13],
$\displaystyle
ds^{2}=\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}dt^{2}-\frac{dr^{2}}{1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}}.$
(8)
This metric can be globally embedded into the $D=4$ dimensional flat metric
as,
$\displaystyle ds^{2}=(dz^{0})^{2}-(dz^{1})^{2}-(dz^{2})^{2}+(dz^{3})^{2}$ (9)
where the coordinate transformations are:
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{e^{2}}{r^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{e^{2}}{r^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\frac{r^{2}(r_{+}+r_{-})+r_{+}^{2}(r+r_{+})}{r^{2}(r-r_{-})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{4r_{+}^{5}r_{-}}{r^{4}(r_{+}-r_{-})^{2}}\Big{]}^{1/2}.$ (10)
Here in this case the surface gravity $\kappa=\frac{r_{+}-r_{-}}{2r_{+}^{2}}$
and $r_{\pm}=m\pm\sqrt{m^{2}-e^{2}}$. The black hole event horizon is given by
$r_{H}=r_{+}$. Note that for $e=0$, the above transformations reduce to the
Schwarzschild case (3). The Hawking detector moving in the curved space
outside the horizon, following a constant $r$ trajectory, maps to the Unruh
detector on the constant ($z^{2},z^{3}$) surface. The trajectory of the Unruh
detector is given by
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\Big{(}\frac{r_{+}-r_{-}}{2r_{+}^{2}}\Big{)}^{-2}\Big{(}1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(11)
This, according to Unruh [2], immediately leads to the local Hawking
temperature $T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar(r_{+}-r_{-})}{4\pi
r_{+}^{2}\sqrt{1-2m/r+e^{2}/r^{2}}}$ which was also obtained from the full
global embedding [4]. Again, since in this case
$g_{0_{tt}}=1-2m/r_{0}+e^{2}/r_{0}^{2}$ which reduces to unity at
$r_{0}=\infty$ and $g_{tt}=1-2m/r+e^{2}/r^{2}$, use of Tolman formula (6)
leads to the standard Hawking temperature
$T_{0}=\sqrt{g_{tt}}~{}T=\frac{\hbar(r_{+}-r_{-})}{4\pi r_{+}^{2}}$.
### 2.3 Schwarzschild-AdS metric
Near the event horizon the relevant effective metric is [13],
$\displaystyle
ds^{2}=\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}dt^{2}-\frac{dr^{2}}{\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}},$
(12)
where $R$ is related to the cosmological constant $\Lambda=-1/R^{2}$. This
metric can be globally embedded in the flat space (9) with the following
coordinate transformations:
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{r^{2}}{R^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2m}{r}-\frac{r^{2}}{R^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\Big{(}\frac{R^{3}+Rr_{H}^{2}}{R^{2}+3r_{H}^{2}}\Big{)}^{2}\frac{r^{2}r_{H}+rr_{H}^{2}+r_{H}^{3}}{r^{3}(r^{2}+rr_{H}+r_{H}^{2}+R^{2})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{(R^{4}+10R^{2}r_{H}^{2}+9r_{H}^{4})(r^{2}+rr_{H}+r_{H}^{2})}{(r^{2}+rr_{H}+r_{H}^{2}+R^{2})(R^{2}+3r_{H}^{2})^{2}}\Big{]}^{1/2}$
(13)
where the surface gravity $\kappa=\frac{R^{2}+3r_{H}^{2}}{2r_{H}R^{2}}$ and
the event horizon $r_{H}$ is given by the root of the equation
$1-\frac{2m}{r_{H}}+\frac{r^{2}_{H}}{R^{2}}=0$. Note that in the
$R\rightarrow\infty$ limit these transformations reduce to those for the
Schwarzschild case (3). We observe that the Unruh detector on the
($z^{2},z^{3}$) surface (i.e. the Hawking detector moving outside the event
horizon on a constant $r$ surface) follows the hyperbolic trajectory:
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\Big{(}\frac{R^{2}+3r_{H}^{2}}{2r_{H}R^{2}}\Big{)}^{-2}\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}$
(14)
leading to the local Hawking temperature
$T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\Big{(}1-\frac{2m}{r}+\frac{r^{2}}{R^{2}}\Big{)}^{1/2}}$.
This result was obtained earlier [4], but with more technical complexities,
from the embedding of the full metric.
It may be pointed out that for the present case, the observer must be at a
finite distance away from the event horizon, since the space-time is
asymptotically AdS. Therefore, if the observer is far away from the horizon
($r_{0}>>r$) where $g_{0_{tt}}=1-2m/r_{0}+r_{0}^{2}/R^{2}$, then use of (6)
immediately leads to the temperature measured at $r_{0}$:
$\displaystyle
T_{0}=\frac{\hbar\kappa}{2\pi\sqrt{1-2m/r_{0}+r_{0}^{2}/R^{2}}}.$ (15)
Now, this shows that $T_{0}\rightarrow 0$ as $r_{0}\rightarrow\infty$; i.e. no
Hawking particles are present far from horizon.
## 3 Kerr-Newman metric
So far we have discussed a unified picture of Unruh and Hawking effects using
our reduced global embedding approach for spherically symmetric metrics,
reproducing standard results. However, our approach was technically simpler
since it involved the embedding of just the two dimensional near horizon
metric. Now we shall explore the real power of this new embedding.
The utility of the reduced embedding approach comes to the fore for the Kerr-
Newman black hole which is not spherically symmetric. The embedding for the
full metric, as far as we are aware, is not done in the literature.
The effective $2$-dimensional metric near the event horizon is given by [15,
16],
$\displaystyle
ds^{2}=\frac{\Delta}{r^{2}+a^{2}}dt^{2}-\frac{r^{2}+a^{2}}{\Delta}dr^{2},$
(16)
where
$\displaystyle\Delta=r^{2}-2mr+a^{2}+e^{2}=(r-r_{+})(r-r_{-});\,\,\,\
a=\frac{J}{m};$ $\displaystyle r_{\pm}=m\pm\sqrt{m^{2}-a^{2}-e^{2}}.$ (17)
The event horizon is located at $r=r_{+}$. This metric can be embedded in the
following $D=5$-dimensional flat space:
$\displaystyle
ds^{2}=\Big{(}dz^{0}\Big{)}^{2}-\Big{(}dz^{1}\Big{)}^{2}-\Big{(}dz^{2}\Big{)}^{2}+\Big{(}dz^{3}\Big{)}^{2}+\Big{(}dz^{4}\Big{)}^{2},$
(18)
where the coordinate transformations are
$\displaystyle
z^{0}_{out}=\kappa^{-1}\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}^{1/2}\textrm{sinh}(\kappa
t),\,\,\,\
z^{1}_{out}=\kappa^{-1}\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}^{1/2}\textrm{cosh}(\kappa
t),$ $\displaystyle
z^{0}_{in}=\kappa^{-1}\Big{(}\frac{2mr}{r^{2}+a^{2}}-\frac{e^{2}}{r^{2}+a^{2}}-1\Big{)}^{1/2}\textrm{cosh}(\kappa
t),\,\,\,\
z^{1}_{in}=\kappa^{-1}\Big{(}\frac{2mr}{r^{2}+a^{2}}-\frac{e^{2}}{r^{2}+a^{2}}-1\Big{)}^{1/2}\textrm{sinh}(\kappa
t),$ $\displaystyle z^{2}=\int
dr\Big{[}1+\frac{(r^{2}+a^{2})(r_{+}+r_{-})+r_{+}^{2}(r+r_{+})}{(r^{2}+a^{2})(r-r_{-})}\Big{]}^{1/2},$
$\displaystyle z^{3}=\int
dr\Big{[}\frac{4r_{+}^{5}r_{-}}{(r^{2}+a^{2})^{2}(r_{+}-r_{-})^{2}}\Big{]}^{1/2},$
$\displaystyle z^{4}=\int
dra\Big{[}\frac{r_{+}+r_{-}}{(a^{2}+r_{-}^{2})(r_{-}-r)}+\frac{4(a^{2}+r_{+}^{2})(a^{2}-r_{+}r_{-}+(r_{+}+r_{-})r)}{(r_{+}-r_{-})^{2}(a^{2}+r^{2})^{3}}$
$\displaystyle+\frac{4r_{+}r_{-}(a^{2}+2r_{+}^{2})}{(r_{+}-r_{-})^{2}(a^{2}+r^{2})^{2}}+\frac{rr_{-}-a^{2}+r_{+}(r+r_{-})}{(a^{2}+r_{-}^{2})(a^{2}+r^{2})}\Big{]}^{1/2}.$
(19)
Here the surface gravity $\kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}$. For
$e=0,a=0$, as expected, the above transformations reduce to the Schwarzschild
case (3) while only for $a=0$ these reduce to the Reissner-
Nordstr$\ddot{\textrm{o}}$m case (10).
As before, the trajectory adopted by the Unruh detector on the constant
($z^{2},z^{3},z^{4}$) surface corresponding to the Hawking detector on the
constant $r$ surface is given by the hyperbolic form,
$\displaystyle\Big{(}z^{1}_{out}\Big{)}^{2}-\Big{(}z^{0}_{out}\Big{)}^{2}=\kappa^{-2}\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}=\frac{1}{{\tilde{a}}^{2}}.$
(20)
Hence the local Hawking temperature is
$\displaystyle
T=\frac{\hbar{\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\sqrt{\Big{(}1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}\Big{)}}}.$
(21)
Finally, since $g_{tt}=1-\frac{2mr}{r^{2}+a^{2}}+\frac{e^{2}}{r^{2}+a^{2}}$
(corresponding to the near horizon reduced two dimensional metric) and
$g_{0_{tt}}=\frac{r_{0}^{2}-2mr_{0}+a^{2}+e^{2}-a^{2}{\textrm{sin}}^{2}\theta}{r_{0}^{2}+a^{2}{\textrm{cos}}^{2}\theta}$
(corresponding to the full four dimensional metric), use of the Tolman
relation (6) leads to the Hawking temperature
$\displaystyle
T_{0}=\frac{\sqrt{g_{tt}}}{\sqrt{(g_{0}{{}_{tt}})_{r_{0}\rightarrow\infty}}}~{}T=\frac{\hbar\kappa}{2\pi}=\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})},$
(22)
which is the well known result [15].
## 4 Conclusion
We provide a new approach to the study of Hawking/Unruh effects including
their unification, initiated in [3, 4, 5], popularly known as global embedding
Minkowskian space-time (GEMS). Contrary to the usual formulation [3, 4, 5, 9,
10, 11], the full embedding was avoided. Rather, we required the embedding of
just the two dimensional ($t-r$)-sector of the theory. This was a consequence
of the fact that the effective near horizon theory is basically two
dimensional. Only near horizon theory is significant since Hawking/Unruh
effects are governed solely by properties of the event horizon.
This two dimensional embedding ensued remarkable technical simplifications
whereby the treatment of more general black holes (e.g. those lacking
spherical symmetry like the Kerr-Newman) was feasible. Also, black holes in
any dimensions were automatically considered since the embedding just required
the ($t-r$)-sector.
## References
* [1] S. W. Hawking, Nature 248, 30 (1974).
S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206
(1976)].
* [2] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
* [3] S. Deser and O. Levin, Class. Quant. Grav. 14, L163 (1997) [arXiv:gr-qc/9706018].
S. Deser and O. Levin, Class. Quant. Grav. 15, L85 (1998) [arXiv:hep-
th/9806223].
* [4] S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159].
* [5] H. Narnhofer, I. Peter and W. E. Thirring, Int. J. Mod. Phys. B 10 (1996) 1507.
* [6] H. F. Goenner, “General Relativity and Gravitation, edited by A. Held (Plenum, Newyork,1980), p.441.
* [7] J. Rosen, Rev. Mod. Phys. 37 (1965) 204.
* [8] R. C. Tolman, “Relativity, Thermodynamics and Cosmology”, New York: Dover Publication, (1987), p.318.
* [9] Y. W. Kim, Y. J. Park and K. S. Soh, Phys. Rev. D 62, 104020 (2000) [arXiv:gr-qc/0001045].
S. T. Hong, Y. W. Kim and Y. J. Park, Phys. Rev. D 62, 024024 (2000)
[arXiv:gr-qc/0003097].
S. T. Hong, W. T. Kim, Y. W. Kim and Y. J. Park, Phys. Rev. D 62, 064021
(2000) [arXiv:gr-qc/0006025].
S. T. Hong, W. T. Kim, J. J. Oh and Y. J. Park, Phys. Rev. D 63, 127502 (2001)
[arXiv:hep-th/0103036].
H. Z. Chen, Y. Tian, Y. H. Gao and X. C. Song, JHEP 0410, 011 (2004)
[arXiv:gr-qc/0409107].
N. L. Santos, O. J. C. Dias and J. P. S. Lemos, Phys. Rev. D 70, 124033 (2004)
[arXiv:hep-th/0412076].
H. Z. Chen and Y. Tian, Phys. Rev. D 71, 104008 (2005).
* [10] Y. Tian, JHEP 0506, 045 (2005) [arXiv:gr-qc/0504040].
* [11] E. J. Brynjolfsson and L. Thorlacius, JHEP 0809, 066 (2008) [arXiv:0805.1876 [hep-th]]. Y. W. Kim, J. Choi and Y. J. Park, arXiv:0909.3176 [gr-qc].
* [12] S. T. Hong, Gen. Rel. Grav. 36, 1919 (2004) [arXiv:gr-qc/0310118].
* [13] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074].
* [14] S. Carlip, Phys. Rev. Lett. 82, 2828 (1999) [arXiv:hep-th/9812013].
* [15] S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74, 044017 (2006) [arXiv:hep-th/0606018].
* [16] K. Umetsu, arXiv:0907.1420 [hep-th].
|
arxiv-papers
| 2010-02-04T12:48:02 |
2024-09-04T02:49:08.260100
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rabin Banerjee, Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1002.0985"
}
|
1002.0995
|
# Granular Gases under Extreme Driving
W. Kang wfkang@gmail.com Department of Physics, University of Massachusetts,
Amherst, Massachusetts 01003 USA J. Machta machta@physics.umass.edu
Department of Physics, University of Massachusetts, Amherst, Massachusetts
01003 USA E. Ben-Naim ebn@lanl.gov Theoretical Division and Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico
87545 USA
###### Abstract
We study inelastic gases in two dimensions using event-driven molecular
dynamics simulations. Our focus is the nature of the stationary state attained
by rare injection of large amounts of energy to balance the dissipation due to
collisions. We find that under such extreme driving, with the injection rate
much smaller than the collision rate, the velocity distribution has a power-
law high energy tail. The numerically measured exponent characterizing this
tail is in excellent agreement with predictions of kinetic theory over a wide
range of system parameters. We conclude that driving by rare but powerful
energy injection leads to a well-mixed gas and constitutes an alternative
mechanism for agitating granular matter. In this distinct nonequilibrium
steady-state, energy cascades from large to small scales. Our simulations also
show that when the injection rate is comparable with the collision rate, the
velocity distribution has a stretched exponential tail.
###### pacs:
45.70.Mg, 47.70.Nd, 05.40.-a, 81.05.Rm
Granular materials are ubiquitous in nature, but nevertheless, fundamental
understanding of the properties of granular materials presents many challenges
jaeger96 ; kadanoff99 ; gennes99 ; ig ; bp . Underlying these challenges are
structural inhomogeneities, macroscopic particle size, and energy dissipation,
all of which are defining features of granular matter.
Equilibrium gases have Maxwellian velocity distributions. Due to the
irreversible nature of the dissipative collisions, granular gases are out of
equilibrium. Indeed, non-Maxwellian velocity distributions are observed in a
wide range of experiments in driven granular matter including in particular
shaken grains olafsen98 ; gzb ; losert99 ; cafiero2000 ; rm ; blair01 ;
aranson02 ; prevost02 ; wp ; fm ; tmhs ; kohlstedt05 . In such experiments,
energy is injected over a wide range of scales and the measured velocity
distribution has a stretched exponential form. To a large extent, a kinetic
theory where energy injection through the system boundary is modeled by a
thermostat successfully describes these nonequilibrium steady-states ep ;
brey03 ; ve ; krb .
Furthermore, theoretical studies suggest that the steady-state is controlled
primarily by the ratio between the energy injection rate and the collision
rate mackintosh04 ; mackintosh05 . When the injection rate is much larger than
the collision rate, the velocity distribution is Maxwellian. However, when the
injection rate is smaller than the collision rate, the velocity distribution
is non-Maxwellian, and has a stretched-exponential tail.
In this study, we focus on the limiting case where the injection rate is
vanishingly small and energy is injected at extremely large velocity scales bm
; bmm . Under such extreme driving, the injected energy cascades down from
large velocity scales to small scales and thereby counters the dissipation by
collisions. Kinetic theory shows that in the stationary state, the velocity
distribution has a power-law high energy tail. These theoretical predictions
were supplemented by Monte-Carlo simulations of the homogeneous Boltzmann
equation where spatial correlations are ignored. However, such nonequilibrium
steady-states have yet to be observed using more realistic molecular dynamics
simulations. In this work we carry out extensive molecular dynamics
simulations to investigate the behavior of inelastic hard disks under extreme
driving.
The goal of this investigation is to establish whether extreme driving is a
feasible mechanism for driving granular matter. Our main result is that under
rare but powerful injection of energy, a granular gas indeed reaches a
stationary state that is characterized by a power-law velocity distribution.
Moreover, our simulations quantitatively confirm the predictions of the
kinetic theory as the exponent characterizing the tail of the distribution is
validated over a wide range of parameters. Our results show that extreme
driving is a feasible mechanism for agitating granular matter.
Kinetic Theory. Our starting point is the observation that the purely-
collisional homogeneous Boltzmann equation supports stationary solutions bm ;
bmm ; etb ; bz . The evolution equation for the velocity distribution $f({\bf
v})$ of inelastic hard disks takes the form
$\displaystyle\frac{\partial f({\bf v})}{\partial t}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\iiint d\hat{\bf n}\,d{\bf u}_{1}\,d{\bf u}_{2}\left|({\bf
u}_{1}-{\bf u}_{2})\cdot\hat{\bf n}\right|f({\bf u}_{1})f({\bf u}_{2})$ (1)
$\displaystyle\times$ $\displaystyle[\delta({\bf v}-{\bf v}_{1})-\delta({\bf
v}-{\bf u}_{1})],$
with $\hat{\bf n}$ the impact direction. This Boltzmann equation is
supplemented by the inelastic collision rule which specifies the post-
collision velocities ${\bf v}_{1,2}$ as a linear combination of the pre-
collision velocities ${\bf u}_{1,2}$,
${\bf v}_{1,2}={\bf u}_{1,2}-\frac{1+r}{2}({\bf u}_{1,2}-{\bf
u}_{2,1})\cdot\hat{\bf n}\,\hat{\bf n}.$ (2)
In an inelastic collision, the normal component of the relative velocity
reverses sign and is scaled down by the restitution coefficient $0\leq r\leq
1$, $({\bf v}_{1}-{\bf v}_{2})\cdot\hat{\bf n}=-r({\bf u}_{1}-{\bf
u}_{2})\cdot\hat{\bf n}$. The energy loss equals $\Delta
E=-\frac{1-r^{2}}{4}|({\bf u}_{1}-{\bf u}_{2})\cdot\hat{\bf n}|^{2}$.
The collision rule (2) simplifies to a “fragmentation” rule ${\bf u}\to({\bf
w}_{1},{\bf w}_{2})$ for collisions involving one extremely energetic particle
with velocity ${\bf u}$ and a second, implicit, particle with speed much less
than $|{\bf u}|$. The post-collision velocities ${\bf w}_{1}=\frac{1+r}{2}{\bf
u}\cdot\hat{\bf n}\,\hat{\bf n}$ and ${\bf w}_{2}={\bf u}-\frac{1+r}{2}{\bf
u}\cdot\hat{\bf n}\,\hat{\bf n}$ follow by substituting ${\bf u}_{1}=0$ and
${\bf u}_{2}={\bf u}$, respectively, into (2). In the limit $|{\bf
v}|\to\infty$, the nonlinear Boltzmann equation (1) becomes linear and its
stationary form is
$\displaystyle 0\\!=\\!\iint d\hat{\bf n}\,d{\bf u}\left|{\bf u}\cdot\hat{\bf
n}\right|f({\bf u})\left[\delta({\bf v}\\!-\\!{\bf w}_{1})\\!+\\!\delta({\bf
v}\\!-\\!{\bf w}_{2})\\!-\\!\delta({\bf v}\\!-\\!{\bf u})\right].$ (3)
For arbitrary dimension and for arbitrary collision parameters, this linear
and homogeneous equation admits the power-law solution
$f(v)\sim v^{-\sigma}.$ (4)
In two-dimensions, the subject of our investigation, the exponent $\sigma$
obeys the transcendental equation bm
$\frac{1-\\!{}_{2}F_{1}\left(\frac{3-\sigma}{2},1,\frac{3}{2},1-(\frac{1-r}{2})^{2}\right)}{\left(\frac{1+r}{2}\right)^{\sigma-3}}=\frac{\Gamma(\frac{\sigma-1}{2})\Gamma(\frac{3}{2})}{\Gamma(\frac{\sigma}{2})},$
(5)
where ${}_{2}F_{1}$ is the hypergeometric function. The exponent $\sigma$
grows monotonically with the restitution coefficient. The limiting values are
$\sigma=4.14922$ for completely inelastic collisions ($r=0$) and $\sigma=5$ in
the elastic limit ($r\rightarrow 1$).
The power-law distribution (4) is a stationary solution of the linear
Boltzmann equation (3). Yet, Monte Carlo methods show that the full nonlinear
Boltzmann equation (1) does admit a stationary solution with a tail given by
(4). These numerical solutions are computed by injecting energy at a rate that
is much smaller than the collision rate. In an individual injection event a
randomly chosen particle is given a velocity much larger than the typical
velocity. Such extreme driving maintains a steady-state in which energy
injection balances energy dissipation.
The physical mechanism underlying these driven steady-states is a cascade in
which a high energy particle collides with a particle of typical energy
yielding two high energy particles, each with energies less than that of the
original high energy particle. These two high energy particles produce two
more high energy particles, again by collisions with the much more numerous
particles with typical energies, and so on.
Scaling Analysis. Interestingly, there is a family of steady-states generated
by extreme driving. If $f(v)$ is a stationary solution of (1), then
$v_{0}^{-2}f(v/v_{0})$ with the arbitrary typical velocity $v_{0}$ is also a
stationary solution because the collision rule (2) is invariant under the
scale transformation ${\bf v}\to{\bf v}/v_{0}$. The energy injection rate
$\gamma$, the velocity injection scale $V$, and the typical velocity $v_{0}$
are related by the energy balance requirement.
We relate these three quantities by a heuristic argument and first note that
the energy injection rate is simply $\gamma V^{2}$. Also, we anticipate that
the velocity distribution is truncated at the injection scale $V$. The energy
dissipation is dominated by the tail of the distribution and is controlled by
the upper cut-off $V$. We estimate the dissipation rate $\Gamma$ as follows
norm ,
$\displaystyle\Gamma\sim\rho\\!\int^{V}\\!\\!\\!v\cdot
v^{2}\frac{1}{v_{0}^{2}}f\left(\frac{v}{v_{0}}\right)\,v\,dv\sim\rho
V^{3}(V/v_{0})^{2-\sigma},$ (6)
where $\rho$ is the particle density. In the integrand, the first term $v$
accounts for the collision rate, and the second term $v^{2}$ accounts for the
energy dissipation in an inelastic collision. The integration is performed
using the velocity distribution (4), and since $\sigma<5$ the dissipation is
indeed dominated by the high velocity tail of the distribution. Balancing
energy injection with dissipation we obtain a relationship between the
injection rate $\gamma$, the injection scale $V$, and the typical velocity
$v_{0}$,
$\gamma\sim\rho\,V(V/v_{0})^{2-\sigma}.$ (7)
Since the collision rate is proportional to $\rho\,v_{0}$, the dimensionless
ratio $\psi$ of the injection rate to the collision rate scales as a power of
the velocity ratio $V/v_{0}$,
$\psi\sim(V/v_{0})^{3-\sigma}.$ (8)
Since $\sigma>3$, we expect a wide power-law range, $V\gg v_{0}$, when the
injection rate is much smaller than the collision rate, $\psi\ll 1$. There is
no lower cutoff on the injection rate $\psi$, below which power-law is not
observed; the smaller is $\psi$, the broader the power-law range. When $\psi$
is order one, the velocity distribution no longer has a power-law tail, and of
course, when $\psi\gg 1$, the velocity distribution should simply mirror the
distribution of injected velocities.
Molecular Dynamics Simulations. We used molecular dynamics alder59 to
simulate inelastic hard disks in a square box with elastic walls. In these
event driven simulations, upon impact, the velocities of the colliding
particles are updated according to the collision rule (2). Subsequent to each
collision, we identify the time and location of the next collision. The
particles undergo purely ballistic motion between two successive collisions.
We implemented the following velocity-dependent restitution coefficient
goldman98
$r(\delta_{n})=\begin{cases}1-(1-r)(\delta_{n}/v_{c})^{3/4}&\text{$\delta_{n}<v_{c}$},\\\
r&\text{$\delta_{n}\geq v_{c}$},\end{cases}$ (9)
where $\delta_{n}=(\mathbf{v_{1}}-\mathbf{v_{2}})\cdot\mathbf{\hat{n}}$ is the
normal component of the relative velocity. Here, $r$ is the nominal value of
the restitution coefficient, valid at large velocities, and $v_{c}$ is the
cutoff velocity, below which collisions become elastic. With this realistic
restitution coefficient luding96 ; bizon98 , we avoid inelastic collapse where
an infinite number of collisions can occur in a finite time my . Typically, we
set $v_{c}$ much smaller than the typical velocity $v_{0}$, but for small
restitution coefficients, we must set $v_{c}$ comparable to $v_{0}$ to avoid
inelastic collapse.
To maintain a steady-state, we periodically boost a single randomly-selected
particle to a large, random velocity. These injection events are rare and they
are governed by a Poisson process with rate $\gamma$, that is, with
probability $\gamma\,dt$ injection is implemented during the time interval
$[t,t+dt]$. The injection speed is selected from a Gaussian distribution with
zero mean and standard deviation $V$. By taking a long-time average, we
confirmed that the total energy approaches a constant, and hence, that the
system reaches a statistical steady-state where energy injection and energy
dissipation balance. Moreover, the velocity distributions were produced by
sampling particle velocities at a very large number ($10^{8}$) of equally
spaced time intervals. We stress that the velocities are sampled at time
intervals that are completely uncorrelated with either collision events or
injection events. We tested that our sampling produces robust velocity
distributions, and that the velocity distribution, representing an average
over the entire system, is truly stationary. In particular, the system does
not enter the homogeneous cooling state pkh in between the rare injection
events.
Figure 1: The velocity distribution $f(v)$ versus the velocity $v$. Molecular
dynamics simulation results (bullets) are compared with the power-law tail
predicted by kinetic theory (solid line).
We performed numerical simulations using a system of $N=10^{3}$ identical
particles with diameter $2R=1$ in a square box of size $L=400$, corresponding
to the low area fraction $\phi=N\pi\,(R/L)^{2}=4.9\times 10^{-3}$. Unless
noted otherwise, these parameters are used throughout this study. Energy was
injected at rate $\gamma=5\times 10^{-7}$ and the injection scale was $V=850$.
First, we considered weakly-inelastic particles, $r=0.9$, with the cutoff
$v_{c}=0.1$. With these parameters, the injection rate is much smaller than
the measured collision rate and their ratio is $\psi=1.5\times 10^{-5}$.
Consequently, there is substantial scale separation between the typical
velocity $v_{0}$ and the injection velocity $V$. Over this range, the steady-
state velocity distribution obtained by the molecular dynamics simulations has
a power-law high energy tail as in (4) and the exponent $\sigma=4.74$ is in
very good agreement with the kinetic theory prediction given by (5),
$\sigma=4.74104$ (see figure 1). We also confirmed that the ratio
$V/v_{0}\approx 10^{2}$ is consistent with the scaling estimate (8).
Next, we varied the restitution coefficient and repeated the simulations. Over
the entire parameter range $0.1\leq r\leq 0.9$, we find stationary velocity
distributions with a power-law tail. In general, the ratio $V/v_{0}$ is
consistent with the scaling relation (8). Moreover, the exponent $\sigma$
obtained from the molecular dynamics simulations is in excellent agreement
with the kinetic theory predictions (5) for all restitution coefficients
(figure 2). We thus arrive at our main result that at least for dilute gases,
extreme driving in the form of rare but powerful energy injection generates a
steady-state with a broad distribution of velocities. The tail of the velocity
distribution is power-law and the characteristic exponent is nonuniversal as
it depends on the restitution coefficient.
Figure 2: The exponent $\sigma$ versus the restitution coefficient $r$. The
molecular dynamics results (bullets) are compared with the kinetic theory
predictions (solid line).
The fact that kinetic theory holds shows that, to good approximation, the gas
is well-mixed. We comment that it is remarkable that extreme driving results
in a well-mixed gas. On short time scales, energy injection clearly generates
spatial correlations because a just-energized particle transfers much of its
energy to nearby particles by inelastic collisions. Yet, on larger time scales
energetic particles break coherent structures which are known to be the
consequence of inelastic collisions gz . While these two mechanisms have
opposite effects, the simulations indicate that when a long time average is
taken, the latter effect dominates. Thus, energy injected at extreme velocity
scales in a tightly localized region of space, ends up evenly distributed
throughout the system.
Snapshots of the time evolution of the system following energy injection
demonstrate how the inelastic cascade works (figure 3). In the initial stages
of the cascade, the injection affects only a small region in space and,
moreover, there are strong spatial correlations between the velocities of the
particles (figure 3 a-c). However, after many inelastic collisions, the
injected energy ends up evenly distributed throughout the system (figure 3 d).
When a long time average is taken over many energy injection events at
different locations, the system is maintained in a homogeneous, well-mixed
state. Spatial correlations induced by inelasticity and the injection
mechanism do not affect the predicted power-law velocity distributions.
(a) (b)
(c) (d)
Figure 3: The inelastic energy cascade. Shown are four time-ordered snapshots
of the gas shortly after an injection event. The top three figures show a
small window around the injection event in the early stages of the cascade:
(a) the initial energetic particle (red online) (b) two energetic particles
after one collision, (c) four energetic particles after three collisions.
Figure (d) shows the entire system, with energetic particles shown larger (red
online), in a late stage of the cascade. After many collisions, the injected
energy is evenly distributed throughout the system. The simulation parameters
are $r=0.8$, $N=1000$, $L=400$, $V=707$ and $\gamma=2\times 10^{-6}$.
Therefore, rare, powerful, and spatially localized energy injection is a
unique mechanism of agitating granular gases. This mechanism induces an
extended energy cascade which distributes the injected energy to the rest of
the system. This physical mechanism is different than energy injection by
walls gzb ; losert99 or by an effective thermostat ve or by multiplicative
driving cafiero2000 where the injected energy directly affects only a small
region of space. In this sense, the energy cascade represents a novel
mechanism for agitating granular matter.
Figure 3 also illustrates that the velocity distribution is correlated with
injection times for our system so that the predicted power-law distribution
arises only after averaging over many measurements taken at times that are
uncorrelated with the injection times. However, for very large systems driven
with a fixed but small injection rate per particle there would be many
temporally overlapping but spatially well-separated injection events. At any
instant of time in a very large system cascades at all stages of development
would be present somewhere in the system and the power law tail would be time-
independent.
We performed additional simulations to test whether the results are robust
with respect to change of parameters. In particular, we varied the area
fraction by fixing the number of particles and varying the system size. The
results shown in figure 4 are for three different area fractions:
$\phi=7.7\times 10^{-5}$, $\phi=4.9\times 10^{-3}$, and $\phi=7.9\times
10^{-2}$. The corresponding values of $\psi$ are $4.7\times 10^{-4}$,
$5.4\times 10^{-5}$, and $3.4\times 10^{-6}$, respectively. Note that in all
cases $\psi\ll 1$. We find the same power-law tail in all three cases, and the
exponent is in good quantitative agreement with the kinetic theory prediction
(figure 4). Thus, the energy cascade mechanism can be realized even at area
fractions as high as $\phi\approx 10^{-1}$ and with $\psi$ as small as order
$10^{-6}$.
Figure 4: The velocity distribution at three different densities (solid
lines). The simulation parameters are: $r=0.8$, $\gamma=2\times 10^{-6}$, and
$V=707$. Eq. (4) with $\sigma=4.57246$ is also shown as a reference (dashed
line). A best fit to a power-law yields $\sigma=4.6$, $4.6$, and $4.7$ for
$\phi=7.7\times 10^{-5}$, $4.9\times 10^{-3}$, and $7.9\times 10^{-2}$,
respectively.
We also studied the dependence on the ratio $\psi$ between the injection rate
and the collision rate by varying the injection rate $\gamma$ and the
injection velocity $V$. In accord with (8), we find that the range $[v_{0},V]$
of power-law behavior shrinks as $\psi$ increases. As long as $\psi$ is
sufficiently small, the distribution has a power-law tail (figure 5). When
this ratio becomes sufficiently large, the tail is no longer algebraic and a
sharper decay occurs. For $\psi=5.8\times 10^{-2}$, we find the stretched
exponential tail $f(v_{x})\sim\exp\big{(}-{\rm
const.}\times|v_{x}|^{\zeta}\big{)}$ with $\zeta=1.52$ (figure 6). This value
is consistent with the theoretical value $\zeta=3/2$ for inelastic gases
driven by white noise ve ; krb and the experimental value observed in
vigorously shaken beads rm . Indeed, in this intermediate injection rate
regime, the energy cascade becomes localized, and frequent, small injections
are similar to white noise driving. On the other hand, these stretched
exponential tails do not relate to those observed in bmm as the system cools
down after injection is turned-off. We find stretched exponential tails for
the range $10^{-1}\lessapprox\psi\lessapprox 1$. When the injection rate
exceeds the collision rate ($\psi\gg 1$) the entire distribution becomes
Maxwellian as the velocity distribution simply mirrors the distribution of
injected velocities.
Figure 5: The velocity distribution $f(w)$ versus the normalized velocity
variable $w=v/v_{0}$ with different ratios of injection rate to collision
rate: $\psi=1.4\times 10^{-3}$ (bullets), $2.7\times 10^{-3}$ (squares),
$1.1\times 10^{-2}$ (diamonds), $2.3\times 10^{-2}$ (up-triangles), and
$4.4\times 10^{-2}$ (down-triangles). The restitution coefficient is $r=0.8$.
Also shown is the reference theoretical curve (4) with $\sigma=4.57246$
(dashed line).
Finally, we mention that we even varied the energy injection mechanism itself.
In particular, to implement injection strictly at large energy scales, we used
an “energy loss counter” to keep track of the total energy dissipated by
collisions since the last injection. When the dissipated energy equals a fixed
large value, we inject this amount of energy into a single randomly chosen
particle bm ; bmm . Using this variant of extreme driving to maintain the
steady-state, we found similar power-law velocity distributions. From these
studies, we conclude that the parameter $\psi$ controls the velocity
distribution, and that different energy injection mechanisms lead to a power-
law distribution with the very same exponent $\sigma$, as long as $\psi\ll 1$.
Conclusions. Extensive event-driven simulations show that under extreme
driving in the form of rare but powerful energy injections, an inelastic gas
reaches a steady-state with a broad distribution of energies. Such driven
steady-states are observed for a wide range of collision parameters,
densities, and energy injection rates, as long as the injection rate is much
smaller than the collision rate. The velocity distributions have a power-law
tail and the characteristic exponent is in good agreement with the kinetic
theory predictions. When the ratio between the energy injection rate and the
collision rate becomes sufficiently large, the velocity distribution has a
much sharper stretched exponential tail.
Figure 6: The distribution $f(v_{x})$ of the horizontal component of the
velocity $v_{x}$ at a moderate $\psi=5.8\times 10^{-2}$. These simulations are
performed with $r=0.8$, $\gamma=6\times 10^{-4}$, and $V=1.41$. The solid line
is a best-fit to the stretched exponential $f(v_{x})\sim\exp\big{(}-{\rm
const.}\times|v_{x}|^{\zeta}\big{)}$ with $\zeta=1.52$.
We conclude that extreme driving where energy is injected only at very large
scales presents an alternative mechanism for agitating granular matter and
that such driving leads to a fundamentally different steady-state compared
with traditional driving where energy is injected over all scales. Realizing
this driving in experiments is a challenge because the agitation must be
applied only at very large velocities. One possible mechanism is shooting very
fast particles into the system. While such a system would involve a growing
number of particles, the injection of energetic particles should lead to
transfer of energy from large scales to small scales by a cascade of
collisions.
Figure 7: The time development, left to right, of a cascade in a denser
system ($\phi=7.9\times 10^{-2}$ and the same parameters as in Fig. 4.)
Particles with speeds greater than a fixed threshold are shown larger (red
online).
Interestingly, the power-law velocity distribution appears to hold in systems
that include dense clusters. We have observed that when the density is
increased, there is a tendency for clustering near the walls with no
measurable deviation from the predicted power law (figure 7). In this case
injection leads to explosive breakup of dense regions. It is intriguing that
the power-law tail is quite robust, and extends to situations where the
assumptions underlying the kinetic theory approach can no longer be justified.
In our simulations, inelastic collapse does not play a role because collisions
become elastic at small relative velocities. Yet, if the collisions are purely
inelastic, there should be a competition between the formation of high-density
regions by inelastic collisions and the destruction of such clusters by high
energy particles. Elucidating this competition is another possibility for
further investigation. Nonetheless, our simulations suggest that extreme
driving generates well-mixed steady-states despite the fact that this driving
is very inhomogeneous in both space and time.
Acknowledgments. We thank Narayanan Menon, Felix Werner for useful discussions
and Hong-Qiang Wang for sharing the molecular dynamics code. We acknowledge
Financial support from NSF grant DMR-0907235 (JM and WK) and DOE grant DE-
AC52-06NA25396 (EB).
## References
* (1) H. M. Jaeger, S. R. Nagel and R. P. Behringer, Rev. Mod. Phys. 68, 1259 (1996).
* (2) L. P. Kadanoff Rev. Mod. Phys. 71, 435 (1999).
* (3) P. G. de. Gennes, Rev. Mod. Phys. 71, S374 (1999).
* (4) I. Goldhirsch, Ann. Rev. Fluid. Mech. 35, 267 (2003).
* (5) Kinetic theory of granular gases, N. Brilliantov and T. Pöschel, (Oxford, Oxford, 2003).
* (6) J. S. Olafsen and J. S. Urbach, Phys. Rev. Lett. 81, 4369 (1998); Phys. Rev. E 60, R2468 (1999).
* (7) E. L. Grossman, T. Zhou, and E. Ben-Naim, Phys. Rev. E 55, 4200 (1997).
* (8) W. Losert, D. G. W. Cooper, J. Delour, A. Kudrolli and J. P. Gollub, Chaos 9, 682, 1999.
* (9) R. Cafiero, S. Luding, and H. J. Herrmann, Phys. Rev. Lett. 84, 6014 (2000).
* (10) F. Rouyer and N. Menon, Phys. Rev. Lett. 85, 3676 (2000).
* (11) D. L. Blair and A. Kudrolli, Phys. Rev. E 64, R050301 (2001).
* (12) I. S. Aranson, and J. S. Olafson, Phys. Rev. E 66, 061302 (2002).
* (13) K. Kohlstedt, A. Snezhko, M. V. Sakozhnikov, I. S. Aranson, J. S. Olafson, E. Ben-Naim, Phys. Rev. Lett. 95, 068001 (2005).
* (14) A. Prevost and et al, Phys. Rev. Lett. 89, 084301 (2002).
* (15) R. D. Wildman and D. J. Parker, Phys. Rev. Lett. 88, 064301 (2002).
* (16) K. Feitosa and N. Menon, Phys. Rev. Lett. 88, 198301 (2002).
* (17) S. Tatsumi, Y. Murayama, H. Hayakawa, M. Sanoi, J. Fluid Mech. 641, 521 (2009).
* (18) S. E. Esipov and T. Pöschel, J. Stat. Phys. 86, 1385 (1997).
* (19) J. J. Brey and M. J. Ruiz-Montero, Phys. Rev. E 67, 021307 (2003).
* (20) T. P. C. van Noije and M. H. Ernst, Gran. Matt. 1, 57 (1998).
* (21) P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010).
* (22) J. S. van Zon, and F. C. MacKintosh, Phys. Rev. Lett. 93, 038001 (2004).
* (23) J. S. van Zon, and F. C. MacKintosh, Phys. Rev. E 72, 051301 (2005).
* (24) E. Ben-Naim and J. Machta, Phys. Rev. Lett. 94, 138001 (2005).
* (25) E. Ben-Naim, B. Machta, and J. Machta, Phys. Rev. E 72, 021302 (2005).
* (26) M. H. Ernst, E. Trizac, and A. Barrat, Europhys. Lett. 76, 56 (2006).
* (27) E. Ben-Naim and A. Zippelius, J. Stat. Phys. 129 677 (2007).
* (28) Throughout this paper we use the normalization $2\pi\int_{0}^{\infty}f(v)\,v\,dv=1$ where $v\equiv|{\bf v}|\equiv\sqrt{v_{x}^{2}+v_{y}^{2}}$.
* (29) B. J. Alder and T. E. Wainwright, J. Chem. Phys. 31, 459 (1959).
* (30) D. Goldman, M. D. Shattuck, C. Bizon, W. D. McCormick, J. B. Swift, and H. L. Swinney, Phys. Rev. E 57, 4831 (1998).
* (31) C. Bizon, M. D. Shattuck, J. B. Swift, W. D. McCormick, and H. L. Swinney, Phys. Rev. Lett. 80, 57 (1998).
* (32) S. Luding, E. Clement, J. Rajchenbach, and J. Duran, Europhys. Lett. 34, 247 (1996).
* (33) S. McNamara and W. R. Young, Phys Fluids A 4, 496 (1992).
* (34) P. K. Haff, J. Fluid. Mech. 134, 401 (1983).
* (35) I. Goldhirsch, and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993).
|
arxiv-papers
| 2010-02-04T13:26:30 |
2024-09-04T02:49:08.264398
|
{
"license": "Public Domain",
"authors": "W. Kang, J. Machta and E. Ben-Naim",
"submitter": "Jon Machta",
"url": "https://arxiv.org/abs/1002.0995"
}
|
1002.1124
|
# Elastic properties and electronic structures of antiperovskite-type InNCo3
and InNNi3
Z. F. Hou Department of Physics, Fudan University, Shanghai, 200433, P. R.
China
###### Abstract
We have performed the first-principles calculations to study the elasticity,
electronic structure, and magnetism of InNCo3 and InNNi3. The independent
elastic constants are derived from the second derivative of total energy as a
function of strain, and the elastic modulus are predicted according to the
Voigt-Reuss-Hill approximation. Our calculations show that the bulk modulus of
InNCo3 is slightly larger than that of InNNi3 due to a smaller lattice
constant for InNCo3. For InNCo3 the ferromagnetic state is energetically
preferable to the paramagnetic state, while the ground state of InNNi3 is a
stable paramagnetic (non-magnetic) state. This is due to the different
strength of 2p-3d hybridization for the N-Co and N-Ni atoms in InNCo3 and
InNNi3.
## I Introduction
The ternary nitrides or carbides with the general formula AXM3 ( A: divalent
or trivalent element; X: carbon or nitrogen; and M: transition metal) are
already known for several decades Goodenough70 ; Chern92 ; Jager93 . These
compounds crystalize in a cubic anti-perovskite structure (A: cube-corner
position; X: body-center position; M: face-center position) and exhibit a wide
range of interesting physical properties Goodenough70 , such as giant magneto-
resistance Kim2001 and nearly zero temperature coefficient resistivity
Chi2001 . They have renewedly attracted considerable attention due to the
discovery of superconductivity at $\sim$8 K in intermetallic compound MgCNi3
He01 .
Considering the Ni-rich composition, it is expected that the ferromagnetism
could exist in MgCNi3. However, the absence of ferromagnetism was observed in
experiment He01 for MgCNi3. From the electronic structures obtained by the
first-principles calculations Singh01 ; Shim01 ; Rosner02 ; Wu2009251 , the
non-ferromagnetic ground state of MgCNi3 is ascribed to a reduced Stoner
factor that results from a strong hybridization between the Ni-3$d$ and C-2$p$
electrons. For other Ni-based ternary carbides ACNi3 (e.g., A = Al, Ca, In,
Zn, and Cd), the first-principles calculations Wu2009251 ; Okoye05 ;
Sieberer07 ; Zhong07 ; Wu20084232 also show that the ground states of these
compounds are non-magnetic and the C-Ni bonding exhibits nearly same
characteristics as the one in MgCNi3. Therefore, this indicates that the
change of composition A could not induce the ferromagnetism in ACNi3. On the
other side, it raises a question whether the change of composition X or M can
lead to the appearance of ferromagnetism in AXM3 or not.
Very recently, the antiperovskite-type compounds InNyCo3 and InNyNi3 ($y\sim$
1.0 and 0.8, respectively) have been synthesized by solid-gas reactions of
metal powders with NH3 and they have been reported to have spin-glass-like
properties based on the measurements of temperature dependence magnetization.
Cao093353 The recent first-principles calculations Sieberer07 showed that
the non-stoichiometry could affect the magnetic properties of ACNi3 (e.g.,
AlCNi3 and GaCNi3) and suggested that the tendencies toward magnetism found in
experiments Dong05 ; Tong06Al ; Tong06Ga ; Tong07 for these compounds should
be explained by the deviation of the Ni/C atomic ratio from the ideal
stoichiometry. To shed more light on the understanding on the magnetic
properties of InNCo3 and InNNi3 reported in experiment Cao093353 , it is of
great importance to theoretically study the electronic structures of these two
compounds as well as the nature of the N-Co and N-Ni bondings.
In order to completely understand the electronic structures and magnetic
ground states of InNCo3 and InNNi3 with cubic anti-perovskite structure, we
carried out the first-principles calculations on these compounds using the
pseudopotentials method with plane-wave basis set within the local density
approximation and the generalized gradient approximation. Since the elastic
properties of a solid are highly associated with various fundamental solid-
state properties such as phonon spectra, specific heat, Debye temperature, and
so on, we have calculated the independent elastic constant and the elastic
moduli of InNCo3 and InNNi3.
## II Computational details
All calculations on antiperovskite-type InNCo3 and InNNi3 were performed using
the Quantum ESPRESSO code Baroni , which is based on the density functional
theory (DFT) Kohn65 . The electronic exchange-correlation potential was
calculated within the local density approximation (LDA) Ceperley80 ; Perdew81
and the generalized gradient approximation using the scheme of Perdew-Burke-
Ernzerhof (PBE) Perdew96 . The spin polarization was also considered in the
calculation in order to assess the magnetic properties of these compounds.
Electron-ion interaction was represented by the norm-conserving optimized
Rappe90 designed nonlocal pseudopotentials. The 4 _d_ electrons are
explicitly included in the valence of In. The electronic wavefunctions were
expanded by the plane waves up to a kinetic energy cutoff of 55 Ry. The
k-point sampling in Brillouin zone (BZ) of simple cubic lattice was treated
with the Monkhorst-Pack scheme Monkhorst76 and a 20$\times$20$\times$20
k-point mesh (i.e., 286 irreducible points in the first BZ) was used. The
chosen plane-wave cutoff and number of k points were carefully checked to
ensure that the total energy was converged to be better than 1 mRy/cell. The
total energies are obtained as a function of volume and they are fitted with
the Birch-Murnaghan 3rd-order equation of states (EoS) Birch47 to give the
equilibrium lattice constant and other ground state properties. During the
calculation of density of states (DOS), a dense k-point mesh of
30$\times$30$\times$30 is used, the total DOS is computed by the tetrahedron
method Lehmann77 , and the atomic-projected DOS is calculated by the Löwdin
populations Portal95 .
For a cubic crystal, its independent elastic constants are $c_{11}$, $c_{12}$,
and $c_{44}$. To determine the elastic constants of InNCo3 and InNNi3 by means
of the curvature of the internal energy versus the strain curves Mehl93 ;
Wu07425 , three strain modes Hou081651 are adopted and their nonzero strains
are as follows: (1) $\epsilon_{11}=\epsilon_{22}=\delta$,
$\epsilon_{33}=(1+\delta)^{-2}-1$; (2)
$\epsilon_{11}=\epsilon_{22}=\epsilon_{33}=\delta$; and (3)
$\epsilon_{12}=\epsilon_{21}=\delta/2$,
$\epsilon_{33}=\delta^{2}/(4-\delta^{2})$. The deformation magnitudes $\delta$
from -0.012 to 0.012 in the step of 0.03 are applied in the first and second
strain modes, and $\delta$ from -0.04 to 0.04 in the step 0.01 are adopted in
the third strain mode. Once the independent elastic constants for single
crystal properties are obtained through the above procedure, the elastic
moduli (e.g., the shear modulus and the bulk modulus) of polycrystalline
aggregates can be estimated according to the Voigt-Reuss-Hill approximation
Voigt28 ; Reuss29 ; Hill52 . In the Voigt average Voigt28 , the shear modulus
and the bulk modulus of cubic lattice are given by
$G_{V}=\frac{1}{5}\left[(c_{11}-c_{12})+3c_{44}\right]$ (1)
and
$B_{V}=\frac{1}{3}(c_{11}+2c_{12}),$ (2)
while in the Reuss average Reuss29 they are given by
$G_{R}=\frac{5}{4(s_{11}-s_{12})+3s_{44}}$ (3)
and
$B_{R}=\frac{1}{3s_{11}+6s_{12}}$ (4)
with the relations $c_{44}=s_{44}^{-1}$, $c_{11}-c_{12}=(s_{11}-s_{12})^{-1}$,
and $c_{11}+2c_{12}=(s_{11}+2s_{12})^{-1}$ in the cubic lattice, where
$s_{ij}$ are the elastic compliance constants. Therefore, $G_{R}$ and $B_{R}$
in the cubic lattice can be rewritten as
$G_{R}=\left[\frac{4}{5}(c_{11}-c_{12})^{-1}+\frac{3}{5}c_{44}^{-1}\right]^{-1},$
(5)
and
$\displaystyle B_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left[(c_{11}+2c_{12})\right]$ (6) $\displaystyle=$
$\displaystyle B_{V}.$
In the Hill empirical average Hill52 , the shear modulus and the bulk modulus
are taken as $G=\frac{1}{2}(G_{V}+G_{R})$ and $B=\frac{1}{2}(B_{V}+B_{R})$,
respectively. Knowing $G$ and $B$, the Young’s modulus $E$ and Poisson’s ratio
$\nu$, which are frequently measured for polycrystalline materials when
investigating their hardness, can be calculated from the isotropic relations:
$E=\frac{9BG}{3B+G}$ (7)
and
$\nu=\frac{3B-2G}{2(3B+G)}.$ (8)
## III Results and Discussions
### III.1 Structural properties
In experiment with the powder X-ray diffraction patterns, Cao et al Cao093353
have reported that InNyCo3 and InNyNi3 ($y\sim$ 1.0 and 0.8, respectively)
have the cubic anti-perovskite structure with the space group
221($Pm\bar{3}m$) and the corresponding lattice parameters were 3.854 Å and
3.844 Å, respectively. Starting from the experimental data, we have calculated
the total energies of unit cell at a series of volumes for each compound in
the paramagnetic (PM) and ferromagnetic (FM) states. The results are presented
in Fig. 1. It is found that the energy difference between the FM and PM states
is -0.0397 eV (-0.226 eV) in the LDA (GGA) calculations for InNCo3 and 0.0 eV
for InNNi3. The total magnetic moment of InNCo3 is about 2.14 $\mu_{B}$ (2.91
$\mu_{B}$) and the local magnetic moment of each In ion is about 0.69
$\mu_{B}$ (0.94 $\mu_{B}$) in LDA (GGA) calculations. The total magnetic
moment of InNNi3 and the local magnetic moment of each Ni atom are zero. These
indicate that the ferromagnetic state is energetically favorable to InNCo3 and
the ground state of InNNi3 is paramagnetic state (non-magnetic). The obtained
equilibrium lattice constant ($a_{0}$), bulk modulus ($B$), and first pressure
derivative of bulk modulus ($B^{\prime}$) of InNCo3 and InNNi3 are listed in
Table 1. In our calculations, the predicted lattice constant of InNCo3 is
slightly larger than that of InNNi3, which is oppsite to the trend reported in
experiment Cao093353 . This may be due to the deviation of the Ni/N atomic
ratio from the ideal stoichiometry in experiment for InNyNi3. In addition, it
can be seen that the deviations of the LDA (GGA) lattice constants of both the
InNCo3 and InNNi3 with respect to the experimental values are less than 2.6%
(1.0%). That is to say, the calculated equilibrium lattice constants of InNCo3
and InNNi3 are in excellent agreement with the experimental data Cao093353 .
### III.2 Elastic properties
The calculated independent elastic constants for single crystal of InNCo3 and
InNNi3 are listed in Table 2. Based on the Voigt-Reuss-Hill approximation
Voigt28 ; Reuss29 ; Hill52 , the elastic moduli of InNCo3 and InNNi3 are
estimated and the results are listed in Table 2. For the bulk moduli of InNCo3
and InNNi3, the estimations based on the independent elastic constants agree
well those obtained by the fit of the Birch-Murnaghan 3rd-order EoS. To the
best of our knowledge, no experimental data or theoretical results for the
elasticity of InNCo3 and InNNi3 compounds have been reported up to now.
Considering that the elastic properties of ZnNNi3, InNSc3, and InCNi3 with
cubic anti-perovskite structure have been studied recently and the theoretical
results are available in literature Maurizio09 ; Wu20084232 ; lichong09 ;
Shein10 , it will be meaningful to compare them with those of InNCo3 and
InNNi3 compounds. The order of bulk moduli of these five compounds from low to
high is: B(InNSc3) $<$ B(InNNi3) $<$ B(InCNi3) $<$ B(InNCo3) $<$ B(ZnNNi3).
This could be understood from the trend in lattice constants ($a$) of these
compounds (i.e., $a$(InNSc3) $>$ a(InNNi3) $>$ a(InCNi3) $>$ a(InNCo3) $>$
a(ZnNNi3)) as well as the relationship between bulk modulus and equilibrium
volume (i.e., $B\sim V^{-1}$) Cohen85 . The GGA lattice constants of InNSc3,
ZnNNi3, and InCNi3 are 4.411 Å Maurizio09 , 3.77 Å lichong09 , and 3.880 Å
Wu20084232 , respectively. The order of shear modulus from low to high is:
G(ZnNNi3 and InNNi3) $<$ G(InCNi3) $<$ G(InNCo3). Pugh Pugh54 has proposed
that a high $B/G$ ratio may be associated with better ductility whereas a low
value would correspond to a more brittleness, and the critical value
separating ductile and brittle materials is around 1.75. From the results of
$B/G$ ratio for InNCo3, InNNi3, InNSc3, ZnNNi3, and InCNi3, it is found that
only InNSc3 can be classified as brittle materials and others may be ductile
materials. Furthermore, InNNi3 seems to be more ductile than InNCo3.
For a cubic crystal, its mechanical stability requires that its three
independent elastic constants should satisfy the following relations Wallace72
:
$(c_{11}-c_{12})>0,c_{11}>0,c_{44}>0,(c_{11}+2c_{12})>0.$ (9)
These conditions also lead to a restriction on the magnitude of $B$:
$c_{12}<B<c_{11}.$ (10)
The predicted $c_{ij}$ values (see Table 2) for InNCo3 and InNNi3 satisfy
these conditions, indicating that cubic antiperovskite-type compounds InNCo3
and InNNi3 are mechanically stable.
### III.3 Electronic structures
In order to understand the different magnetic ground states for InNCo3 and
InNNi3, we examined the electronic structures of these two compounds. For the
simplicity in discussion, only the GGA results are presented below. The
calculated electronic band structures along the high symmetry directions in
the Brillouin zone are shown in Fig. 2. As discussed above, for InNCo3 the
ferromagnetic state is energetically preferable to the paramagnetic state, and
hence it is clearly seen that the spin-splitting occurs in the bands around
the $E_{F}$. For InNCo3 the profile of majority spin bands looks roughly
similar to the one of minority spin bands, however only one band in the
majority spin bands crosses the Fermi level and the corresponding band with
same dispersion in the minority spin bands is unoccupied. In addition, two
bands in the minority spin bands across the Fermi level of InNCo3 (see Fig.
2(a)). For InNNi3 the ferromagnetic state is not energetically preferable to
the paramagnetic state, consequently, the majority and minority spin bands are
degenerated, that is to say, no spin-splitting occurs in the band structure
(see Fig. 2(b)). It is interesting to note that the whole feature of majority
spin bands of InNNi3 is very similar to the one of InNCo3. In order to reveal
the detailed character of band structure, the total density of states (DOS)
and the angular-momentum-projected DOS of each atom in InNCo3 and InNNi3 are
presented in Fig. 3. For both InNCo3 and InNNi3, the four bands from -9 eV to
-5 eV come mainly from the N 2 _p_ states and In 5 _s_ states, and the five
bands roughly from -5 eV to -2.5 eV have significant contribution from 3 _d_
-$t_{2g}$ states of transition metal atoms (Co/Ni) and the 5 _p_ state of In
atom. For the bands from -2.5 eV to 0 eV (i.e., the Fermi level), they are
dominated by the 3 _d_ states of transition metal atoms (Co/Ni) and have small
contribution from the 2 _p_ states of N atom. Because the number of 3 _d_
electrons in Co is one less than that of Ni, two minority spin bands composed
of the 3 _d_ -$t_{2g}$ states around the Fermi level are unoccupied in InNCo3,
while the counterpart in the InNNi3 are occupied. This also results in
different behavior of spin-splitting in InNCo3 and InNNi3. Due to the
significant spin-plitting around the Fermi level in InNCo3, the hybridization
between the Co 3d and N 2p states in InNCo3 are slightly weaker than the one
between Ni 3d and N 2p states in InNNi3(see Fig. 3).
The contributions of each kind of atoms to the DOS at the $E_{F}$ of InNCo3
and InNNi3 are listed in Table 3. The total DOS at the $E_{F}$ of InNCo3 is
about 2.761 states/eV per formula unit (f.u.) in the GGA calculations, which
is larger than that of InNNi3, and its main contribution comes from Co 3$d$
states which accounts for 87%. For InNNi3 in the GGA calculations, the
contribution of Ni 3d states to the the total DOS at the $E_{F}$ (i.e., 1.803
states/eV.f.u.) accounts for 72%. These indicate the 3d states of transition
metal atoms in InNCo3 and InNNi3 play dominant roles in the total dnesity of
states of these compounds.
In order to understand the bonding nature among the ions in InNCo3 and InNNi3,
we analyzed the charge density contours of InNCo3 and InNNi3 in the(110)
plane, as shown in Fig. 4. From Fig. 4, it is found that a certain amount of
charges are accumulated in the intermediate region between N and Co atoms in
InNCo3, and slightly more charges are accumulated intermediate region between
N and Ni atoms in InNNi3. This gives an evidence for the strong hybridization
between N and transition metal (Co/Ni) atoms, indicating that the N-Co and
N-Ni bondings exhibit strong covalent characteristics and the latter is
slightly stronger than the former. The similar bonding characteristics for
Ni-N atoms or Ni-C atoms were also found in other Ni-based ternary nitrides or
carbides AXNi3 Wu2009251 ; Wu20084232 ; lichong09 ; Shein10 . Therefore, our
results suggest that the magnetic properties of InNNi3 reported in experiment
Cao093353 are very likely due to the non-stoichiometry effect, which was also
found in the cases of AlCNi3 and GaCNi3 Dong05 ; Tong06Al ; Tong06Ga ; Tong07
; Sieberer07 .
## IV Conclusions
In summary, we performed the first-principles calculations to study the
elastic and electronic properties of cubic antiperovskites InNCo3 and InNNi3.
Based on the Voigt, Reuss and Hill bounds, the shear, Young’s moduli and
Poisson’s ratio have also been estimated for the InNCo3 and InNNi3
polycrystals. The theoretically predicted equilibrium lattice parameters are
in good agreement with the available experimental data. Our calculations show
that the 3d states of transition metal atoms in InNCo3 and InNNi3 play
dominant roles near the Fermi levels. InNCo3 energetically prefers to the
ferromagnetic state. The magnetic ground state of InNNi3, which is same to
other Ni-based ternary nitrides or carbides with a cubic anti-perovskite
structure, is a stable paramagnetic (non-magnetic) state. This could be
understood from that the hybridization between Ni-3$d$ and N-2$p$ states in
InNNi3 is slightly stronger than the one between Co-3$d$ and N-2$p$ states in
InNCo3 because of the more 3d electrons in Ni.
## Acknowledgments
The author acknowledges support from National Natural Science Foundation of
China under Grant No. 10674028.
## References
* (1) J. B. Goodenough, J. M. Longo, Magnetic and other properties of oxides and related compounds, in: K.-H. Hellwege, O. Madelung (Eds.), Landolt-Bornstein, New Series, Group III, Vol. 4a, Springer-Verlag, Berlin, 1970.
* (2) M. Y. Chern, D. A. Vennos, F. J. DiSalvo, J. Solid State Chem. 96 (1992) 415.
* (3) J. Jäger, D. Stahl, P. C. Schmidt, R. Kniep, Angew. Chem. 105 (1993) 738.
* (4) W. S. Kim, E. O. Chi, J. C. Kim, H. S. Choi, N. H. Hur, Solid State Commun. 119 (2001) 507.
* (5) E. O. Chi, W. S. Kim, N. H. Hur, Solid State Commun. 120 (2001) 307.
* (6) T. He, Q. Huang, A. P. Ramirez, Y. Wang, K. A. Regan, N. Rogado, M. A. Hayward, M. K. Haas, J. S. Slusky, K. Inumara, H. W. Zandbergen, N. P. Ong, R. J. Cava, Nature 411 (2001) 54.
* (7) D. J. Singh, I. I. Mazin, Phys. Rev. B 64 (2001) 140507.
* (8) J. H. Shim, S. K. Keson, B. I. Min, Phys. Rev. B 64 (2001) 180510.
* (9) H. Rosner, R. Weht, M. D. Johannes, W. E. Pickett, E. Tosatti, Phys. Rev. Lett. 88 (2002) 027001.
* (10) S. Q. Wu, Z. F. Hou, Z. Z. Zhu, Solid State Sci. 11 (2009) 251.
* (11) C. M. I. Okoye, Solid State Commun. 136 (2005) 605.
* (12) M. Sieberer, P. Mohn, J. Redinger, Phys. Rev. Lett. 75 (2007) 024431.
* (13) G. H. Zhong, J. L. Wang, Z. Zeng, X. H. Zheng, H. Q. Lin, J. Appl. Phys. 101 (2007) 09G520.
* (14) S. Q. Wu, Z. F. Hou, Z. Z. Zhu, Physica B 403 (2008) 4232.
* (15) W. H. Cao, B. He, C. Z. Liao, L. H. Yang, L. M. Zeng, C. Dong, J. Solid State Chem. 182 (2009) 3353.
* (16) A. F. Dong, G. C. Che, W. W. Huang, S. L. Jia, H. Chen, Z. X. Zhao, Physica C 422 (2005) 65.
* (17) P. Tong, Y. P. Sun, X. B. Zhu, W. H. Song, Phys. Rev. B 74 (2006) 22441.
* (18) P. Tong, Y. P. Sun, X. B. Zhu, W. H. Song, Phys. Rev. B 73 (2006) 245106.
* (19) P. Tong, Y. Sun, X. Zhu, W. Song, Solid State Commun. 141 (2007) 336.
* (20) S. Baroni, A. D. Corso, S. de Gironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pasquarello, K. Laasonen, A. Trave, R. Car, N. Marzari, A. Kokalj, http://www.pwscf.org/.
* (21) W. Kohn, L. J. Sham, Phys. Rev. 140 (1965) A1133.
* (22) D. M. Ceperley, B. J. Alder, Phys. Rev. Lett. 45 (1980) 566.
* (23) J. P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048.
* (24) J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.
* (25) A. M. Rappe, K. M. Rabe, E. Kaxiras, J. D. Joannopoulos, Phys. Rev. B 41 (1990) 1227\.
* (26) H. J. Monkhorst, J. D. Pack, Phys. Rev. B 13 (1976) 5188.
* (27) F. Birch, Phys. Rev. 71 (1947) 809.
* (28) G. Lehmann, M. Taut, Phys. Stat. Sol. (b) 54 (1972) 469.
* (29) D. Sanchez-Portal, E. Artacho, J. M. Soler, Solid State Commun. 95 (1995) 685.
* (30) M. J. Mehl, Phys. Rev. B 47 (1993) 2493.
* (31) S. Q. Wu, Z. F. Hou, Z. Zhu, Solid State Commun. 143 (2007) 425.
* (32) Z. F. Hou, Solid State Sci. 10 (2008) 1651.
* (33) D. W. Voigt, Lehrbuch der Kristallphysik, Taubner, Leipzig, 1928.
* (34) A. Reuss, Z. Angew, Math. Mech 9 (1929) 55.
* (35) R. Hill, Proc. Phys. Soc. London A 65 (1952) 349.
* (36) M. Mattesini, M. Magnuson, F. Tasnádi, C. Höglund, I. A. Abrikosov, L. Hultman, Phys. Rev. B 79 (2009) 125122.
* (37) C. Li, W. G. Chen, F. Wang, S. F. Li, Q. Sun, S. Wang, Y. Jia, J. Appl. Phys. 105 (2009) 123921.
* (38) I. R. Shein, V. V. Bannikov, A. L. Ivanovskii, Phys. Stat. Sol. (b) 247 (2010) 72\.
* (39) M. L. Cohen, Phys. Rev. B 32 (1985) 7988.
* (40) S. F. Pugh, Philos. Mag. 45 (1954) 823.
* (41) D. C. Wallace, Thermodynamics of Crystals, Wiley, New York, 1972.
Table 1: Calculated lattice constants ($a$, in Å), bulk modulus ($B$, in GPa), and the first pressure derivative $B^{\prime}$ of bulk modulus for InNCo3 and InNNi3. $\Delta E_{\mathrm{tot}}^{\mathrm{FM-PM}}$ (in eV per formula unit)is the difference between the total energies of ferromagnetic (FM) and paramagnetic (PM) states for InNCo3 and InNNi3. The available experimental values are also listed. | InNCo3 | InNNi3
---|---|---
| LDA | GGA | Expt. Cao093353 | LDA | GGA | Expt. Cao093353
| PM | FM | PM | FM | | PM/FM | PM/FM |
$a$ | 3.744 | 3.753 | 3.835 | 3.855 | 3.8541 | 3.784 | 3.882 | 3.8445
$B$ | 255.78 | 243.06 | 211.29 | 194.17 | | 226.91 | 179.93 |
$B^{\prime}$ | 4.497 | 4.483 | 5.570 | 5.568 | | 4.761 | 4.281 |
$\Delta E_{\mathrm{tot}}^{\mathrm{FM-PM}}$ | - | -0.0397 | - | -0.226 | | 0.0 | 0.0 |
Table 2: Calculated elastic constants ($c_{11}$, $c_{12}$, and $c_{44}$, in GPa), shear modulus ($G$, in GPa), Young’s modulus ($E$, in GPa), and Poisson’s ratio ($\nu$) of InNCo3 and InNNi3. The Voigt shear modulus ($G_{V}$, in GPa) and the Reuss shear modulus ($G_{R}$, in GPa) are also presented. For comparison, the elastic properties of InNSc3, ZnNNi3, and InCNi3 compounds with a cubic anti-perovskite structure are listed. Compound | Method | $c_{11}$ | $c_{12}$ | $c_{44}$ | $G_{V}$ | $G_{R}$ | $B$ | $G$ | $E$ | $\nu$ | $B/G$
---|---|---|---|---|---|---|---|---|---|---|---
InNCo3 | LDA | 389.11 | 171.12 | 102.55 | 105.13 | 105.03 | 243.78 | 105.08 | 275.63 | 0.311 | 2.320
| GGA | 317.54 | 126.76 | 94.98 | 95.14 | 95.14 | 190.35 | 95.14 | 224.67 | 0.286 | 2.001
InNNi3 | LDA | 356.77 | 164.23 | 69.06 | 80.35 | 78.12 | 228.41 | 79.24 | 212.94 | 0.344 | 2.895
| GGA | 274.08 | 131.20 | 60.01 | 64.58 | 64.11 | 178.83 | 64.35 | 172.37 | 0.339 | 2.779
InNSc3 | GGA Maurizio09 | 238.57 | 54.28 | 90.76 | 91.31 | 91.31 | 115.71 | 91.31 | 216.88 | 0.188 | 1.267
ZnNNi3 | GGA lichong09 | 354.28 | 134.01 | 48.06 | 72.89 | 62.05 | 207.43 | 67.47 | 182.61 | 0.353 | 3.074
| GGA Shein10 | 364.20 | 124.90 | 32.69 | 67.47 | 46.09 | 204.67 | 56.78 | 155.92 | 0.373 | 3.604
InCNi3 | LDA Wu20084232 | 414.67 | 135.72 | 68.02 | 96.60 | 85.56 | 228.71 | 91.08 | 241.22 | 0.324 | 2.511
| GGA Wu20084232 | 344.20 | 106.30 | 62.68 | 85.19 | 77.31 | 185.60 | 81.25 | 212.70 | 0.309 | 2.284
Table 3: Total and partial density of states at the Fermi level ($N$($E_{F}$), in states/eV.f.u.) for InNCo3 and InNNi3. | | Total density | Partial density of states
---|---|---|---
| | of states | In | N | Co3/Ni3
InNCo3 | LDA | 3.811 | 0.073 | 0.337 | 3.401
| GGA | 2.761 | 0.144 | 0.169 | 2.448
InNNi3 | LDA | 1.580 | 0.160 | 0.212 | 1.208
| GGA | 1.803 | 0.180 | 0.256 | 1.367
## Figure Captions
* •
FIG. 1: Total energy (in eV per formula unit) versus the atomic volume (in Å3)
for InNCo3 (upper panel) InNNi3 (lower panel).
* •
FIG. 2: (Color online) Electronic band structures obtained with GGA for the
majority (red solid line) and minority (blue dashed line) spins of (a) InNCo3
and (b) InNNi3.
* •
FIG. 3: Total and partial density of states (DOS) of InNCo3 (left panel) and
InNNi3 (right panel) obtained with the spin-polarized GGA calculations.
* •
FIG. 4: Charge density contours of the (110) plane for (a) InNCo3 and (b)
InNNi3 obtained with the spin-polarized GGA calculations.
Figure 1:
Figure 2:
|
---|---
Figure 3:
---
Figure 4:
|
arxiv-papers
| 2010-02-05T05:33:12 |
2024-09-04T02:49:08.271748
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z.F.Hou",
"submitter": "Z.F. Hou",
"url": "https://arxiv.org/abs/1002.1124"
}
|
1002.1334
|
# Anisotropic phase diagram of the frustrated spin dimer compound Ba3Mn2O8
E. C. Samulon1, K. A. Al-Hassanieh2, Y.-J. Jo3, M. C. Shapiro1, L. Balicas3,
C. D. Batista2, I. R. Fisher1 1Geballe Laboratory for Advanced Materials and
Department of Applied Physics, Stanford University, Stanford, California
94305, USA 2Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA 3National High Magnetic Field Laboratory, Florida State
University, Tallahassee, Florida 32306, USA
###### Abstract
Heat capacity and magnetic torque measurements are used to probe the
anisotropic temperature-field phase diagram of the frustrated spin dimer
compound Ba3Mn2O8 in the field range from 0T to 18T. For fields oriented along
the $c$ axis a single magnetically ordered phase is found in this field range,
whereas for fields oriented along the $a$ axis two distinct phases are
observed. The present measurements reveal a surprising non-monotonic evolution
of the phase diagram as the magnetic field is rotated in the [001]-[100]
plane. The angle dependence of the critical field ($H_{c1}$) that marks the
closing of the spin gap can be quantitatively accounted for using a minimal
spin Hamiltonian comprising superexchange between nearest and next nearest Mn
ions, the Zeeman energy and single ion anisotropy. This Hamiltonian also
predicts a non-monotonic evolution of the transition between the two ordered
states as the field is rotated in the $a$-$c$ plane. However, the observed
effect is found to be significantly larger in magnitude, implying that either
this minimal spin Hamiltonian is incomplete or that the magnetically ordered
states have a slightly different structure than previously proposed.
###### pacs:
75.30.Kz, 75.40.-s, 75.30.-m, 75.45.+j
††preprint: APS/123-QED
## I Introduction
Ba3Mn2O8 is a novel layered spin-dimer compound, comprising magnetic dimers of
Mn5+ ions arranged on triangular planesWeller_1999 . The Mn ions occupy
equivalent sites with a distorted tetrahedral coordination, resulting in a
quenched orbital angular momentum and a total spin $\mathbf{S}=1$ Uchida_2002
. Antiferromagnetic intradimer exchange leads to a singlet ground state with
excited triplet and quintuplet states. Weaker interdimer exchange leads to
dispersion of these excitations, as previously revealed by inelastic neutron
scattering Stone_2008 . The rhombohedral R$\bar{3}$m crystal structure of
Ba3Mn2O8 comprises staggered hexagonal planes (Fig. 1(a)), leading to
geometric frustration both within individual planes, and also between adjacent
planes. Magnetic fields can be used to close the spin gap, and the competition
between interdimer interaction on this highly frustrated lattice, and the
uniaxial single ion anisotropy associated with the $\mathbf{S}=1$ ions, leads
to a very complex phase diagram with at least three distinct ordered states
Samulon_2008 ; Samulon_2009 .
Despite the complexity of the phase diagram of Ba3Mn2O8, we have previously
shown that for fields oriented along the principal crystalline axes the high-
field behavior of this material can be described by a remarkably simple spin
Hamiltonian, including terms representing superexchange between nearest and
next-nearest Mn ions within and between planes, the Zeeman energy and uniaxial
single ion anisotropy Samulon_2008 . These terms have been determined through
a combination of inelastic neutron scattering (INS) and electron paramagnetic
resonance (EPR). INS studies revealed the dominant exchange within a dimer as
$J_{0}$ = 1.642(3) meV; the nearest and next nearest out-of-plane exchanges,
$J_{1}$ = 0.118(2) meV and $J_{4}$ = 0.037(2) meV; and the dominant in-plane
exchanges $J_{2}-J_{3}$ = 0.1136(7) meV (Fig. 1) Stone_2008 . EPR measurements
in the diluted compound Ba3(V1-xMnx)2O8 revealed a nearly isotropic
$g$-tensor, with $g_{aa}$ = 1.96 and $g_{cc}$ = 1.97, and an easy axis single
ion anisotropy $D$ = -0.024 meV Whitmore_1993 . Similar measurements in the
pure Ba3Mn2O8 compound revealed a zero field splitting of the triplet states
characterized by $D$ =- 0.032 meV Hill_2007 , the difference reflecting the
additional effect of intradimer dipolar coupling in the undiluted compound.
Figure 1: (Color online) (a) Schematic diagram illustrating the exchange
interactions between Mn ions (red spheres) in Ba3Mn2O8 (Ba and O ions are not
shown). Values for the exchange constants are given in the main text (b, c)
Phase diagram of singlet-triplet ordered states for fields aligned along the
$c$ and $a$ axes, respectively, taken from our previous workSamulon_2008 . I,
II and PM mark phase I, phase II and the paramagnetic phase as described in
the main text. Solid and open symbols represent data points obtained via heat
capacity and MCE, respectively.
At low temperatures, the field dependence of the magnetization of Ba3Mn2O8
reveals two regions of linearly increasing moment as first the $S^{z}=1$
triplet (between $H_{c1}=8.7$ T and $H_{c2}=26.5$ T) and then $S^{z}=2$
quintuplet (between $H_{c3}=32.5$ T and $H_{c4}=47.9$ T) states are polarized,
separated by a plateau at half the saturation magnetization Uchida_2002 ;
Samulon_2009 . Here we concentrate solely on the first region of increasing
magnetization, which can be accessed by moderate laboratory fields. In this
field range the ground state can be approximated as a coherent mixture of
singlet and $S^{z}$=1 triplet states. Previous measurements revealed a
distinct anisotropy in the phase diagram depending on the field direction
Samulon_2008 . For fields applied along the $c$ axis, a single ordered state
was found in this regime (fig. 1(b)). Analysis of the minimal spin Hamiltonian
suggests that this state is an incommensurate XY antiferromagnet with
wavevector shifted slightly away from the 120∘ structure favored for a 2D
triangular layerSamulon_2008 ; Diep_2005 . NMR measurements appear to confirm
this conclusionSuh_2009 , but to date the magnetic structure has not been
explicitly solved. Alternatively, fields applied perpendicular to the $c$ axis
revealed two ordered states in this field regime (Fig. 1(c)). Analysis of the
minimal spin Hamiltonian for this field direction yields two distinct
incommensurate modulated states - an Ising phase stabilized closed to $H_{c1}$
and $H_{c2}$, as well as a canted XY state in between. The modulation in these
two phases is stabilized by the uniaxial anisotropy associated with the zero
field splitting of the triplets. Similar to the situation for fields oriented
along the $c$ direction, the exact form of the magnetic structure has not been
explicitly solved, and it remains to be seen whether the minimal spin
Hamiltonian that has been used to describe this system so far is complete.
To this end, careful measurements of the phase diagram of Ba3Mn2O8 for fields
at intermediate angles between the $a$ and $c$ axes have the potential to
determine the presence or absence of additional terms. In this paper we
present results of heat capacity and torque magnetometry measurements
revealing how the two distinct ordered states for fields perpendicular to the
$c$ axis evolve into a single phase for fields along the $c$ axis. Through
analysis of the previously established minimal spin Hamiltonian we can
quantitatively account for the angular dependence of $H_{c1}$ solely via
consideration of the triplet dispersion. However, the same analysis,
incorporating the predicted magnetic structures, fails to quantitatively
account for the angular dependence of the transition between the two ordered
states. We discuss the implications of this observation.
## II Experimental Methods
Single crystals of Ba3Mn2O8 were grown from a NaOH flux according to our
previously published procedureSamulon_2008 . Heat capacity ($C_{p}$) studies
were performed on a Quantum Design physical properties measurement system
(PPMS) using standard thermal relaxation-time calorimetry. These measurements
were performed in fields up to 14T and temperatures down to 0.35 K. The sample
was mounted on angled brackets made from oxygen-free high conductivity copper
and the field was oriented in the [100]-[001] plane.
Cantilever torque magnetometry experiments were performed at the National High
Magnetic Field Laboratory (NHMFL) in a superconducting magnet for fields up to
18T in a dilution refrigerator. A crystal was mounted on one face of a
capacitance cantilever which was attached to a rigid plate rotatable about an
axis parallel to the torque axis and perpendicular to the magnetic field. The
magnetic field was aligned away from the principal crystalline axes yielding a
finite torque.
## III Results
Figure 2: (Color Online) Representative heat capacity data taken at 12T for
for fields in the $a$-$c$ plane. Labels indicate the angle between the field
and the $c$ axis. Successive data sets are offset vertically by 1.6 J/molK for
clarity.
Representative heat capacity measurements, taken at 12 T for several angles,
are shown in Figure 2. These data show a single peak for fields aligned along
the $c$ axis, a peak and a shoulder for fields 15∘ from the $c$ axis and two
peaks for larger angles. Significantly, comparison of the data at 75∘ and 90∘
degrees shows that the 75∘ data has both a slightly higher critical
temperature between the paramagnetic phase and phase II ($T_{c_{II}}$) and
also a substantially lower critical temperature between phase II and phase I
($T_{c_{I}}$) than the 90∘ data.
The phase diagram derived from the complete set of $C_{p}$ measurements, shown
in Figure 3(a), reveals the evolution as a function of angle of the two
distinct singlet-triplet ordered states for fields in the [100]-[001] plane.
The data show a single transition for all fields for $H\|c$ and two
transitions for $H\|a$. The extent in temperature of phase II,
$\Delta_{T}=T_{c_{II}}-T_{c_{I}}$, is shown in Figure 3(b). $\Delta_{T}$
increases as a function of angle as the field is rotated away from the $c$
axis, reaches a maximum at 75∘, and decreases at the $a$ axis (90∘). For
example, for a field of 11T, $\Delta_{T}$ is $\sim$ 0.07 K larger at 75∘ than
at 90∘.
Figure 3: (Color online) (a) Phase diagram showing the transitions between the
paramagnetic state and phase II ($T_{II}$), and between phase II and phase I
($T_{I}$), as a function of temperature and angle in the [100]-[001] plane for
various fields, where $\theta$ indicates the angle between the field and the
$c$ axis. (b) Width of phase II, $\Delta_{T}=T_{c_{II}}-T_{c_{I}}$, as a
function of angle for 11T, 12T, 13T and 14T (black circles, red up triangles,
green down triangles, and blue squares, respectively.)
Torque magnetometry measurements, taken at 25 mK, are shown in Figure 4 for
three representative angles. The sample was inclined slightly so that the
field did not exactly rotate within the $a$-$c$ plane such that a finite
torque was generated for all angles studied. Angles are quoted in terms of the
angular position with respect to the closest approach to the $c$ axis, but it
is important to note that the field was never less than $\sim 10^{\circ}$ from
the $c$ axis. Consequently two phase transitions are observed for all angles
studied. Critical fields, marking the transition between the paramagnetic
phase and phase II ($H_{c1}$), and between phase II and phase I ($H_{II-I}$),
were determined from maxima and minima in the second field derivative of the
torque divided by field Samulon_2008 and are marked with dashed vertical
lines. The sign of the peak in the second derivative changed with the
evolution of angle, reflecting a change in anisotropy for the two different
phases. This leads to minor discontinuities in the determination of the phase
boundary (dashed lines in Fig. 5).
Figure 4: Torque scaled by field, and its first two derivatives with respect
to field, plotted versus field for representative angles at 5.5∘, 55.0∘ and
88.0∘ with respect to the nearest approach to the $c$ axis (see main text).
The critical fields were estimated from peaks in the 2nd derivative as shown
by the dashed lines.
Similar to the phase diagram obtained from heat capacity measurements (Fig.
3), the phase diagram obtained from torque measurements at 25mK reveals a non-
monotonic angle dependence (Fig. 5). The maximum value of $H_{II-I}$ occurs
between 65-75∘ from the closest approach to the $c$ axis. This is in agreement
with the heat capacity data, for which the smallest $T_{c_{I}}$ occurs at 75∘
from the $c$ axis. Additionally, the field extent of phase II,
$\Delta_{H}=H_{II-I}-H_{c1}$, is largest at 75∘, and decreases by $\sim$0.05 T
from the maximum at the highest angles.
## IV Discussion
The previously established minimal spin Hamiltonian for arbitrarily oriented
field direction in Ba3Mn2O8 is:
$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{i,j,\mu,\nu}\frac{J_{i\mu
j\nu}}{2}\textbf{S}_{i\mu}\cdot\textbf{S}_{j\nu}+D\sum_{i,\mu}\left(S^{z}_{i\mu}\cos{\theta}-S^{x}_{i\mu}\sin{\theta}\right)^{2}$
(1)
$\displaystyle-\mu_{B}H\sum_{i\mu\alpha\beta}\left({\tilde{g}}_{zz}S^{z}_{i\mu}+{\tilde{g}}_{xz}S^{x}_{i\mu}\right),$
where ${\tilde{g}}_{zz}=g_{aa}\sin^{2}{\theta}+g_{cc}\cos^{2}{\theta}$,
${\tilde{g}}_{xz}=(g_{cc}-g_{aa})\sin{\theta}\cos{\theta}$, $g_{\alpha\beta}$
is the diagonal gyromagnetic tensor with components $g_{cc}$, $g_{aa}=g_{bb}$,
and $\theta$ is the angle between the applied field and the $c$-axis. The
quantization $z$ axis is set along the field direction. Here $i$, $j$
designate the dimer coordinates, $\alpha,\beta=\\{x,y,z\\}$,
$\mu,\nu=\\{1,2\\}$ denote each of the two S=1 spins in each dimer. The
various exchange constants are shown in Fig. 1(a) and are defined as follows:
the exchange within a dimer is $J_{0}=J_{i,1,i,2}$; the dominant out-of-plane
exchange is $J_{1}=J_{i,2,j,1}$ for $i,j$ nearest neighbor dimers between
planes; the dominant in-plane exchanges between dimers is
$J_{2}=J_{i,\mu,j,\mu}$ and $J_{3}=J_{i,\mu,j,\nu}$ for $i,j$ in plane nearest
neighbor dimers and $\mu\neq\nu$; and finally the second largest out-of-plane
exchange is $J_{4}=J_{i,2,j,1}$ for $i,j$ next nearest neighbor dimers between
planes.
Figure 5: (Color online) Phase diagram at 25 mK determined from torque
magnetometry measurements. Circles (triangles) mark transitions between the
disordered phase and phase II (phase II and phase I). Open (closed) symbols
signify that the transition was determined from a peak (trough) in the second
derivative. Angles are measured relative to the closest approach to the $c$
axis as described in the main text. Red dotted line shows the calculated
transition between the paramagnetic and ordered phases as described in the
main text.
Using this spin Hamiltonian and the measured values of $J_{0}$-$J_{4}$ and $D$
given in the introduction, we have previously been able to quantitatively
account for the observed critical fields and magnetization of the ordered
states of Ba3Mn2O8 for fields oriented along the principal axes Samulon_2008 ;
Samulon_2009 ; Suh_2009 . The calculation is based on a generalized spin-wave
approach in which we only keep the singlet and the three triplet states of
each dimer Khaled2010 . The critical field $H_{c1}(\theta)$ corresponds to the
value for which the energy of lowest energy triplet mode becomes equal to
zero. The softening of this triplet mode signals the onset of the magnetic
instability towards an ordered state (phase II). For field directions along
the principal axes, it is possible to obtain simple analytical expressions for
the critical field. The expression for $H\|c$ is:
$(g_{cc}\mu_{B}H_{c1})^{2}=\left(J_{0}-\frac{D}{3}\right)^{2}+\frac{8}{3}\left(J_{0}-\frac{D}{3}\right)\mathcal{J}_{min},$
(2)
while for $H\perp c$ the expression is:
$\displaystyle(g_{aa}\mu_{B}H_{c1})^{2}$ $\displaystyle=$
$\displaystyle\left(J_{0}+\frac{D}{6}\right)^{2}+\frac{8}{3}\left(J_{0}+\frac{D}{6}\right)\mathcal{J}_{min}$
(3) $\displaystyle-\frac{D^{2}}{4}-\frac{4}{3}|D||\mathcal{J}_{min}|$
where $\mathcal{J}_{min}$ is the minimum of the interdimer exchange portion of
the dispersion and is fully described in equation (3) of reference
[Stone_2008, ]. The difference in the first two terms between these
expressions stems from a change in the zero field splitting of an isolated
dimer depending on quantization direction expressed in the reduced basis of
dimer states. For the quantization axis along the $c$ axis, the zero field gap
of an isolated dimer between the $S^{z}=1$ triplet and the singlet is
$J_{0}-D/3$, while for the quantization axis along the $a$ axis the equivalent
zero field gap is $J_{0}+D/6$. The two additional terms of $H_{c1}$ for fields
perpendicular to the $c$ axis arise from a second order process mixing
singlets and triplets described in our previous work Samulon_2008 . This state
mixing causes the gap between the $S^{z}=1$ triplet and singlet states to
close as $\sqrt{H-H_{c1}}$, as expected for an Ising-like QCP. Evaluating
these expressions using the values of the exchanges and single ion anisotropy
described earlier yield values for the critical fields which are in good
accord with the measured data. Numerical calculation of $H_{c1}$ for arbitrary
field orientations in the [001]-[100] plane yields the red dotted curve shown
in Fig. 5. The calculation was performed using the values of $D$ and the
interdimer couplings given in the introduction while $J_{0}$ was allowed to
vary leading to a fit value of 1.567 meV. The calculated values agree well
with the measured data up to the inherent uncertainty associated with the
misalignment of the sample in the torque measurements described above.
The single-ion anisotropy term of ${\cal{H}}$ is:
$\displaystyle
D\bigg{[}\left(S^{z}\right)^{2}\cos^{2}\left(\theta\right)+\left(S^{x}\right)^{2}\sin^{2}\left(\theta\right)$
$\displaystyle-\left(S^{z}S^{x}+S^{x}S^{z}\right)\cos\left(\theta\right)\sin\left(\theta\right)\bigg{]}.$
(4)
The last term of eq. (4) is zero for fields along the $a$ and $c$ axes, but
adds a small contribution for intermediate angles. In particular, this term
grows linearly in small deviations of $\theta$ from $\pi/2$,
$\delta\theta=\pi/2-\theta$, while the other two terms vary quadratically in
$\delta\theta$. Thus, the last term of Eq. (4) determines the shape of the
boundary between phase I and phase II slightly away from $\theta=\pi/2$, and
could be responsible for the striking non-monotonic behavior observed for
$H_{II-I}$ in Fig. 5 as we describe in greater detail below.
To understand the effect that the last term of eq. (4) has on the ground
state, it is convenient to analyze the effective low-energy Hamiltonian,
${\cal H}_{\rm eff}$, introduced in ref. [Samulon_2008, ]. The low-energy
Hamiltonian results from projecting the original Hamiltonian ${\cal H}$ onto
the low-energy subspace generated by the singlet and the $S^{z}=1$ triplet
states of each dimer. This two-level Hilbert space is described by a local
pseudo-spin $\frac{1}{2}$ in each dimer, ${\bf s}_{i}$, such that
$s^{z}_{i}=\frac{1}{2}$ if the dimer $i$ is in the $S^{z}=1$ triplet state and
$s^{z}_{i}=-\frac{1}{2}$ if it is in the singlet state. In our earlier work we
provided the expression of ${\cal H}_{\rm eff}$ for $\theta=0$ and
$\theta=\pi/2$ Samulon_2008 . In particular, for $\theta=\pi/2$, we showed
that the second term of Eq. (4) generates an effective exchange anisotropy
that is responsible for the emergence of phase II. According to our analysis,
this exchange anisotropy favors an Ising-like phase in which the transverse
spin components (perpendicular to the applied field) are aligned along the
easy $c$-axis. In contrast, phase I is an elliptical spiral phase in which the
transverse spin components of adjacent dimers rotate around the field axis.
The effect the last term of Eq. (4) has on ${\cal H}_{\rm eff}$ at
intermediate angles can be determined using second order degenerate
perturbation theory. Such an analysis yields an effective Dzyaloshinskii-
Moriya (DM) interaction between dimers on adjacent bilayers connected by the
$J_{1}$ and $J_{4}$ exchange constants:
$\sum_{\langle\langle i\rightarrow
j\rangle\rangle}{\bf{\tilde{D}}}_{1}\cdot{\bf s}_{i}\times{\bf
s}_{j}+\sum_{\langle\langle i\rightarrow
j\rangle\rangle^{\prime}}{\bf{\tilde{D}}}_{4}\cdot{\bf s}_{i}\times{\bf
s}_{j},$ (5)
with ${\bf{\tilde{D}}}_{l}={\tilde{D}}_{l}{\bf{\hat{y}}}$,
${\tilde{D}}_{l}={\cal O}(DJ_{l}/J_{0})$ and $l=1,4$. The arrow indicates how
the bonds $\langle\langle i\rightarrow j\rangle\rangle$ are oriented ($i$
always denotes the dimer in the lower bilayer). Microscopically, this process
turns one singlet into an $S^{z}=1$ triplet or vice versa.
This effective DM coupling between pseudo-spins results from two important
symmetry considerations. First, the singlet and the triplet states of a given
dimer have opposite parity under exchange of the two sites of the dimer:
$1\leftrightarrow 2$. Second, the $J_{1}$ and $J_{4}$ terms of ${\cal H}$ are
invariant under the inversion symmetry transformation around the center of the
corresponding bonds: $1\leftrightarrow 2$ and $i\leftrightarrow j$ (see Fig.
6(a)). This implies that a DM term is allowed between pseudo-spins connected
by the $J_{1}$ and $J_{4}$ exchanges. In contrast, the effective DM
interaction cannot occur for pairs of dimers within a plane because the
$J_{2}$ and $J_{3}$ terms of ${\cal H}$ are invariant under the symmetry
transformation $i\leftrightarrow j$, while the effective DM interaction
changes sign under such transformation. These symmetries are fundamentally
equivalent to the selection rules governing the DM vector in the effective
lattice, where each dimer constitutes a single, unique site. In such a
lattice, there is a center of inversion symmetry at the midpoint of the
effective $J_{2}$ and $J_{3}$ exchange, precluding a DM term by the normal
selection rules, while no such inversion symmetry exists at the midpoints of
the effective $J_{1}$ and $J_{4}$ exchanges. This is contrast to the
symmetries of the real lattice, where there are inversion symmetries at the
midpoint of the $J_{1}$ and $J_{4}$ exchanges but none at the midpoint of the
$J_{2}$ and $J_{3}$ exchanges.
Figure 6: (a) Schematic diagram illustrating the two dimers involved in
processes of order $J_{l}D/J_{0}$ that leads to Eq. (5). The $c$ axis is
vertical, and the two dimers belong to adjacent planes. $J_{l}$ represents
either $J_{1}$ or $J_{4}$. (b) (i), (ii) Schematic diagram for two pseudo-
spins on adjacent dimers connected by the $J_{l}$ interaction for arbitrary
field direction in the $a$-$c$ plane. Red arrows show uniform moment, while
green arrows show ordered moment. Field direction along $z$ axis,
${\bf{\tilde{D}}}_{l}$ along $y$ axis, and $c$ axis vertical.
This effective DM interaction between dimers on adjacent layers is frustrated
in both ordered states at a mean field level (the mean value of Eq. (5) is
zero for the semi-classical states associated with phases I and II).
Therefore, the small contribution of the effective DM interaction to the
ground state energy must be produced by quantum fluctuations. Because the DM
vectors point along the $y$ direction (perpendicular to the applied field and
to the easy $c$ axis) this contribution term will favor phase II, for which
the pseudo-spins only have $x$ and $z$ components (Fig. 6(b)(ii)), as opposed
to phase I for which the pseudo-spins have an additional third component along
the $y$ direction (Fig. 6(b)(i)) Samulon_2008 . Thus this contribution
strengthens phase II relative to phase I near $\theta=\pi/2$ and leads to a
small non-monotonic behavior of the $H_{II-I}$ curve (see Fig. 5).
Although this simple analysis captures the qualitative non-monotonic behavior
of the $H_{II-I}$ curve, it cannot account for the magnitude of the observed
effect. The amplitude ${\cal D}(\theta)=D\sin{\theta}\cos{\theta}$ of the
effective DM interaction is of order 100 mK for ${\theta}\simeq 75^{\circ}$.
Because the interaction mixes the singlet and $S^{z}=1$ triplet dimer states,
the mean value of the DM term is less than $\sqrt{m}{\cal D}(\theta)$ for any
state with magnetization $m$ (the mean density of $S^{z}=1$ triplets). Noting
that $m\simeq 0.06$ at $H_{II-I}$, the upper bound on the DM term of $\simeq
25$ mK is the order of magnitude of the observed non-monotonic effect of 5-10
mK in the $H_{II-I}$ curve. Given that ${\cal D}(\theta)$ is much weaker than
the dominant terms of ${\cal H}$, it is clear that the effective DM term can
only explain the magnitude of the non-monotonic effect if it gives a first
order contribution to the energy of Phase II. However, as established above
for the proposed ordered states, the effective DM interaction contributes via
a second order correction and must therefore be considerably smaller. This
leaves us with two possibilities: a) The magnetic structure of Phase II is
different from the simple Ising phase proposed in Ref. [Samulon_2008, ] in
such a way that the mean value of the effective DM term is non-zero, or b) The
non-monotonic effect is caused by a term that has not been included in ${\cal
H}$. At present it is impossible to distinguish between these possibilities,
but ongoing efforts to experimentally determine the magnetic structure have
the potential to directly address option (a), while EPR experiments should, at
least in principle, be able to determine the energy scale of additional
interactions not considered in the minimal spin Hamiltonian (eq. (1)).
## V Conclusion
In summary, via heat capacity and torque magnetometry measurements we have
established the angular dependence of the phase boundary for singlet-triplet
ordered states of Ba3Mn2O8. The data reveal a striking non-monotonicity of the
phase boundary as the field is rotated between the principal axes. The angle-
dependence of $H_{c1}$ can be quantitatively understood in terms of the
original minimal spin Hamiltonian that we had proposed for this material. This
quantity does not depend on details of the magnetically ordered states but
only on the minimum of the triplet dispersion. However, the observed non-
monotonicity in $H_{I-II}$ is at least an order of magnitude larger than
anticipated based on this model and assuming the magnetic structures
previously proposed. This indicates that a complete theoretical description of
this material requires either subtle changes in the proposed magnetic ordered
structures or an additional low-energy term in the Hamiltonian.
## VI Acknowledgements
Work at Stanford University is supported by the Division of Materials
Research, National Science Foundation under Grant No. DMR-0705087. LB is
supported by DOE-BES and the NHMFL-UCGP program. A portion of this work was
performed at the National High Magnetic Field Laboratory, which is supported
by the National Science Foundation under Cooperative Agreement No.
DMR-0084173, by the state of Florida, and the Department of Energy.
## References
* (1) M. T. Weller and S. J. Skinner, Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 55, 154 (1999).
* (2) M. Uchida, H. Tanaka, H. Mitamura, F. Ishikawa, and T. Goto, Phys. Rev. B 66, 054429 (2002).
* (3) M. B. Stone, M. D. Lumsden, S. Chang, E. C. Samulon, C. D. Batista, and I. R. Fisher, Phys. Rev. Lett. 100, 237201 (2008).
* (4) E. C. Samulon, Y.-J. Jo, P. Sengupta, C. D. Batista, M. Jaime, L. Balicas, and I. R. Fisher, Phys. Rev. B 77, 214441 (2008).
* (5) E. C. Samulon, Y. Kohama, R. D. McDonald, M. C. Shapiro, K. A. Al-Hassanieh, C. D. Batista, M. Jaime, and I. R. Fisher, Phys. Rev. Lett. 103, 047202 (2009).
* (6) M. H. Whitmore, H. R. Verdún, and D. J. Singel, Phys. Rev. B 47, 11479 (1993).
* (7) S. Hill, private communication.
* (8) Frustrated Spin Systems Ed H. T. Diep (World Scientific, Singapore, 2004).
* (9) S. Suh, K. A. Al-Hassanieh, E. C. Samulon, J. S. Brooks, W. G. Clark, P. L. Kuhns, L. L. Lumata, A. Reyes, S. E. Brown, C. D. Batista, arXiv:0905.0718.
* (10) While the $g$ factor will also play a role in the angle dependence of phase II, it is nearly isotropic and will vary monotonically as a function of angle.
* (11) K. A. Al-Hassanieh and C. D. Batista, in preparation.
|
arxiv-papers
| 2010-02-06T00:03:04 |
2024-09-04T02:49:08.278157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. C. Samulon, K. A. Al-Hassanieh, Y.-J. Jo, M. C. Shapiro, L.\n Balicas, C. D. Batista, I. R. Fisher",
"submitter": "Eric Samulon",
"url": "https://arxiv.org/abs/1002.1334"
}
|
1002.1339
|
eurm10 msam10 119–126
# Prandtl-Blasius temperature and velocity boundary layer profiles in
turbulent Rayleigh-Bénard convection
Quan ZHOU1 Richard J. A. M. STEVENS2 Kazuyasu SUGIYAMA2,3 Siegfried GROSSMANN4
Detlef LOHSE2 Ke-Qing XIA5 1Shanghai Institute of Applied Mathematics and
Mechanics, Shanghai University, Shanghai 200072, China
2Physics of Fluids Group, Department of Science and Technology, J. M. Burgers
Centre for Fluid Dynamics, and Impact-Institute, University of Twente, 7500 AE
Enschede, The Netherlands
3 Department of Mechanical Engineering, School of Engineering, The University
of Tokyo, Tokyo, Japan
4Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany
5Department of Physics, The Chinese University of Hong Kong, Shatin, Hong
Kong, China
(1996; ?? and in revised form ??)
###### Abstract
The shape of velocity and temperature profiles near the horizontal conducting
plates in turbulent Rayleigh-Bénard convection are studied numerically and
experimentally over the Rayleigh number range $10^{8}\lesssim Ra\lesssim
3\times 10^{11}$ and the Prandtl number range $0.7\lesssim Pr\lesssim 5.4$.
The results show that both the temperature and velocity profiles well agree
with the classical Prandtl-Blasius laminar boundary-layer profiles, if they
are re-sampled in the respective dynamical reference frames that fluctuate
with the instantaneous thermal and velocity boundary-layer thicknesses.
###### keywords:
Rayleigh-Bénard Convection, kinematic and thermal boundary layers, Prandtl-
Blasius boundary layer theory, turbulent thermal convection
††volume: 538
## 1 Introduction
The turbulent motion in a fluid layer sandwiched by two parallel plates and
heated from below, i.e. Rayleigh-Bénard (RB) convection, has become a fruitful
paradigm for understanding the physical nature of a wide range of complicated
convection problems occurring in nature and in engineering problems (Siggia
1994; Ahlers, Lohse $\&$ Grossmann 2009; Lohse $\&$ Xia 2010). A key issue in
the study of turbulent RB system is to understand how heat is transported
upwards by turbulent flow across the fluid layer. It is measured in terms of
the Nusselt number $Nu$, defined as $Nu=J/(\kappa\Delta/H)$, which depends on
the turbulent intensity and the fluid properties. These are characterized,
respectively, by the Rayleigh number $Ra$ and the Prandtl number $Pr$, namely
$Ra=\alpha gH^{3}\Delta/\nu\kappa$ and $Pr=\nu/\kappa$. Here $J$ is the
temperature current density across the fluid layer with a height $H$ and with
an applied temperature difference $\Delta$, $g$ the gravitational
acceleration, and $\alpha$, $\nu$, and $\kappa$ are, respectively, the thermal
expansion coefficient, kinematic viscosity, and thermal diffusivity of the
convecting fluid, for which the Oberbeck-Boussinesq approximation is
considered as valid. As heat transport is controlled by viscosity and thermal
diffusion in the immediate vicinity of the solid boundaries, $Nu$ is
intimately related to the physics of the boundary layers.
In thermal convective turbulent flow two types of boundary layers (BL) exist
near the top and bottom plates, both of which are generated and stabilized by
the viscous shear of the large-scale mean flow: One is the kinematic boundary
layer and the other is the thermal boundary layer. The two layers are not
isolated but are coupled dynamically to each other. They both play an
essential role in turbulent thermal convection, especially for the global heat
flux across the fluid layer. Almost all theories proposed to predict the
relation between $Nu$ and the control parameters $Ra$ and $Pr$ are based on
some kind of assumptions for the BLs, such as the stability assumption of the
thermal BL from the early marginal stability theory Malkus (1954), the
turbulent-BL assumption from the theories of Shraiman & Siggia (1990) and
Siggia (1994) and of Dubrulle (2001, 2002), and the Prandtl-Blasius laminar-BL
assumption of the Grossmann $\&$ Lohse (GL) theory Grossmann & Lohse (2000,
2001, 2002, 2004). Because of the complicated nature of the problem, different
theories based on different assumptions for the BL may yield the same
predictions for the global quantities, such as the $Nu$-$Ra$ scaling relation
Castaing et al. (1989); Shraiman & Siggia (1990). Therefore, direct
characterization of the BL properties is essential for the differences between
and the testing of the various theoretical models and will also provide
insight into the physical nature of turbulent heat transfer in RB system.
In the GL theory, the kinetic energy and thermal dissipation rates have been
decomposed into boundary layer and bulk contributions. Scaling wise and in a
time averaged sense a laminar Prandtl-Blasius boundary layer has been assumed.
This theory can successfully describe and predict the Nusselt and the Reynolds
number dependences on $Ra$ and $Pr$ (see e.g. the recent review in Ahlers et
al., 2009). As the Prandtl-Blasius laminar BL is a key ingredient of the GL
theory, it is important to make direct experimental verification of it. We
note that also the (experimentally verified) calculation of the mean
temperature in the bulk in both liquid and gaseous non-Oberbeck-Boussinesq RB
flows Ahlers et al. (2006, 2007, 2008) is based on the Prandtl-Blasius theory.
In a recent high-resolution measurement of the properties of the velocity
boundary layer, Sun, Cheung $\&$ Xia (2008) have found that, despite the
intermittent emission of plumes, the Prandtl-Blasius-type laminar boundary
layer description is indeed a good approximation, in a time-averaged sense,
both in terms of its scaling and its various dynamical properties. However,
because of the intermittent emissions of thermal plumes from the BLs, the
detailed dynamics of both kinematic and thermal BLs in turbulent RB flow are
much more complicated. On the one hand, direct comparison of experimental
velocity (du Puits, Resagk $\&$ Thess 2007) and numerical temperature
Shishkina & Thess (2009) profiles with theoretical predictions has shown that
both the classical Prandtl-Blasius laminar BL profile and the empirical
turbulent logarithmic profile are not good approximations for the time-
averaged velocity and temperature profiles. Furthermore, Sugiyama et al.
(2009) from two-dimensional (2D) and Stevens, Verzicco $\&$ Lohse (2010) from
three-dimensional (3D) numerical simulations found that the deviation of the
BL profile from the Prandtl-Blasius profile increases from the plate’s center
towards the sidewalls, due to the rising (falling) plumes near the sidewalls.
On the other hand, Qiu & Xia (1998) have found near the sidewall and Sun et
al. (2008) near the bottom plate that the velocity BL obeys the scaling law of
the Prandtl-Blasius laminar BL, i.e., its width scales as $\lambda_{v}/H\sim
Re^{-0.5}$, where $\lambda_{v}$ is the kinematic BL thickness, defined as the
distance from the wall at which the extrapolation of the linear part of the
local mean horizontal velocity profile $u(z)=\langle u_{x}(z,t)\rangle$, with
$z$ being the vertical distance from the bottom plate and
$\langle\cdots\rangle$ being the time average at the plate center, meets the
horizontal line passing through the maximum horizontal velocity
$[u(z)]_{max}$, and $Re$ is the Reynolds number based on $[u(z)]_{max}$. These
papers highlight the need to study the nature of the BL profiles, both
velocity and temperature, in turbulent thermal RB convection.
Considerable progress on this issue has recently been achieved by Zhou & Xia
(2009) who have experimentally studied the velocity BL for water ($Pr=4.3$)
with particle image velocimetry (PIV). They found that, since the dynamics
above and below the range of the boundary layer is different, a time-average
at a fixed height $z$ above the plate with respect to the laboratory (or
container) frame will sample a mixed dynamics, one pertaining to the BL range
and the other one pertaining to the bulk, because the measurement position
will be sometimes inside and sometime outside of the fluctuating width of the
boundary layer. To make a clean separation between the two types of dynamics,
Zhou & Xia (2009) studied the BL quantities in a time-dependent frame that
fluctuates with the instantaneous BL thickness itself. Within this dynamical
frame, they found that the mean velocity profile well agrees with the
theoretical Prandtl-Blasius laminar BL profile. In figure 1 we show the
essence of the results, again for the velocity boundary layer but for somewhat
larger $Pr$, now $Pr=5.4$. (For details of the experiment and the apparatus
used, please see Xia, Sun $\&$ Zhou 2003; Zhou $\&$ Xia 2009). Also here the
method of using the time dependent frame works as good as for the $Pr=4.3$
case of Zhou & Xia (2009). While at the large $Ra=1.8\times 10^{11}$ the time
and space averaged velocity profile (triangles) already considerably deviates
from the Prandtl-Blasius profile (solid line), the dynamically rescaled
profile (circles) perfectly agrees with the Prandtl-Blasius profile. Thus a
dynamical algorithm has been established to directly characterize the BL
properties in turbulent RB systems, which is mathematically well-defined and
requires no adjustable parameters.
Figure 1: Comparison between the spatial $x$-interval and time averaged
velocity profiles $u(z)$ (triangles), the dynamically rescaled velocity
profile $u^{*}(z^{*}_{v})$ (circles – for the notation we refer to section 3),
and the Prandtl-Blasius velocity profile (solid line) near the bottom plate
obtained experimentally at $Ra=1.8\times 10^{11}$ and $Pr=5.4$ (working fluid
water).
The questions which immediately arise are: (i) Does this dynamical rescaling
method also work for the temperature field, giving good agreement with the
(Prandtl number dependent) Prandtl-Blasius temperature profile? (ii) And does
the method also work for lower $Pr$, where the velocity field is more
turbulent? Both these questions cannot be answered with the current Hong Kong
experiments, as PIV only provides the velocity field and not the temperature
field, and as PIV has not yet been established in gaseous RB, i.e., at low
$Pr$ number Rayleigh-Bénard flows.
In the present paper we will answer these two questions with the help of
direct numerical simulations (DNS). To avoid the complications of oscillations
and rotations of the large scale convection roll plane and as the Prandtl-
Blasius theory is a 2D theory anyhow we will restrict ourselves to the 2D
simulations of Sugiyama et al. (2009). Our results will show that Zhou & Xia
(2009)’s idea of using time-dependent coordinates to disentangle the mixed
dynamics of BL and bulk works excellently also for the temperature field and
also for low $Pr$ flow. I.e., if dynamically rescaled, both velocity and
temperature BL profiles can be brought into excellent agreement with the
theoretical Prandtl-Blasius BL predictions, for both larger and lower $Pr$.
## 2 DNS of the 2D Oberbeck-Boussinesq equations
The numerical method has been explained in detail in Sugiyama et al. (2009).
In a nutshell, the Oberbeck-Boussinesq equations with no-slip velocity
boundary conditions at all four walls are solved for a 2D RB cell with a
fourth-order finite-difference scheme. The aspect ratio is $\Gamma\equiv
D/L=1.0$, the Rayleigh number $Ra=10^{8}-10^{9}$, and the Prandtl number
either $Pr=4.3$ (water) or $Pr=0.7$ (gas). Sugiyama et al. (2009) have
provided a detailed code validation.
As the governing equations are strictly Oberbeck-Boussinesq, there exists a
top-bottom symmetry. We therefore discuss only the velocity and temperature
profiles near the bottom plate. For the temperature profiles, we introduce the
non-dimensional temperature $\Theta(z,t)$, defined as
$\Theta(z,t)=\frac{\theta^{bot}-\theta(z,t)}{\Delta/2},$ (1)
where $\theta^{bot}$ is the temperature of the bottom plate. In this
definition, $\Theta(H)=2$ and $\Theta(0)=0$ are the temperatures for the top
and bottom plates, respectively, and $\Theta(H/2)=1$ is the mean bulk
temperature.
## 3 Dynamical BL rescaling
Figure 2: Examples of (a) an instantaneous horizontal velocity profile
$u(z,t)$ and (b) a normalized instantaneous temperature profile $\Theta(z,t)$,
averaged over $0.475<x/D<0.525$. The DNS data are obtained at $Ra=10^{9}$ and
$Pr=0.7$. (c) and (d) show enlarged portions of the velocity and temperature
profiles near the bottom plate, respectively. The two tilted dashed lines are
linear fits to the linear parts of the velocity and temperature profiles near
the plate and the two horizontal dashed lines mark the instantaneous maximum
horizontal velocity and the bulk temperature $\Theta=1$, respectively. The
distances of there crossing points from the plate define the instantaneous BL
thicknesses $\delta_{v,th}^{bot}(t)$. The instantaneous profiles are not top-
down symmetric, the time averaged ones are. Within our present statistical
error our data are consistent with zero thermal gradient in the bulk.
The idea of the Zhou & Xia (2009) method is to construct a dynamical frame
that fluctuates with the local instantaneous BL thickness. To do this, first
the instantaneous kinematic and thermal BL thicknesses are determined using
the algorithm introduced by Zhou & Xia (2009). To reduce data scatter, the
horizontal velocity and temperature profiles at each discrete time $t$,
$u(z,t)$ and $\Theta(z,t)$, are obtained by averaging the velocity and
temperature fields along the $x$-direction (horizontal) over the range
$0.475<x/D<0.525$. Figures 2(a) and (b) show examples of $u(z,t)$ and
$\Theta(z,t)$ versus the normalized height $z/H$, respectively, of the DNS
data obtained at $Ra=10^{9}$ and $Pr=0.7$. Both $u(z,t)$ and $\Theta(z,t)$
rise very quickly from 0 to either the instantaneous maximum velocity or to
the bulk temperature within very thin layers above the bottom plate. While
after reaching its maximum value, $u(z,t)$ slowly decreases in the bulk region
of the closed convection cell, $\Theta(z,t)$ reaches and stays nearly constant
at the bulk temperature $\Theta=1$. To see the velocity and the temperature in
the vicinity of plates more resolved, we plot the enlarged near-plate parts of
the $u(z,t)$ and $\Theta(z,t)$ profiles in figures 2 (c) and (d). One sees
that both profiles enjoy a linear portion near the plate. The instantaneous
velocity BL thickness $\delta_{v}(t)$ is then defined as the distance from the
plate at which the extrapolation of the linear part of the velocity profile
meets the horizontal line passing through the instantaneous maximum horizontal
velocity, and the instantaneous thermal BL thickness $\delta_{th}(t)$ is
obtained as the distance from the plate at which the extrapolation of the
linear part of the temperature profile crosses the horizontal line passing
through the bulk temperature. The arrows in figures 2(c) and (d) illustrate
how to determine $\delta_{v}(t)$ and $\delta_{th}(t)$ as the crossing point
distances.
With these measured $\delta_{v}(t)$ and $\delta_{th}(t)$, we can now construct
the local dynamical BL frames at the plate’s center. The time-dependent
rescaled distances $z^{*}_{v}(t)$ and $z^{*}_{th}(t)$ from the plate in terms
of $\delta_{v}(t)$ and $\delta_{th}(t)$, respectively, are defined as
$z^{*}_{v}(t)\equiv z/\delta_{v}(t)\mbox{\ \ and\ \ }z^{*}_{th}(t)\equiv
z/\delta_{th}(t).$ (2)
The dynamically time averaged mean velocity and temperature profiles
$u^{*}(z^{*}_{v})$ and $\Theta^{*}(z^{*}_{th})$ in the dynamical BL frames are
then obtained by averaging over all values of $u(z,t)$ and $\Theta(z,t)$ that
were measured at different discrete times $t$ but at the same relative
positions $z^{*}_{v}$ and $z^{*}_{th}$, respectively, i.e.,
$u^{*}(z^{*}_{v})\equiv\langle u(z,t)|z=z^{*}_{v}\delta_{v}(t)\rangle\mbox{\ \
and\ \
}\Theta^{*}(z^{*}_{th})\equiv\langle\Theta(z,t)|z=z^{*}_{th}\delta_{th}(t)\rangle.$
(3)
Figure 3: Comparison among (a) velocity profiles: dynamical frame based
$u^{*}(z^{*}_{v})$ (circles), laboratory frame based $u(z)$ (triangles), and
the Prandtl-Blasius velocity profile (solid line), and (b) the corresponding
temperature profiles: $\Theta^{*}(z^{*}_{th})$ (circles), $\Theta(z)$
(triangles), and the Prandtl-Blasius temperature profile (solid line) near the
bottom plate. All results obtained numerically at $Ra=10^{8}$ and $Pr=4.3$.
The inset of (b) shows enlarged portions of the profiles around the thermal
boundary layers’ mergers to the bulk.
We first discuss our results from the simulation performed at $Pr=4.3$, the
Prandtl number corresponding to water at 40 ∘C. Figure 3(a) shows the
$u^{*}(z^{*}_{v})$ profile (circles), normalized by its maximum value
$[u^{*}(z^{*}_{v})]_{max}$, obtained at $Ra=10^{8}$. For comparison, we also
plot in the figure the time-averaged horizontal velocity profile $u(z)$
($=\langle u(z,t)\rangle$) (triangles), obtained from the same simulation. The
solid line represents the Prandtl-Blasius velocity BL profile, the initial
slope of which is matched to that of the measured profiles (cf. Ahlers et al.,
2006). For the range $z^{*}_{v}\lesssim 2$ the $u^{*}(z^{*}_{v})$ profile
obtained in the dynamical frame agrees well with the Prandtl-Blasius profile,
while the time-averaged $u(z)$ profile obtained in the laboratory frame
obviously is much lower than the Prandtl-Blasius profile in the region around
a few kinematic BL widths. - Note that for $z^{*}_{v}\gtrsim 2$ the
$u^{*}(z^{*}_{v})$ profile deviates gradually from the Prandtl-Blasius profile
because $u^{*}(z^{*}_{v})$ decreases in the bulk region of the closed
convection system down to 0 in the center and then changes sign. The Prandtl-
Blasius profile, instead, describes the situation of an asymptotically
constant, nonzero flow velocity. - These DNS results are similar to those
found experimentally in a rectangular cell Zhou & Xia (2009).
Figure 3(b) shows a direct comparison among the temperature profiles obtained
from the same simulation: the dynamical frame based $\Theta^{*}(z^{*}_{th})$
(circles), the laboratory frame time-averaged temperature profile $\Theta(z)$
($=\langle\Theta(z,t)\rangle$) (triangles), and the Prandtl-Blasius
temperature profile. At first glance both the $\Theta^{*}(z^{*}_{th})$ and
$\Theta(z)$ profiles are consistent with the Prandtl-Blasius thermal profile.
However, looking more carefully at the region around the thermal BL to bulk
merger (the inset of figure 3(b)), one notes that the $\Theta^{*}(z^{*}_{th})$
profile obtained in the dynamical frame is significantly closer to the
Prandtl-Blasius profile than the time-averaged $\Theta(z)$ profile obtained in
the laboratory frame, indicating that the dynamical frame idea of Zhou & Xia
(2009) works also for the thermal BL. Taken together, figures 3(a) and (b)
illustrate that both the kinematic and the thermal BLs in turbulent RB
convection are of Prandtl-Blasius type, which is a key assumption of the GL
theory Grossmann & Lohse (2000, 2001, 2002, 2004), and the dynamical frame
idea of Zhou & Xia (2009) can achieve a clean separation for both temperature
and velocity fields between their BL and bulk dynamics.
Figure 4: Comparison between (a) velocity profiles: dynamical
$u^{*}(z^{*}_{v})$ (circles), laboratory $u(z)$ (triangles), and the Prandtl-
Blasius laminar velocity profile (solid line), and (b) temperature profiles:
dynamical $\Theta^{*}(z^{*}_{th})$ (circles), laboratory $\Theta(z)$
(triangles), and the Prandtl-Blasius laminar temperature profile (solid line)
near the bottom plate, all obtained numerically at $Ra=10^{9}$ and $Pr=0.7$,
representative for gases.
We next turn to the simulation performed at $Pr=0.7$, a Prandtl number typical
for gases, which is relevant in all atmospheric processes and many technical
applications. Figures 4(a) and (b) show direct comparison between the
temperature and velocity profiles, respectively, at $Ra=10^{9}$. Again, around
the BL-bulk merger range the laboratory frame time-averaged profiles are found
to be obviously lower than the Prandtl-Blasius profile. This once more
indicates that the time-averaged BL quantities obtained in the laboratory
frame are contaminated by the mixed dynamics inside and outside the
fluctuating BLs. On the other hand, within the dynamical frame, both
$u^{*}(z^{*}_{v})$ and $\Theta^{*}(z^{*}_{th})$ are found to agree pretty well
with the Prandtl-Blasius laminar BL profiles, indicating that the dynamical
frame idea works also for the turbulent RB system with working fluids whose
Prandtl numbers are of the same order as those for gases.
## 4 Shape factors of the velocity and temperature profiles
Let us now quantitatively compare the differences between the Prandtl-Blasius
profile and the profiles obtained from both simulations and experiments for
various $Ra$ and various $Pr$. The shapes of the velocity and temperature
(thermal) profiles, labeled by $i=v$ or $i=th$, can be characterized
quantitatively by their shape factors $H_{i}$, defined as Schlichting &
Gersten (2004),
$H_{i}=\frac{\lambda^{d}_{i}}{\lambda^{m}_{i}}~{},~{}~{}~{}i=v,th.$ (4)
$\lambda^{d}_{i}$ and $\lambda^{m}_{i}$ denote, respectively, the displacement
and the momentum thicknesses of the profile, namely,
$\lambda^{d}_{i}=\int_{0}^{\infty}\\{1-\frac{Y(z)}{[Y(z)]_{max}}\\}dz\mbox{\ \
and\ \
}\lambda^{m}_{i}=\int_{0}^{\infty}\\{1-\frac{Y(z)}{[Y(z)]_{max}}\\}\\{\frac{Y(z)}{[Y(z)]_{max}}\\}dz.$
(5)
Here $Y(z)=u(z)$ is the velocity profile if $i=v$ and $Y(z)=\Theta(z)$ the
thermal profile if $i=th$. The deviation of these profiles from the Prandtl-
Blasius profile is then measured by
$\delta H_{i}=H_{i}-H^{PB}_{i},$ (6)
where $H^{PB}_{i}$ is the shape factor for the respective Prandtl-Blasius
laminar BL profile. If a given profile exactly matches the Prandtl-Blasius
profile, $\delta H_{i}$ is zero. Note that the Prandtl-Blasius velocity
profile shape factor $H^{PB}_{v}=2.59$ is independent of $Pr$, while the
thermal Prandtl-Blasius BL profile shape factor $H^{PB}_{th}$ varies with
$Pr$.
Figure 5: (a) The shape factors for the thermal (solid line) and velocity
(dashed line) Prandtl-Blasius BL profiles as function of $Pr$. The asymptotic
value $H^{PB}_{th}(Pr\gg 1)$ is $2.61676...$ and $H^{PB}_{v}=2.59$. Both
Prandtl-Blasius BL profiles for the velocity and for the temperature coincide
for $Pr=1$. (b) The thermal Prandtl-Blasius BL profiles for three (four) $Pr$
numbers and the reference linear and exponential profiles, see (7); in the
figure’s resolution the thermal profile for $Pr=100$ is indistinguishable from
that of $Pr=1$. Note that the shape factor of the thermal Prandtl-Blasius BL
profile decreases with decreasing $Pr$ due to the slower approach to its
asymptotic level $1$.
Figure 5(a) shows the shape factors $H_{i}(Pr)$ of the thermal and the
velocity Prandtl-Blasius BL profiles as functions of $Pr$ and figure 5(b)
shows the corresponding thermal profiles as functions of $z^{*}_{th}$ for
three different $Pr$. Note that the Prandtl-Blasius velocity BL profile is
identical to the thermal one for $Pr=1$. The two figures show that the thermal
shape factor $H_{th}^{PB}$ decreases with decreasing $Pr$. We attribute this
to the decrease of the temperature profiles in the BL range and the
corresponding increase of the tails for lower $Pr$. Thus we expect that the
slower approach to the asymptotic height $1$ of the thermal profiles in the
laboratory frame in figures 3 and 4 should lead to a negative deviation of
their $H_{th}$’s from the respective Prandtl-Blasius values, cf. figure 6. In
contrast, a positive $\delta H_{i}$ is obtained if the profile runs to its
asymptotic level faster than the Prandtl-Blasius profile. To see this more
clearly, we have plotted in figure 5(b) also two extreme cases, the linear and
the exponential profiles. Using (5) one calculates the shape factor $3$ for
the linear profile $\Theta^{*}(z^{*}_{th})=\mbox{min}(1,z^{*}_{th})$ and the
shape factor $2$ for the exponential one
$\Theta^{*}(z^{*}_{th})=1-\mbox{exp}(-z^{*}_{th})$. The $H$-decreasing effect
by lowering the profile can also be demonstrated by analyzing some profiles
analytically. Using a combination of exponential profiles,
$\Theta^{*}(z^{*}_{th})=0.5(1-\mbox{exp}(-(1-n)(z^{*}_{th}))+0.5(1-\mbox{exp}(-(1+n)z^{*}_{th})),$
(7)
with $0\leq n<1$ one can evaluate, using (5), that the shape factor for small
$n$ is
$H(n)\approx H(n=0)-n^{2}=2-n^{2}.$ (8)
As is shown in figure 5(b) the profile for $n>0$ is below the profile for
$n=0$. This analytical example again reflects what we found as the
characteristic difference between the laboratory frame profiles as compared to
the dynamical frame profiles.
Figure 6: The $Ra$-dependence of the deviations of the profile shape factors
from the respective Prandtl-Blasius shape factors. (a) Laboratory frame
$\delta H_{v}$ (open symbols) and dynamical frame $\delta H^{*}_{v}$ (solid
symbols); (b) laboratory frame $\delta H_{th}$ (open symbols) and dynamical
frame $\delta H^{*}_{th}$ (solid symbols); all from simulations performed at
$Pr=0.7$ (circles), $Pr=4.3$ (triangles), and from experiments at $Pr=5.4$
(squares).
Figure 6(a) shows the velocity shape factor deviations $\delta H_{v}$ (open
symbols) and $\delta H^{*}_{v}$ (solid symbols) as obtained from simulations
at $Pr=0.7$ (circles) and $Pr=4.3$ (triangles) as well as from experiments at
$Pr=5.4$ (squares). Here, $\delta H_{v}$ is calculated with the time averaged
profile $u(z)$ in the laboratory frame, while $\delta H^{*}_{v}$ is calculated
with the dynamical, time-dependent frame profile $u^{*}(z^{*}_{v})$. The
laboratoty frame based deviations turn out to be definitely smaller than zero.
In contrast, the shape factor deviations $\delta H^{*}_{v}$ for the dynamical
frame profiles obviously are much closer to zero. A similar result is found
for the thermal BLs: Figure 6(b) shows $\delta H_{th}$ (open symbols) and
$\delta H^{*}_{th}$ (solid symbols), versus $Ra$, for the same $Pr$ number
simulations. Again $\delta H^{*}_{th}$ is nearly zero, whereas $\delta H_{th}$
is significantly off. Thus these quantitative deviation measures again
indicate that the algorithm using the dynamical coordinates can effectively
disentangle the mixed dynamics inside and outside the fluctuating BLs.
## 5 Conclusions
In summary, we have studied the velocity and temperature BL profiles in
turbulent RB convection both numerically and experimentally. We extended
previous results to different Prandtl numbers and in particular to thermal
BLs. The results show that both the velocity and the temperature BLs (at least
in the plates’ center region) are of laminar Prandtl-Blasius type in the co-
moving dynamical frame in turbulent thermal convection for the parameter
ranges studied. However, the fluctuations of the BL widths, induced by the
fluctuations of the large-scale mean flow and the emissions of thermal plumes,
cause measuring probes at fixed heights above the plate to sample a mixed
dynamics, one pertaining to the BL range and the other one pertaining to the
bulk. This is the reason why the time-averaged velocity and temperature
profiles measured in previous work in fixed laboratory (RB cell) frames
deviate from the Prandtl-Blasius profiles. To disentangle that mixed dynamics,
we constructed a dynamical BL frame that fluctuates with the instantaneous BL
thicknesses. Within this dynamical frame, both velocity and temperature
profiles are very well consistent with the classical Prandtl-Blasius laminar
BL profiles, both for lower and larger $Pr$ (from $0.7$ to $5.4$). We have
thus validated the idea and algorithm of using dynamical coordinates over a
range of $Pr$ and $Ra$ for both kinematic and thermal BLs and have shown that
the Prandtl-Blasius laminar BL profile is a valid description for the BLs of
both velocity and temperature in turbulent thermal convection. Laminar
Prandtl-Blasius BL theory in turbulent RB thermal convection has thus turned
out to indeed be valid not only scaling wise, but also in the time average as
seen from the dynamical frame, co-moving with the local, instantaneous BL
widths.
###### Acknowledgements.
We gratefully acknowledge support of this work by the Natural Science
Foundation of Shanghai (No. 09ZR1411200), “Chen Guang” project (No.
09CG41)(Q.Z.), by the Research Grants Council of Hong Kong SAR (Nos.
CUHK403806 and 403807) (K.Q. X), and by the research programme of FOM, which
is financially supported by NWO (R.J.A.M.S. and D.L.).
## References
* Ahlers et al. (2006) Ahlers, G., Brown, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S. & Lohse, D. 2006 Non-Oberbeck-Boussinesq effects in strongly turbulent Rayleigh-Bénard convection. J. Fluid Mech. 569, 409–445.
* Ahlers et al. (2008) Ahlers, G., Calzavarini, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S., Lohse, D. & Sugiyama, K. 2008 Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point. Phys. Rev. E 77, 046302.
* Ahlers et al. (2007) Ahlers, G., Fontenele Araujo, F., Funfschilling, D., Grossmann, S. & Lohse, D. 2007 Non-Oberbeck-Boussinesq effects in gaseous Rayleigh-Bénard convection. Phys. Rev. Lett. 98, 054501\.
* Ahlers et al. (2009) Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503–537.
* Castaing et al. (1989) Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh-Bénard convection. J. Fluid Mech. 204, 1–30.
* Dubrulle (2001) Dubrulle, B. 2001 Logarithmic corrections to scaling in turbulent thermal convection. Eur. Phys. J. B 21, 295–304.
* Dubrulle (2002) Dubrulle, B. 2002 Scaling in large prandtl number burbulent thermal convection. Eur. Phys. J. B 28, 361–367.
* Grossmann & Lohse (2000) Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.
* Grossmann & Lohse (2001) Grossmann, S. & Lohse, D. 2001 Thermal convection for large prandtl numbers. Phys. Rev. Lett. 86, 3316–3319.
* Grossmann & Lohse (2002) Grossmann, S. & Lohse, D. 2002 Prandtl and rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.
* Grossmann & Lohse (2004) Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh-Bénard convection: The role of plumes. Phys. Fluids 16, 4462–4472.
* Lohse & Xia (2010) Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.
* Malkus (1954) Malkus, M. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196–212.
* du Puits et al. (2007) du Puits, R., Resagk, C. & Thess, A. 2007 Structure of the thermal boundary layer in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 99, 234504.
* Qiu & Xia (1998) Qiu, X.-L. & Xia, K.-Q. 1998 Viscous boundary layers at the sidewall of a convection cell. Phys. Rev. E 58, 486–491.
* Schlichting & Gersten (2004) Schlichting, H. & Gersten, K. 2004 Boundary Layer Theory. Springer, 8th ed.
* Shishkina & Thess (2009) Shishkina, O. & Thess, A. 2009 Mean temperature profiles in turbulent Rayleigh-Bénard convection of water. J. Fluid Mech. 633, 449–460.
* Shraiman & Siggia (1990) Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-rayleigh-number convection. Phys. Rev. A 42, 3650–3653.
* Siggia (1994) Siggia, E. D. 1994 High rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–168.
* Stevens et al. (2010) Stevens, Richard J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh-Bénard convection. J. Fluid Mech. 643, 493–507.
* Sugiyama et al. (2009) Sugiyama, K., Calzavarini, E., Grossmann, S. & Lohse, D. 2009 Flow organization in non-Oberbeck-Boussinesq Rayleigh-Bénard convection in water. J. Fluid Mech. 637, 105–135.
* Sun et al. (2008) Sun, C., Cheung, Y.-H. & Xia, K.-Q. 2008 Experimental studies of the viscous boundary layer properties in turbulent Rayleigh-Bénard convection. J. Fluid Mech. 605, 79–113.
* Xia et al. (2003) Xia, K.-Q., Sun, C. & Zhou, S.-Q. 2003 Paricle image velocimetry measurements of the velocity field in turbulent thermal convection. Phys. Rev. E 68, 066303.
* Zhou & Xia (2009) Zhou, Q. & Xia, K.-Q. 2009 Measured instantaneous viscous boundary layer in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. submitted.
|
arxiv-papers
| 2010-02-06T01:17:19 |
2024-09-04T02:49:08.283035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan Zhou, Richard J. A. M. Stevens, Kazuyasu Sugiyama, Siegfried\n Grossmann, Detlef Lohse, and Ke-Qing Xia",
"submitter": "Quan Zhou",
"url": "https://arxiv.org/abs/1002.1339"
}
|
1002.1435
|
# Agegraphic Chaplygin gas model of dark energy
A. Sheykhi 111 sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
We establish a connection between the agegraphic models of dark energy and
Chaplygin gas energy density in non-flat universe. We reconstruct the
potential of the agegraphic scalar field as well as the dynamics of the scalar
field according to the evolution of the agegraphic dark energy. We also extend
our study to the interacting agegraphic generalized Chaplygin gas dark energy
model.
## I Introduction
Over the course of the past decade, evidence for the most striking result in
modern cosmology has been steadily growing, namely the existence of an exotic
dark energy component which has negative pressure and pushes the universe to
accelerated expansion Rie . Of course, a natural explanation to the
accelerated expansion is due to a positive tiny cosmological constant. Though,
it suffers the so-called fine-tuning and cosmic coincidence problems. A great
variety of scenarios have been proposed to explain this acceleration while
most of them cannot explain all the features of universe or they have so many
parameters that makes them difficult to fit. For a recent review on dark
energy proposals see Pad . Many theoretical attempts toward understanding the
dark energy problem are focused to shed light on it in the framework of a
fundamental theory such as string theory or quantum gravity. Although a
complete theory of quantum gravity has not established yet today, we still can
make some attempts to investigate the nature of dark energy according to some
principles of quantum gravity. The holographic dark energy model and the
agegraphic dark energy model are just such examples, which are originated from
some considerations of the features of the quantum theory of gravity. That is
to say, the holographic and agegraphic dark energy models possess some
significant features of quantum gravity. The former, that arose a lot of
enthusiasm recently Coh ; Li ; Huang ; Hsu ; HDE ; Setare ; Seta1 , is
motivated from the holographic hypothesis Suss1 and has been tested and
constrained by various astronomical observations Xin . However there are some
difficulties in holographic dark energy model. Choosing the event horizon of
the universe as the length scale, the holographic dark energy gives the
observation value of dark energy in the universe and can drive the universe to
an accelerated expansion phase. But an obvious drawback concerning causality
appears in this proposal. Event horizon is a global concept of spacetime;
existence of event horizon of the universe depends on future evolution of the
universe; and event horizon exists only for universe with forever accelerated
expansion. In addition, more recently, it has been argued that this proposal
might be in contradiction to the age of some old high redshift objects, unless
a lower Hubble parameter is considered Wei0 . The later (agegraphic dark
energy) is based on the uncertainty relation of quantum mechanics together
with the gravitational effect in general relativity. The agegraphic dark
energy model assumes that the observed dark energy comes from the spacetime
and matter field fluctuations in the universe Cai1 ; Wei2 ; Wei1 . Since in
agegraphic dark energy model the age of the universe is chosen as the length
measure, instead of the horizon distance, the causality problem in the
holographic dark energy is avoided. The agegraphic models of dark energy have
been examined and constrained by various astronomical observations age ; shey1
; setare .
Among the various candidates to explain the accelerated expansion, the
Chaplygin gas dark energy model has emerged as a possible unification of dark
matter and dark energy, since its cosmological evolution is similar to an
initial dust like matter and a cosmological constant for late times. Inspired
by the fact that the Chaplygin gas possesses a negative pressure, the authors
of Gorini have undertaken the simple task of studying a FRW cosmology of a
universe filled with this type of fluid. The equation of state of the
Chaplygin gas dark energy obeys Kam
$\displaystyle p_{D}=\frac{-A}{\rho_{D}},$ (1)
where $\rho_{D}>0$ and $p_{D}$ are, respectively, the energy density and
pressure of Chaplygin gas dark energy, and $A$ is a positive constant. This
equation of state has raised a certain interest Baz because of its many
interesting and, in some sense, intriguingly unique features. Some possible
motivations for this model from the field theory points of view are
investigated in Bil . The Chaplygin gas emerges as an effective fluid
associated with D-branes Bor and can also be obtained from the Born-Infeld
action Ben . The connection between the holographic models of dark energy and
the Chaplygin gas energy density has been established in seta1 ; seta2 .
Our aim in this Letter is to establish a correspondence between the agegraphic
dark energy scenarios and the Chaplygin gas model. We suggest the agegraphic
description of the Chaplygin gas dark energy in FRW universe and reconstruct
the potential and the dynamics of the scalar field which describe the
Chaplygin cosmology. In the next section we study the original agegraphic
Chaplygin gas model. In section III, we establish the correspondence between
the new model of agegraphic dark energy and the Chaplygin gas dark energy. In
section IV, we extend our study to the interacting agegraphic generalized
Chaplygin gas dark energy model. The last section is devoted to conclusions.
## II THE ORIGINAL ADE as a Chaplygin gas
We assume the agegraphic Chaplygin gas dark energy is accommodated in the
Friedmann-Robertson-Walker (FRW) universe which is described by the line
element
$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(2)
where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . The corresponding Friedmann equation takes
the form
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right).$ (3)
We introduce, as usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (4)
thus, the Friedmann equation can be written
$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (5)
Inserting the equation of state (1) into the relativistic energy conservation
equation, leads to a density evolving as
$\displaystyle\rho_{D}=\sqrt{A+\frac{B}{a^{6}}}.$ (6)
where $B$ is an integration constant. We adopt the viewpoint that the scalar
field models of dark energy are effective theories of an underlying theory of
dark energy. The energy density and pressure of the scalar field can be
written as
$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\sqrt{A+\frac{B}{a^{6}}},$
(7) $\displaystyle
p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)=\frac{-A}{\sqrt{A+\frac{B}{a^{6}}}},$
(8)
Then, we can easily obtain the scalar potential and the kinetic energy term as
$\displaystyle V(\phi)=\frac{2Aa^{6}+B}{2a^{6}\sqrt{A+\frac{B}{a^{6}}}},$ (9)
$\displaystyle\dot{\phi}^{2}=\frac{B}{a^{6}\sqrt{A+\frac{B}{a^{6}}}}.$ (10)
Now we are focussing on the reconstruction of the original agegraphic
Chaplygin gas model of dark energy. Let us first review the origin of the
agegraphic dark energy. Following the line of quantum fluctuations of
spacetime, Karolyhazy et al. Kar1 argued that the distance $t$ in Minkowski
spacetime cannot be known to a better accuracy than $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ where $\beta$ is a dimensionless constant of order unity.
Based on Karolyhazy relation, Maziashvili discussed that the energy density of
metric fluctuations of the Minkowski spacetime is given by Maz
$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (11)
where $t_{p}$ is the reduced Planck time. We use the units $c=\hbar=k_{b}=1$
throughout this Letter. Therefore one has $l_{p}=t_{p}=1/m_{p}$ with $l_{p}$
and $m_{p}$ are the reduced Planck length and mass, respectively. The original
agegraphic dark energy density has the form (11) where $t$ is chosen to be the
age of the universe
$T=\int_{0}^{a}{\frac{da}{Ha}},$ (12)
Thus, the energy density of the original agegraphic dark energy is given by
Cai1
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T^{2}},$ (13)
where the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time (since the energy density is derived for Minkowski
space-time), and so on. The dark energy density (13) has the same form as the
holographic dark energy, but the length measure is chosen to be the age of the
universe instead of the horizon radius of the universe. Thus the causality
problem in the holographic dark energy is avoided. Using Eqs. (4) and (13), we
have
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}T^{2}}.$ (14)
We assume the agegraphic dark energy and dark matter evolve according to their
conservation laws
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0,$ (15)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (16)
where $w_{D}$ is the equation of state parameter of agegraphic dark energy.
Taking the derivative with respect to the cosmic time of Eq. (13) and using
Eq. (14) we get
$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{n}\rho_{D}.$ (17)
Inserting this relation into Eq. (15), we obtain the equation of state
parameter of the original agegraphic dark energy
$\displaystyle w_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}.$ (18)
Differentiating Eq. (14) and using relation
${\dot{\Omega}_{D}}={\Omega^{\prime}_{D}}H$, we reach
$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(19)
where the dot is the derivative with respect to the cosmic time and the prime
denotes the derivative with respect to $x=\ln{a}$. Taking the derivative of
both side of the Friedman equation (3) with respect to the cosmic time, and
using Eqs. (5), (13), (14) and (16), it is easy to find
$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1-\Omega_{D})-\frac{\Omega^{3/2}_{D}}{n}-\frac{\Omega_{k}}{2}.$
(20)
Substituting this relation into Eq. (19), we obtain the equation of motion of
agegraphic dark energy
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right)+\Omega_{k}\right].$
(21)
Next, we establish the connection between the original agegraphic dark energy
and Chaplygin gas energy density. Combining Eqs. (6) and (13), we obtain
$\displaystyle B=a^{6}\left(9n^{4}m^{4}_{p}T^{-4}-A\right).$ (22)
Using Eqs. (1), (6) and (18) one can write
$\displaystyle
w_{D}=\frac{-A}{\rho_{D}^{2}}=\frac{-A}{A+\frac{B}{a^{6}}}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}.$
(23)
Substituting $B$ in the above equation, we obtain following relation for $A$:
$\displaystyle
A=9n^{4}m^{4}_{p}T^{-4}\left(1-\frac{2}{3n}\sqrt{\Omega_{D}}\right).$ (24)
Therefore the constant $B$ is given by
$\displaystyle B=6a^{6}n^{4}m^{4}_{p}T^{-4}\frac{\sqrt{\Omega_{D}}}{n}.$ (25)
Finally, we rewrite the scalar potential and kinetic energy term as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
n^{2}m^{2}_{p}T^{-2}\left(3-\frac{\sqrt{\Omega_{D}}}{n}\right)=m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{n}\right),$
(26) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
nm_{p}T^{-1}\sqrt{\frac{2}{n}\Omega^{1/2}_{D}}=m_{p}H\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}}.$
(27)
Using relation $\dot{\phi}=H{\phi^{\prime}}$, we get
$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}}.$ (28)
Consequently, we can easily obtain the evolution behavior of the scalar field
$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}}da},$
(29)
where $a_{0}$ is the present value of the scale factor, and $\Omega_{D}$ can
be obtained through Eq. (21).
## III THE NEW ADE as a Chaplygin gas
Soon after the original agegraphic dark energy model was introduced by Cai
Cai1 , a new model of agegraphic dark energy was proposed in Wei2 , while the
time scale is chosen to be the conformal time $\eta$ instead of the age of the
universe. This new agegraphic dark energy contains some new features different
from the original agegraphic dark energy and overcome some unsatisfactory
points. For instance, the original agegraphic dark energy suffers from the
difficulty to describe the matter-dominated epoch while the new agegraphic
dark energy resolved this issue Wei2 . The energy density of the new
agegraphic dark energy can be written
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (30)
where the conformal time is given by
$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (31)
The fractional energy density of the new agegraphic dark energy is now given
by
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (32)
Taking the derivative with respect to the cosmic time of Eq. (30) and using
Eq. (32) we get
$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{na}\rho_{D}.$ (33)
Inserting this relation into Eq. (15) we obtain the equation of state
parameter of the new agegraphic dark energy
$\displaystyle w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}.$ (34)
Then we obtain, following the approach of the previous section, the evolution
behavior of the new agegraphic dark energy,
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)+\Omega_{k}\right].$
(35)
Next, we construct the new agegraphic Chaplygin gas model, connecting the
Chaplygin gas model with new agegraphic dark energy. Identifying Eq. (6) with
(30) we have
$\displaystyle B=a^{6}\left(9n^{4}m^{4}_{p}\eta^{-4}-A\right).$ (36)
Using Eqs. (1), (6) and (34), we reach
$\displaystyle
w_{D}=\frac{-A}{A+\frac{B}{a^{6}}}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}.$ (37)
Combining this equation with Eq. (36), we obtain
$\displaystyle A$ $\displaystyle=$ $\displaystyle
9n^{4}m^{4}_{p}\eta^{-4}\left(1-\frac{2}{3na}\sqrt{\Omega_{D}}\right),$ (38)
$\displaystyle B$ $\displaystyle=$ $\displaystyle
6a^{6}n^{4}m^{4}_{p}\eta^{-4}\frac{\sqrt{\Omega_{D}}}{na}.$ (39)
Finally, we reconstruct the scalar potential and kinetic energy term as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
n^{2}m^{2}_{p}\eta^{-2}\left(3-\frac{\sqrt{\Omega_{D}}}{na}\right)=m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{na}\right),$
(40) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
nm_{p}\eta^{-1}\sqrt{\frac{2}{na}{\Omega^{1/2}_{D}}}=m_{p}H\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}}.$
(41)
Eq. (41) can also be reexpressed as
$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}}.$ (42)
Therefore, we can obtain the evolutionary form of the scalar field
$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}}da},$
(43)
where $\Omega_{D}$ can be derived from Eq. (35).
## IV Interacting new agegraphic generalized Chaplygin gas
In this section we extend our study to the generalized Chaplygin gas when
there is an interaction between generalized Chaplygin gas energy density and
dark matter. The total energy density is $\rho=\rho_{m}+\rho_{D}$, where
$\rho_{m}$ and $\rho_{D}$ are the energy density of dark matter and dark
energy, respectively. The total energy density satisfies a conservation law
$\dot{\rho}+3H(\rho+p)=0.$ (44)
However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately; they must rather
enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (45)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q,$ (46)
where $Q$ denotes the interaction term and can be taken as $Q=3b^{2}H\rho$
with $b^{2}$ being a coupling constant. In the generalized Chaplygin gas
approach Ben , the equation of state (1) is generalized to
$\displaystyle p_{D}=\frac{-A}{{\rho_{D}^{\alpha}}}.$ (47)
The above equation of state leads to a density evolution as
$\displaystyle\rho_{D}=\left({A+{B}a^{-3\beta}}\right)^{1/\beta},$ (48)
where $\beta=\alpha+1$. Thus we have
$\displaystyle w_{D}=\frac{p_{D}}{\rho_{D}}=\frac{-A}{{A+{B}a^{-3\beta}}}.$
(49)
Inserting relation (33) into Eq. (46) and using Eqs. (4) and (5) , we obtain
the equation of state parameter of the interacting new agegraphic dark energy
$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k}).$
(50)
The evolution behavior of the new agegraphic dark energy is given by
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(51)
We now establish the correspondence between the new agegraphic dark energy and
generalized Chaplygin gas energy density. Identifying Eq. (30) with Eq. (48)
and using Eq. (32) we get
$\displaystyle\left(3m^{2}_{p}H^{2}\Omega_{D}\right)^{\beta}={A+{B}a^{-3\beta}}.$
(52)
Combining Eqs. (49) and (50) we find
$\displaystyle
A=-\left({A+{B}a^{-3\beta}}\right)\left[-1+\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k})\right].$
(53)
Solving Eqs. (52) and (53) we obtain
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(3m^{2}_{p}H^{2}\Omega_{D}\right)^{\beta}\left[1-\frac{2}{3na}\sqrt{\Omega_{D}}+\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k})\right],$
(54) $\displaystyle B$ $\displaystyle=$
$\displaystyle\left(3m^{2}_{p}H^{2}\Omega_{D}a^{3}\right)^{\beta}\left[\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k})\right].$
(55)
Next we regard the scalar field model as an effective description of an
underlying theory of dark energy with energy density and pressure
$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\left({A+{B}a^{-3\beta}}\right)^{1/\beta},$
(56) $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}-V(\phi)=-A\left({A+{B}a^{-3\beta}}\right)^{-\alpha/\beta},$
(57)
where we have identified $\rho_{\phi}$ with $\rho_{D}$. Substituting $A$ and
$B$ into Eqs. (56) and (57) one can easily find the scalar potential and the
kinetic energy term as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{na}+\frac{3b^{2}}{2}\frac{(1+\Omega_{k})}{\Omega_{D}}\right),$
(58) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
m_{p}H\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(59)
Using Eq. (59), Eq. (58) can be reexpressed as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
3m^{2}_{p}H^{2}\Omega_{D}\left(1-\frac{\dot{\phi}^{2}}{6m^{2}_{p}H^{2}\Omega_{D}}\right).$
(60)
We can also rewrite Eq. (59) as
$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(61)
Consequently, we can easily obtain the evolutionary form of the field by
integrating the above equation. The result is
$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})}da},$
(62)
where $\Omega_{D}$ is given by Eq. (51). In this way we connect the
interacting new agegraphic dark energy with a generalized Chaplygin gas energy
density and reconstruct the potential of the agegraphic Chaplygin gas.
## V Conclusions
In this Letter, we have established a correspondence between the agegraphic
dark energy scenarios and the Chaplygin gas model of dark energy in non-flat
FRW cosmology. A so-called agegraphic dark energy model has been proposed
recently by Cai Cai1 , based on the uncertainty relation of quantum mechanics
together with the gravitational effect in general relativity. Since the
original agegraphic dark energy model suffers from the difficulty to describe
the matter-dominated epoch, a new model of agegraphic dark energy was proposed
in Wei2 while the time scale is chosen to be the conformal time $\eta$
instead of the age of the universe. We have adopted the viewpoint that the
scalar field models of dark energy are effective theories of an underlying
theory of dark energy. If we regard the scalar field model as an effective
description of such a theory, we should be capable of using the scalar field
model to mimic the evolving behavior of the agegraphic dark energy and
reconstructing this scalar field model according to the evolutionary behavior
of agegraphic dark energy. We have reconstructed the potential of the
agegraphic scalar field as well as the dynamics of the scalar field which
describe the Chaplygin cosmology. Finally, we have extended our study to the
agegraphic generalized Chaplygin gas dark energy model when there is an
interaction between generalized Chaplygin gas energy density and dark matter.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha.
## References
* (1) A.G. Riess, et al., Astron. J. 116 (1998) 1009;
S. Perlmutter, et al., Astrophys. J. 517 (1999) 565;
S. Perlmutter, et al., Astrophys. J. 598 (2003) 102;
P. de Bernardis, et al., Nature 404 (2000) 955.
* (2) T. Padmanabhan, Phys. Rep. 380 (2003) 235;
P. J. E. Peebles, B. Ratra, Rev. Mod. Phys. 75 (2003) 559;
E. J. Copeland, M. Sami and S. Tsujikawa, hep-th/060305.
* (3) A. Cohen, D. Kaplan, A. Nelson, Phys. Rev. Lett. 82 (1999) 4971\.
* (4) M. Li, Phys. Lett. B 603 (2004) 1.
* (5) Q. G. Huang, M. Li, JCAP 0408 (2004) 013.
* (6) S. D. H. Hsu, Phys. Lett. B 594 (2004) 13.
* (7) E. Elizalde, S. Nojiri, S.D. Odintsov, P. Wang, Phys. Rev. D 71 (2005) 103504;
B. Guberina, R. Horvat, H. Stefancic, JCAP 0505 (2005) 001;
B. Guberina, R. Horvat, H. Nikolic, Phys. Lett. B 636 (2006) 80;
H. Li, Z. K. Guo, Y. Z. Zhang, Int. J. Mod. Phys. D 15 (2006) 869;
Q. G. Huang, Y. Gong, JCAP 0408 (2004) 006;
J. P. B. Almeida, J. G. Pereira, Phys. Lett. B 636 (2006) 75;
Y. Gong, Phys. Rev. D 70 (2004) 064029;
B. Wang, E. Abdalla, R. K. Su, Phys. Lett. B 611 (2005) 21;
J.Y . Shen, B. Wang, E. Abdalla, R. K. Su, Phys. Lett. B 609 (2005) 200.
* (8) M. R. Setare, Phys. Lett. B 642 (2006) 1;
M. R. Setare, S. Shafei, JCAP 09 (2006) 011;
M. R. Setare, E. C. Vagenas, Phys. Lett. B 666 (2008) 111;
H. M. Sadjadi, arXiv:0902.2462;
M. R. Setare, E. N. Saridakis, Phys. Lett. B 671 (2009) 331;
M. Jamil, E. N. Saridakis, M. R. Setare, arXiv: 0906.2847.
* (9) M. R. Setare, Eur. Phys. J. C 50 (2007) 991;
M. R. Setare, JCAP 0701 (2007) 023;
M. R. Setare, Phys. Lett. B 642 (2006) 421.
* (10) G. t Hooft, gr-qc/9310026;
L. Susskind, J. Math. Phys. 36 (1995) 6377\.
* (11) X. Zhang, F. Q. Wu, Phys. Rev. D 72 (2005) 043524;
X. Zhang, F. Q. Wu, Phys. Rev. D 76 (2007) 023502;
Q. G. Huang, Y.G. Gong, JCAP 0408 (2004) 006;
K. Enqvist, S. Hannestad, M. S. Sloth, JCAP 0502 (2005) 004;
J.Y. Shen, B. Wang, E. Abdalla, R.K. Su, Phys. Lett. B 609 (2005) 200;
Q. Wu, et al., arXiv:0705.1006.
* (12) H. Wei and S. N. Zhang, arXiv:0707.2129
* (13) R. G. Cai, Phys. Lett. B 657 (2007) 228.
* (14) H. Wei and R. G. Cai, Phys. Lett. B 660 (2008) 113.
* (15) H. Wei and R. G. Cai, Eur. Phys. J. C 59 (2009) 99.
* (16) H. Wei and R. G. Cai, Phys. Lett. B 663 (2008) 1;
J. Cui, et al., arXiv:0902.0716;
Y. W. Kim, et al., Mod. Phys. Lett. A 23 (2008) 3049;
Y. Zhang, et al. arXiv:0708.1214;
J .P Wu, D. Z. Ma, Y. Ling, Phys. Lett. B 663, (2008) 152;
K. Y. Kim, H. W. Lee, Y. S. Myung, Phys.Lett. B 660 (2008) 118;
X. Wu, et al., arXiv:0708.0349;
J. Zhang, X. Zhang, H. Liu, Eur. Phys. J. C 54 (2008) 303;
I. P. Neupane, Phys. Lett. B 673 (2009) 111.
* (17) A. Sheykhi, arXiv:0907.5144;
A. Sheykhi, 0908.0606.
* (18) M. R. Setare, arXiv:0907.4910;
M. R. Setare, arXiv:0908.0196.
* (19) V. Gorini, A. Kamenshchik, U. Moschella, V. Pasquier, gr-qc/0403062.
* (20) A.Yu. Kamenshchik, U. Moschella, V. Pasquier, Phys. Lett. B 511 (2001) 265.
* (21) D. Bazeia, R. Jackiw, Ann. Phys. 270 (1998) 246;
D. Bazeia, Phys. Rev. D 59 (1999) 085007;
R. Jackiw, A.P. Polychronakos, Commun. Math. Phys. 207 (1999) 107;
N. Ogawa, Phys. Rev. D 62 (2000) 085023.
* (22) N. Bilic, G.B. Tupper, R.D. Viollier, Phys. Lett. B 535 (2002) 17;
N. Bilic, G.B. Tupper, R.D. Viollier, astro-ph/0207423.
* (23) M. Bordemann, J. Hoppe, Phys. Lett. B 317 (1993) 315;
J.C. Fabris, S.V.B. Gonsalves, P.E. de Souza, Gen. Relativ. Gravit. 34 (2002)
53\.
* (24) M.C. Bento, O. Bertolami, A.A. Sen, Phys. Lett. B 575 (2003) 172.
* (25) M. R. Setare, Phys. Lett. B 648 (2007) 329.
* (26) M. R. Setare, Eur. Phys. J. C 52 (2007) 689;
M. R. Setare, Phys. Lett. B 654 (2007) 1;
M. R. Setare, Int. J. Mod. Phys. D 18 (2009) 419.
* (27) D. N. Spergel, Astrophys. J. Suppl. 148 (2003) 175;
C. L. Bennett, et al., Astrophys. J. Suppl. 148 (2003) 1;
U. Seljak, A. Slosar, P. McDonald, JCAP 0610 (2006) 014;
D. N. Spergel, et al., Astrophys. J. Suppl. 170 (2007) 377.
* (28) F. Karolyhazy, Nuovo.Cim. A 42 (1966) 390;
F. Karolyhazy, A. Frenkel and B. Lukacs, in Physics as natural Philosophy
edited by A. Shimony and H. Feschbach, MIT Press, Cambridge, MA, (1982);
F. Karolyhazy, A. Frenkel and B. Lukacs, in Quantum Concepts in Space and Time
edited by R. Penrose and C.J. Isham, Clarendon Press, Oxford, (1986).
* (29) M. Maziashvili Int. J. Mod. Phys. D 16 (2007) 1531;
M. Maziashvili, Phys. Lett. B 652 (2007) 165.
|
arxiv-papers
| 2010-02-07T07:45:24 |
2024-09-04T02:49:08.290286
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1002.1435"
}
|
1002.1551
|
# Magnetotransport in a time-modulated double quantum point contact system
Chi-Shung Tang cstang@nuu.edu.tw Kristinn Torfason Vidar Gudmundsson
Department of Mechanical Engineering, National United University, Miaoli
36003, Taiwan Science Institute, University of Iceland, Dunhaga 3, IS-107
Reykjavik, Iceland
###### Abstract
We report on a time-dependent Lippmann-Schwinger scattering theory that allows
us to study the transport spectroscopy in a time-modulated double quantum
point contact system in the presence of a perpendicular magnetic field.
Magnetotransport properties involving inter-subband and inter-sideband
transitions are tunable by adjusting the time-modulated split-gates and the
applied magnetic field. The observed magnetic field induced Fano resonance
feature may be useful for the application of quantum switching.
###### keywords:
magnetotransport , time-modulated , quantum point contact , magnetic field
††journal: Computer Physics Communications
## 1 Introduction
Electron transport in mesoscale devices smaller than the electron phase
coherence length has received extensive studies [1]. Magnetotransport and
time-dependent transport in gate-controlled semiconducting systems are
essential fundamental entities in mesoscopic physics. Recently, It was
reported that the conductance involving Aharonov-Bohm (AB) interference as a
function of magnetic field exhibits step-like structures [2]. Sigrit et al.
measured the differential conductance of an AB interferometer by varying the
bias voltage [3]. Their results indicate that varying either the magnetic
field or the electrostatic confining potentials allows the interference to be
tuned. In this work, we investigate the magneto-conductance in a double
quantum point contact (DQPC) system by controlling two pairs of split-gate
(SG) voltages for the manipulation of the dynamical electronic transport
properties in the DQPC-confined cavity region.
## 2 Model
The system under investigation is supposed to be a parabolically confined
quantum channel fabricated from a modulation-doped GaAs-based heterostructure
with two-pairs of spit-gates defining the DQPC system treated as a scattering
potential $V_{\rm sc}(x,y,t)$, as depicted in Fig. 1.
Figure 1: Schematic illustration of the double quantum point contact system
constructed by two pairs of split-gates with gate voltages $V_{g}$ described
by $V_{\rm sc}$ in our model.
The Hamiltonian describing the system can be expressed in the form
${\cal
H}(t)=-\frac{\hbar^{2}}{2m^{*}}\left(\nabla^{2}-\frac{2i}{l^{2}}y\partial_{x}-\frac{y^{2}}{l^{4}}\right)+\frac{m^{*}}{2}\Omega_{0}^{2}y^{2}+V_{\rm
sc}\,,$ (1)
in which the effective mass $m^{*}$=$0.067m_{e}$, and the magnetic length
$l$=$\hbar/(eB)$ is related to the perpendicular magnetic field ${\bf
B}$=$B{\hat{\bf z}}$. The characteristic confining energy $\hbar\Omega_{0}$ of
the parabolic confinement is modified by the applied magnetic field leading to
the effective confining energy
$\hbar\Omega_{\omega}=\hbar\left(\omega_{c}^{2}+\Omega_{0}^{2}\right)^{1/2}$
where $\omega_{\rm c}=eB/(m^{*}c)$. The scattering potential
$V_{\rm sc}(x,y,t)=V_{s}(x,y)+\sum_{i=1}^{2}V_{t}(x,y)\cos(\omega t+\phi_{i})$
(2)
contains a static part $V_{s}$ as well as a time-dependent part with strength
$V_{t}$ and driving frequency $\omega$. The time-modulated SGs may have an
arbitrary phase $\phi_{i}$.
We employ the mixed momentum-coordinate representation [4] to transform the
total wave function $\Psi(x,y,t)=\sum_{n}\phi_{n}(y,p)\psi_{n}(p,t)$ into the
wave function $\Psi(p,y,t)$ in terms of the eigenfunctions $\phi_{n}(y,p)$ of
the unperturbed quantum channel. Performing the expansion allows us to obtain
a coupled nonlocal time-dependent integral equation in the momentum space:
$\displaystyle i\hbar\partial_{t}\psi_{n}(p,t)$ $\displaystyle=$
$\displaystyle\left[E_{n}(0)+K(p)\right]\psi_{n}(p,t)$ (3)
$\displaystyle+\sum_{n^{\prime}}\int\frac{q}{2\pi}V_{n,n^{\prime}}(p,q,t)\psi_{n^{\prime}}(q,t)\,.$
This equation describes the electron propagation of an asymptotic state
occupying subband $n$ along the $\mathbf{x}$-direction from the source
electrode. Here the subband threshold
$E_{n}(0)=\left(n+1/2\right)\hbar\Omega_{\omega}$ is determined by the lateral
confinement and the effective kinetic energy
$K(p)=\hbar^{2}p^{2}(\hbar\Omega_{0})^{2}/[2m^{*}(\hbar\Omega_{\omega})^{2}]$.
The matrix elements of the scattering potential
$V_{n,n^{\prime}}(p,q,t)=\int dy\
dxe^{-i(p-q)x}\phi_{n}^{*}(y,p)V(x,y,t)\phi_{n^{\prime}}(y,q)$ (4)
indicates the electrons in the subband $n$ may be making inter-subband
transitions to the intermediate subband $n^{\prime}$.
To proceed, it is convenient to transform the time-dependent wave function
from the time domain to the frequency domain
$\psi_{n}(p,t)=\sum_{m=-\infty}^{\infty}e^{-iE_{m}t/\hbar}\psi_{n}^{m}(p)\,,$
(5)
where the quasi-energy $E_{m}=E_{0}+m\hbar\omega$. Similarly, we have
$V_{n,n^{\prime}}(p,q,t)=\sum_{m^{\prime}=-\infty}^{\infty}e^{-im^{\prime}\omega
t}V_{nn^{\prime}}^{m^{\prime}}(p,q)$ with $m^{\prime}$ indicating the photon
sideband index. Defining the wave number of an electron occupying the subband
$n$ and the sideband $m$ intermediate state
$\frac{1}{2}\left(\frac{k_{n}^{m}}{\beta}\right)^{2}\frac{(\hbar\Omega_{0})^{2}}{\hbar\Omega_{\omega}}=E_{m}-E_{n}(0)$
(6)
and
$\widetilde{V}_{n,n^{\prime}}^{m-m^{\prime}}(q,p)\equiv
2\frac{(\hbar\Omega_{\omega})^{2}}{(\hbar\Omega_{0})^{2}}\frac{\beta}{\hbar\Omega_{\omega}}V_{n,n^{\prime}}^{m-m^{\prime}}(q,p)$
(7)
allows us to obtain the multiple scattering identity
$\psi_{n}^{m}(q)=\left[\left(\frac{k_{n}^{m}}{\beta}\right)^{2}-\left(\frac{q}{\beta}\right)^{2}\right]^{-1}\sum_{m^{\prime}n^{\prime}}\int\frac{dp}{2\pi}\widetilde{V}_{n,n^{\prime}}^{m-m^{\prime}}(q,p)\psi_{n^{\prime}}^{m^{\prime}}(p).$
(8)
Taking all the intermediate states $(n^{\prime},m^{\prime})$ into account, we
can obtain the Lippmann-Schwinger equation in the momentum space
$\displaystyle\psi_{n}^{m}(q)$ $\displaystyle=$
$\displaystyle\psi_{n}^{m,0}(q)+G_{n}^{m}(q)$ (9)
$\displaystyle\times\frac{1}{2\pi}\sum_{n^{\prime},m^{\prime}}\int
d\left(\frac{p}{\beta}\right)\,\widetilde{V}_{n,n^{\prime}}^{m-m^{\prime}}(q,p)\psi_{n^{\prime}}^{m^{\prime}}(p)$
in terms of the unperturbed Green function $G_{n}^{m}(q)$. Since the incident
wave $\psi_{n}^{m,0}(q)$ is of the delta-function type, to achieve exact
numerical computation one has to define the $T$ matrix
$\displaystyle T_{n^{\prime},n}^{m^{\prime},m}(q,p)$ $\displaystyle=$
$\displaystyle V_{n^{\prime},n}^{m^{\prime}-m}(q,p)$ $\displaystyle+$
$\displaystyle\sum_{r,s}\int\frac{dk}{2\pi}V_{n^{\prime},r}^{m^{\prime}-s}(q,k)G_{r}^{s}(k)T_{r,n}^{s,m}(k,p)$
that couples all the intermediate states $(r,s)$. The potential is expanded in
the Fourier series yields a connection between the sidebands for constructing
the $T$ matrix
$\displaystyle
T_{n^{\prime},n}^{m^{\prime},m}(q,p)=V_{s,n^{\prime}n}(q,p)\delta_{m^{\prime}-m,0}$
$\displaystyle+\frac{1}{2}V_{t,n^{\prime}n}(q,p)(\delta_{m^{\prime}-m,-1}+\delta_{m^{\prime}-m,1})$
$\displaystyle+\sum_{r}\int\frac{dk}{2\pi}V_{s,n^{\prime}r}(q,k)G_{r}^{m^{\prime}}(k)T_{r,n}^{m^{\prime},m}(k,p)$
$\displaystyle+\frac{1}{2}\sum_{r}\int\frac{dk}{2\pi}V_{t,n^{\prime}r}^{+}(q,k)G_{r}^{m^{\prime}+1}(k)T_{r,n}^{(m^{\prime}+1),m}(k,p)$
$\displaystyle+\frac{1}{2}\sum_{r}\int\frac{dk}{2\pi}V_{t,n^{\prime}r}^{-}(q,k)G_{r}^{m^{\prime}-1}(k)T_{r,n}^{(m^{\prime}-1),m}(k,p)$
(11)
where $V_{t,n^{\prime}r}^{\pm}(q,k)=\sum_{i}V_{t,n^{\prime}r}(q,k)e^{\pm
i\phi_{i}}$ coupling the adjacent sidebands. In terms of the $T$ matrix, we
can obtain the momentum-space wave function
$\displaystyle\psi_{n^{\prime}}^{m^{\prime}}(q)$ $\displaystyle=$
$\displaystyle\psi_{n^{\prime}}^{m^{\prime},0}(q)+G_{n^{\prime}}^{m^{\prime}}(q)$
(12)
$\displaystyle\times\sum_{n,m}\int\frac{dk}{2\pi}T_{n^{\prime},n}^{m^{\prime},m}(q,k)\psi_{n}^{m,0}(k)\,.$
Performing the inverse Fourier transform to the real space and the residue
integration allows us to obtain the transmission amplitude of the electron
wave along the $\mathbf{x}$-direction
$\mathbf{t}_{n^{\prime},n}^{m^{\prime},0}=\delta_{n^{\prime},n}\delta_{m^{\prime},0}-\frac{i}{2k_{n^{\prime}}^{m^{\prime}}}T_{n^{\prime},n}^{m^{\prime},0}\left(k_{n^{\prime}}^{m^{\prime}},k_{n}^{0}\right)\,.$
(13)
The time-average conductance can be obtained based on the Landauer-Büttiker
framework [5, 6]
$G=G_{0}\sum_{m^{\prime}=-\infty}^{\infty}{\rm
Tr}\left[\,\mathbf{t}_{n^{\prime},n}^{m^{\prime},0}\,\left(\mathbf{t}_{n^{\prime},n}^{m^{\prime},0}\right)^{*}\right]$
(14)
with $G_{0}=2e^{2}/h$. This indicates that the transmission matrix connecting
the contribution from all the photon sideband $m^{\prime}$ of propagating
modes has to be taken into account for the electrons occupying arbitrary
subbands below the Fermi energy.
## 3 Numerical Results
We assume that the system is fabricated in a high-mobility GaAs-based
heterostructure such that the effective Rydberg energy
$E_{\mathrm{Ryd}}\approx 5.9$ meV and the Bohr radius $a_{\rm B}\approx 9.8$
nm. The confining parameter of the quantum channel is $\hbar\Omega_{0}=1$ meV,
the length is scaled by $\beta_{0}^{-1}$ $\approx 33.7$ nm, and the energy is
either in $\mathrm{meV}$ or in units of $\hbar\Omega_{\omega}$. The
$\hbar\Omega_{\omega}$ = $1.0148$ meV for the magnetic field $B$ = $0.1$ T.
The time-modulated DQPC system is described by the scattering potential
$V_{\rm sc}(\mathbf{r},t)=V_{\rm SG1}(\mathbf{r},t)+V_{\rm
SG2}(\mathbf{r},t)\,,$ (15)
where
$V_{\rm
SG1}=V_{1}(t)\left[e^{-\alpha_{x}(x+x_{0})^{2}}+e^{-\alpha_{x}(x+x_{0})^{2}}\right]e^{-\alpha_{y}(y+y_{0})^{2}}$
(16)
and
$V_{\rm
SG2}=V_{2}(t)\left[e^{-\alpha_{x}(x-x_{0})^{2}}+e^{-\alpha_{x}(x-x_{0})^{2}}\right]e^{-\alpha_{y}(y+y_{0})^{2}}$
(17)
with $V_{i}(t)$ = $V_{s}+V_{t}\cos(\omega t+\phi_{i})$ and $i$=$1,2$.
Moreover, we select $(\alpha_{x},\alpha_{y})$ = $(0.5,0.3)\beta_{0}^{2}$, and
$(x_{0},y_{0})$ = $(8,3)\beta_{0}$ such that the gate-width $\sim 80$ nm and
the SG-confined cavity area $\sim 540\times 200$ nm2.
Figure 2: Conductance as a function of incident energy for the cases of
$B=0.0$ T (red dashed) and $B=0.1$ T (blue solid). The other parameters are
$V_{s}=6.0$ meV, $V_{t}=1.5$ meV, $\phi=\pi$, and
$\omega=0.17\Omega_{\omega}$.
In Fig. 2, we show the conductance as a function of incident energy for the
time-modulated DQPC with applied magnetic field $B=0.1$ T (blue solid) in
comparison with the zero magnetic field situation (red dashed). The DQPC
system is confined by $V_{s}$ = $6.0$ meV, and the time-modulation with
strength $V_{t}=1.5$ meV and frequency $\omega=0.17\Omega_{\omega}$. In
addition, we have assumed that the phase difference between the two split-
gates SG1 and SG2 is $\phi=\phi_{1}-\phi_{2}=\pi$. In general, the electron
kinetic energy turns out to play a role of suppressing the quasibound state
feature, namely suppressing the side-peak structures beneath a main resonance
peak in conductance. However, in the high kinetic energy regime, an
appropriate magnetic field may induce the time-modulated Fano antiresonance
features at energies $E/\hbar\Omega_{\omega}\approx$ 1.75, 1.92, 2.16 as well
as the time-modulated Breit-Winger dip feature at
$E/\hbar\Omega_{\omega}\approx$ 2.38. Below, we focus on the robust Fano line-
shape feature: The Fano-peak is at $E/\hbar\Omega_{\omega}=1.918$ and the
Fano-dip is at $E/\hbar\Omega_{\omega}=1.920$, as depicted in Fig. 2.
In order to get better understanding of the magnetic-field induced time-
nodulated Fano antiresonance feature, we explore the electronic probability
density with energies around the Fano line-shape. It is clearly shown in Fig.
3 that the electron occupying the first subband with higher kinetic energy
favors to form a long-lived (4,2) localized bound state in the cavity formed
by the DQPC system. However, the electron occupying the second subband with
lower kinetic energy is not fitting to the characteristic energies in the
cavity and hence forming a short-lived (4,1) extended state. The interference
of the $n=0$ localized state and the $n=1$ extended state induces the Fano
peak at the energy $E/\hbar\Omega_{\omega}=1.918$.
Figure 3: Probability density with magnetic field $B=0.1$ T at the Fano peak
with electronic energy $E/\hbar\Omega_{\omega}=1.918$ for the electron
incident from the subband $n=0$ (left) and $n=1$ (right).
The probability density features for the electron with incident energy at
$E/\hbar\Omega_{\omega}=1.920$ are demonstrated in Fig. 4. The electrons
occupying the lowest subband ($n=0$) can also form a long-lived (4,2)
localized state, but with higher coupling to source-lead thus forming the
Fano-dip line-shape. For the electron occupying the second subband ($n=1$) at
$E/\hbar\Omega_{\omega}=1.920$, the extended (4,1) state is weaker than the
electron with $E/\hbar\Omega_{\omega}=1.918$.
Figure 4: Probability density with magnetic field $B=0.1$ T at the Fano dip
with electronic energy $E/\hbar\Omega_{\omega}=1.920$ for the electron
incident from the subband $n=0$ (left) and $n=1$ (right).
The energy difference $\delta E_{\rm Fano}\approx 2.03$ $\mu{\rm eV}$ between
the Fano-peak and the Fano-dip should be within the observable resolution via
the current transport measurement technique.
## 4 Summary
In summary, we have presented coherent magnetotransport numerical calculation
on a time-modulated double QPC system and demonstrated dynamical control of
the magnetic-field induced Fano interference by manipulating the applied
magnetic field. It was reported that the anti-symmetric ac split-gate voltage
can be utilized to induce the Fano resonance [7]. Differently, we have
reported here by tuning an appropriate magnetic field in the DQPC system with
symmetric ac split-gates to induce the Fano resonance that becomes non-
resonant by switching off the applied magnetic field. This robust magnetic
field induced dynamic Fano resonance feature may be useful for the magneto-
control of quantum switching in arbitrary time-modulated mesoscopic systems.
## Acknowledgments
This work was supported by the Research and Instruments Funds of the Icelandic
State; the Research Fund of the University of Iceland; the Icelandic Science
and Technology Research Programme for Postgenomic Biomedicine, Nanoscience and
Nanotechnology; and the National Science Council of the Republic of China
through Contract No. NSC97-2112-M-239-003-MY3.
## References
* [1] C. W. J. Beenakker and H. van Houten, in Solid State Physics: Advances in Research and Applications, edited by H. Ehrenreich and D. Turnbull (Academic, New York, 1991), Vol. 44, pp. 1-228.
* [2] F. E. Camino, W. Zhou, and V. J. Goldman, Phys. Rev. B 72 (2005) 155313.
* [3] M. Sigrist, T. Ihn, K. Ensslin, M. Reinwald, and W. Wegscheider, Phys. Rev. Lett. 98 (2007) 036805.
* [4] S. A. Gurvitz, Phys. Rev. B 51 (1995) 7123.
* [5] R. Landauer, IBM J. 1 (1957) 223.
* [6] M. Büttiker and R. Landauer, Phys. Rev. Lett. 49 (1982) 1739.
* [7] M. Yang and S.-S. Li, Phys. Rev. B 70 (2004) 045318.
|
arxiv-papers
| 2010-02-08T08:54:46 |
2024-09-04T02:49:08.296278
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chi-Shung Tang, Kristinn Torfason, Vidar Gudmundsson",
"submitter": "Chi-Shung Tang",
"url": "https://arxiv.org/abs/1002.1551"
}
|
1002.1693
|
# Sensitivity analysis of random two-body interactions
Calvin W. Johnson Department of Physics, San Diego State University, 5500
Campanile Drive, San Diego, CA 92182-1233 Plamen G. Krastev Department of
Physics, San Diego State University, 5500 Campanile Drive, San Diego, CA
92182-1233 Lawrence Livermore National Laboratory, P.O. Box 808, L-414,
Livermore, CA 94551
###### Abstract
The input to the configuration-interaction shell model includes many dozens or
hundreds of independent two-body matrix elements. Previous studies have shown
that when fitting to experimental low-lying spectra, the greatest sensitivity
is to only a few linear combinations of matrix elements. Here we consider
interactions drawn from the two-body random ensemble, or TBRE, and find that
the low-lying spectra are also most sensitive to only a few linear
combinations of two-body matrix elements, in a fashion nearly
indistinguishable from an interaction empirically fit to data. We find in
particular the spectra for both the random and empirical interactions are
sensitive to similar matrix elements, which we analyze using monopole and
contact interactions.
††preprint: INT-PUB-10-008††preprint: LLNL-JRNL-423216
## I Introduction and Motivation
The configuration-interaction shell model is a useful framework for a detailed
understanding of low-energy nuclear structure BG77 ; br88 ; ca05 . The many-
body basis is a large dimension ($10^{3-10}$) set of Slater determinants,
which are antisymmeterized products of single-particle states. One must
truncate the single-particle states, corresponding to one or a few shells
(typically using the harmonic oscillator as an approximation to the mean-
field); the many-body basis may be further truncated. For phenomenological
calculations one writes the Hamiltonian in terms of single-particle energies
and two-body matrix elements, while for ab initio calculations one may extend
this to three-body interactions NO03 .
The two-body matrix elements are the matrix elements of the residual
interaction in the lab frame,
$V_{JT}(ij,kl)=\langle ij;JT|\hat{V}|kl;JT\rangle$ (1)
where $|ij;JT\rangle$ is the normalized, antisymmeterized product of particles
in orbits labeled by $i$ and $j$ and coupled to good angular momentum $J$ and
isospin $T$. If one starts from a translationally invariant interaction
between particles, one can either compute the integral in the lab frame or
start in the relative frame and then transform to the lab frame; in either
case there are correlations between the matrix elements, although they are not
obvious to the casual observer.
Often for semi-phenomenological calculations, one starts from a “realistic”
interaction, and then adjusts the two-body matrix elements until the rms error
on a set of experimentally known energy levels is minimized BG77 . In the
$1s$-$0d$ shell, such a semi-phenomenological interaction has been recently
derived br06 , improving on an earlier interactionWildenthal .
It has been found that the fits are empirically dominated by a few linear
combinations of matrix elements. The physical meaning of those dominant
combinations is not immediately obvious. One might naively guess the linear
correlations are due to an underlying translationally invariant interaction
(although a density dependence would destroy this). Somewhat more
phenomenologically, it has been argued by appealing to mean-field properties
that one can improve fits primarily through adjusting the monopole-monopole
part of the interaction, that is, interaction terms that look like
$n_{a}(n_{b}-\delta_{ab})$, where $n_{a}$ is the number of particles in the
$a$th orbit. This protocol for shifting monopole strengths has been
successfully applied to several semi-empirical interactions po81 ; ma97 ; ut99
; ho04 ; su06
A related study BJ09 , investigating the origin of many-body forces from
truncation of the model space, also found an empirical fit dominated by a few
linear combinations of matrix elements. While much of the fit was dominated by
the monopole interactions, even better agreement was brought about using a
contact interaction motivated by its usage in mean-field calculations sk56 ;
be03 and effective field theoryvk99 .
In investigating the character and origin of the dominant matrix elements, it
is useful to ask if there is anything special about the nuclear interaction.
One way to ask this question is to compare with interactions drawn from the
two-body random ensemble (TBRE), which despite their arbitrary nature are
known to echo some features of real nuclear spectra JBD98 ; ZV04 ; ZAY04 ;
PW07 . In this paper we conduct a sensitivity analysis of the low-lying
spectra of random interactions and compare against a standard empirical
interaction, USDB. We find that for more measures all the interactions are
nearly indistinguishable, at least on a statistical level.
## II Methodology and results
Our methodology follows previous work BG77 ; br06 ; BJ09 ; we work in the
$1s$-$0d$ valence space with an inert 16O core. Given an input set of two-body
matrix elements (we leave aside single-particle energies and any $A$-dependent
scaling), which we write as a vector $\vec{v}$, we can calculate the
eigenvalues $E_{\alpha}(\vec{v})$ of the many-body Hamiltonian. For this work
the label $\alpha$ ranged over all nuclides with $0\leq
Z_{\mathrm{valence}}\leq N_{\mathrm{valence}}\leq 10$ and took the ground
state binding energy and the first five excitation energies.
If one has a target spectrum $E_{\alpha}^{0}$, say from experiment, then the
goal of the fit is to minimize
$\sum_{\alpha}\left(E_{\alpha}(\vec{v})-E_{\alpha}^{0}\right)^{2}$ (2)
(for simplicity we leave off the experimental uncertainty in each state).
Expanding to first order
$E_{\alpha}(\vec{v}+\delta\vec{v})\approx E_{\alpha}(\vec{v})+\sum_{i}\delta
v_{i}\frac{\partial E_{\alpha}}{\partial v_{i}}$ (3)
then minimizing (2) yields
$\sum_{\alpha}\left(E_{\alpha}(\vec{v})-E_{\alpha}^{0}\right)\frac{\partial
E_{\alpha}}{\partial v_{i}}=\sum_{j}\sum_{\alpha}\frac{\partial
E_{\alpha}}{\partial v_{i}}\frac{\partial E_{\alpha}}{\partial v_{j}}\delta
v_{j}.$ (4)
This equation is in the form $\vec{b}=\mathbf{A}\vec{x}$ where
$A_{ij}=\sum_{\alpha}\frac{\partial E_{\alpha}}{\partial v_{i}}\frac{\partial
E_{\alpha}}{\partial v_{j}}.$ (5)
The derivatives come via the Hellmann-Feynman theorem F39
$\frac{\partial E_{\alpha}}{\partial
v_{i}}=\left\langle\Psi_{\alpha}\left|\hat{H}_{i}\right|\Psi_{\alpha}\right\rangle$
(6)
where $\hat{H}_{i}$ is the Hamitonian operator whose strength is $v_{i}$.
We then find the eigenvalues of $\mathbf{A}$ (which is equivalent to finding
the squares of the eigenvalues in the singular-value decomposition of
$\partial E_{\alpha}/\partial v_{i}$). We do this for both the USDB
interaction and for an ensemble of 100 sets of random two-body interactions,
also called the two-body random ensemble (TBRE). The results are shown for the
TBRE in Fig. 1, where we have separated out the sensitivity just for the
binding energies (ground state energies) and the excitation energies. Although
not shown, the equivalent SVD eigenvalues for USDB are completely within the
TBRE results.
Figure 1: (Color online)The spectra of eigenvalues from a singular-value
decomposition of the sensitivity matrix $\mathbf{A}$ (Eq. 5) for the two-body
random ensemble. The lower curve is for ground state energies, while the upper
curve is for excitation energies.
The lower curve is for ground states only, while the upper curve is for
excitations energies relative to the ground state. Clearly, and perhaps
unsurprisingly, the ground state energies are predominantly sensitive to just
a few linear combinations of matrix elements–significantly fewer than
excitations energies.
Fig. 1 characterizes eigenvalues of $\mathbf{A}$. The next step is to
characterize the eigenvectors associated with the dominant eigenvalues,
specifically by comparing with monopole and contact interactions. To do so we
first discuss a method for quantifying the overlap between two vector
subspaces MC ; SK08 . Consider two vectors subspaces, $S_{1}$ and $S_{2}$. Let
$\mathbf{V}_{1}$ be a matrix whose column vectors are the (orthonormal) basis
vectors of $S_{1}$, and similarly with $\mathbf{V}_{2}$. From these one
constructs the overlap matrix
$\mathbf{\Omega}=\mathbf{V}_{1}^{\dagger}\mathbf{V}_{2}$. Note that if the
subspaces are not of equal dimension then $\mathbf{\Omega}$ is not a square
matrix. In any case we do a singular value decomposition of $\mathbf{\Omega}$;
the SVD eigenvalue spectrum then is a measure of the overlap of the two
spaces. If the two spaces perfectly overlap then all eigenvalues are 1, if
just $N$ of the dimensions perfectly overlap than $N$ eigenvalues will be 1
and the rest zero. Note that this method is invariant under arbitrary choice
of orthonormal bases.
We begin with the monopole-monopole interaction of the form
$n_{a}(n_{b}-\delta_{ab})$, which has six unique terms, and thus six vectors
or linear combinations of matrix elements, in the $sd$-shell. These we combine
with the $k$ most dominant linear combinations that arise from the previous
analysis; somewhat arbitrarily we chose $k=6$ (our results do not change
qualitatively for other small values of $k$). The results, the SVD eigenvalues
of $\Omega$, are shown in Fig. 2. It is important to note we compute the
spectrum separately for each randomly generated interaction and then compute
the distribution.
Figure 2: (Color online) The SVD spectrum that measures the overlap of the
subspace defined by the six largest eigenvectors from Fig. 1, with the
subspace defined by the monopole-monopole interaction. (Black) squares are for
the TBRE, while (red) diamonds are for USDB.
The results for ground states and for excited states are similar, so we
combine all states into a single calculation. The eigenvalues for USDB are
roughly $50\%$ higher than for the TBRE, but otherwise qualitatively very
similar.
We also compared for contact interactions; we took only two terms, the
$S=0,T=1$ channel and $S=1,T=0$ channel (there being only the $s$-wave channel
in relative coordinates). These results we summarize in Table 1.
Table 1: Leading eigenvalues from SVD of subspace overlaps of two-term contact interaction Interaction | ground states | excited states | all states
---|---|---|---
USDB | 0.60 | 0.58 | 0.62
TBRE | $0.55\pm 0.04$ | $0.41\pm 0.06$ | $0.55\pm 0.04$
For comparison, the leading eigenvalue for the overlaps of USDB versus the
six-term monopole is 0.94, while that of the TBRE versus monopole is $0.66\pm
0.03$. There is somewhat more sensitivity to the monopole interaction than the
contact interaction; however, the reader should keep in mind that is not the
whole story. Recall that when fitting an interaction, the linearized equations
are cast in the form $\mathbf{A}\vec{x}=\vec{B}$. Our analysis in this paper
is entirely with the eigenvalues of $\mathbf{A}$, but in any fit one must also
look to $\vec{b}$ (which in practice is the deviation of the theoretical
spectra from experiment). For example, in BJ09 it was found that using a
contact interaction brought better agreement than a monopole interaction. One
can understand this in terms of conjugate gradient methods NR : the direction
of local steepest gradient may not in fact point towards the global minimum.
Figure 3: SVD spectrum from the overlap of the dominant eigenvectors from
USDB and the TBRE.
By our measures so far, both the TBRE and the empirically-fit USDB look
qualitatively similar. Therefore we take a final analysis by comparing the
dominant linear combinations of the USDB with those from the TBRE. This is
show in Fig. 3, using the same analysis as for Fig. 2. For comparison with the
previous results, the leading eigenvalue is $0.72\pm 0.06$.
## III Conclusion
We have analyzed the sensitivity of the low-lying spectra of the random two-
body ensemble of interactions to variations of the Hamiltonian matrix
elements; by using singular value decomposition, we find the dominant linear
combinations, which would be important in any fit to experimental data. We
found the SVD eigenvalues follow a pattern remarkably similar to that shown by
semi-realistic/semi-phenomenological interactions such as USDB. We also
analyzed the most dominant linear combinations of matrix elements by computing
the overlap with monopole and contact interactions. Overall, both the TBRE and
the empirical USDB had qualitatively similar results.
The U.S. Department of Energy supported this investigation through contracts
DE-FG02-96ER40985 and DE-FC02-09ER41587, and through subcontract B576152 by
Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.
## References
* (1) P.J. Brussard and P.W.M. Glaudemans, Shell-model applications in nuclear spectroscopy (North-Holland Publishing Company, Amsterdam, 1977).
* (2) B. A. Brown and B. H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38, 29 (1988).
* (3) E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys. 77, 427 (2005).
* (4) P. Navrátil and W. E. Ormand, Phys. Rev. C 68, 034305 (2003).
* (5) B.A. Brown and W.A. Richter, Phys. Rev. C 74 034315 (2006).
* (6) B.H. Wildenthal, Prog. Part. Nucl. Phys. 11, 5 (1984).
* (7) A. Poves and A. Zuker: Phys. Rep. 70, 235 (1981).
* (8) G. Martínez-Pinedo, A. P. Zuker, A. Poves, and E. Caurier, Phys. Rev. C55 187 (1997).
* (9) Y. Utsuno, T. Otsuka, T. Mizusaki, and M. Honma: Phys. Rev. C 60, 054315 (1999).
* (10) M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Phys. Rev. C 69, 034335 (2004).
* (11) T. Suzuki, S. Chiba, T. Yoshida, T. Kajino, and T. Otsuka Phys.Rev. C 74, 034307 (2006).
* (12) G. F. Bertsch and C. W. Johnson, Phys. Rev. C 80, 027302 (2009).
* (13) T.H.R. Skyrme, Philos. Mag. 1, 1043 (1956).
* (14) M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003)
* (15) U. van Kolck, Prog. Part. Nucl. Phys. 43, 337 (1999); P. F. Bedaque and U. van Kolck, Annu. Rev. Nucl. Part. Sci. 52, 339 (2002); E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006).
* (16) C. W. Johnson, G. F. Bertsch, and D. J. Dean Phys. Rev. Lett. 80, 2749 (1998).
* (17) V. Zelevinsky and A. Volya, Phys. Rep. 391, 311 (2004).
* (18) Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rep. 400, 1 (2004).
* (19) T. Papenbrock and H. A. Weidenmüller, Rev. Mod. Phys. 79, 997 (2007)
* (20) H. Hellman, Einführung in die Quantenchemie (Franz Deuticke, Leipzig, 1937), p. 285; R. P. Feynman, Phys. Rev. 56, 340, (1939).
* (21) G. H. Golub and C. F. van Loan, Matrix Computations, Second Edition (The Johns Hopkins University Press, Baltimore, 1989)
* (22) S. Kvaal, Phys. Rev. C 78, 044330 (2008).
* (23) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and Brian P. Flannery, Numerical Recipes in Fortran, Second Edition (Cambridge University Press, Cambridge, 1992).
|
arxiv-papers
| 2010-02-08T19:33:00 |
2024-09-04T02:49:08.301910
|
{
"license": "Public Domain",
"authors": "Calvin W. Johnson and Plamen G. Krastev",
"submitter": "Calvin W. Johnson",
"url": "https://arxiv.org/abs/1002.1693"
}
|
1002.1738
|
# The Isospin Dependence Of The Nuclear Equation Of State Near The Critical
Point
M. Huang Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou,
730000,China. Graduate University of Chinese Academy of Sciences, Beijing,
100049, China. A. Bonasera bonasera@lns.infn.it Cyclotron Institute, Texas
A$\&$M University, College Station, Texas 77843 Laboratori Nazionali del Sud,
INFN,via Santa Sofia, 62, 95123 Catania, Italy Z. Chen Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843 Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou, 730000,China. R. Wada
wada@comp.tamu.edu Cyclotron Institute, Texas A$\&$M University, College
Station, Texas 77843 K. Hagel Cyclotron Institute, Texas A$\&$M University,
College Station, Texas 77843 J.B. Natowitz Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843 P.K. Sahu Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843 L. Qin Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843 T. Keutgen
FNRS and IPN, Université Catholique de Louvain, B-1348 Louvain-Neuve, Belgium.
S. Kowalski Institute of Physics, Silesia University, Katowice, Poland. T.
Materna Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843 J. Wang Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, 730000,China. M.Barbui Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843 C.Bottosso Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843 M.R.D.Rodrigues
Cyclotron Institute, Texas A$\&$M University, College Station, Texas 77843
###### Abstract
We discuss experimental evidence for a nuclear phase transition driven by the
different concentration of neutrons to protons. Different ratios of the
neutron to proton concentrations lead to different critical points for the
phase transition. This is analogous to the phase transitions occurring in
4He-3He liquid mixtures. We present experimental results which reveal the N/A
(or Z/A) dependence of the phase transition and discuss possible implications
of these observations in terms of the Landau Free Energy description of
critical phenomena.
Intermediate Heavy ion reactions, phase transition, Landau approach, symmetry
energy
###### pacs:
25.70Pq,21.65.Ef,24.10.-i
Nuclei are quantum Fermi systems which exhibit ma-
ny interesting features which depend on temperature and density. At zero
temperature and ground state density, nuclei are charged quantum drops, i.e.
they have a Fermi motion pres due to their quantum nature, and nucleons
interact through a short range attractive force and the long range Coulomb
repulsions among the constituent protons. In the absence of the Coulomb force,
the Nuclear Hamiltonian is perfectly symmetric for exchange of protons and
neutrons apart from a small but NOT insignificant difference between the
proton and neutron masses. This symmetry is revealed by similar energy levels
in mirror nuclei, i.e. nuclei with the same mass number A but opposite numbers
of neutrons, N, and protons, Z. Of course this feature is observed for
relatively small systems because the Coulomb energy is small pres . Analogous
to the properties of mirror nuclei, we could expect that if we study nuclei at
finite temperatures, T, and low densities, $\rho$, then, if the Coulomb force
is not important, the invariance under exchange of protons to neutrons might
lead to important and interesting consequences. In fact, since the fundamental
Hamiltonian of nuclei is invariant under exchange of N with Z (apart from
Coulomb effects), we could expect that such an invariance should be manifested
only at high T (disordered state), while there is a spontaneous symmetry
breaking at lower T (ordered state). That means that in symmetric nuclear
matter at high T, the state with fragments having $N=Z$ defines the minimum of
the free energy, i.e. symmetric fragments such as deuterons and alphas would
be favored at low density huang ; land . On the other hand, there could be a
symmetry breaking favoring $N\neq Z$ at lower T. In this case fragments near a
(first order) phase transition might prefer either a neutron or proton rich
configuration. There might even be a more interesting situation, suggested by
the present data, the existence of a line of first order phase transitions
huang which terminates in a tri-critical point. For such a line the free
energy has three equal minima: one with N=Z and the other two for $N\neq Z$.
Thus a phase transition is driven by the difference in isospin concentration
of the fragments $m=(N-Z)/A$. In this paper we will discuss data which clearly
demonstrate that m is an order parameter of the phase transition. Its
conjugate field huang which we indicate with H, is due to the chemical
potential difference between protons and neutrons of the emitting source at
the density and temperature reached during a collision between heavy ions prl
; huang10_1 . We also note that the phase transition has a strong resemblance
to that observed in superfluid mixtures of liquid 4He-3He near the $\lambda$
point. In both systems, changing the concentration of one of the components of
the mixture, changes the characteristics of the Equation Of State (EOS) huang
; land .
In recent times a large body of experimental evidence has been interpreted as
demonstrating the occurrence of a phase transition in finite nuclei at
temperatures (T) of the order of 6 MeV and at densities, $\rho$, less than
half of the normal ground state nuclear density wci . Even though strong
signals for a first and a second-order phase transition have been found wci ;
bon00 , there remain a number of open questions regarding the Equation of
State of nuclear matter near the critical point. In particular the roles of
Coulomb, symmetry, pairing and shell effects have yet to be clearly
delineated.
Theoretical modeling indicates that a nucleus excited in a collision expands
nearly adiabatically until it is close to the instability region thus the
expansion is isentropic siem . At the last stage of the expansion the role of
the Coulomb force becomes very important. In fact, without the Coulomb force,
the system would require a much larger initial compression and/or temperature
in order to enter the instability region and fragment. The Coulomb force acts
as an external piston, giving the system an ‘extra push’ to finally fragment.
These features are clearly seen in Classical Molecular Dynamics (CMD)
simulations of expanding drops with and without a Coulomb field belk95 ; dorso
. The expansion with the Coulomb force included is very slow in the later
stage and nearly isothermal.
Even though at high T and small $\rho$ the nucleus behaves as a classical
fluid, the analogy to classical systems should not be overemphasized as, in
the (T,$\rho$) region of interest, the nucleus is still a strongly interacting
quantum system. In particular the ratio of T to the Fermi energy at the
(presumed) critical point is still smaller than 1 which suggests that the EOS
of a nuclear system is quite different from the classical one. To date this
expected difference has not been well explored mor ; wci ; dago ; mas ; nat ;
ala ; maria .
The paper is organized as follows: in the next section we discuss the
experimental setup in detail. This is followed by a description of the data
analysis and a discussion in terms of the Landau O($m^{6}$) free energy. We,
then derive some critical exponents and the EOS corresponding to possible
scenarios suggested by our data in terms of the Fisher model of fragmentation.
Finally we draw some conclusions and suggest possible future work.
## I Experimental Details
The experiment was performed at the K-500 superconducting cyclotron facility
at Texas $A\&M$ University. 64,70Zn and 64Ni beams were incident on 58,64Ni,
112,124Sn, 197Au and 232Th targets at 40 A MeV. Intermediate mass fragments
(IMF) were detected by a detector telescope placed at 20o. The telescope
consisted of four Si detectors. Each Si detector had 5cm $\times$ 5cm area.
The thicknesses were 129, 300, 1000 and 1000 $\mu$m. All Si detectors were
segmented into four sections and each quadrant had a 5o acceptance in polar
and azimuthal angles. The fragments were detected at average angles of 17.5o
$\pm$ 2.5o and 22.5o $\pm$ 2.5o. Typically 6-8 isotopes were clearly
identified for a given Z up to Z=18 with an energy threshold of 4-10 A MeV,
using the $\Delta$E-E technique for any two consecutive detectors. The
$\Delta$E-E spectrum was linearized by an empirical code based on a range-
energy table. In the code, isotopes are identified by a parameter $Z_{Real}$.
For the isotopes with A=2Z, $Z_{Real}$ = Z is assigned and other isotopes are
identified by interpolating between them. The energy spectrum of each isotope
was extracted by gating on lines corresponding to the individual identified
isotopes. In order to compensate for the imperfectness of the linearization,
actual gates for isotopes were made on the 2D plot of $Z_{Real}$ versus
energy. The multiplicity of each isotope was evaluated from the extracted
energy spectra, using a moving source fit at the two given angles. Since the
energy spectra of some isotopes have very low statistics, the following
procedure was adopted for the fits. Using a single source with a smeared
source velocity around half of the beam velocity, the fit parameters were
first determined from the energy spectrum summed over all isotopes for a given
Z, assuming A=2Z. Then assuming that the shape of the velocity spectrum is the
same for all isotopes for a given Z. All parameters except the normalizing
multiplicity parameter were assumed to be the same as for the summed spectrum.
The multiplicity for a given isotope was then derived by normalizing the
standard spectrum to the observed spectrum for that isotope.
In order to evaluate the back ground contribution to the extracted
multiplicity a two Gaussian fit to each isotope peak was used with a linear
background. The second Gaussian (about 10$\%$ of the height of the first one)
was added to reproduce the valleys between isotopes. This component was
attributed to the reactions of the isotope in the Si detector. The centroid of
the main Gaussian was set to the value calculated from the range-energy table
within a small margin. The final multiplicity of an isotope with Z $>$ 2 was
obtained by correction of the multiplicity evaluated form the moving source
fit for the ratio between the sum of the two Gaussian yields and the linear
background.
The yields of light charged particles (Z $\leq$ 2) in coincidence with IMFs
were also measured using 16 single crystal CsI(Tl) detectors of 3cm thickness
set around the target. The light output from each detector was read by a photo
multiplier tube. The pulse shape discrimination method was used to identify p,
d, t, h and $\alpha$ particles. The energy calibration for these particles
were performed using Si detectors (50 -300 $\mu$m) in front of the CsI
detectors in separate runs. The yield of each isotope was evaluated, using a
moving source fit. Three sources (projectile-like(PLF), nucleon-nucleon-
like(NN) and target-like (TLF)) were used. The NN-like sources have source
velocities of about a half of the beam velocity. The parameters were searched
globally for all 16 angles. Detailed procedures of the data analysis are also
given in refs. huang10_2 ; chen10 .
Special care has been taken for 8He identification. All He isotopes are
identified in the Si telescope, using the $\Delta$E-E technique, in a narrow
energy range. When a proton and an $\alpha$ hit the same quadrant and when
both of them stop in the E detector, their $\Delta$E-E points overlap with
those of 8He. Since the multiplicities of protons and alphas are about three
orders of magnitude larger than that of 8He, the contribution of accidental
events becomes significant, especially for the reaction systems with lower
numbers of neutrons, in which 8He production is suppressed. Since Z=1
$\Delta$E-E spectra are not available in this experiment, $\Delta$E-E for Z
$\leq 3$ were measured in a separate run. Using the light charged particle
multiplicity extracted from the 16 CsI detectors, the accidental events were
simulated for each reaction for the observed $\alpha$ yield in the $\Delta$E-E
spectra in this experiment, the solid angle of the quadrant and the
multiplicity of Z=1 particles. In order to minimize the accidental events, the
runs with a low beam intensity were selected in each reaction. Typical
linearized $Z_{Real}$ spectra with these accidentals are shown in Fig.(1) for
70Zn + 232Th (N/Z $\sim$ 1.5) and 64Ni + 112Sn ( N/Z $\sim$ 1.25). As one can
see, the $Z_{real}$ values for the accidental events of proton and $\alpha$
pileup is nearly identical to that of 8He, while 6He is clearly identified.
The contributions from d + $\alpha$ and t + $\alpha$ are also reasonably
consistent with the observed background yields. A significant excess of 8He
yield beyond the accidentals is only observed for the reaction systems with
the 124Sn, 197Au and 232Th targets. After the correction of the accidental
contributions, the multiplicities of 6He and 8He were calculated using the
source fit parameters obtained for Li isotopes.
Figure 1: Typical $Z_{Real}$ spectra for He and Li isotopes. Accidental events
are generated only for p, d, t +$\alpha$ and shown separately by shaded
histograms as indicated.
## II Data Analysis
The key factor in our analysis is the value $I=N-Z$ of the detected fragments.
A plot of the yield versus mass number when I=0 displays a power law behavior
with yields decreasing as $A^{-\tau}$ min ; prl . This is shown in Fig.(2) for
the 64Ni+124Sn case at 40 MeV/nucleon. In the figure we have made separate
fits for odd-odd (open symbols) or even-even(filled symbols) nuclei. As seen,
different exponents $\tau$ appear which suggests that pairing is playing a
role in the dynamics pres , leading to higher yields for even-even nuclei.
Figure 2: Mass distribution for the 64Ni+124Sn system at 40 MeV/nucleon for
I=0. The lines are power law fits with exponents $2.3\pm 0.02$ (odd-odd
nuclei, dashed line), and $3.4\pm 0.06$ (even-even nuclei, full line)
respectively.
The observation of the power law behavior suggests that the mass distributions
may be discussed in terms of a modified Fisher model bon00 ; min :
$Y=y_{0}A^{-\tau}e^{-\beta\Delta\mu A},$ (1)
where y0 is a normalization constant, $\tau=2.3$ is a critical exponent bon00
, $\beta$ is the inverse temperature and $\Delta\mu=F(I/A)$ is the free energy
per particle, $F$, near the critical point. Recall that in general, the free
energy is a function of the mass A (volume), $A^{2/3}$(surface) and the
chemical composition $m$ of the fragments and possibly pairing. The region we
are studying in this paper seems near the critical point for a liquid gas
phase transition (volume and surface equal to zero) but modified by
$m=\frac{I}{A}$. Because of this modification we can observe different
features of the transition such as a first order phase transition driven by
$m$, the order parameter.
We begin our analysis by noting that the Fisher free energy is usually written
in terms of the volume and the surface of a drop undergoing a (second order)
phase transition fisher . Our data indicate that those terms are not important
in the present case bon00 as we will show more in detail below. If they are
negligible this suggests that we are near the critical point for a liquid gas
phase transition. Because we have two different interacting fluids, neutrons
and protons, the transition becomes more complex and more interesting than in
a single component liquid. Experiments at different energies might display a
free energy which depends on all these factors. If we accept that $F$ is
dominated by the symmetry energy we can make the approximation that
$F(I/A)=E_{sym}=25(I/A)^{2}$ MeV/A, i.e. the symmetry energy of a nucleus in
its ground state pres . We will use this relationship in order to infer an
approximate value of the temperature of the system. However, we stress that in
actuality, F(I/A) is a function of density, temperature and all other relevant
quantities near the critical point. According to the Fisher equation given
above, we can compare all systems on the same basis by normalizing the yields
and factoring out the power law term. For this purpose we have chosen to
normalize the yield data for each system to the 12C yield ($I=0$) in that
system, i.e. we define a ratio:
$R=\frac{YA^{\tau}}{Y(^{12}C)12^{\tau}}.$ (2)
The normalized ratios for the system 64Ni + 64Ni at 40 MeV/nucleon are plotted
as a function of the (ground state) symmetry energy in Fig.(3), bottom panel.
The data display an exponential decrease with increasing symmetry energy,
except for the isotopes for which $I=0$. The yields of these I = 0 isotopes
are of course not sensitive to the symmetry energy but rather to the Coulomb
and pairing energies and possibly to shell effects. A fit to the exponentially
decreasing portion of the data using the ground state symmetry energy gives an
‘apparent temperature’ T of 6.0 MeV. This value of T would be the real one if
only the symmetry energy is important, if entropy can be neglected, if
$a_{sym}=25$ MeV (the g.s. symmetry energy coefficient value) and if secondary
decay effects are negligible. In general we expect that the symmetry energy
coefficient is density and temperature dependent. Further, secondary decay
processes may modify the primary fragment distributions huang10_2 ; chen10 .
we will discuss these questions in the framework of the Landau free energy
approach below. We stress that the appearance of two branches in Fig.3
(bottom), indicates that the total free energy must contain an odd power term
in (I/A) at variance with the common expression for the ground state symmetry
energy. For reference in the top part of Fig.(3) we have plotted the ratio
versus the total ground state binding energy of the fragments. No clear
correlations are observed which might suggest that the symmetry energy
dominates the process.
Figure 3: Ratio versus fragments ground state binding energy (top panel) and
symmetry energy (bottom panel) for the 64Ni + 64Ni case at 40 MeV/nucleon.
$a_{s}ym$=25 MeV is used. The $I<0$ and $I>0$ ($I=0$)isotopes are indicated by
the open and full circles respectively (full squares). The dashed lines
(bottom panel) are fits using a ground state symmetry energy, Eq.1, and a
‘temperature’ of 6 MeV. Notice that the given experimental 8He yield is the
upper limit.
It is surprising that such a scaling appears as a function of the symmetry
energy only. In fact we might wonder about the role of the Coulomb energy if
we accept that surface and volume terms give negligible contribution. In
figure 4 we have plotted the same normalized ratios as a function of the
quantity $\alpha E_{coul}+\beta E_{sym}$, $\alpha$ and $\beta$ are arbitrary
parameters given in the figure and $E_{coul}=0.7Z(Z-1)A^{-1/3}$ is the Coulomb
contribution to the ground state energy of the nucleus. We see from the figure
that by decreasing the relative contribution of the Coulomb energy compared to
the symmetry energy the scaling appears. This implies that the Coulomb energy
is much less important than the symmetry energy near the critical point, which
suggests that either the density dependence of those two terms is different or
that, at the time of formation the fragments are strongly deformed, reducing
the Coulomb effect. Such deformations have been seen in CMD calculations of
fragmentation bon00 .
Figure 4: Ratio versus symmetry energy + Coulomb energy for the 64Ni + 124Sn
case at 40 MeV/nucleon. The panels from top to bottom are for different
combinations of the symmetry and Coulomb energy. The $I<0$ and $I>0$
($I=0$)isotopes are indicated by the open and full circles respectively (full
squares).
To further explore the role of the relative nucleon concentrations we plot in
Fig.(5) the quantity $\frac{F}{T}=-\frac{ln(R)}{A}$ versus $m=(I/A)$, the
difference in neutron and proton concentration of the fragment. As expected
the normalized yield ratios depend strongly on m.
Pursuing the question of phase transition we can perform a fit to these data
within the generalized Landau free energy description huang . In this approach
the ratio of the free energy to the temperature is written in terms of an
expansion:
$\frac{F}{T}=\frac{1}{2}am^{2}+\frac{1}{4}bm^{4}+\frac{1}{6}cm^{6}-m\frac{H}{T},$
(3)
where $m$ is an order parameter, $H$ is its conjugate variable and $a-c$ are
fitting parameters huang . We observe that the Free energy is even in the
exchange of $m\rightarrow-m$, reflecting the invariance of the nuclear forces
when exchanging N and Z. This symmetry is violated by the conjugate field $H$
which arises when the source is asymmetric in chemical composition. We stress
that m and H are related to each other through the relation $m=-\frac{\delta
F}{\delta H}$.
Figure 5: Free energy versus m for the case 64Ni+232Th. The full line is a
free fit based on Landau O($m^{6}$) free energy. The dashed-dotted-dotted-
dotted line is obtained imposing in the fit $b=-\sqrt{16/3ac}$ and it is
located on a line of first order phase transitions. The short dashed line
corresponds to $b=-\sqrt{4ac}$, i.e. superheating. The $O(m^{2})$ case,
$F/T=a(m-m_{s})^{2},$i.e. $b=c=0$, $m_{s}=0.1$, is given by the long dashed
line.
The use of the Landau approach is for guidance only. While the approximation
to $O(m^{4})$ does not work prl , the $O(m^{6})$ case is in good agreement
with the data. This is not surprising since, if fluctuations are important, a
higher order approximation to the free energy is better, i.e. gives critical
exponents closer to those seen in the data and satisfies the Ginzburg
criterion huang . A free fit using Eq.3 is displayed in Fig.5 (full line).
Notice the change of curvature near $m=0.3$, which incidentally is close to
$m_{cn}$ of the compound nucleus. For comparison in the same figure we have
displayed the $O(m^{2})$ case, i.e., $F/T=a(m-m_{s})^{2}$ ($b=c=0$)
111$F/T=a(m-m_{s})^{2}=(a/2)m^{2}-H/Tm+(a/2)m_{s}^{2};H/T=am_{(}s)$. The last
term is dropped out when the yields are normalized by 12C. As seen in the plot
last assumption also produces a reasonable fit, although it does not reproduce
shoulders near m $\sim\pm$0.3. As we will discuss in more detail below the
appearance of two minima for $m\neq 0$ (when $H/T=0$) might be a signature for
the existence of a first order phase transition occurring in these reactions.
In general the coefficients entering the Landau free energy Eq.(3), depend on
temperature, pressure or density of the source. Usually one assumes $c>0$,
$a=a_{0}(\rho)(T-T_{0})$ and $b=b(T,\rho)$, where $T_{0}$ is some ‘critical’
temperature discussed below. The precise determination of these parameters
determines the nuclear equation of state (NEOS) near the critical point. The
data we have do not allow such a complete constraining of the NEOS but do
suggest some interesting possible scenarios which we discuss below.
We begin by noting that the conjugate variable H which appears in equation (3)
is determined by the chemical composition of the source. Since, in general,
the source has $N\neq Z$, the extreme of $F/T$ are displaced from the values
obtained when H=0. In fact if we take the first derivative of the free energy
we get:
$(\frac{F}{T})^{\prime}=am+bm^{3}+cm^{5}-\frac{H}{T}.$ (4)
When $H/T=0$ the first derivative is zero for the following values of m huang
:
$m_{0}=0;m^{2}_{\pm}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2c}.$ (5)
If we now assume $H\neq 0$ but small, we can expand the solutions above as
$m=m_{0\pm}+\eta$ with $\eta$ small. Equating the first derivative to zero,
Eq.(4), and neglecting terms $O(\eta^{2})$ we get:
$\eta=\frac{H/T}{a+3bm_{0\pm}^{2}+5cm_{0\pm}^{4}}.$ (6)
The shift of the minimum from $m_{0}=0$ should be given by the equation above
and should be proportional to $m$ of the emitting source. We can easily check
this feature in our data. In Fig.(6) we plot the values of $H/T$ obtained from
the fits to our data for all systems using Eq.(3) versus $m_{cn}=(I/A)_{cn}$.
Figure 6: H/T versus (I/A) of the compound nucleus obtained from the data fit
to the Landau free energy, Eq.(3). The full circles are for 64Ni, the full
triangles are for 70Zn and the squares for 64Zn projectiles impinging on
various targets, see text.
The linear fit in Fig.6 is given by $H/T=0.47+1.6(I/A)_{cn}$ which agrees with
the linear dependence of Eq.(6). However, for this fit $H/T\neq 0$ for
$I_{cn}=0$ which could indicate the favoring of $N>Z$ fragments by the Coulomb
field. Another possibility is that $(I/A)_{source}\propto(I/A){cn}$ which then
gives $H=0$ when $I_{source}=0$. Finally we should consider that together with
H also the temperature may also be changing some since the collisions are
between different target-projectile combinations at the same beam energy. If
the temperature were the same then the coefficients of the free energy, Eq.3,
should be independent of the source size, only H/T should change. In Fig.(7)
we plot the parameters a, b and c as a function of the compound nucleus
$m_{cn}$. As we see there is some dependence which may reflect differences in
temperature. However, we note that the error bars and fluctuations are large
which may also indicate important secondary decay effects. Thus is not too so
easy to draw definite conclusions.
Figure 7: The parameters a,b and c versus (I/A) of the compound nucleus
obtained from the data fit to the Landau free energy, Eq.(3). The symbols are
like in figure 6.
Given the information on the parameters of the Landau free energy contained in
figures 6 and 7 we can discuss some features regarding the NEOS. In particular
for each reaction system we can estimate F/T when $H/T=0$. In figure 8 we plot
this quantity versus m of the fragments for various reactions.
Figure 8: F/T (H/T=0) versus m of the fragments obtained from the a, b and c
parameters fit to the Landau free energy, Eq.(3). Results for all
experimentally investigated reactions are displayed.
The curves do not differ much suggesting that temperatures are quite similar.
The fits exhibit curvature near $m=\pm 0.4$ which may suggest the presence of
additional minima at larger absolute values of m. This could indicate either a
first order phase transition or superheating (see below). The lack of data at
very large $m$ makes it difficult to constrain the fit. However, we can study
other situations of particular physical interest which arise when the
relationships among the parameters a, b and c are constrained. huang ; land ;
prl :
We have considered four such cases as follows:
1) superheating. This case corresponds to $b=-\sqrt{4ac}$ and gives two minima
at $m\neq 0$ and is plotted in figure 5 for the 64Ni+232Th system with a short
dashed line. These are not absolute minima, which occur only at $m=0$, and
they correspond to metastable states. They might be observed in high quality
data for collisions of more neutron rich or proton rich systems making a hot
source with $m_{source}\approx\pm 0.4$. In fact if the system could be gently
brought to the right temperature $T_{s}$, with the correct isotopic
composition, it might stay in the minimum, i.e. more fragments of that $m$
should appear;
2) line of first order phase transition. This corresponds to the condition
$b=-\sqrt{16ac/3}$ at a temperature $T_{3}$, which, if imposed on the fit of
the free energy, results in the dashed-dotted-dotted-dotted line of Fig.5.
This fit is of similar quality to the previous cases. Now the minima are at
$m\approx 0.6$, i.e. for more neutron rich fragments due to the fact that
$H/T\neq 0$. This suggests that in this situation we might produce a large
number of neutron rich fragments. However, most of those fragments are
probably unstable, thus coincidence measurements may be required to determine
their yields. Of course this feature should become important in neutron rich
stars.
3) first order phase transition. Corresponds to the case $a=0$ and determines
the ‘critical temperature’ $T_{0}$ where the minimum at $m=0$ disappears and
only the ones at $m\neq 0$ survive. This case is excluded by our present data.
However, the fit in Fig.(5) suggests an intermediate situation between this
and case (2) above.
4) line of second order phase transition, tri-critical point. Corresponds to
$a=0$ and $b>0$ ($T=T_{c}$). When $b=0$ as well we have a tricritical point
($T=T_{3c}$), i.e. the point where the line of first order phase transition
terminates into a second order phase transition. This case is also excluded by
our data.
Figure 9: F/T (H/T=0) versus m of the fragments obtained from the a, b and c
parameter fit to the Landau free energy, Eq.(3) for 64Ni+232Th. The four
curves correspond to vapor(full line), superheating(short dashed), line of
first order transition(3 critical dashed-dotted-dotted-dotted line), first
order phase transition(long dashed line), see text.
We can extrapolate the cases discussed above to $H/T=0$ as was done for Fig.8.
In Fig.9 we plot F/T (H/T=0) (extrapolated from the data) vs. $m$. Purists
will not call this the EOS but reserve that for the pressure vs. $m$ case
(that we discuss below). Since H/T is zero, the curves are symmetric with
respect to $m$. We see in the plot: vapor (dashed-dotted-dotted line)
$T>T_{s}$, superheating $(T=T_{s})$(dotted line), a point in the line of a
first order phase transition $(T=T_{3})$(dashed-dotted line) which displays
three equal minima at $m_{0}$ and $m_{\pm}$, see Eq.5, the experimental data
fit line(full line)$T<T_{3}$. We have also added the case $a=0$ which should
be obtained at $T=T_{0}$, where the minimum at $m=0$ becomes a maximum. There
is a series of cases not displayed in the figure, corresponding to the
temperatures between $T_{0}$ and $T_{3}$ where $m=0$ is still a minimum but
$\it{not}$ an absolute minimum. This corresponds to supercooling and might be
observed in gentle collisions of $N=Z$ nuclei similarly to the superheating
case.
The features in Fig.9 are reminiscent of the superfluid $\lambda$ transition
observed as some 3He is added to 4He huang . Pure 4He has a critical
temperature of 2.18 K. The critical temperature for the second-order
transitions decreases with increasing 3He concentration until at temperature,
$T=0.867$K, a first-order transition appears. This point is known as the tri-
critical point for this system. In a similar fashion, a nucleus, which can
undergo a liquid-gas phase transition, should be influenced by the different
neutron to proton concentrations. Thus the discontinuity observed in Fig.5
($m=0$) could be a signature for a tri-critical point as in the 4He-3He case.
We believe that our data, analyzed in terms of the the Landau O($m^{6}$) free
energy, suggest such a feature but are not sufficient to clearly demonstrate
this. Some other work campi ; gulm , also suggests that a line of critical
points might be found away from its ‘canonical’ position, i.e. at the end of a
first-order phase transition and, for small systems, even extending into the
coexistence region.
## III Critical exponents
In the fits discussed above the parameters a, b and c were left free since we
do not have any particular values to fix the scale. Nevertheless, we saw in
figure 8 that the free energy (H/T=0) looks very similar for the different
systems. Thus the values of the fitting parameters are similar apart from a
scaling factor. We can avoid unnecessary factors by defining suitable
dimensionless quantities. This can be accomplished by looking at the solutions
of the minima of the free energy, cf. Eq.5. In particular from the value at
the minimum, $m_{+}$ we can define the following quantities ($b\neq 0$):
$x=\frac{4ac}{b^{2}}.$ (7)
Recalling that $a$ is related to the distance from the critical temperature
while b and c should only depend on density huang , we deduce that x is a
measure of the distance $T-T_{3}$ from the critical temperature in a suitable
dimensionless fashion. Similarly we can define a reduced order parameter from
Eq.5:
$y=\frac{2cm^{2}}{|b|}.$ (8)
Thus Eq.5 can be rewritten as:
$y=1+\sqrt{1-x}.$ (9)
Near the critical point we know that the order parameter has a singular part
that behaves in a power law fashion, thus we can define the singular part as:
$M=\pm\sqrt{y-1}=\pm{(1-x)}^{1/4},$ (10)
defining the temperature ‘distance’ from the critical point, $|t|=|1-x|$,
immediately gives the value of a critical exponent: $\beta=\frac{1}{4}$. This
exponent is very close to the accepted experimental value as is well known in
the O($m^{6}$) Landau theory huang .
Figure 10: Order parameter versus reduced temperature for all studied
systems. The dashed line is given by Eq.(10), the vertical line indicates the
critical temperature $T_{3}$. To the right of this line the system is in a
superheated state. Supercooling occurs on the left of the vertical line and
M=0 huang .
In Fig.(10) the experimental values of M and x obtained within the Landau
theory are plotted together with the equilibrium condition given by Eq.(10).
Supercooling and Superheating regions, as discussed in the previous section,
can be identified as well huang .
As is the case for macroscopic systems we can now ‘turn ’ the external field H
on and off. In our case this is done with a suitable choice of the colliding
systems. In this way we can study the ‘EOS’ at the critical point by turning
on H:
$M=H^{1/\delta},$ (11)
which defines the critical exponent $\delta$. In the Landau theory this
exponent can be determined at the critical point where $a=b=0$. From equation
(4) we easily get $\delta=5$, which is the accepted value for such a critical
exponent huang . In order to exactly determine this exponent we need to bring
the system to the critical point. This does not appear to be the case for our
data as we saw in figure 10. Nevertheless a plot of the order parameter versus
H should display a power law behavior as it is well known in macroscopic
systems huang . A precise determination of the critical exponent requires the
knowledge of the temperature T both above and below the critical point. This
is feasible but requires precise experimental data. ¿From Eq.(6) assuming the
only minimum is at m=0 we get
$\eta=\frac{H/T}{a}.$ (12)
The temperature (a) dependence of the order parameter shows that we are away
from the critical point. Nevertheless we can study the behavior close to the
critical point by suitably defining scaling forms huang :
$\frac{M}{|t|^{\beta}}=\frac{\eta}{|t|^{\beta}}$ vs.
$\frac{H/T}{|t|^{\beta\delta}}$. These quantities are plotted in figure 11 and
compared to magnetization data for nickel metal. The scaled magnetization is
plotted versus the scaled external magnetic field huang . The nuclear data
have been shifted in the region near the crossing of data above and below the
critical temperature where we expect our data to be, see Fig.10. Of course it
is not possible at this stage to directly compare to the macroscopic data
since we have no information for the absolute values of the temperatures.
Furthermore the role of the density (or pressure) is not clear since we expect
that the parameter $a$ (or equivalently x) depends on the ‘distance’ from the
critical temperature and critical pressure. These quantities could however be
obtained in $4\pi$ experiments where charges, masses and their velocities are
carefully determined.
Figure 11: Scaling form for magnetization M vs. external field for nickel
huang , open symbols. The corresponding quantities for nuclei normalized to
the metal case are given by the full symbols.
Once we have derived the ‘reduced’ parameters of the Landau $O(m^{6})$ theory,
we can write a ‘reduced’ free energy as ($b\neq 0$):
$\frac{f}{T}=\frac{1}{2}xm^{2}-|z|m^{4}+\frac{2}{3}z^{2}m^{6}-\frac{{\it
h}}{T}m,$ (13)
where: $\frac{f}{T}=\frac{4c}{b^{2}}\frac{F}{T}$, $z=\frac{c}{|b|}$,
$\frac{h}{T}=\frac{4c}{b^{2}}\frac{H}{T}$. These quantities together with the
temperature, Eq.(7), and the reduced order parameter y,of Eq.(8), constitute
the Landau $O(m^{6})$ theory in dimensionless form. It is instructive to study
how these quantities change with the reaction system as we did in figures (7)
and (8).
Figure 12: The parameters x, z and h/T versus (I/A) of the compound nucleus
obtained from the data fit to the Landau free energy, Eq.(3). The symbols are
like in figure 6.
In Fig.(12) we plot these normalized quantities vs. difference in neutron
proton concentration of the compound nucleus. Compare to figure 7. A feature
worth noticing is the following, while the parameter $a$ is decreasing with
increasing (I/A) of the compound nucleus, the opposite holds for the parameter
$x$ which gives the ‘distance’ from the critical temperature, see Fig.(12).
This is very important since only normalized quantities should be used when
inferring the properties of the EOS (i.e. temperature, density etc.) near the
critical point.
## IV Symmetry and Pairing compared to the Coulomb Energy.
In the previous sections we have seen that the Coulomb energy might become
important especially for large values of the charges. We can now try to derive
some qualitative understanding of when and why Coulomb corrections might
become important and might even hinder a possible phase transition. From the
mass formula we can write the Coulomb energy for large Z as pres :
$\frac{E_{c}}{A}=0.77\frac{Z^{2}}{A^{2}}A^{2/3}=\frac{0.77}{4}(1-m)^{2}A^{2/3},$
(14)
which explicitly introduces the order parameter $m$ in the Coulomb energy. We
can define an ‘effective’ symmetry energy (per particle) as:
$\frac{E_{eff}}{A}=(a_{sym}+\frac{0.77}{4}A^{2/3})m^{2}-\frac{0.77}{2}A^{2/3}m+\frac{0.77}{4}A^{2/3},$
(15)
where the symmetry energy coefficient $a_{sym}=25MeV$. Ignoring for a moment
density corrections we see that $O(m^{2})$ term should be affected by Coulomb
corrections for large fragment mass numbers. Furthermore, a linear term in $m$
is introduced which will then modify the ‘external’ field even in collisions
where the source $m_{s}=0$ as we discussed in Fig.(6). Finally there is a term
not dependent on $m$ that will destroy the scaling for large mass (charge)
numbers. We should also notice that assuming a spherical expansion, at low
densities the Coulomb energy will decrease as $\rho^{1/3}$ while contributions
to the symmetry energy should depend both on $\rho^{2/3}$ reflecting the Fermi
energy of the nuclei and on $\rho$, the latter coming from different n-p
interactions. At low densities we would expect Coulomb to be stronger than it
appears to be in the data. This may be indicative that the fragments must be
highly deformed, reducing the Coulomb energy. Coulomb corrections should
become more important when $m=0$ for the detected fragment. We have plotted
the yields of $m=0$ nuclei in Fig.(2) and pointed out that pairing appears to
be playing a role. From Eq.(15) above we should expect that, if Coulomb is
dominant for such fragments, the free energy should depend on $A^{2/3}$. In
Fig.(13) (top panel) we plot F/T versus $A$ for $m=0$ fragments. The expected
dependence with mass number in the free energy suggested from ‘effective’
symmetry energy Eq.(15) is not seen in the figure.
Figure 13: Free energy versus mass for $m=0$ isotopes for the 70Zn+124Sn
system (top panel), the dashed line is a fit using Coulomb and pairing
contributions. Free energy times $\delta$ (see text-bottom panel) versus mass
for $m=0$ isotopes. The lines are separate fits suggested by the
Coulomb(dashed line) and pairing(full line) energy mass number dependence.
Rather, a staggering between odd-odd and even-even nuclei is clearly visible.
To better clarify these arguments we can write the pairing energy from the
mass formula as pres :
$\frac{E_{p}}{A}=12\frac{\delta}{A^{3/2}},$ (16)
where $12MeV$ is the ground state pairing energy coefficient and $\delta$:
$\delta=\frac{(-1)^{N}+(-1)^{Z}}{2}.$ (17)
The suggested mass dependence from pairing, Eq.16, is completely different
from the Coulomb one when $m=0$, see Eq.14.
Notice that it is the $\delta$ factor from pairing that changes the sign of
the contribution for odd-odd to even-even nuclei. A combined fit to the data
using Coulomb plus pairing contributions results in the dashed line in
Fig.(13), top panel. The agreement with data is very good. If we multiply the
pairing energy by the factor $\delta$ we should get no discontinuities when
plotting this quantity versus mass number. Similarly if the properties of the
free energy depend on the pairing term, as for the ground state case, then it
should be a monotonic function of $A$ after multiplying it by $\delta$. In
Fig.(13) (bottom panel), we plot the quantity $\frac{F}{T}\delta$ versus mass
number for the same system of Fig.(13) (top panel). The fit using the pairing
mass dependence is also good. The Coulomb mass dependence fails especially for
small mass number. From the values of the fit, using the ground state
coefficients we can derive a temperature for the Coulomb case of
$T=9.2(\frac{\rho}{\rho_{0}})^{1/3}MeV$ where we have explicitly indicated a
possible density correction. For the pairing case we get $T=6.45MeV$. Notice
that in this case we have not suggested any density correction since the fate
of the pairing energy at low density and finite temperature is ‘terra
incognita’. When making a combined fit using pairing and Coulomb energy we get
a good reproduction of the data (dashed line in Fig.(13),top panel). While the
fitting value for pairing results in a ‘temperature’ T=5.13 MeV, we get an
increase of the Coulomb contribution to T=12.1 MeV. Assuming that pairing is
independent on density, we could derive a density from the Coulomb result. A
simple calculation give $\frac{\rho}{\rho_{0}}=(6.45/9)^{3}=0.34$ which could
be a reasonable indication of the density of the system when it breaks into
fragments.
In summary in this section we have shown that the role of the Coulomb energy
appears to be rather reduced in the reactions analyzed in this paper. We
expect it to become more important for large nuclei. On the other hand large
nuclei have smaller symmetry and pairing energies per nucleon, thus a precise
determination of the EOS can be obtained from measurements of isotopes having
relatively small masses.
## V dynamics of the phase transition
As we have seen we have been able to discuss some observables in the
fragmentation of nuclei using a language common to macroscopic systems
undergoing a phase transition. In the nuclear case we have a finite system
composed at most of hundreds of particles which evolves in time under the
influence of a long range Coulomb force. This poses many questions on why
techniques of statistical mechanics should apply in such evolving nuclear
systems. This also offers the possibility of dealing with statistical
mechanics of open systems and the problem of extending the description of a
phase transition to such a system.
We start by observing that even though we are dealing with a dynamical system,
the order parameter defined in this work, $m$, is confined between -1 and +1.
In this sense we have a somewhat ‘closed’ system. Also the density at which
the transition occurs should be smaller then normal density and thus Coulomb
effects are reduced. However, if we deal with larger sources, such as in U+U
collisions the phase transition might be washed out by the strong Coulomb
field. We expect our current considerations to be valid for small sources
only.
From statistical mechanics we know that in a first order phase transition
huang a small seed increases in size depending upon the surface tension at a
given T and density $\rho$. If the pressure of the surrounding matter is
smaller than the internal pressure of the drop, the drop will grow by
capturing surrounding matter. On the other hand if the opposite is true then
the drop will decrease in size to balance the external pressure. The entire
process is driven by surface tension. Drops of a given size will survive only
when their internal pressure balances the external pressure. If the system is
at a very low density the interaction between different parts might take a
relatively long time. In these conditions a big nuclear drop whose internal
pressure is larger than that of the surroundings could be considered to be a
nucleus which is evaporating particles in order to balance the external (zero
in the case of an isolated nucleus) pressure. If we accept this picture than
the evaporation step is part of the dynamics of the phase transition. Thus a
very low density system might be thought of as many isolated drops evaporating
particles and reaching their equilibrium conditions before they collide with
other parts of the system or as small fragments being evaporated by other
drops. In a finite system this does not happen, but we might think of a
process where at some point the finite system becomes unconfined and an
infinite system is approximated by an infinite number of repetitions or
‘events’. Of course in a statistically equilibrated system we know that time
averages and event averages are the same. Here we are extending this concept
to finite systems where only event averages can be used. A major question here
is whether the properties of the phase transition are decided very early, i.e.
when the system ‘enters’ the instability region. As we said above if we have
an infinite system at a very low density undergoing a first order phase
transition, then the drops can explode, evaporate, and fuse with other
particles over a very long time. Our finite system might behave similarly but
without the fusion at later times. If this were the case than the detected
fragments carry all the information of the phase transition, if not then we
need to reconstruct the primary fragment distributions coming out of the
instability region.
Figure 14: Free energy vs. time in AMD calculations (see text) for 64Zn+112Sn
system at 40MeV/A and central collisions, i.e. impact parameter less than 3
fm. Different picture correspond to T=200, 300, 500, 1000, 1500 and 2000 fm/c
respectively.
We can try to clarify some of these questions by means of microscopic models
such as Antisymmetrized Molecular Dynamics (AMD) or similar approaches where
the time evolution of the system is followed ono . However, we have to stress
that in such microscopic models some assumptions are made in order to
recognize the fragments at particular times during the time evolution. In
simpler approaches, fragments are recognized if particles are close in
coordinate space (of the order of the range of the attractive nuclear forces)
bon00 . In such a case the recognized fragments are ‘excited’ and they evolve
in time until a final state is reached after a long time of the order of
thousands of fm/c. A more refined approach for fragment recognitions is given
by defining clusters when its components are within a given distance in phase
space. The naive expectation would be that in this case we should recognize
fragments earlier than the previous case and this is the method that we will
adopt here for simplicity following ref. ono . In an ambitious approach dorso
the claim is that fragments are recognized very early during the time
evolution, of the order of tens of fm/c, if one searches for particles
connected in phase space to form fragments and minimize the energy. This case
probably corresponds to minimizing the entropy of the fragmenting system. If
this last picture will hold true then a picture of an infinite system at low
density will be equivalent to an ‘infinite’ repetition of events. Finally in
all the considerations above we have to add the necessary and interesting
complication that we have a mixture and not a single fluid, thus we can have
more situations to explore than discussed in the previous sections and we can
‘turn on and off’ an external field as well.
Figure 15: Fit parameters a, b and c vs. time (see text) for the same system
of Fig.14. Solid circles refer to AMD calculations while the open symbol in
the M vs. x plot is the experimental value for this system.
We have performed AMD calculations for the same systems investigated
experimentally. After some time t, fragments are separated enough in phase
space that we can recognize within a simple phase space coalescence approach
as discussed in ono . In this way we can define a yield at a given time and
from this derive the free energy exactly as we did with the experimental data.
Characteristic results for the free energy versus time is given in Fig.(14)
together with a Landau $O(m^{6})$ fit. Some time evolution is observed. Using
a more sophisticated fragment recognition approach dorso might even decrease
the time over which this evolution occurs. We can study the time evolution in
more detail by plotting the variables a, b and c. M defined in the previous
sections versus time. The results of the fits to the free energy at different
times is given in Fig.(15). While the quantities a, b and H/T change somewhat
during the time evolution, smaller changes are observed in the time evolution
of normalized quantities, x, z and h/T. Nevertheless the time evolution of the
fitting parameters influences the time evolution of the order parameter M
versus reduced temperature x as seen in the bottom right of Fig.(15). It is
very interesting to see that in these units the system is initially very hot
(superheated) and cools down when coming to equilibrium below the critical
temperature. The final result is very close to the observed values given by
the open points.
Thus in this model most qualitative features of the phase transition are
decided very early during the time evolution. This might correspond to an
entropy saturation early during the evolution. However, different models and
fragment recognition approaches might change the picture somewhat.
## VI Equation of state
Once we know the free energy (at least in some cases) we can calculate the
NEOS by means of the Fisher model fisher . Since we do not have at present
experimental information on the density $\rho$, temperature T and pressure P
of the system we can only estimate the ‘reduced pressure’ paolo :
$\frac{P}{\rho T}(m)=\frac{M_{0}}{M_{1}},$ (18)
where $M_{i}$ are moments of the mass distribution given by:
$\displaystyle
M_{k}=\sum_{A}A^{k}Y(A,m)=Y_{0}\sum_{A}A^{k}A^{-\tau}e^{-F/T(m)A};$
$\displaystyle k=0,...n.$ (19)
Notice that the quantities above are now dependent on the order parameter m.
From the knowledge of $F/T$ $(H/T=0)$ from the previous section we can easily
calculate the reduced pressure near the critical point. In particular given
the simple expression for the moments we can also derive some analytical
formulas following paolo :
$\frac{P}{\rho
T}(m)=\frac{3.072|F/T|^{4/3}+1.417-3.631|F/T|+...}{-4.086|F/T|^{1/3}+3.631+0.966|F/T|+...},$
(20)
which gives at the critical point a critical compressibility factor ($F/T=0$):
$\frac{P}{\rho T}|_{c}=\frac{1.417}{3.631}=0.39$.
This value is essentially that derived from the Van-der-Waals gas equation but
is well above the values observed for real gases. Using the relations above we
can calculate the NEOS for the situations illustrated in Fig.9. The results
are displayed in Fig.16 where the reduced pressure is plotted versus m for
vaporization, superheating and first order phase transitions on the tri-
critical line. Notice that there is not a large difference between the first
two cases, while the last case displays two critical points (a third one is on
the negative m axis).
We have seen in Fig.2 that $N=Z$ nuclei display a power law. We can also
estimate the critical reduced pressure for this case noticing that the sums in
Eq.(VI) above must be restricted to $A=2Z$ nuclei. This leads to a critical
compressibility factor $\frac{P}{\rho T}_{c}=0.20$ which is a value closer to
that estimated from other multi-fragmentation studies before mor .
Figure 16: Reduced pressure versus m of the fragments obtained from the a, b
and c parameters fit to the Landau free energy, Eq.(3) for the 64Ni+232Th. The
curves correspond to vapor(open circles), superheating(open squares), first
order (3 critical line-solid stars), see text. The solid circle is for $N=Z$
nuclei at the critical point.
We can compare our analytical result given in Eq.(20) with the numerical
values obtained above. This is displayed in Fig.(17) and we see that the
numerical approximation is especially good near the critical point(s) as
expected. If, from detailed comparison to experimental data, we are able to
extract the temperature and pressure dependence of the parameters entering the
Landau free energy, then Eq.(18) would be the Nuclear equation of state near a
critical point. From the actual data at our disposal we can only estimate the
behavior of the reduced pressure as function of the order parameter m.
Figure 17: Comparison to the analytical result, Eq.(20)(solid circles) for a
first order phase transition.
On similar ground we can define a reduced compressibility as
$\chi\rho T(m)=\frac{M_{2}}{M_{1}}.$ (21)
Figure 18: Reduced compressibility versus m of the fragments as in Fig.(16)
Its behavior is displayed in Fig.(18) for the cases outlined above.
Divergences near the critical point(s) are obtained.
## VII conclusions
In conclusion, in this paper we have presented and discussed experimental
evidence for the observation of a quantum phase transition in nuclei, driven
by the neutron/proton asymmetry. Using the Landau approach, we have derived
the free energies for our systems and found that they are consistent with the
existence of a line of first-order phase transitions terminating at a point
where the system undergoes a second-order transition. The properties of the
critical point depend on the symmetry. This is analogous to the well known
superfluid $\lambda$ transition in 3He-4He mixtures. We suggest that a
tricritical point, observed in 3He-4He systems may also be observable in
fragmenting nuclei. These features call for further vigorous experimental
investigation using high performance detector systems with excellent isotopic
identification capabilities. Extension of these investigations to much larger
asymmetries should be feasible as more exotic radioactive beams become
available in the appropriate energy range.
It is important to stress that the observables discussed here represents only
$necessary$ conditions for a critical behavior. A definite proof of a phase
transition and a tricritical point could be given by a precise determination
of yields of fragments whose $m\approx\pm 0.5$, i.e. very unstable nuclei
which, most probably, decay before reaching the detectors. Thus fragment-
particle correlation measurements for exotic primary fragments such as
${}^{4}Li$, ${}^{5}Be$ (proton rich) or extremely neutron rich ${}^{10}He$ are
needed. More generally, such correlation experiments can also shed light on
the effects of secondary decay on the fragment observables. This remains a key
question in many equation of state studies and model calculations differ in
their assessment of these effects wci ; bon00 . Higher quality data over a
wider range of beam energies and colliding systems should also help in
clarifying the role of other energy terms, such as surface, Coulomb etc.,
which are important at lower excitation energies. In particular the role of
pairing and the possibility of Bose-Einstein condensation, should be more
deeply investigated. Our data for $I=0$ fragments already show that pairing is
important. This might be due to its importance during the phase transition or
to its role during secondary decay of the excited primary fragments.
Exploration of quantum phase transitions in nuclei is important to our
understanding of the nuclear equation of state and can have a significant
impact in nuclear astrophysics, helping to clarify the evolution of massive
stars, supernovae explosions and neutron star formation.
###### Acknowledgements.
We thank the staff of the Texas A$\&$M Cyclotron facility for their support
during the experiment. We thank L. Sobotka for letting us to use his spherical
scattering chamber. This work is supported by the U.S. Department of Energy
under Grant No. DE-FG03-93ER40773 and the Robert A. Welch Foundation under
Grant A0330. One of us(Z. Chen) also thanks the “100 Persons Project” of the
Chinese Academy of Sciences for the support.
## References
* (1) M. A. Preston, Physics of the nucleus, Addison-Wesley pub., Reading-Mass. (1962).
* (2) K. Huang, Statistical Mechanics, second edition, ch.16-17, J. Wiley and Sons, New York, 1987.
* (3) L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd edition, Pergamon press, New York,1989.
* (4) A.Bonasera et al., Phys. Rev. Lett. 101, 122702 (2008).
* (5) M. Huang et al.,arXiv:1002.0311 [nucl-ex]
* (6) WCI proceedings, Ph. Chomaz et al., eds., EPJ A30 (2006), numb.1.
* (7) A. Bonasera et al., Rivista Nuovo Cimento, 23 (2000) 1.
* (8) H. Muller and B. D. Serot, Phys. Rev. C52 (1995) 2072; P.J. Siemens and G. Bertsch, Phys. Lett. B126 (1983) 9; P. J. Siemens, Nature 336 (1988) 109.
* (9) M. Belkacem, V. Latora, and A. Bonasera, Phys. Rev.C52 (1995) 271; A. Bonasera et al., Phys. Lett. B244 (1990) 169.
* (10) C. O. Dorso, V. C. Latora, and A. Bonasera, Phys. Rev. C60 (1999) 034606; M. Belkacem et al., Phys Rev. C54 (1996) 2435.
* (11) J. B. Elliott et al., Phys. Rev. Lett. 88 (2002) 042701; L. G. Moretto et al., Phys. Rev. Lett. 94 (2005) 202701.
* (12) M. D’Agostino et al., Nucl. Phys. A650 (1999) 329.
* (13) P. F. Mastinu et al., Phys. Rev. Lett. 76 (1996) 2646.
* (14) K. Hagel et al., Phys. Rev. C62 (2000) 034607.
* (15) J. Pochodzalla et al., Phys. Rev. Lett. 75 (1995) 1040.
* (16) V.Baran et al., Phys.Rep.410(2005)335.
* (17) M. Huang et al.,arXiv:1001.3621 [nucl-ex]
* (18) Z.Chen et al.,arXiv:1002.0319 [nucl-ex]
* (19) R.W. Minich et al., Phys.Lett.B118(1982)458.
* (20) N. Marie et al., Phys. Rev. C58 256 (1998).
* (21) M. E. Fisher, Rep. Prog. Phys. 30 (1967) 615.
* (22) A. Ono and H. Horiuchi, Progr. Part. Nucl. Phys. 53 (1996) 2958; A. Ono, Phys. Rev. C59 (1999) 853.
* (23) X. Campi, H. Krivine and N. Sator, Nucl. Phys. A681 (2001)458.
* (24) F. Gulminelli, Ph. Chomaz, M. Bruno, and M. D Agostino, Phys. Rev.C65 051601(R) (2002).
* (25) P. Finocchiaro et al. Nucl.Phys.A600(1996)236; T.Kubo et al. Zeit.Phys.A352(1995)145.
|
arxiv-papers
| 2010-02-09T00:18:50 |
2024-09-04T02:49:08.306926
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Huang, A. Bonasera, Z. Chen, R. Wada, K. Hagel, J.B. Natowitz,\n P.K.Sahu, L. Qin, T. Keutgen, S. Kowalski, T. Materna, J. Wang, M.Barbui,\n C.Bottosso, and M.R.D.Rodrigues",
"submitter": "Meirong Huang",
"url": "https://arxiv.org/abs/1002.1738"
}
|
1002.1751
|
0.120450pt black (461,226)(420,226) $10^{-3}$black (461,578)(420,578)
$10^{-2}$black (461,685)(420,685) $0.02$black (461,931)(420,931) $0.1$black
(461,1037)(420,1037) $0.2$black (461,1177)(420,1177) $0.5$black
(505,226)(505,185) $0$black (778,226)(778,185) $10$black (1104,226)(1104,185)
$10^{2}$black (1452,226)(1452,185) $10^{4}$black (1802,226)(1802,185)
$10^{5}$black black (461,1259)(461,226)(1876,226) black [origin=c]90black
black vertex in-degreeblack black red black SingleRWred
$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$$\scriptstyle\Diamond$blue
black FS ($m=1000$)blue
$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$$\scriptstyle+$green
black MutipleRW ($m=1000$)green
$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$$\scriptstyle\Box$black
(461,1259)(461,226)(1876,226) black
|
arxiv-papers
| 2010-02-09T15:00:13 |
2024-09-04T02:49:08.316229
|
{
"license": "Public Domain",
"authors": "Bruno Ribeiro and Don Towsley",
"submitter": "Bruno Ribeiro",
"url": "https://arxiv.org/abs/1002.1751"
}
|
1002.1864
|
# The standard model of star formation applied to massive stars: accretion
disks and envelopes in molecular lines
Eric Keto1 and Qizhou Zhang1
1Harvard-Smithsonian Center for Astrophysics, 160 Garden St, Cambridge, MA
02420, USA E-mail: keto@cfa.harvard.edu (EK); qzhang@cfa.harvard.edu (QZ)
(May 8, 2009)
###### Abstract
We address the question of whether the formation of high-mass stars is similar
to or differs from that of solar-mass stars through new molecular line
observations and modeling of the accretion flow around the massive protostar
IRAS20126+4104. We combine new observations of NH3 (1,1) and (2,2) made at the
Very Large Array, new observations of CH3CN(13-12) made at the Submillimeter
Array, previous VLA observations of NH3(3,3), NH3(4,4), and previous Plateau
de Bure observations of C34S(2-1), C34S(5-4), and CH3CN(12-11) to obtain a
data set of molecular lines covering 15 to 419 K in excitation energy. We
compare these observations against simulated molecular line spectra predicted
from a model for high-mass star formation based on a scaled-up version of the
standard disk-envelope paradigm developed for accretion flows around low-mass
stars. We find that in accord with the standard paradigm, the observations
require both a warm, dense, rapidly-rotating disk and a cold, diffuse
infalling envelope. This study suggests that accretion processes around 10 M⊙
stars are similar to those of solar mass stars.
###### keywords:
Keywords.
## 1 Introduction
Does the formation of massive stars differ significantly from that of solar
mass stars? As far as we know, stars of all masses form in gravitationally
unstable regions of molecular clouds and gain their mass by accretion. A
standard model developed for accretion flows around low-mass stars consists of
two-components, a rotationally-supported disk inside a freely-falling envelope
(Shu, Adams, Lizano, 1987; Hartmann, 2001). This model has been particularly
successful in explaining infrared observations of low-mass star formation. The
disk and envelope produce an excess of long-wavelength infrared emission that
has been adopted as the identifying signature of accreting protostars in
Galactic (Whitney et al., 2003) and extragalactic (Whitney et al., 2008) star-
forming regions. Furthermore, because the disk and envelope have different
densities and temperatures, the evolutionary state of the protostars can be
identified by the shape of the infrared spectral energy distribution: class 0
(envelope dominated) and class I (disk dominated) (Lada, 1987; Andre et al.,
1993).
Does this standard two component accretion model developed for low-mass stars
also describe the accretion flows around massive stars? There are some doubts.
The more massive stars are luminous enough to generate radiation pressure and
hot enough to ionize their own accretion flows such that the outward radiative
and thermal pressures rival the inward pull of the stellar gravity (Larson &
Starrfield, 1971; Kahn, 1974; Keto, 2002; Keto & Wood, 2006). Do these outward
pressures result in accretion flows that are different around more massive
stars? In this paper we compare new and previous molecular line observations
of an accretion flow around one massive star against the standard disk-
envelope paradigm for accretion flows around low-mass stars.
The previous molecular line observations of the massive protostar
IRAS20126+4104 111 IRAS20126 is located in the Cygnus-X region at a distance
of 1.7 kpc (Wilking et al. 1989). , suggest an accretion disk and bipolar
outflow around a 7 to 15 M⊙ protostar embedded in a dense molecular envelope
(Cesaroni et al., 1997; Zhang, Hunter, & Sridharan, 1998; Cesaroni et al.,
1999; Kawamura et al., 1999; Zhang et al., 1999; Shepherd et al., 2000;
Cesaroni et al., 2005; Lebron et al., 2006; Su et al., 2007; Qiu et al.,
2008). Previous infrared observations of absorption and scattering also reveal
a disk and outflow cavity immediately around the star (Sridharan, Williams, &
Fuller, 2005; deBuizer, 2007).
In this study, we assemble a suite of observations of molecular lines of
different excitation temperature in order to compare with molecular spectra
predicted from the disk-envelope model. Lines of different excitation
temperature are useful because massive stars heat the surrounding molecular
gas to observationally significant temperatures ($>100K$) at observationally
significant distances from the star, ($T_{gas}\sim
T_{\ast}(R/R_{\ast})^{\alpha}$), and we can exploit the relationship between
temperature and radius to distinguish emission from gas at different radii
around the star. We expect to identify the emission from the higher excitation
temperature lines with gas in the flow closer to the star and also to separate
the emission from the disk and envelope components. In previous observations,
Keto, Ho & Haschick (1987) and Cesaroni et al. (1994) used this technique,
observing several lines of NH3 to study the accretion flows around very high
mass stars associated with HII regions.
We present new observations of the NH3 (1,1) and (2,2), inversion transitions
made with the National Radio Astronomy Observatory’s Very Large Array
(VLA)222The National Radio Astronomy Observatory is a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc. and new observations of CH3CN(13-12) made with the
Submillimeter Array (SMA). The new observations of NH3(1,1) and NH3(2,2) have
a factor of 3 better sensitivity than the earlier observations of Zhang,
Hunter, & Sridharan (1998). Combined with previous observations of NH3(3,3)
and NH3(4,4) (Zhang et al., 1999) the 4 NH3 lines span a range of excitation
temperatures from 23 K to 200 K. We also have additional lines of lower and
higher excitation temperature from previous observations of the C34S(2-1) and
C34S(5-4) lines with energies of 7 K and 35 K respectively, and the ladder of
CH3CN (J=12-11) lines with energies ranging from 69 K for (K=0) to 419 K for
(K=7). The C34S and CH3CN observations were previously presented in Cesaroni
et al. (1999) and Cesaroni et al. (2005).
We specify the two-component accretion model in terms of 6 parameters: the
scale factors for the density and temperature of the envelope and of the disk,
the angular momentum of the envelope, and the stellar mass. We use our
molecular line emission code MOLLIE to predict molecular line spectra from the
parameterized model, and we use a least-squares fitting procedure to adjust
the parameters to fit the observations.
We find that the two-component, disk-envelope model can successfully describe
the observations, but a single component model cannot. A warm, dense
rotationally-supported disk is required to obtain sufficient brightness and
width in the high-excitation lines, and a cold, large-scale envelope is
required to match the emission from the lower excitation lines. We find no
evidence that the accretion flow around IRAS20126 is profoundly altered by the
outward force of radiation pressure or by ionization. At 10 M⊙, the star might
simply not be luminous enough or hot enough for its radiation pressure or
ionization to significantly affect the accretion process. Based on this
example, the accretion flows around 10 M⊙ stars are quite similar to flows
around lower-mass stars.
## 2 The Data
### 2.1 Ammonia inversion lines
We observed the IRAS 20126+4104 region, R.A. (J2000) = 20:14:26.06, decl.
(J2000) = 41:13:31.50, in the inversion transitions of NH3 (J,K) = (1,1),
(2,2), (3,3) and (4,4) on 1999 March 27 and 1999 May 29 with the VLA in its D
configuration. We used the 2IF correlator mode to sample both the right and
left polarizations, a spectral bandwidth of 3.13 MHz, and a channel width of
24 kHz or 0.3 kms-1. The primary beam of the VLA at the NH3 line frequencies
is about $2^{\prime}$. Quasars 3C48 and 3C286 were used for flux calibration,
3C273 and 3C84 for bandpass calibration, and quasar 2013+370 for phase
calibrations. The NH3(3,3) and NH3(4,4) observations were previously discussed
in Zhang et al. (1999).
The VLA data were processed using the NRAO Astronomical Image Processing
System (AIPS) package. Data from the two days were combined to achieve an rms
of 3 mJy per 0.3 km s-1 channel for the NH3 (1,1), (2,2) lines and 6 mJy and 4
mJy per 0.6 km s-1 channel for the NH3 (3,3) and (4,4) lines respectively.
Figure 1 shows two images of IRAS20126 in the integrated intensity of NH3(1,1)
and (2,2). The protostar is located within the bright core of the larger scale
cloud. Also shown in figure 1 is the image of the bipolar outflow in SiO(2-1)
from Cesaroni et al. (1999). The NH3 spectra used in the analysis are taken at
positions offset from the phase center by -0.23, 1.00 (center), -3.10, 3.25
(left), and 2.50,-0.94 (right) in arc seconds of RA and dec. The corresponding
absolute positions are R.A. (J2000) = 20:14:26.02, decl. (J2000) = 41:13:32.8
(center) , R.A. (J2000) = 20:14:25.79, decl. (J2000) = 41:13:30.8 (right) ,
and R.A. (J2000) = 20:14:26.28, decl. (J2000) = 41:13:35.0 (left) , The
locations of the spectra are shown as red stars on figure 1.
### 2.2 Methyl Cyanide and Carbon Sulfide lines
The CH3CN(13-12) observations were made with the SMA on July 10, 2006 with a
maximum baseline of $\sim 0.5$ km, a beam size of $0.40\times 0.36$”, and a
phase center of R.A. (J2000) = 20:14:26.02, decl. (J2000) = 41:13:32.7. The
channel width is 0.8 kms-1 at the observing frequency of 239 GHz. The rms
noise of the data is 0.05 Jy beam-1 channel-1 or 8 K channel-1 in brightness
temperature.
Data in the CH3CN J = 12-11, and C34S J = 5-4 and J = 2-1 transitions were
obtained from the IRAM interferometer between 1996 December through 1997
March, and 2002 January through March. The phase center of the observations is
R.A. (J2000) = 20:14:26.03, decl. (J2000) = 41:13:32.7. The primary beam of
the array at the frequency of the observed CH3CN transitions (230 GHz) and
C34S(5-4) (241 GHz) is approximately $20^{\prime\prime}$ with a spatial
resolution of about $0.8^{\prime\prime}$. At the lower frequency of the
C34S(2-1) transtion (96 GHz) the primary beam is about $50^{\prime\prime}$ and
the spatial resolution is and $2.4^{\prime\prime}$. The full details of these
observations can be found in Cesaroni et al. (1999, 2005).
## 3 The Standard Model for Star Forming Accretion Flows
### 3.1 The density of the accretion flow
The standard model for a star-forming accretion flow consists of an inflowing
envelope around a rotationally supported disk. Similar to the approach in
Whitney et al. (2003), we use the model of Ulrich (1976) to describe the gas
density in the infalling envelope and standard thin-disk theory (Pringle,
1981) to describe the disk,
The model for the envelope assumes that the gas flows toward a gravitational
point source along ballistic trajectories. The flow conserves angular momentum
and ignores the pressure and self-gravity of the gas. This is the same
envelope model that Whitney et al. (2003) adopted for low-mass stars and
similar to the model used to describe the accretion flow onto a cluster of
high-mass stars, G10.6-0.07 (Keto, 1990; Keto & Wood, 2006). Thus despite, or
perhaps because of, its simplicity, the model has found application to
accretion flows from low-mass stars to star clusters. The model is
particularly useful in analyzing observational data because only 3 parameters
are required to describe individual cases: 1) the density of the gas at a
single radius (the gas density elsewhere follows from mass conservation), 2)
the mass of the point source, and 3) the specific angular momentum.
The model for the envelope is fully described in Ulrich (1976). We use the
equations as presented in Keto (2007), and Mendoza, Canto, Raga (2004). The
envelope density is (equation 10 of Keto (2007)),
$\displaystyle\rho_{\rm
env}=\rho_{e0}(r/R_{d})^{-3/2}\bigg{(}1+{{\cos{\theta}}\over{\cos{\theta_{0}}}}\bigg{)}^{-1/2}$
$\quad\quad\quad\times\quad(1+(r/R_{d})^{-1}(3\cos^{2}{\theta_{0}}-1))^{-1}$
(1)
where $\rho_{e0}=\rho_{\rm env}(R_{d})$ is the density in the mid-plane,
$\theta=\pi/2$, at radius, $R_{d}$, $\theta_{0}$ is the initial polar angle of
the streamline, and $\theta$ and $r$ are the polar angle and radius at each
point along a streamline. The angle, $\theta$ is related to the polar radius,
$r$, and the initial angle $\theta_{0}$ (equation 7 of Keto (2007)),
$r={R_{d}\cos{\theta_{0}}\sin^{2}{\theta_{0}}\over{\cos{\theta_{0}}-\cos{\theta}}}$
(2)
The density is related to the mass accretion rate (equation 11 of Keto (2007))
$\dot{M}=\rho_{e0}(R_{d})4\pi R_{d}^{2}v_{k}$ (3)
where $v_{k}$ is the Keplerian velocity at $R_{d}$.
The model for the disk assumes that a rotationally-supported disk forms at the
radius where the centrifugal force in the rotating envelope equals the
gravitational force of the point mass,
${{\Gamma^{2}}\over{R^{3}_{d}}}={{GM_{\ast}}\over{R^{2}_{d}}}$ (4)
and $\Gamma=v_{\phi}R$. The disk is truncated at $R_{d}$.
The gas density in the disk is (equation 3.14 of Pringle (1981)),
$\rho_{\rm disk}(z,R)=\rho_{0}(R)\exp(-z^{2}/2H^{2})$ (5)
where $\rho_{0}(R)$ is the density in the mid-plane, $\theta=\pi/2$, at any
radius $R$. We use $R=\sqrt{x^{2}+y^{2}}$ to denote the cylindrical radius,
and $r=\sqrt{x^{2}+y^{2}+z^{2}}$ for the polar radius. In the thin disk
theory, the scale height, $H$, is
$H^{2}=c_{s}^{2}R^{3}/GM_{\ast}$ (6)
where $c_{s}$ is the sound speed. We follow Whitney et al. (2003) and use a
modified version of this equation,
$H=H_{0}(R/R_{\ast})^{1.25}$ (7)
with $H_{0}=0.01R_{\ast}$. The density in the mid-plane of the accretion disk,
$\rho_{0}(R)$, is
$\rho_{0}=\rho_{d0}({{R_{d}}/{R}})^{2.25}$ (8)
where $\rho_{d0}$ is the density in the mid-plane at $R_{d}$. Again following
Whitney et al. (2003), we use an exponent of 2.25 in equation 8 rather than 2
which is derived in the thin-disk theory. As explained in Whitney et al.
(2003) the modifications to the equations for the scale height and mid-plane
densities are based on fits to numerical models of disk structure.
The densities in the disk and envelope at radius $R_{d}$ are related by a
factor, $A_{\rho}$,
$\rho_{d0}=A_{\rho}\rho_{e0}$ (9)
For example, in the steady-state flow, the disk would have a higher density
if, as should be the case, the inward velocities in the rotationally-supported
disk were lower than in the freely-falling envelope. Because the inward
velocities in the disk are not described by the thin-disk theory, we leave
$A_{\rho}$ as an adjustable parameter. The total gas density at any point in
the model is the sum of the densities in the disk and envelope,
$\rho_{\rm gas}=\rho_{\rm disk}+\rho_{\rm env}$ (10)
### 3.2 The temperature
We assume that the envelope is heated by the star. The envelope temperature is
(equation 7.36 of Lamers & Cassinelli (1999))
$T_{\rm env}=T_{\ast}(R_{\ast}/2r)^{2/(4+p)}$ (11)
where $p$ in our model is an adjustable parameter related to the dust opacity
and the geometry of the flow. If the density structure were spherically
symmetric, then $p$ would be the exponent in the frequency dependence,
$\nu^{p}$, of the Planck mean opacity of the dust. In a flattened flow, $p$
can be negative if the geometrical dilution of the radiation in the accretion
flow is greater than $r^{-2}$. This can happen if the dust at each radius
absorbs and isotropically re-emits the outward flowing radiation. Radiation
that is emitted perpendicular to the disk escapes from the flow. Only the
radiation that is emitted in the direction along the flattened flow continues
to heat the dust at larger radii. The result is a decrease in the radiation in
the disk faster than $r^{-2}$.
In the thin-disk theory, the disk is heated by dissipation related to the
accretion rate. The disk temperature is (equation 3.23 of Pringle (1981))
$T_{\rm disk}=B_{T}\bigg{[}\bigg{(}{{3GM_{\ast}\dot{M}}\over{4\pi
R^{3}\sigma}}\bigg{)}\bigg{(}1-\sqrt{{{R_{\ast}}\over{R}}}\bigg{)}\bigg{]}^{1/4}$
(12)
We include an adjustable factor, $B_{T}$, to allow for additional, or possibly
less, disk heating. For example, the observations constrain the gas density
through the observed optical depth of NH3. This means that the density and
therefore the accretion rate, $\dot{M}$, are dependent on the assumed NH3
abundance which is not well known. The adjustable factor, $B_{T}$, decouples
the disk temperature from the assumed molecular abundance. Also the disk
temperature may be raised by stellar radiation (passive heating). The factor,
$B_{T}$, allows for these effects in an approximate way.
The gas temperature in the model is the density weighted average of the
envelope and disk temperatures,
$T_{\rm gas}={{T_{\rm disk}\rho_{\rm disk}+T_{\rm env}\rho_{\rm
env}}\over{\rho_{\rm disk}+\rho_{\rm env}}}$ (13)
### 3.3 The velocity
The 3 components of the gas velocity in the envelope are given in spherical
coordinates by equations 4,5,6 of Keto (2007), but there are errors in earlier
papers. In particular, equations 5 and 6 of Keto (2007) and equation 8 of
Ulrich (1976) are incorrect. Equation 8 of Ulrich (1976) (same as equation 5
of Keto (2007)) does not result in energy conservation, $mv^{2}/2=GM/r$, when
combined with the other velocity components. The error is small, 1 part in
$10^{4}$. The following equations, same as in Mendoza, Canto, Raga (2004), are
exact.
$v_{r}(r,\theta)=-\bigg{(}{{GM_{\ast}}\over{r}}\bigg{)}^{1/2}\bigg{(}1+{{\cos{\theta}}\over{\cos{\theta_{0}}}}\bigg{)}^{1/2}$
(14)
$v_{\theta}(r,\theta)=\bigg{(}{{GM_{\ast}}\over{r}}\bigg{)}^{1/2}\bigg{(}{{\cos{\theta_{0}}-\cos{\theta}}\over{\sin\theta}}\bigg{)}\bigg{(}1+{{\cos\theta}\over{\cos{\theta_{0}}}}\bigg{)}^{1/2}$
(15)
$v_{\phi}(r,\theta)=\bigg{(}{{GM_{\ast}}\over{r}}\bigg{)}^{1/2}{\sin{\theta_{0}}\over{\sin{\theta}}}\bigg{(}1-{{\cos{\theta}}\over{\cos{\theta_{0}}}}\bigg{)}^{1/2}$
(16)
The velocity in the disk is simply the Keplerian velocity,
$v_{\rm disk}(R)=\sqrt{GM_{\ast}/R}$ (17)
where the velocity in the disk is purely azimuthal. We assume that the radial
velocity in the rotationally-supported disk is comparatively small. The gas
velocity in the model is the density weighted average of the envelope and disk
velocities,
$\vec{v}_{\rm gas}={{\vec{v}_{\rm disk}\rho_{\rm disk}+\vec{v}_{\rm
env}\rho_{\rm env}}\over{\rho_{\rm disk}+\rho_{\rm env}}}$ (18)
where $\vec{v}_{\rm env}$ is given by equations 14 through 16 and
$\vec{v}_{\rm disk}$ is given by 17.
### 3.4 The density singularities in the accretion model
The density in the Ulrich model is singular in the mid-plane of the disk at
the centrifugal radius and also at the origin. These singularities are caused
by the convergence of the streamlines in the simple mathematical description
of the flow. This convergence is not expected on physical grounds because gas
pressure, neglected in the Ulrich model, would prevent it. We handle the
singularities in two ways. We define the computational grid to have an even
number of cells so that the centers of the middle cells are above and below
the midplane and around the origin. Second, we smooth the density in the
radial direction with a Gaussian with a width of $R_{d}/2$.
### 3.5 The model parameters
Based on the analyses of the previous observations cited in the introduction,
we assume that the disk-envelope is viewed edge-on. The model contains 6
adjustable parameters: 1) $\rho_{0}$ sets the density of the envelope
(equation 1) and the mass accretion rate (equation 3), 2) $p$ sets the
exponent of the power law decrease of the temperature in the envelope
(equation 11), 3) $\Gamma=v_{\phi}r$ is the specific angular momentum
(equation 4) of the envelope flow, 4) $M_{\ast}$ is the stellar mass 5)
$A_{\rho}$ sets the ratio (equation 9) of the disk density to the envelope
density at $R_{d}$, 6) $B_{T}$ is factor multiplying the disk temperature
(equation 12).
## 4 Fitting the model to the NH3 data
The disk enters into the model additively, and we can test for the presence of
a disk by fitting models to the data with and without the disk. In the first
case, with the disk, we adjust all 6 model parameters for both the envelope
and the disk, and in the second case, without the disk, we adjust only the
first 4 parameters for the envelope. The fitting is done independently in each
case, and the 4 envelope parameters are therefore different in the 2 cases.
The procedure for fitting the data is the same as described in Keto et al.
(2004). We use a fast simulated annealing algorithm to adjust the model
parameters to minimize the summed squared difference ($\chi^{2}$) between the
data and the model spectra.
The model spectra for each particular set of model parameters are generated by
our radiative transfer code MOLLIE (Keto, 1990; Keto et al., 2004). We assume
LTE conditions for the NH3 and CH3CN lines. The LTE approximation is
appropriate for NH3 because in the absence of strong infrared radiation the
level populations are expected to be mostly in the ”metastable” states which
are the lowest J-state of each K-ladder. Since radiative transitions between
K-ladders are forbidden, the coupling between the metastable states is purely
collisional and the population approximates a Boltzmann distribution as in LTE
(Ho & Townes, 1983). CH3CN is also a symmetric top and transitions across the
K-ladders are similarly forbidden; however, unlike NH3, the upper states are
easily populated in warm gas ($>100$ K). While the justification for the LTE
approximation is not as strong for CH3CN as for NH3, we find that CH3CN always
traces hot, very dense gas where collisional transitions should be important.
For C34S we use the accelerated lambda iteration algorithm (ALI) of Rybicki &
Hummer (1991) to solve for the non-LTE level populations.
In comparing the model to the data, we take into account the spatial averaging
of the brightness by the width of the observing beam. We compute spectra over
a grid of positions and smooth the result by convolution with a Gaussian with
the FWHM equivalent to the observing beam of each observation, §2.
We simultaneously fit 12 NH3 spectra, 4 transitions at 3 locations. The 3
locations are marked on figure 1. We ran twenty thousand trial models for each
of the two cases with and without the disk. The spectra of the 2 best-fit
models (with and without the disk) are shown in figures 2 and 3. Parameters
for both cases are listed in table 1. Figure 4 shows the temperature and
density in the mid-plane of our best fitting models. Figure 5 shows the
density and velocity of the disk-envelope model on planes parallel and
perpendicular to the rotation.
We also have CH3CN and C34S data from the observations of Cesaroni et al.
(1999). The way the radiative transfer simulation program operates, we cannot
use our automated search algorithm on more than one molecule at a time. The
program is recompiled for each molecule. We also have not implemented the
automated search for CH3CN. We could fit the model to the C34S data, but this
line does not have the hyperfine structure of NH3 that is so useful in
constraining the optical depth and temperature. Therefore we opted for a
different strategy. We do not use the CH3CN or the C34S data in the fitting.
We use the data on these other lines as a check on the model derived from the
NH3 data. We simulate the CH3CN and C34S emission using the same two models
(with and without the disk) previously derived from the NH3 observations for
comparison with the predicted spectra against the data. The models derived
from the NH3 fitting fix the temperature, densities, and velocities, but we
still need to assume abundances for CH3CN and C34S. For CH3CN we assume an
abundance of $6\times 10^{-8}$. The assumed abundance of C34S is $9.0\times
10^{-11}$, chosen to match the brightness of the (2-1) line.
### 4.1 Comparison of the model with the observed spectra
The comparison of figures 2 and 3 shows that the higher temperature and
density disk and the lower temperature and density envelope are both required
to fit both the high-excitation and low-excitation NH3 spectra. Without the
disk, there is not enough warm, dense gas to reproduce the (4,4) line
brightness. Without the large-scale envelope, there is not enough cold gas to
fit the observed ratios of the NH3(1,1) hyperfine lines.
Further evidence for the disk component comes from the CH3CN spectrum. The
results again show that the warm, high-density, disk component is required to
get enough brightness in lower frequency (higher velocity) K transitions.
Based on the observed line width, the CH3CN(12-11) emission comes from gas
very close to the star. The observed CH3CN(12-11) lines that are not blended
with other molecular lines, have line widths of 10.0 kms-1. The 2 lines that
appear much broader in the data contain contaminating emission from
transitions of CH13CN(12-11) and HNCO (Cesaroni et al., 1999). Rotational
velocities $>10$ kms-1 are only found at radii closer than 600 AU
($<GM/v^{2}$) around a 10 M⊙ star. Figure 4 shows that the gas in the disk has
a temperature of greater than 150 K within this radius. Thus the CH3CN
emission derives mostly from warm gas that is very close to the star and is
thus a good molecular tracer of the inner disk.
The CH3CN(13-12) spectra is too noisy to help discriminate between the two
cases. The peak signal-to-noise ratio is about 6 after smoothing by every
other channel. Nonetheless, the models are at least consistent with these
observations.
The brightness ratio of the C34S (5-4) and (2-1) lines also suggests the
presence of a warm, dense disk (figure 7). The model with the disk reproduces
the observed brightness ratio, 6:1 for the 2 lines, although line width is
less than observed. The model without the disk is not able to generate
sufficient brightness in the higher excitation line, and the shapes of the
line profiles do not match the data. In particular, the strong splitting that
is seen in the model profiles without the disk is not seen in the data. This
difference is also seen in the NH3 and CH3CN spectra of the 2 models, although
not as prominently. We usually associate split line profiles with spatially
unresolved observations of Keplerian disks (Beckwith & Sargent, 1993) whereas
here the disk produces a triangular profile. What happens here is that the
disk component is spatially resolved and has a very high density. Thus within
the beam through the center of the model there is a lot of high density gas
from the outer part of the disk with very low velocity projected along the
line-of-sight. This gas disk creates the peak in the spectrum around zero
velocity. In the envelope-only model without the high density disk, the model
is optically thin in C34S. In this case, we get the usual split line profiles
from the rotation and infall in the envelope.
### 4.2 Goodness of fit
How well do the data constrain the model parameters? Some appreciation can be
gained by plotting $\chi^{2}$ obtained from all the trial models versus each
model parameter. Each panel shows the fits obtained (ordinate) for each value
of a single parameter (abscissa) as all the other parameters are varied. The
lowest value of $\chi^{2}$ (ordinate) at each parameter value (abscissa) is
the best fit that can be obtained for that parameter value, for any
combination of all the other parameters. The formal error of a model parameter
is proportional to the second derivative of $\chi^{2}$ with respect to the
model parameter. Thus the curvature or width of the lower boundary of the
collection of points in each figure is a qualitative measure of the
sensitivity of the model to the parameter.
## 5 Discussion
### 5.1 The density
The gas density is constrained by the optical depths of the NH3 lines, which
can be determined from the brightness ratios of the hyperfine lines. The
density, $\rho_{0}$, at $R_{d}$ is $7.9\times 10^{4}$ (table 1) assuming an
NH3 abundance of $10^{-7}$. With this abundance, the mass of the disk is 2.5
M⊙, the mass of the envelope is 12.6 M⊙, and the accretion rate (equation 3),
asssumed to be the same in the envelope and the disk, is $\dot{M}=7.6\times
10^{-5}$ M⊙ yr-1. The total mass in the accretion flow is mass of the disk and
envelope together which is 15.1 M⊙. This mass estimate is the total mass
within the model boundary of 26000 AU. Cesaroni et al. (1999) estimates that
there is between 0.6 and 8.0 M⊙ within a radius of 5000 AU.
The mass estimate derived from molecular line observations is subject to a
large uncertainty. The radiative transfer modeling determines the column
density of NH3 rather than H2. Therefore, the gas density, the masses of the
disk and envelope, and the accretion rate depend inversely on the assumed
abundance of NH3 which is not well known. Estimates from models and
observations of similar clouds range from $10^{-9}$ to $10^{-6}$ (Herbst &
Klemperer, 2003; Keto, 1990; Estalella et al., 1993; Caproni et al., 2000;
Galvan-Madrid et al., 2009). Thus the abundance of NH3 may be uncertain by
more than an order of magnitude, yet a factor of 2 in the masses of the disk
and envelope is significant in an interpretation of the accretion dynamics.
Furthermore, the abundance of NH3 could be different in the disk and envelope.
Estimates of the NH3 abundance are generally lowest in colder clouds and
highest in warm gas around massive stars. Some NH3 may be frozen onto dust
grains in colder gas and sublimated into the gas phase as the temperature
rises. Therefore, it is possible that the NH3 abundance is higher in the warm
disk than the cold envelope. If so, the mass of the envelope could be higher
than 12.6 M⊙, assuming a lower abundance, without necessarily implying a
higher mass for the disk.
### 5.2 The disk density factor
In the model with the flared disk, the density in the mid-plane of the disk at
the radius of the disk boundary $R_{d}$ is 5 times more dense than the
smoothed density of the envelope at the same point. Considering that the gas
density changes by 6 orders of magnitude across the model, there is little
discontinuity between the envelope and disk densities (figure 4). In the model
without the disk the smoothed density is approximately constant from $R_{d}$
to the origin (figure 4).
### 5.3 The stellar mass
The stellar mass, 10.7 M⊙, is constrained primarily by the disk velocities
(equation 17 ) required to match the observed linewidths (about 10 kms-1) and
the gas temperature that determines the brightness ratios of the low and high
excitation lines. The gas velocities and therefore line widths due to
unresolved motions within the beam depend on the stellar mass because the
velocities are proportional to the square root of the mass. The gas
temperature depends on the mass of the star because the stellar temperature is
a strong function of the stellar mass and because the accretion rate depends
on the stellar mass. In the disk, heating by both the star and by accretion
(dissipation) are important (equations 11, 12 and 13 ). In the best-fit model,
the disk temperature would be about 1/3 lower without the passive heating from
the star, The temperature of the infalling envelope is determined entirely by
the stellar temperature (equation 11).
### 5.4 The angular momentum of the envelope
The angular momentum of the envelope is determined from the widths of the
spectral lines and from the VLSR of the NH3 spectra at the locations left and
right of the center. If there were too much rotation then both of the off-
center spectra would have an incorrect VLSR; too little, and all the NH3 line
widths would be too narrow. The disk radius, $R_{d}=6900$, is set by the
angular momentum in the envelope, $\Gamma$, and the stellar mass (equation 4).
The velocity at $R_{d}$, $v_{k}=1.2$ kms-1, increases inward as $R^{-1/2}$. We
assume a microturbulent broadening of 1 kms-1, added in quadrature to the
thermal broadening. Since the observed line widths are about 10 kms-1 most of
the width of the spectral lines comes from rotation and infall in the model.
The model radius of 6900 AU seems large, and we would regard this as an upper
limit. First, the Ulrich flow conserves angular momentum whereas real
accretion flows probably involve some braking of the spin-up. If the spin-up
were slower, then the disk would be found at a smaller radius. Second, the
high density of the disk is helpful in providing optical depth to strengthen
the NH3(1,1) outer hyperfines. The Ulrich flow has a density singularity in
the mid-plane which we have handled as desribed in §3.4 by a combination of
gridding and smoothing. Maybe our mid-plane density in the envelope is too
low, and to compensate, the thin disk is bigger.
### 5.5 The exponent of the temperature power law
The modelling suggests that the temperature of the envelope falls off very
quickly away from the star, as $r^{-1}$. This implies that the dilution of the
radiation is faster than spherical, $r^{-2}$. The rapid dilution is consistent
with the rotationally flattened geometry. At each radius, radiation is
absorbed and re-emitted by dust in the flow. In a flattened flow, much of this
reprocessed radiation escapes vertically out of the flow. In contrast, in a
spherical flow, all the reprocessed radiation continues to interact with the
flow at larger radii. If the gas temperature decreases outward fast enough
that most of the envelope is at the minimum gas temperature, then a larger
value of the exponent (more negative value of the parameter $p$ in equation
11) would have no further effect. We assume a minimum temperature of 10 K, a
typical temperature of molecular gas, We derive an upper limit $p<-1$.
### 5.6 The disk temperature multiplier
The disk temperature multiplier determines the relative importance of active
disk heating, owing to accretion and dissipation (equation 12), and passive
heating from the protostar (equation 11). With active disk heating, the
temperature in the disk is about 50% above that from passive heating alone.
## 6 Conclusions
This investigation shows that a standard model of a freely-falling,
rotationally-flattened accretion envelope around a rotationally supported disk
is able to reproduce the NH3, CH3CN, and C34S spectral line observations of
IRAS20126. Both the disk and envelope components are required to fit the
observed brightness of both the low and high excitation lines, and to fit the
observed line widths.
This disk-envelope model was developed for low-mass stars and is quite
successful in explaining many of their observable characteristics. The success
of this model in explaining the molecular line observations of the massive
star IRAS20126 suggests that at least up to 10 M⊙, the accretion processes of
massive stars are similar to those of solar mass stars.
There are some differences. Although the mass of the disk is uncertain owing
to its dependence on the NH3 abundance, the disk mass is a significant
fraction of the mass of the star. Furthermore, the extent of the disk is quite
large, $\sim$6900 AU. This suggests that self-gravity in the disk is
dynamically important, and therefore, the disk may be unstable to local
fragmentation and the formation of companion stars.
The physical model in combination with the molecular line radiative transfer
presented in this paper has a further application to other massive star
forming regions. The spatial resolution and sensitivity of the present
generation of interferometers cannot spatially resolve accretion disks in
massive protostellar objects at multi-kpc distances. However, spectral lines
sampling a wide range of densities and temperatures can still provide
constraints to the physical structure of the core and disk. As shown in
IRAS20126, the spectral lines formed in the higher density and temperature
regions confirm the presence of an accretion disk. While future observatories
such as ALMA will be able to spatially resolve the flow on the disk scale,
before the science commissioning of ALMA, this method is a promising technique
to probe the spatially unresolved regions of the flow and disk using data from
the current interferometers.
## Acknowledgments
The authors thank Riccardo Cesaroni and T.K. Sridharan for the use of their
observational data.
## References
* Andre et al. (1993) Andre, P., Ward-Thompson, D., & Barsony, M., 1993, 406, 122
* Caproni et al. (2000) Caproni, A., Abraham, Z., & Vilas-Boas, J.W.S., 2000, AA, 361, 685
* Cesaroni et al. (1997) Cesaroni, R., Felli, M., Walmsley, C.M., & Olmi, L, 1997, AA, 325, 725
* Cesaroni et al. (1999) Cesaroni, R., Felli, M., Jenness, T., Neri, R., Olmi, L., Robberto, M., Testi, L., Walmsley, C.M., 1999, AA, 345, 949
* Cesaroni et al. (2005) Cesaroni, R., Neri, R., Olmi, L., Testi, L., Walmsley, C.M., & Hofner, P., 2005, AA, 434, 1039
* Estalella et al. (1993) Estalella, R., Mauersberger, R., Torrelles, J.M., Anglada, G., Gomez, J.F., Lopez, R., Murders, D., 1993, ApJ, 419,698
* deBuizer (2007) deBuizer J.M., 2007, ApJ, 654, L147
* Galvan-Madrid et al. (2009) Galvan-Madrid, R., Keto, E., Zhang, Q., Kurtz, S., Rodriguez,L.F., Ho, P.T.P., 2009, ApJ, submitted
* Herbst & Klemperer (2003) Herbst, E. & Klemperer, W., 1973, ApJ, 616, 301
* Keto, Ho & Haschick (1987) Keto, E., Ho, P., & Haschick, A., 1987, ApJ, 318, 712
* Cesaroni et al. (1994) Cesaroni, R., Churchwell, E., Hofner, P., Walmsley, C.M., & Kurtz, S., 1994, AA, 288, 903
* Hartmann (2001) Hartmann, L., Accretion Processes in Star Formation Cambridge, UK: Cambridge University Press
* Ho & Townes (1983) Ho, P.T.P., Townes, C.H., 1983, ARAA, 21, 239
* Kahn (1974) Kahn, F.D., 1974, AA, 37, 149
* Kawamura et al. (1999) Kawamura, J.H., Hunter, T.R. , Tong, C.-Y.E., Blundell, R., Zhang, Q., Katz, C., Papa, D.C. & T.K. Sridharan, 1999, PASP, 111, 1088
* Keto (1990) Keto, E., 1990, ApJ, 355, 190
* Keto (2002) Keto, E. 2002b, ApJ, 580, 980
* Keto et al. (2004) Keto, E., Rybicki, G.B., Bergin, E.A., Plume, R., 2004, ApJ, 613, 355
* Keto & Wood (2006) Keto, E. & Wood, K., 2006, ApJ, 637, 850
* Keto (2007) Keto, E., 2007, ApJ, 666, 976
* Kumar & Grave (2007) Kumar M.S.N. & Grave, J.M.C., 2007, AA, 472, 155
* Lada (1987) Lada, C.J., in IAU Symp. 115, Star Forming Regions, ed. M Peimbert & J.Jugaku (Dordrecht:Kluwer), 1
* Lamers & Cassinelli (1999) Lamers, H.J.G.L.M. & Cassinelli, J.P., 1999, Introduction to Stellar Winds Cambridge, UK: Cambridge University Press
* Larson & Starrfield (1971) Larson, R.B. & Starrfield, S., 1971, AA, 13, 190
* Lebron et al. (2006) Lebron, M., Beuther, H., Schilke, P. & Stanke, T. 2006, AA, 448, 1037
* Mendoza, Canto, Raga (2004) Mendoza, S., Canto, J., Raga, A.C., 2004 Rev. Mex. AA, 40, 147
* Pringle (1981) Pringle J., 1981, ARAA, 19, 137
* Qiu et al. (2008) Qiu, Keping, Zhang, Qizhou, Megeath, S Thomas, Gutermuth, Robert A., Beuther, Henrik, Shepherd, Debra S., Testi, L., and De Pree, C. G. 2008, ApJ, 685, 1005
* Rybicki & Hummer (1991) Rybicki, G.B. & Hummer, D.G., 1991, AA, 245, 171
* Beckwith & Sargent (1993) Beckwith, S.V.W. & Sargent, A.I., 1993, ApJ, 402, 280
* Shepherd et al. (2000) Shepherd, D. S., Yu, K. C., Bally, J. & Testi, L. 2000, ApJ, 535, 833
* Shu, Adams, Lizano (1987) Shu, F.H., Adams, F.C., Lizano, S., 1987, ARAA, 25, 23
* Sridharan, Williams, & Fuller (2005) Sridharan, T.K., Williams, S.J., & Fuller, G.A., 2005, ApJ, L73
* Su et al. (2007) Su, Yu-Nung, Liu, Sheng-Yuan, Chen, Huei-Ru, Zhang, Qizhou, and Cesaroni, Riccardo, 2007, ApJ, 671, 571
* Ulrich (1976) Ulrich J., 1976, ApJ, 210, 377
* Whitney et al. (2003) Whitney, B.A., Wood, K., Bjorkman, J.E., Wolff, M.J., 2003, ApJ, 591, 1049
* Whitney et al. (2008) Whitney, B.A., et al., 2008, AJ, 136, 18
* Wilking, Lada, Young (2007) Wilking, Bruce A, Lada, Charles J., & Young, Eric T. 1989, ApJ, 340, 823
* Wolfile & Cassinelli (1987) Wolfire, M.G., Cassinelli, J.P., 1987, ApJ, 319, 850
* Zhang, Hunter, & Sridharan (1998) Zhang, Q., Hunter, T.R., Sridharan, T.K., 1998, ApJ, 505, L151
* Zhang et al. (1999) Zhang, Q., Hunter, T. R., Sridharan, T. K., & Cesaroni, R., 1999, Astrophysical Journal, 527, L117
Table 1: Model Parameters. Parameter | Symbol | Disk | No Disk
---|---|---|---
Env. Density at $R_{d}$ (cm-3) | $\rho_{0}$ | $7.9\times 10^{4}$ | $7.9\times 10^{5}$
Temperature power law exp. | $p$ | $<-1$ | 0.4
Angular Momentum (AU kms-1) | $\Gamma$ | 8100 | 3500
Stellar Mass (M⊙) | M∗ | 10.7 | 7.3
Disk Density Ratio | $A_{\rho}$ | 5.1 | …
Disk Temperature factor | $B_{T}$ | 15.0 | …
Centrifugal radius (AU) | $R_{d}$ | 6900 | 1900
Velocity at $R_{d}$ (kms-1) | $v_{k}$ | 1.2 | 1.8
Total mass1 within 0.128 pc ($M_{\odot}$) | | 12.6 | 10.1
Disk mass ($M_{\odot}$) | | 2.5 | …
Envelope accretion rate ($M_{\odot}$ yr-1) | | $7.6\times 10^{-5}$ | $1.0\times 10^{-4}$
1 The mass estimates include the envelope and the disk for the case | | |
with a disk, and the envelope only for no disk. | | |
Both cases assume an NH3 abundance of $1.0\times 10^{-7}$. | | |
Figure 1: Images of IRAS20126 in NH3(1,1), NH3(2,2), and SiO(2-1). The red
stars in the NH3 images show the locations of the NH3 spectra in figures 2 and
3. See text for absolute coordinates. The single star in the SiO image
corresponds to the middle star in the other images. The angular resolution
(FWHM) is indicated by the shaded ellipse, lower-left corner of each panel.
The contour interval and first contour is 35 mJy kms-1 for the NH3(1,1) and
NH3(2,2) emission, and 130 mJy kms-1 for the SiO(2-1). The SiO image is from
Cesaroni et al. (1999). Figure 2: Data (red) and model (blue) spectra of
NH3(1,1) (top row) NH3(2,2) (2nd row) NH3(3,3) (3rd row) and NH3(4,4) (bottom
row) at 3 positions across the mid-plane of IRAS20126 for the accretion flow
including both an infalling envelope and a flared disk. The spectra in the
middle column are toward the center of the flow, and the the columns to the
left and right show spectra on either side of the center, 6200 AU to the left,
and 5700 au to the right. The 3 locations are shown in figure 1.
Figure 3: Spectra in the same format as figure 2 for the accretion flow with
an envelope only without a flared disk.
Figure 4: The temperature (red) and density (blue) for the model with both an
envelope and flared disk (solid line) and for the model without a disk (dashed
line). The temperatures are about the same in both models, but the disk adds
considerable density. The increase in density in the envelope-only model
(dashed) at 3300 AU is the smoothed density singularity at $R_{d}$ in the
Ulrich (1976) model.
Figure 5: Density (color) and velocities (arrows) for the model with both an
envelope and flared disk. Left: The XZ plane (perpendicular to the rotation)
and right: the XY plane (the mid-plane of the disk and rotationally-flattened
envelope). In the left panel, the color scale is logarithmic between
$9.2\times 10^{4}$ and $3.7\times 10^{6}$ cm-3. Contours (white) indicate the
density at 0.006, 0.01, and 0.03, of the peak density. The longest arrow
corresponds to 2.5 kms-1 In the right panel the color scale is logarithmic
between $1.1\times 10^{5}$ and $8.1\times 10^{7}$ cm-3. Contours (white) show
the density at 0.03, 0.10, and 0.3 of the peak density. The linear scale in
both panels is $R_{d}=6900$ AU.
Figure 6: Data (red) and model (blue) spectra of CH3CN(12-11). The left panel
shows the spectrum for the model with both an accretion envelope and flared
disk. The model is the same as shown in figures 2, 4, 5. The right panel is
for the model with only an accretion envelope and without a disk (figures 3,
4). In the observed spectrum, the line that is anomalously wide (K=6, 2nd from
right) is contaminated by HNCO emission. Additionally, contamination from weak
CH${}_{3}^{13}$CN emission shows up to the red (right) of 4 of the lines. The
velocity of the K=6 hyperfine line has been set to zero. The observed spectra
have been shifted in velocity to match.
Figure 7: C34S(2-1) (blue) and C34S(5-4)(red) spectra of the data (solid
lines) and the model (dashed lines). Left: Model with both the accretion
envelope and flared disk. Right: Model with only an accretion envelope and
without a flared disk. The brightness of the C34S(5-4) spectra, both observed
and modeled, have been divided by 6. The comparison shows that the model with
the disk produces the correct brightness ratio. Without the disk, the
C34S(5-4) line is not bright enough and both lines have the wrong profile
shape. The zero velocity refers to the model. The observed spectra have been
shifted to match.
Figure 8: Spectra of CH3CN(13-12) in the same format as figure 6 except that
we have only the 4 lowest K transitions. The K=0 and K=1 transitions are
blended together in the bluest peak. The left panel shows the spectrum for the
model with both an accretion envelope and flared disk. The right panel is for
the model with only an accretion envelope and without a disk (figures 3, 4).
The signal-to-noise ratio of the observed spectra, about 6 at the peak, is not
high enough to discriminate between the two models. At least the models are
consistent with the data. The data are from T.K. Sridharan (private
communication).
Figure 9: The value of $\chi^{2}$ as a function of a single parameter. The
plots show $\chi^{2}$ for the better fitting disk-envelope models as a
function of a single parameter. The width of the collection of points in each
figure is a qualitative measure of the sensitivity of the models to each
parameter. Values of the parameters outside the range shown by the lower
$\chi^{2}$ values produce very poor fits. The units of the density are the log
of $cm^{-3}$, the units of stellar mass are $M_{\odot}$, and the units of
angular momentum are AU $\times$ kms-1 divided by 1000. The parameters are
listed in Table 1.
|
arxiv-papers
| 2010-02-09T19:18:25 |
2024-09-04T02:49:08.323019
|
{
"license": "Public Domain",
"authors": "Eric Keto and Qizhou Zhang",
"submitter": "Eric Keto",
"url": "https://arxiv.org/abs/1002.1864"
}
|
1002.2140
|
# Theoretical overview of $b\to s$ hadronic decays
University of Sussex, Department of Physics and Astronomy, Falmer, Brighton
BN1 9QH, UK
E-mail
###### Abstract:
A wealth of data on hadronic $b\to s$ transitions is available from the
$B$-factories and will be improved at the LHCb experiment and possible future
super-$B$-factories. I review the theory of these decays as it pertains to the
search for physics beyond the Standard Model and various puzzles in the
present data.
## 1 Introduction
Charmless hadronic $b\to s$ transitions are a rich source of information about
the physics of the weak and/or TeV scales. Their sensitivity to short-distance
physics derives from the CKM hierarchy and a GIM cancellation which combine to
suppress contributions at tree-level in the weak interaction or through light-
quark loops. As a consequence, the Standard-Model (SM) amplitudes are governed
by the combination
$V_{ts}^{*}V_{tb}\times\frac{1}{16\pi^{2}}\times\frac{m_{B}^{2}}{M_{W}^{2}}\sim
10^{-6}.$ (1)
The resulting rareness of these modes makes them sensitive to contributions of
new particles with TeV-scale masses, so we should expect deviations from the
Standard Model. The task is to disentangle SM and new-physics (NP)
contributions in a given mode, such that a possible NP signal can be
recognized, and to identify those observables, or combinations of them, where
this is best possible. More ambitiously, one may want to quantify a signal in
terms of NP-model parameters.
It is worth contrasting the $b\to s$ transitions with the $b\to d$ ones. Here,
the CKM hierarchy is different, such that tree-level contributions involving
$V_{ub}\propto(\bar{\rho}-i\bar{\eta})$ can compete with or dominate over loop
contributions involving $V_{td}\propto(1-\bar{\rho}-i\bar{\eta})$. Indeed,
$b\to d$ hadronic decays, together with $b\to u$ semileptonic and, by now,
purely leptonic $B^{+}\to\tau\nu_{\tau}$ decays, provide the main input to the
global CKM fit (Figure 1).
Figure 1: Global CKM fit as shown at this conference [1]
The two dominant inputs are the ratio of $B_{d}$ and $B_{s}$ mass differences
(orange ring) and the mixing-induced CP violation in $B_{d}\to J/\psi K_{S}$
(blue wedge) derive from the $B-\bar{B}$ mixing amlitudes, which are again
loop processes. Two constraints in a plane will generically intersect in a
discrete set of points, and the most significant consistency check is through
the “$\alpha$” measurements in $B_{d}\to{\pi\pi,\pi\rho,\rho\rho}$ transitions
(shown as light blue “half moon” in the Figure). Hence the consistency of the
CKM fit at present allows ${\cal O}(10\%)$ NP effects. Beyond this level, NP
contributions to different observables would have to conspire to maintain the
observed level of agreement.111 The picture may change as progress in lattice
QCD makes more precise predictions for $B$ meson mixing, $B\to\tau\nu$, and
$\epsilon_{K}$ possible. Interestingly, a significantly improved calculation
of $B_{K}$ [2] indicates a tension with the aforementioned CKM determinations
at about the $2\sigma$ level [3]. On the other hand, the combination
$V_{ts}^{*}V_{tb}$ relevant to $b\to s$ transitions is very weakly dependent
on $\bar{\rho}$ and $\bar{\eta}$. Hence these processes are determined, in
principle, with a small parametric uncertainty in the SM. Moreover, the
consistency of the CKM fit has little to say about new physics in $b\to s$
transitions. Indeed, several puzzles have shown up in recent years in the
data, notably
1. 1.
time-dependent CP violation in $b\to s$ decays of $B^{0}_{d}$ mesons to a CP
eigenstate. In the SM, one expects to measure $-\eta_{CP}S\approx\sin 2\beta$,
but some modes show a deviation (Figure 2). None of these is very significant
at the moment, but this might change with more precise data becoming available
from LHCb and, eventually, a super-$B$ factory.
2. 2.
The time-dependent CP violation in $B_{s}\to J/\psi\phi$, in combination with
lifetime difference and semileptonic asymmetry, determines the phase of the
mixing amplitude to be [4]
$\phi_{B_{s}}\in(-168,-102)^{\circ}\cup(-78,-11)^{\circ},$ (2)
about $2.2\sigma$ from the SM, with much better statistics ahead at Tevatron
and LHCb. The theory is reviewed in a separate talk at this conference [5].
3. 3.
Direct CP asymmetries in $B\to\pi K$ decays. These modes have received
attention for several years. It has been stressed that $A_{\rm
CP}(B^{+}\to\pi^{0}K^{+})\not=A_{\rm CP}(B^{0}\to\pi^{-}K^{+})$ at $5\sigma$
significance [6]. The verdict is less clear, since the SM does not predict
identical asymmetries.
## 2 Hadronic decay amplitudes
Interpreting items 1 and 3 requires knowledge about hadronic decay amplitudes,
which always involve nonperturbative QCD. As the latter is generally under
limited control, approximations are necessary, either neglecting some small
parameter or expanding in it.
For any $b\to s$ transition to a final state $f$, we can write
$\displaystyle{\cal A}_{f}$ $\displaystyle\equiv$ $\displaystyle{\cal A}(B\to
f)=V_{us}V_{ub}^{*}T_{f}+V_{cs}V_{cb}^{*}P_{f}+P_{f}^{\rm NP},$ (3)
$\displaystyle\bar{\cal A}_{\bar{f}}$ $\displaystyle\equiv$
$\displaystyle{\cal
A}(\bar{B}\to\bar{f})=V_{us}^{*}V_{ub}T_{f}+V_{cs}^{*}V_{cb}P_{f}+P_{\bar{f}}^{\rm
NP},$ (4)
where $T_{f}$ and $C_{f}$ are CP-even “strong” amplitudes and $P_{f}^{\rm
NP}$, $P_{\bar{f}}^{\rm NP}$ are new-physics contributions. CKM unitarity has
been used to eliminate the combination $V_{ts}V_{tb}^{*}$
($V_{ts}^{*}V_{tb}$). Branching fractions and CP asymmetries are functions of
the magnitudes and relative phases of the strong amplitudes, as well as
magnitudes and phases of the CKM elements. For instance, if $f$ is a CP
eigenstate, $|\bar{f}\rangle=\eta_{\rm CP}(f)|f\rangle$, then the time-
dependent CP asymmetry is given as
$A_{\rm CP}(f;t)\equiv\frac{\Gamma(\bar{B}(t)\to f)-\Gamma(B(t)\to
f)}{\Gamma(\bar{B}(t)\to f)+\Gamma(B(t)\to f)}\equiv-C_{f}\cos\Delta
m_{d}\,t+S_{f}\sin\Delta m_{d}\,t,$ (5)
$C_{f}=\frac{1-|\xi|^{2}}{1+|\xi|^{2}},\qquad S_{f}=\frac{2{\rm
Im}\xi}{1+|\xi|^{2}},\qquad\xi=e^{-i2\beta}\frac{{\cal A}(\bar{B}\to f)}{{\cal
A}(B\to f)}=-\eta_{\rm
CP}(f)e^{-i2\beta}\frac{V_{cs}^{*}V_{cb}+\dots}{V_{cs}V_{cb}^{*}+\dots}.$ (6)
Here the dots are proportional to the ratio $T_{f}/P_{f}$, multiplied by CKM
factors of ${\cal O}(\lambda^{2})$. If the tree amplitudes are neglected, then
$-\eta_{\rm CP}(f)S_{f}=\sin(2\beta)$ results to very good approximation.
While experimentally (Figure 2) the various modes are in reasonable agreement
with each other and the determination of $\sin 2\beta$ from $b\to c\bar{c}s$
transitions, the suggestive pattern of the central values begs the question
whether it could be caused by the neglected SM tree amplitudes, or one has to
invoke NP terms $P_{f}^{\rm NP}$. Quantitative information on the amplitudes
derives from (i) flavour-$SU(3)$ (and isospin) relations [8] together with
measurements of $b\to d$ transitions and (ii) the heavy-quark expansion in
$\Lambda_{\rm QCD}/m_{b}$ (QCDF [9] and its effective-field-theory formulation
in SCET [10, 11, 12], and the somewhat different “pQCD” approach [13]).
Guidance on the relative importance of amplitudes follows from (iii) Cabibbo
counting and (iv) the large-$N$ expansion [14]. (i), (ii), and (iv) involve
the subdivision of the “physical” tree and penguin amplitudes into several
“topological” amplitudes,
$\displaystyle T_{M_{1}M_{2}}$ $\displaystyle=$
$\displaystyle\Big{[}A_{M_{1}M_{2}}(\alpha_{1}(M_{1}M_{2})+\alpha_{2}(M_{1}M_{2})+\alpha_{4}^{u}(M_{1}M_{2}))$
$\displaystyle+B_{M_{1}M_{2}}(b_{1}(M_{1}M_{2})+b_{2}(M_{1}M_{2})+b_{3}^{u}(M_{1}M_{2})+b_{4}^{u}(M_{1}M_{2}))+{\cal
O}(\alpha)\Big{]}\;+(M_{1}\leftrightarrow M_{2})\;,$ $\displaystyle
P_{M_{1}M_{1}}$ $\displaystyle=$
$\displaystyle\Big{[}A_{M_{1}M_{2}}\alpha_{4}^{c}(M_{1}M_{2})+B_{M_{1}M_{2}}(b_{3}^{c}(M_{1}M_{2})+b_{4}^{c}(M_{1}M_{2}))+{\cal
O}(\alpha)\Big{]}+(M_{1}\leftrightarrow M_{2})\;,$ (8)
where we employ the notation of [15, 9], which is general but is particularly
suited for the heavy-quark expansion. $A_{M_{1}M_{2}}$ and $B_{M_{1}M_{2}}$
are normalization factors which by convention contain certain form factors and
decay constants. The $\alpha_{i}$ and $b_{i}$ denote the different topological
amplitudes. Often, $A\alpha_{1}$ and $A\alpha_{2}$ are written as $T$ and $C$,
$A\alpha_{4}^{c}$ as $P_{ct}$, etc., or variations thereof. Table 1 summarizes
the counting in the various small parameters.
Figure 2: Left: Measurements of mixing-induced CP asymmetries in $b\to s$ penguin transitions as compiled by the HFAG [7]. Right: Constraints from decay rate data in the $(A,S)$ plane for $B\to\pi^{0}K_{S}$ ($A=-C$) [23] Table 1: Hierarchies among topological amplitudes from expansions in the Cabibbo angle $\lambda$, in $1/N_{c}$, and in $\Lambda_{\rm QCD}/m_{b}$. (Some amplitudes, such as electroweak penguins, are omitted from the list.) | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{4}^{u}$ | $\alpha_{4}^{c}$ | $\alpha_{3EW}$ | $\alpha_{4EW}$ | $b_{3}^{c}$ | $b_{4}^{c}$ | $b_{1}$ | $b_{2}$
---|---|---|---|---|---|---|---|---|---|---
| $(T)$ | $(C)$ | $(P_{ut})$ | $(P_{ct})$ | $(P_{\rm EW})$ | $(P_{\rm EW}^{\rm C})$ | | | $(E)$ | $(A)$
Cabibbo $(b\to d)$ | all amplitudes are ${\cal O}(\lambda^{3})$
Cabibbo $(b\to s)$ | $\lambda^{4}$ | $\lambda^{4}$ | $\lambda^{4}$ | $\lambda^{2}$ | $\lambda^{2}$ | $\lambda^{2}$ | $\lambda^{2}$ | $\lambda^{2}$ | $\lambda^{4}$ | $\lambda^{4}$
$1/N$ | $1$ | $\frac{1}{N}$ | $\frac{1}{N}$ | $\frac{1}{N}$ | $1$ | $\frac{1}{N}$ | $\frac{1}{N}$ | $\frac{1}{N}$ | $\frac{1}{N}$ | $1$
$\Lambda/m_{b}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\Lambda/m_{b}$ | $\Lambda/m_{b}$ | $\Lambda/m_{b}$ | $\Lambda/m_{b}$
At the quantitative level, the leading-power amplitudes $\alpha_{1}$,
$\alpha_{2}$, …can be factorized [9] into products of “hard kernels” that can
be computed order by order in perturbation theory and include all strong
(rescattering) phase information, and nonperturbative normalization factors
such as $f_{+}^{B\pi}(0)f_{K}$ or $f_{B}f_{K}f_{\pi}$ (usually factored out
into $A_{M_{1}M_{2}}$ and $B_{M_{1}M_{2}}$). This statement holds up to
generally incalculable $\Lambda/m_{b}$ corrections. Certain amplitudes
(annihilation amplitudes $b_{i}$) are altogether power-suppressed and not
calculable. See [16] for more details. Over the last years, a number of
higher-order (NNLO) calculations of the kernels have been performed [17]. The
main phenomenological findings can be summarized as follows.
* •
The colour-allowed trees $\alpha_{1}$ are well behaved in perturbation theory,
with overall uncertainties at the few-percent level (not counting the
nonperturbative normalization).
* •
The colour-suppressed trees $\alpha_{2}$ show cancellations within the (well-
behaved) perturbative part, and suffer from a large uncertainty in the
normalization and sensitivity to power corrections. Attaching an ${\cal O}(1)$
uncertainty to the (small) theoretical prediction using the power-correction
model of [9] would still imply $|\alpha_{2}/\alpha_{1}|<{\cal O}(1/2)$.
* •
The topological (QCD) penguin amplitudes are also under good control (but only
a subset of NNLO corrections is known), but are phenomenologically
indistinguishable from the incalculable (formally power-suppressed) penguin
annihilation amplitudes.
* •
The colour-allowed and colour-suppressed electroweak penguins amplitudes
behave qualitatively like the colour-allowed and colour-suppressed trees,
respectively.
Further recent work focussing on phenomenological issues can be found in [18],
and a new take on long-distance charm penguins in [19].
## 3 Phenomenological applications
Table 2: Predictions for $\Delta S$ defined in the text for several penguin-dominated modes. From [16]; see therein for details, in particular the meaning and comparison of errors. mode | QCDF/BBNS [20] | SCET/BPRS [11, 21] | pQCD [22] | experiment [7]
---|---|---|---|---
$\phi K_{S}$ | $0.01$ …$0.05$ | $0$ / $0$ | $0.01$ …$0.03$ | $-0.23\pm 0.18$
$\omega K_{S}$ | $0.01$ …$0.21$ | $-0.25$ …$-0.14$ / $0.09$ …$0.13$ | $0.08$ …$0.18$ | $-0.22\pm 0.24$
$\rho^{0}K_{S}$ | $-0.29$ …$0.02$ | $0.11$ …$0.20$ / $-0.16$ …$-0.11$ | $-0.25$ …$-0.09$ | $-0.13\pm 0.20$
$\eta K_{S}$ | $-1.67$ …$0.27$ | $-0.20$ …$0.13$ / $-0.07$ …$0.21$ | |
$\eta^{\prime}K_{S}$ | $0.00$ …$0.03$ | $-0.06$ …$0.10$ / $-0.09$ …$0.11$ | | $-0.08\pm 0.07$
$\pi^{0}K_{S}$ | $0.02$ …$0.15$ | $0.04$ …$0.10$ | | $-0.10\pm 0.17$
Several authors have estimated the tree “pollution” in the hadronic $b\to s$
penguins combining experimental data and heavy-quark-expansion calculations in
different ways. Their results, compared in Table 2, are in general agreement
with each other (as they should) and can be compared to Figure 2. Clearly, the
SM does not produce the pattern of experimental (central) values. While the
significance of the measured $\Delta S$ values is low for all modes, in the
case of $\pi^{0}K_{S}$ one can perform a combined analysis of all $B\to\pi K$
decay data to get a somewhat stronger “signal”. The method discussed here [23]
(see also [24]) invokes the well-known isospin symmetry relation
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{0})+{\cal
A}(B^{0}\to\pi^{-}K^{+})=-\Big{[}(\hat{T}+\hat{C})e^{i\gamma}+\hat{P}_{\rm
ew}\Big{]}\equiv 3A_{3/2}.$ (9)
This relation, and a similar one for the CP conjugates, allows to fix all four
complex decay amplitudes from the four decay rates if the isospin-$3/2$
amplitudes are known, up to a four-fold ambiguity. The latter can indeed be
obtained as
$3A_{3/2}=-R_{T+C}|V_{us}/V_{ud}|\sqrt{2}|A(B^{+}\to\pi^{+}\pi^{0})|\Big{(}e^{i\gamma}-0.66\frac{0.41}{R_{b}}R_{q}\Big{)},$
(10)
where $R_{b}$ is a side of the unitarity triangle and
$R_{T+C}=1.23^{+0.02}_{-0.03}$ and
$R_{q}=(1.02^{+0.27}_{-0.22})e^{i(0^{+1}_{-1})^{\circ}}$ quantify $SU(3)$
breaking, with uncertainties obtained in a QCDF calculation. Fixing the
ambiguity by a (minimal) usage of either QCDF or $SU(3)$, one obtains a
prediction of $S_{\pi^{0}K_{S}}$ (Figure 2) from the remaining data. This is
one of many ways of visualizing the tension in the $\pi K$ system,
distinguished perhaps by a particularly limited use of uncertain theoretical
predictions or assumptions. A future perspective on the uncertainty is also
indicated (thin band). For more on NP in $B\to\pi K$, see [25].
One can also attempt to compute directly the difference in direct CP
asymmetries. Unfortunately, this involves the uncertain colour-suppressed tree
amplitude, and the significance of this discrepancy is currently difficult to
quantify. Making no assumptions about $C$, one still has the relation [26]
$A_{\rm CP}(K^{+}\pi^{-})+A_{\rm CP}(K^{0}\pi^{+})\approx A_{\rm
CP}(K^{+}\pi^{0})+A_{\rm CP}(K^{0}\pi^{0})$, which is satisfied by the current
experimental data [24], and expected to hold (in general) to few-percent
level.
## Acknowledgment
I thank the organizers for a superb conference experience within and outside
the lecture theatre.
## References
* [1] CKMfitter Group (J. Charles et al.), Eur. Phys. J. C 41 (2005) 1 [arXiv:hep-ph/0406184], updated results and plots available at http://ckmfitter.in2p3.fr; K. Trabelsi, talk at this conference.
* [2] D. J. Antonio et al. [RBC and UKQCD Collaborations], Phys. Rev. Lett. 100 (2008) 032001 [arXiv:hep-ph/0702042].
* [3] E. Lunghi and A. Soni, Phys. Lett. B 666 (2008) 162 [arXiv:0803.4340 [hep-ph]]; A. J. Buras and D. Guadagnoli, Phys. Rev. D 78 (2008) 033005 [arXiv:0805.3887 [hep-ph]].
* [4] C. Amsler et al. [Particle Data Group], Phys. Lett. B 667 (2008) 1. The average is from the 2009 update and has been performed by the Heavy Flavour Averaging Group, see [7] and http://www.slac.stanford.edu/xorg/hfag/osc/PDG_2009/.
See also the talks by K. Gibson and I. Bertram at this conference.
* [5] P. Ball, talk at this conference.
* [6] The Belle Collaboration, Nature 452 (2008) 332.
* [7] E. Barberio et al. [HFAG], arXiv:0808.1297 [hep-ex], http://www.slac.stanford.edu/xorg/hfag/index.html.
* [8] D. Zeppenfeld, Z. Phys. C 8 (1981) 77; M. Gronau, O. F. Hernandez, D. London and J. L. Rosner, Phys. Rev. D 50 (1994) 4529 [arXiv:hep-ph/9404283].
* [9] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914 [arXiv:hep-ph/9905312]; Nucl. Phys. B 591 (2000) 313 [arXiv:hep-ph/0006124]; Nucl. Phys. B 606 (2001) 245 [arXiv:hep-ph/0104110].
* [10] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63 (2001) 114020 [arXiv:hep-ph/0011336]; J. g. Chay and C. Kim, Phys. Rev. D 68 (2003) 071502 [arXiv:hep-ph/0301055]; C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Phys. Rev. D 70 (2004) 054015 [arXiv:hep-ph/0401188].
* [11] A. R. Williamson and J. Zupan, Phys. Rev. D 74, 014003 (2006) [Erratum-ibid. D 74, 03901 (2006)] [arXiv:hep-ph/0601214].
* [12] M. Beneke and S. Jäger, Nucl. Phys. B 768, 51 (2007) [arXiv:hep-ph/0610322].
* [13] Y. Y. Keum, H. n. Li and A. I. Sanda, Phys. Lett. B 504, 6 (2001) [arXiv:hep-ph/0004004].
* [14] A. J. Buras, J. M. Gerard and R. Rückl, Nucl. Phys. B 268, 16 (1986); M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 (1987) 103.
* [15] M. Beneke and M. Neubert, Nucl. Phys. B 675 (2003) 333 [arXiv:hep-ph/0308039].
* [16] M. Antonelli et al., arXiv:0907.5386 [hep-ph], and references therein.
* [17] T. Becher and R. J. Hill, JHEP 0410, 055 (2004) [arXiv:hep-ph/0408344]; M. Beneke and D. Yang, Nucl. Phys. B 736, 34 (2006) [arXiv:hep-ph/0508250]; G. G. Kirilin, arXiv:hep-ph/0508235; N. Kivel, JHEP 0705, 019 (2007) [arXiv:hep-ph/0608291]; M. Beneke and S. Jäger, Nucl. Phys. B 768, 51 (2007) [arXiv:hep-ph/0610322]; A. Jain, I. Z. Rothstein and I. W. Stewart, arXiv:0706.3399 [hep-ph]; V. Pilipp, Nucl. Phys. B 794, 154 (2008) [arXiv:0709.3214 [hep-ph]]; G. Bell, Nucl. Phys. B 795, 1 (2008) [arXiv:0705.3127 [hep-ph]]; Nucl. Phys. B 822, 172 (2009) [arXiv:0902.1915 [hep-ph]]; M. Beneke, T. Huber and X. Q. Li, arXiv:0911.3655 [hep-ph].
* [18] M. Duraisamy and A. L. Kagan, arXiv:0812.3162 [hep-ph]; H. n. Li and S. Mishima, arXiv:0901.1272 [hep-ph]; G. Bell and V. Pilipp, Phys. Rev. D 80 (2009) 054024 [arXiv:0907.1016 [hep-ph]]. H. Y. Cheng and C. K. Chua, Phys. Rev. D 80 (2009) 074031 [arXiv:0908.3506 [hep-ph]].
* [19] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Eur. Phys. J. C 61 (2009) 439 [arXiv:0902.4446 [hep-ph]].
* [20] M. Beneke, Phys. Lett. B 620, 143 (2005) [arXiv:hep-ph/0505075].
* [21] W. Wang, Y. M. Wang, D. S. Yang and C. D. Lu, Phys. Rev. D 78, 034011 (2008).
* [22] H. n. Li and S. Mishima, Phys. Rev. D 74, 094020 (2006) [arXiv:hep-ph/0608277].
* [23] R. Fleischer, S. Jäger, D. Pirjol and J. Zupan, Phys. Rev. D 78 (2008) 111501 [arXiv:0806.2900 [hep-ph]].
* [24] M. Gronau and J. L. Rosner, Phys. Lett. B 666 (2008) 467 [arXiv:0807.3080 [hep-ph]].
* [25] It is difficult to do justice to the enormous literature on NP in $B\to\pi K$ in a conference proceedings. Some classic works are: T. Yoshikawa, Phys. Rev. D 68 (2003) 054023 [arXiv:hep-ph/0306147]; M. Gronau and J. L. Rosner, Phys. Lett. B 572 (2003) 43 [arXiv:hep-ph/0307095]; A. J. Buras, R. Fleischer, S. Recksiegel and F. Schwab, Phys. Rev. Lett. 92 (2004) 101804 [arXiv:hep-ph/0312259]; Nucl. Phys. B 697 (2004) 133 [arXiv:hep-ph/0402112]; S. Baek, P. Hamel, D. London, A. Datta and D. A. Suprun, Phys. Rev. D 71 (2005) 057502 [arXiv:hep-ph/0412086]; Y. L. Wu and Y. F. Zhou, Phys. Rev. D 72 (2005) 034037 [arXiv:hep-ph/0503077]; see also [15]. Recent work includes: T. Feldmann, M. Jung and T. Mannel, JHEP 0808 (2008) 066 [arXiv:0803.3729 [hep-ph]]; A. Soni, A. K. Alok, A. Giri, R. Mohanta and S. Nandi, Phys. Lett. B 683 (2010) 302 [arXiv:0807.1971 [hep-ph]]; M. Ciuchini, E. Franco, G. Martinelli, M. Pierini and L. Silvestrini, Phys. Lett. B 674 (2009) 197 [arXiv:0811.0341 [hep-ph]]; N. Mahajan, arXiv:0812.0230 [hep-ph]; R. Mohanta and A. K. Giri, Phys. Rev. D 79 (2009) 057902 [arXiv:0812.1842 [hep-ph]]; V. Barger, L. Everett, J. Jiang, P. Langacker, T. Liu and C. Wagner, Phys. Rev. D 80 (2009) 055008 [arXiv:0902.4507 [hep-ph]]; Q. Chang, X. Q. Li and Y. D. Yang, JHEP 0905 (2009) 056 [arXiv:0903.0275 [hep-ph]]; S. Baek, C. W. Chiang, M. Gronau, D. London and J. L. Rosner, Phys. Lett. B 678 (2009) 97 [arXiv:0905.1495 [hep-ph]]; V. Barger, L. L. Everett, J. Jiang, P. Langacker, T. Liu and C. E. M. Wagner, JHEP 0912 (2009) 048 [arXiv:0906.3745 [hep-ph]]; S. Khalil, A. Masiero and H. Murayama, Phys. Lett. B 682 (2009) 74 [arXiv:0908.3216 [hep-ph]]; M. Bauer, S. Casagrande, U. Haisch and M. Neubert, arXiv:0912.1625 [hep-ph]; A. Soni, A. K. Alok, A. Giri, R. Mohanta and S. Nandi, arXiv:1002.0595 [hep-ph].
* [26] M. Gronau, Phys. Lett. B 627 (2005) 82 [arXiv:hep-ph/0508047].
|
arxiv-papers
| 2010-02-10T17:21:18 |
2024-09-04T02:49:08.332122
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sebastian Jager",
"submitter": "Sebastian J\\\"ager",
"url": "https://arxiv.org/abs/1002.2140"
}
|
1002.2275
|
# Statistical and geometrical properties of thermal plumes in turbulent
Rayleigh-Bénard convection
Quan Zhou1,2 and Ke-Qing Xia1 1Department of Physics, The Chinese University
of Hong Kong, Shatin, Hong Kong, China
2Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University,
Shanghai 200072, China kxia@phy.cuhk.edu.hk
###### Abstract
We present a systematic experimental study of geometric and statistical
properties of thermal plumes in turbulent Rayleigh-Bénard convection using the
thermochromic-liquid-crystal (TLC) technique. The experiments were performed
in three water-filled cylindrical convection cells with aspect ratios 2, 1,
and 0.5 and over the Rayleigh-number range $5\times 10^{7}\leq Ra\leq
10^{11}$. TLC thermal images of horizontal plane cuts at various depths below
the top plate were acquired. Three-dimensional images of thermal plumes were
then reconstructed from the two-dimensional slices of the temperature field.
The results show that the often-called sheetlike plumes are really one-
dimensional structures and may be called rodlike plumes. We find that the
number densities for both sheetlike/rodlike and mushroomlike plumes have
power-law dependence on $Ra$ with scaling exponents of $\sim 0.3$, which is
close to that between the Nusselt number $Nu$ and $Ra$. This result suggests
that it is the plume number that primarily determines the scaling exponent of
the $Nu$-$Ra$ scaling relation. The evolution of the aspect ratio of
sheetlike/rodlike plumes reveals that as $Ra$ increases the plume geometry
changes from more-elongated to less-elongated. Our study of the plume area
fraction (fraction of coverage over the surface of the plate) further reveals
that the increased plume numbers with $Ra$ mainly comes from increased plume
emission, rather than fragmentation of plumes. In addition, the area,
perimeter, and the shape complexity of the two-dimensional horizontal cuts of
sheetlike/rodlike plumes were studied and all are found to obey log-normal
distributions.
## 1 Introduction
As an important class of turbulent flows, turbulent thermal convection is
ubiquitous in nature, ranging from those in the planets and stars, in the
Earth’s mantle and its outer core, and in the atmosphere and oceans to the
convection in heat transport and mixing in engineering applications. Turbulent
Rayleigh-Bénard (RB) convection, a fluid layer sandwiched between two parallel
plates and heated from below, has long been used as a model system to study
natural convections [1, 2, 3]. Two of the important issues in the study of
turbulent RB convection are heat transport and coherent structures such as
thermal plumes. In the first one, one tries to understand how heat transported
upwards across the fluid layer, which is characterized by the Nusselt number
$Nu=J/(\chi\bigtriangleup/H)$, depends on the turbulent intensity, which is
characterized by the Rayleigh number $Ra=\alpha g\Delta H^{3}/\nu\kappa$.
Here, $\Delta$ is the applied temperature difference across the fluid layer,
$g$ is the gravitational acceleration, and $\alpha$, $\nu$, $\chi$ and
$\kappa$ are, respectively, the volume expansion coefficient, kinematic
viscosity, thermal conductivity, and thermal diffusivity of the convecting
fluid. In many theories and experiments, it is often assumed that there is a
simple power law relation between $Nu$ and $Ra$, i.e. $Nu\sim Ra^{\beta}$. In
the second issue one wants to understand the statistical and geometrical
properties of thermal plumes, which is a localized thermal structure and has
been shown to play a key role in many natural phenomena and engineering
applications, such as in mantle convection where mantle plumes just below
Earth’s crust are responsible for the formation of volcanoes (see, for
example, Ref. [4, 5]), in nuclear explosions and stellar convection where
plumes dominate both the dynamics and the energy transport (see, for example,
Ref. [6]). The two issues are not independent of each other. It has been shown
recently that for turbulent RB system most heat is carried and transported by
thermal plumes [7, 8, 9]. Accordingly, it remains a challenge to establish a
quantitative relationship between thermal plumes and heat transport. Although
a theoretical effort [10] has recently focused on this connection, few such
experimental studies have been made.
Thermal plumes generated by a localized hot/cold spot have a well organized
structure, which consists of a mushroom cap with sharp temperature gradient
and a stem that is relatively diffusive [11, 12]. Plumes with such a structure
are often referred to as mushroomlike plumes, which are also observed in
turbulent RBC when viewed from the side [13, 14, 15, 11, 16, 17, 18, 19, 7,
20]. By extracting plumes from temperature time series measured locally near
the cell sidewall, Zhou and Xia [21, 22] showed that the size of mushroomlike
plumes obey log-normal statistics. However, the morphology of the plumes is
totally different when one observes from above (or below): thermal plume is
extended in one horizontal direction but concentrated in the orthogonal
horizontal direction [23, 13, 15, 24, 25, 26, 27, 28, 29, 30, 31]. Plumes with
such a structure are often assumed to have significant vertical extent and
thus are called sheetlike plumes. Puthenveettil and Arakeri [27] studied near-
wall structures in turbulent natural convection driven by concentration
differences across a membrane and found that the plume spacings show a common
log-normal probability density function (PDF). Zhou _et al._ [29] further
revealed that both the area and the heat content of sheetlike plumes are log-
normal distributed. Shishkina and Wagner [32] investigated quantitatively
geometric properties of sheetlike plumes using direct numerical simulations.
However, systematical experimental studies of geometrical structures of
sheetlike plumes are still missing. Recently, Funfschilling _et al._ [30]
suggested that when viewed from above thermal plumes near the two plates
should be better referred to as linelike plumes, as the vertical extent of
these structures do not seem to be established and it appears more likely that
they are one dimensional excitations in the marginally-stable boundary layers.
As two different configurations of thermal plumes coexist simultaneously in
turbulent RB system, it is natural to ask how sheetlike and mushroomlike
plumes transform from each other. By using thermochromic-liquid-crystal (TLC)
technique, Zhou _et al._ [29] showed, in a cylindrical cell with unity aspect
ratio ($\Gamma=D/H$ with $D$ and $H$ as the inner diameter and the height of
the convection cell, respectively.), that hot fluids (plumes) move upwards,
imping on the top plate from below, then spread horizontally along the top
plate and form waves or sheetlike plumes. As they travel horizontally along
the plate, these sheetlike plumes collide with each other or with the
sidewall, convolute and form swirls. As these swirls are cooler than the bulk
fluid, they move downwards, merge and cluster together. By the symmetry of the
system, the same process is expected for the morphological evolution of
thermal plumes occurring near the bottom plate. However, in Ref. [29] the
morphological evolution was visualized only at a single Rayleigh number
($Ra=2.0\times 10^{9}$), and it is not clear whether this process is universal
for much higher $Ra$ or for cells with different aspect ratios. The
understanding of this evolution process is of great importance, as the
characterization of coherent structures is essential to the understanding of
turbulent flows in many systems.
In the present paper, we report new experiments of the temperature and
velocity fields measured at varying depths from the top plate over the
Rayleigh number range $5\times 10^{7}\leq Ra\leq 10^{11}$ and the Prandtl
number range $4.1\leq Pr=\nu/\kappa\leq 5.3$ and in three water-filled
cylindrical sapphire cells with aspect ratios 0.5, 1, and 2. We present
quantitative results on the relationship between plume number density and $Nu$
and results that relate the evolution of the plume morphology to the heat
transport. The remainder of the paper is organized as follows. We give a
detailed description about the experimental setup and data analysis method in
section 2. In section 3, we study the $z$\- and $Ra$-dependence of plume
number and discuss the relationship between thermal plumes and the $Nu$-$Ra$
scaling. Section 4 presents study on the geometric properties of sheetlike
plumes, which are mainly based on data from the aspect ratio 1 cell. We
summarize our findings and conclude in section 5.
## 2 Experimental setup and procedures
### 2.1 The convection cell and the experimental parameters
Figure 1: A schematic drawing of the convection cell used in the experiments.
The top and bottom plates are made of sapphire so that the flow in a
horizontal plane can be visualized and captured. A: the cooling chamber cover,
B: the top sapphire plate, C: the Plexiglas sidewall, D: the bottom sapphire
plate, E: the heating chamber cover, F: thermistors, G: nozzles for cooling or
heating water, H: stainless steel rings as part of the heating and cooling
chambers, I: nozzle for filling fluid into the cell, J: nozzle for letting air
out of the cell, K: a square-shaped glass jacket.
The experiments were carried out in three cylindrical sapphire cells [33, 29].
The schematic diagram of the cells is shown in figure 1. To capture and study
the horizontal temperature and velocity fields from top, two sapphire discs
(Almaz Optics, Inc) with diameter $19.5$ cm and thickness $5.0$ mm were chosen
as the top and bottom plates for their good thermal conductivity ($35.1$ W/mK
at 300 K) compared with other transparent materials. Two chambers, constructed
by stainless steel rings $H$ and Plexiglas discs $A$ and $E$, are used to heat
the bottom plate and cool the top plate. Each chamber is connected to a
separate refrigerated circulator by four nozzles, water flows into the chamber
by two nozzles in two opposite direction and leaves from the other two nozzles
perpendicular to the inlets. The sidewall of the cell is a vertical tube made
of Plexiglas ($C$) with inner diameter $D=18.5$ cm and wall thickness $8$ mm,
respectively. The separations between the top and bottom plates $H$ are $9.3$,
$18.5$, and $37$ cm so that the aspect ratios of the cells are respectively
$\Gamma=2$, 1, and 1/2. Draining ($I$) and filling ($J$) tubes are fitted on
the Plexiglas tube at a distance of $1.5$ cm from the top and bottom plates. A
square-shaped jacket ($K$) made of flat glass plates and filled with water is
fitted to the outside of the sidewall [20], which greatly reduces the spatial
variation in the intensity of the white lightsheet caused by the curvature of
the cylindrical sidewall. Four rubber O-rings (not shown in the figure) are
placed between the two sapphire plates and the steel rings and between the
rings and the two Plexiglas discs to avoid fluid leakage. Four thermistors $F$
(model 44031, Omega Engineering Inc.) are used to measure the temperature
difference between the two sapphire plates. To keep good contact between the
plates and the thermistors, the thermistors are wrapped by the heat transfer
compound (HTC10s from Electrolube Limited). It is found that the measured
relative temperature difference between the two thermistors in the same plate
is less than $3\%$ of that across the convection cell for both plates and for
all $Ra$ investigated. This indicates the uniform distribution of the
temperature across the horizontal plates.
### 2.2 Liquid crystal measurements
Figure 2: The optical setup for thermochromatic liquid crystal visualization
and measurements: $S$, halogen lamp; $L1$, concave mirror; $L2$ and $L3$,
condensing lenses; $L4$, diverging cylindrical lens.
The visualization technique employing thermochromatic liquid crystal (TLC)
particles has been widely used and documented in fluid visualization
experiments and was used in the present work to visualize the temperature and
velocity fields in horizontal fluid layers of varying depth from the cell’s
top plate. Two types of TLC microspheres (Hallcrest, Ltd.) were used in the
experiments. One type (model R29C4W) was used for low-$\Delta$ experiments
with the mean bulk temperature $T_{0}=30$ ∘C and $Pr=5.3$, and the other type
(model R40C5W) was used for high-$\Delta$ measurements with $T_{0}=42$ ∘C and
$Pr=4.1$. Both these particles have a mean diameter of $50$ $\mu$m and density
of $1.03-1.05$ g/cm3, and were suspended in the convection fluid in very low
concentrations (about $0.01\%$ by weight), at which the influence of TLC
particles on the fluid can be neglected. The peak wavelength of light
scattered by these particles changes from red to green and then to blue within
a temperature window of 4 ∘C from about 29 to 33 ∘C for R29C4W, and of 5 ∘C
from about 40 to 45 ∘C for R40C5W. In the experiments, the mean bulk
temperature was set to 30 ∘C for R29C4W (Pr $=5.3$) and to 42 ∘C for R40C5W
(Pr $=4.1$), so that the background fluid appears blue and the red and green
regions correspond to cold fluid, i.e. cold plumes. Figure 2 shows a schematic
diagram of the optical setup for the experiments. A halogen photo optic lamp
($S$) with a power of 650 W was used as the light source. One concave mirror
$L1$ and two condensing lenses $L2$ and $L3$ were used to collect the light
from $S$ and focus it onto the central section of the cell. A horizontal sheet
of white light, generated by a diverging cylindrical lens $L4$ and then
projected onto an adjustable slit, passed through the cell parallel to the top
plate. The thickness of the lightsheet inside the cell is approximately 3 mm.
A Nikon D1X camera, with a resolution $2000\times 1312$ pixels and 24 bit
dynamic range, was placed on the top of the cell to take photographs of the
TLC microspheres. With short camera exposure time ($0.02$ s), the captured
photographs give the instantaneous temperature field, and with long camera
exposure time (0.77s) they will in addition show the the trajectories of the
particles.
Figure 3: Images of TLC microspheres with the camera exposure time of 0.02 s
taken at $2$ mm from the top plate at (a) $Ra=1.1\times 10^{9}$, (b)
$3.0\times 10^{9}$, and (c) $1.0\times 10^{10}$. (d), (e) and (f), which
correspond to (a), (b) and (c), respectively, are extracted sheetlike plumes
with background in the original images removed.
To study thermal plume properties quantitatively, we counted plume numbers and
extracted sheetlike plumes. For sheetlike plumes, we took $150$ to $300$
consecutive images (at $z=2$ mm and with camera exposure time of $0.02$ s) at
$30$ to $60$ s intervals for a given $Ra$, so that two successive images in
each sequence are statistically independent. A cold sheetlike plume is
extracted by first manually drawing a contour around its perimeter and then
using a software to collect all pixels enclosed by the contour. A program is
then used to calculate the perimeter $P$ and the area $A$ of each extracted
plume. To ensure that plumes are identified correctly the operator uses
knowledge gained from viewing movies of plume motions. A total of $3000$ to
$6000$ plumes are identified from the $150$ to $300$ images, which are not
large numbers but should be indicative of the statistical properties. Figures
3(d)-(f) show three examples of extracted plumes with background removed,
which correspond to images of TLC microsperes in figures 3(a)-(c),
respectively, for three different values of $Ra$ obtained from the unity-
aspect-ratio cell. For mushroomlike plumes, we took sequences of images at
varying depth $z$ from the top plate and with camera exposure time of $0.77$ s
at $30$ s intervals for each $Ra$. Each sequence consists $200$ images for a
given depth. A cold mushroomlike plume was identified as an object with
nonzero vorticity and with temperature (color) much lower than that of the
background fluid [29] (see, e.g., figure 4(b)). As mushroomlike plumes are
partly entangled together, they cannot be separated completely. Nevertheless,
we can identify them individually as swirls with cold temperature and thus
count their numbers. The cold sheetlike plumes were also identified and
counted from these images, as objects with a linelike shape and lower
temperature.
Figure 4: Images of TLC microspheres with the camera exposure time of 0.77 s
taken at (a) $2$ mm and (b) $2$ cm from the top plate at $Ra=1.0\times
10^{9}$.
## 3 Plume number statistics
### 3.1 Morphological evolution of thermal plumes
Figure 5: The process of morphological evolution between sheetlike and
mushroom plumes.
The process of morphological evolution between sheetlike and mushroomlike
plumes has been revealed by Zhou _et al._ [29] in a cylindrical cell. Here,
the same process was also observed for all values of $Ra$ and $\Gamma$
investigated in the experiment. Figure 4(a), taken at 2 mm from the top plate,
shows how this evolution comes about. One sees that near the top plate the
motion of TLC microspheres appear to emanate from certain regions or “sources”
with bluish color, suggesting that hot fluids (plumes) are moving upwards,
impinging on and spreading horizontally along the top plate. Along the
particle traces, the color turns from blue to green and red, implying that the
wave fronts are cooled down gradually by the top plate (and the top thermal
boundary layer) as they spread. As they travel along the plate’s surface,
sheetlike plumes collide with each other or with the sidewall. As different
plumes carry momenta in different directions, they merge, convolute and form
swirls (hence generating vorticity). As these swirls are cooler than the bulk
fluid, they move downwards, merge and cluster together (see, e.g, figure
4(b)). We note that, by symmetry of the system, the same morphological
evolution of thermal plumes should occur near the bottom plate. The physical
picture of this evolution process is illustrated in figure 5.
### 3.2 Depth-dependent properties
Figure 6: Mean numbers of (a), (c), and (e) (cold) sheetlike plumes
$N_{pl}^{sheet}$ and (b), (d), and (f) (cold) mushroomlike plumes
$N_{pl}^{mush}$ as functions of the distance $z$ for all measured $Ra$. (a)
and (b) for $\Gamma=2$, (c) and (d) for $\Gamma=1$, and (e) and (f) for
$\Gamma=0.5$.
Because of the collision and convolution of sheetlike plumes and the
transformation from sheetlike to mushroomlike ones, it is expected that the
number of sheetlike plumes should decrease from the top plate, while that of
mushroomlike ones should increase, and this is indeed the case as shown in
figure 6. Figures 6(a), (c), and (e) show the mean numbers of (cold) sheetlike
plumes $N_{pl}^{sheet}$ versus the normalized height $z/\delta_{th}$ obtained
from the cells with $\Gamma=2$, 1, and 0.5, respectively. (Here, $\delta_{th}$
is the thickness of thermal boundary layer, which was measured in [34, 35].)
One sees that $N_{pl}^{sheet}$ decays quickly away from the top plate and the
behavior is similar for all $Ra$ studied. Beyond the depth $z\approx
15\delta_{th}$, one can hardly identify any sheetlike plumes from the acquired
images. Note that, in addition to transforming into mushroomlike plumes, the
rapid reduction of $N_{pl}^{sheet}$ with depth could also be attributed to the
limited vertical extent of sheetlike plumes themselves, i.e. plumes, whose
vertical extents are shorter than a certain height $h$, could not be captured
by the images that are taken farther than $h$ from the top plate. We further
note that the $z$-dependence of plume numbers $N_{pl}^{sheet}$ may be well
described by a decreasing exponential function, i.e.
$N_{pl}^{sheet}=N_{0}e^{-b_{r}z/\delta_{th}}$ with fitting parameters $N_{0}$
and $b_{r}$ for nearly all measured $Ra$. Hence $z_{e}=1/b_{r}$ may be used as
a typical height of vertical extent of sheetlike plumes. Figure 7(a) shows the
scaled height $z_{e}/\delta_{th}$ as a function of $Ra$. It is seen that
$z_{e}/\delta_{th}$ increases with $Ra$ for the $\Gamma=2$ and 0.5 cells,
while keeps nearly constant for the $\Gamma=1$ cell. We can not judge the
significance of the different behaviors of $z_{e}/\delta_{th}$ varying with
$Ra$ in cells with different $\Gamma$. However, $z_{e}$ varies only in the
range of a few times the thickness of thermal boundary layer
($2\delta_{th}<z_{e}<5\delta_{th}$). This suggests that the vertical extent of
the so-called sheetlike plumes does not have enough spatial extension in the
vertical direction to form sheets.
Figure 7: (a) $z_{e}/\delta_{th}$ and (b) $z_{C}/\delta_{th}$ vs $Ra$ for
$\Gamma=2$ (up-triangles), 1 (circles), and 0.5 (down-triangles). The dashed
line in (b) marks the mean value of $z_{c}$ for $Ra>3\times 10^{8}$.
For mushroomlike plumes, the mean numbers of (cold) mushroomlike plumes
$N_{pl}^{mush}$ as a function of the scaled depth $z/H$ is shown in figures
6(b), (d), and (f). With increasing $z$, $N_{pl}^{mush}$ for all $Ra$ first
increases rapidly, then decreases and finally remains approximately constant
for positions $z>0.2H$. When $N_{pl}^{mush}$ first increases with $z$, it
crosses over with $N_{pl}^{sheet}$ at some crossover-depth $z_{C}$. The
normalized crossover depths $z_{C}$ are shown in figure 7(b). It is seen that
$z_{c}\approx 2.5\delta_{th}$ when $Ra\leq 3\times 10^{8}$. For $Ra>3\times
10^{8}$, although the data points seem to be somewhat scattered, they exhibit
no clear dependence on $Ra$ and have a mean value of $\sim 4.84\delta_{th}$.
This implies that the transformation occurs within a region, whose height is
only associated with the thermal boundary layer thickness. If one takes this
value as the typical vertical extent of sheetlike plumes, these results again
suggest that the vertical extent of what is called sheetlike plumes is only
the order of several $\delta_{th}$, not much larger than it, and thus not
large enough to have a sheetlike shape. This will be further discussed in
section 4.4. After $N_{pl}^{mush}$ attains its maximum value around $z=z_{p}$,
it then drops sharply beyond the depth $z\simeq 0.2H$, because of the mixing,
merging, and clustering of mushroomlike plumes. As found by Zhou _et al._
[29], this region corresponds to the region of full width at half maximum of
vertical vorticity profile and can be used as a quantitative definition and
measure of the mixing zone [36, 21]. It is further found that the peak
position in the profiles of $N_{pl}^{mush}$, $z_{p}$, has no obvious
$Ra$-dependence for each $\Gamma$, but decreases slightly from $0.08H$ for
$\Gamma=2$ to $0.03H$ for $\Gamma=0.5$.
### 3.3 The Rayleigh number dependency
Table 1: The fitted parameters $\alpha_{s}$, $\beta_{s}$, $\alpha_{m}$, and $\beta_{m}$ for all three cells. $\Gamma$ | $\alpha_{s}$ | $\beta_{s}$ | $\alpha_{m}$ | $\beta_{m}$
---|---|---|---|---
0.5 | 1.2 | $0.29\pm 0.03$ | 0.45 | $0.31\pm 0.03$
1 | 1.5 | $0.28\pm 0.03$ | 0.37 | $0.33\pm 0.03$
2 | 1.6 | $0.30\pm 0.03$ | 1.86 | $0.28\pm 0.03$
Figure 8: $Ra$-dependency of sheetlike plume number density
$N_{pl}^{sheet}/(\pi D^{2}/4)$ (dark-green symbols), counted just outside the
thermal boundary layer, and mushroomlike plume numbers $N_{pl}^{mush}/(\pi
D^{2}/4)$ (red symbols), counted at the peak of the height-profile of
$N_{pl}^{mush}$, for the $\Gamma=2$ (up-triangles), 1 (circles), and 0.5
(down-triangles) cells. For both $N_{pl}^{sheet}$ and $N_{pl}^{mush}$, the
$\Gamma=2$ (0.5) data have been shifted down (up), so that they agree with
corresponding data for $\Gamma=1$ in their overlap regions of $Ra$. Solids
lines are power-law fittings to the data shown in the same color. The fitted
scaling exponents are $0.29\pm 0.03$ and $0.32\pm 0.03$ for $N_{pl}^{sheet}$
and $N_{pl}^{mush}$, respectively.
We next focus on the $Ra$-dependency of plume number density. For sheetlike
plumes, the number was counted just outside the thermal boundary layer and the
counted number, $N_{pl}^{sheet}$, for each cell can be well described by a
power law $N_{pl}^{sheet}/(\pi
D^{2}/4)=\alpha_{s}(\Gamma)Ra^{\beta_{s}(\Gamma)}$, with two fitting
parameters $\alpha_{s}$ and $\beta_{s}$ as functions of $\Gamma$. Here, the
fitted values of $\alpha_{s}$ and $\beta_{s}$ for all three aspect ratios are
listed in table 1. It is seen that $\alpha_{s}(\Gamma)$ varies with the cell’s
aspect ratio $\Gamma$ while $\beta_{s}(\Gamma)$ is approximately the same for
all three cells. As the range of $Ra$ is limited for each aspect ratio, we
shifted the plume number densities for the $\Gamma=2$ and 0.5 cells to agree
with data for the $\Gamma=1$ cell in their respective overlapping range of
$Ra$. We then fitted a single power-law to the combined data set from the
three cells that span almost three decades of $Ra$. Figure 8 shows the shifted
numbers, normalized by the area of the top plate $\pi D^{2}/4$, as well as the
plume number from the $\Gamma=1$ cell as dark-green symbols. It is seen that a
single power law
$N_{pl}^{sheet}/(\pi D^{2}/4)=\alpha_{s}Ra^{\beta_{s}}\mbox{\ \ with\ \
}\alpha_{s}=1.4\mbox{\ \ and\ \ }\beta_{s}=0.29\pm 0.03$ (1)
can be used to well describe the data from all three cells.
For mushroomlike plumes, one sees from figures 6(b), (d), and (f) that there
is a peak on the height-profile of mushroomlike plume number $N_{pl}^{mush}$
for all three cells and for all measured $Ra$. We use the value at the peak of
the height-profile of mushroomlike plume number to examine the $Ra$-dependence
of $N_{pl}^{mush}$. It is found that a power law relation $N_{pl}^{mush}/(\pi
D^{2}/4)=\alpha_{m}(\Gamma)Ra^{\beta_{m}(\Gamma)}$ can also be used to
describe the $Ra$-dependence of $N_{pl}^{mush}$. Again, the fitted
$\beta_{m}(\Gamma)$ is nearly $\Gamma$-independent (see table 1). Figure 8
shows the $Ra$-dependence of $N_{pl}^{mush}/(\pi D^{2}/4)$ (red symbols).
Here, the $\Gamma=2$ and 0.5 data have also been shifted to agree with the
$\Gamma=1$ in their respective overlap regions of $Ra$. It is seen that
$N_{pl}^{mush}$ shows similar trend with $Ra$ as $N_{pl}^{sheet}$. Again, the
relationship between $N_{pl}^{mush}$ and $Ra$ can be well represented by
power-law fits
$N_{pl}^{mush}/(\pi D^{2}/4)=\alpha_{m}Ra^{\beta_{m}}\mbox{\ \ with\ \
}\alpha_{m}=0.41\mbox{\ \ and\ \ }\beta_{m}=0.32\pm 0.03.$ (2)
To make a quantitative connection between thermal plumes and the heat flux, we
use the previous experimental finding from both Eularian and Lagrangian
measurements that the heat flux are mainly carried by thermal plumes [7, 9].
As mushroomlike plumes are evolved morphologically from sheetlike plumes [29],
the total heat flux carried by either type should be the same, i.e.,
$Nu\simeq N_{pl}^{sheet}F_{pl}^{sheet}\simeq N_{pl}^{mush}F_{pl}^{mush}\sim
Ra^{\beta}.$ (3)
Here, $F_{pl}^{sheet}$ and $F_{pl}^{mush}$ are the mean heat flux carried by
individual sheetlike and mushroomlike plumes, respectively, and $\beta\simeq
0.3\pm 0.02$ in the $Ra$ and $Pr$ ranges of the experiment [37, 38]. Recall
that the $Ra$-dependence of $N_{pl}^{sheet}$ [Eq. (1)] and
$(N_{pl}^{mush})_{peak}$ [Eq. (2)] discussed above, one sees that the scaling
exponents of the $Nu$-$Ra$ relation and of the $Ra$-dependence of plume
numbers are the same within experimental uncertainty. This indicates that the
heat flux transported by individual plumes is nearly independent of the
turbulent intensity, i.e. $Ra$, and hence the $Nu$-$Ra$ scaling relation is
determined primarily by the number of thermal plumes.
## 4 Geometric properties of sheetlike plumes
### 4.1 Aspect ratio of sheetlike plumes
Figure 9: $Ra$-dependence of the mean aspect ratio of sheetlike plumes
$\langle\gamma_{pl}^{sheet}\rangle$ vs. $Ra$ in a log-log plot. Figure 10:
$Ra$-dependence of (a) the mean normalized area $\langle 4A/\pi D^{2}\rangle$
and (b) the mean normalized perimeter $\langle P/D\rangle$ of sheetlike
plumes.
A striking feature that one can observe from figure 3 is that, as the Rayleigh
number is increased, the morphology or geometry of sheetlike plumes change
from more elongated shape to less elongated and more fragmented. The extracted
horizontal cuts of sheetlike plumes may be characterized by a typical length
$l$ and a typical width $w$. With increasing $Ra$, the length of sheetlike
plumes seems to decrease, while the plume’s width seems to increase. To
describe these geometric properties quantitatively, we define the aspect ratio
of sheetlike plumes, $\gamma_{pl}^{sheet}$, as
$\gamma_{pl}^{sheet}=l/w\mbox{,\ with\
}\left\\{\begin{array}[]{ll}P=2(l+w),\\\\[8.0pt] A=lw,\\\\[8.0pt] l\geq
w.\end{array}\right.$ (4)
Figure 9 shows the $Ra$-dependency of the mean aspect ratio of sheetlike
plumes $\langle\gamma_{pl}^{sheet}\rangle$ in a log-log plot. It is seen that
$\langle\gamma_{pl}^{sheet}\rangle$ decreases with increasing $Ra$ and the
relation can be described by a power-law with a scaling exponent $-0.23$. We
note that $\langle\gamma_{pl}^{sheet}\rangle$ can also be described by a
logarithmic function of $Ra$ and we cannot definitively conclude which one,
between power-law and logarithm, is a better choice. Figures 10(a) and (b)
show $Ra$-dependence of (a) the mean normalized area $\langle 4A/\pi
D^{2}\rangle$ and (b) the mean normalized perimeter $\langle P/D\rangle$ of
sheetlike plumes. Again, these two quantities are found to decrease with
increasing $Ra$. All these behaviors are well illustrated in figure 3, and
suggest that when the flow becomes more turbulent, the increased mixing,
collision, and convolution can more easily fragment large-sized sheetlike
plumes into smaller ones.
Figure 11: The area fraction $f_{pl}$ of sheetlike plumes (area coverage of
plumes over the top plate) as a function of $Ra$ in a log-log plot. The solid
line represents a power-law fit to the data.
Figure 11 shows the plume area fraction $f_{pl}$ as a function of the Rayleigh
number in a log-log plot. It is seen that $f_{pl}$ increases with increasing
$Ra$ and a power-law function with a scaling exponent 0.23 can be used to
describe well the relationship between $f_{pl}$ and $Ra$. Using the
shadowgraph technique, [30] studied $Ra$-dependence of the plume area
fraction. Although the scaling range is limited, they found a power-law
scaling of $f_{pl}$ with $Ra$ for $f_{pl}<0.1$ and the obtained scaling
exponent is around 2.
Figure 12: (a) Scatter plot of the normalized perimeter $P/D$ vs the
normalized size $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes at $Ra=3.0\times
10^{9}$. (b) The conditional average $\langle P|A\rangle/D$ on the normalized
size $(4A/\pi D^{2})^{1/2}$ for the same data as (a). Inset is the plot of
fractal dimensions of sheetlike plumes’ boundary $d_{A}$ vs Ra. Solid lines in
both (a) and (b) are power-law fits.
Next we examine the relationship between plume perimeter and area. Figure
12(a) shows a scatter plot of the normalized perimeter $P/D$ and the
normalized size $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes, which contains a
total of $6071$ sheetlike plumes, extracted from a sequence of $260$ images at
Ra $=3.0\times 10^{9}$. Here, the perimeter and area are normalized by the
diameter and the area of the conducting plate, respectively. Although the data
look somewhat scatter, one sees that the perimeter increases with area and all
data points can be fitted by a power-low function,
$(P/D)\sim(\sqrt{4A/\pi D^{2}})^{d_{A}}\mbox{\ \ with\ \ }d_{A}=1.50.$ (5)
To better explore this feature, the conditional average $\langle P|A\rangle/D$
on the normalized size $(4A/\pi D^{2})^{1/2}$ is shown in figure 12(b). In the
figure, a good scaling range can be seen and the solid line is a power-law fit
$(P/D)=28.1\times(4A/\pi D^{2})^{1.50\pm 0.02}$. We find that such a scaling
behavior exists for all $Ra$ investigated. The inset of figure 12(b) shows the
$Ra$-dependence of $d_{A}$. One sees that these $d_{A}$ have a mean value of
$1.50$, but decrease from $1.53$ to $1.46$ when $Ra$ increases from $6.7\times
10^{8}$ to $6.3\times 10^{9}$. These measured values of $d_{A}$ are further
found to be larger than the value of 1.29 found for isoconcentration contours
of passive scalars measured in the same system [40].
Mathematically, as introduced by Mandelbrot [41] and Lovejoy [42], $d_{A}$ is
the fractal dimension of the boundary of sheetlike plumes and satisfies $1\leq
d_{A}<2$. Fractal dimensions in turbulence have been widely studied (see, for
example, Ref. [43]). In the present case, our results seem to suggest that the
boundary of sheetlike plumes is fractal with the dimension of around $1.50$
and the slight decrease of $d_{A}$ may suggest that the shape of sheetlike
plumes becomes smoother when the flow becomes more turbulent, as a result of
the increased mixing. However, the extracted sheetlike plumes, as shown in
figure 3, do not look like fractal objects. To understand this and to see if
applying the machinery of fractal analysis to the geometric properties of
plume can shed some light on the problem, we study the shape complexity of the
plumes.
### 4.2 Shape complexity of sheetlike plumes
Figure 13: (a) Scatter plot of the shape complexity $\Omega_{2}$ vs the
normalized size $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes at $Ra=3.0\times
10^{9}$. (b) The conditional average $\langle\Omega_{2}|A\rangle$ on the
normalized size $(4A/\pi D^{2})^{1/2}$ with a solid line as a power-law fit.
Inset: $d_{\Omega}$ vs $Ra$. (same data as in figure 9)
The geometric complexity of an object can be characterized by its shape
complexity, which is a dimensionless ratio between its area and volume. The
shape complexity was previously used to study the shape of isocontours for
passive scalar fields in turbulence [39]. We introduce it here to describe the
shape of sheetlike plumes. For a two-dimensional (2D) closed contour, the
shape complexity is defined as follows,
$\Omega_{2}=\frac{P}{2\sqrt{\pi A}},$ (6)
where $P$ and $A$ are the perimeter of the contour and area inclosed by the
contour, respectively, and the subscript 2 refers to 2D. As a circle has the
minimum perimeter $2\sqrt{\pi A}$ among all 2D objects with the same area $A$,
the shape complexity $\Omega_{2}$ satisfies $1\leq\Omega_{2}<\infty$ and can
be used to describe the departure of an object from the shape of a circle. For
a fractal object, a larger $\Omega_{2}$ implies a rougher contour (or surface)
the object. For a non-fractal object, a larger $\Omega_{2}$ signifies that the
object is more different from a circle (e.g. it is more elongated).
Figure 13(a) shows a scatter plot of the shape complexity $\Omega_{2}$ and the
normalized dimension (or size) $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes,
obtained from the same data set as those in figure 12. It is seen that the
mean trend of the relation between the size and shape complexity can be
captured by an increasing function, i.e., the plume with larger size has a
higher probability of possessing a larger $\Omega_{2}$. To illustrate this
mean trend more clearly, the conditional average $\langle\Omega_{2}|A\rangle$
on the normalized size $(4A/\pi D^{2})^{1/2}$ is plotted in figure 13(b). The
solid line in the figure is a power-law fit, i.e.,
$\Omega_{2}\sim(\sqrt{4A/\pi D^{2}})^{d_{\Omega}}\mbox{\ \ with\ \
}d_{\Omega}=0.50\pm 0.01$ (7)
and the reasonable scaling range can be found. Here, $d_{\Omega}\simeq
d_{A}-1$ is an immediate consequence of the definition of shape complexity
[Eq. (6)] together with the power-law relation [Eq. (5)]. The inset of figure
13(b) shows the $Ra$-dependence of $d_{\Omega}$. One sees that $d_{\Omega}$
decreases slightly with increasing $Ra$, which may be understood from the
decrease of $d_{A}$.
Figure 14: (a) $Ra$-dependence of the mean shape complexity
$\langle\Omega_{2}\rangle$ of sheetlike plumes. (b) The ratio between
$\langle\Omega_{2}\rangle$ and
$\frac{1}{\sqrt{\pi}}(\sqrt{\langle\gamma_{pl}^{sheet}\rangle}+\frac{1}{\sqrt{\langle\gamma_{pl}^{sheet}\rangle}})$,
$C_{R}$, vs $Ra$.
Figure 14(a) shows the $Ra$-dependence of the mean shape complexity
$\langle\Omega_{2}\rangle$. It is seen that $\langle\Omega_{2}\rangle$
decreases with increasing $Ra$. This suggests that, because of the increased
mixing, collision and convolution of sheetlike plumes, the shape of sheetlike
plumes become closer to the shape of a circle when the flow becomes more
turbulent. This also implies the decreases of plume’s aspect ratio with
increasing $Ra$. Indeed, with the definitions of $\gamma_{pl}^{sheet}$ [Eq.
(4)] and $\Omega_{2}$ [Eq. (7)], one can obtain the relation between of
$\Omega_{2}$ and $\gamma_{pl}^{sheet}$, i.e.,
$\Omega_{2}=\frac{1}{\sqrt{\pi}}(\sqrt{\gamma_{pl}^{sheet}}+\frac{1}{\sqrt{\gamma_{pl}^{sheet}}}),$
(8)
which implies that $\Omega_{2}$ decreases with decreasing
$\gamma_{pl}^{sheet}$ when $\gamma_{pl}^{sheet}\geq 1$. Furthermore,
$\Omega_{2}$ is proportional to $\gamma_{pl}^{sheet}$ when
$\gamma_{pl}^{sheet}\gg 1$. Accordingly, the decrease of
$\langle\Omega_{2}\rangle$, shown in figure 14(a), may be understood as the
result of the change in aspect ratio of the sheetlike plumes, i.e. the
decrease of $\langle\gamma_{pl}^{sheet}\rangle$ (see figure 9), as
$\gamma_{pl}^{sheet}$ is much larger than 1 for most of situations. To see
this more clearly, we study the ratio $C_{R}$ between
$\langle\Omega_{2}\rangle$ and
$\frac{1}{\sqrt{\pi}}(\sqrt{\langle\gamma_{pl}^{sheet}\rangle}+\frac{1}{\sqrt{\langle\gamma_{pl}^{sheet}\rangle}})$.
Note that Eq. (8) is valid only for an individual plume, while $C_{R}$ is in
an average sense. Figure 14(b) shows $C_{R}$ as a function of $Ra$. It is seen
that $C_{R}\simeq 1$ for nearly all $Ra$ investigated. This result
demonstrates clearly that the decrease of $\Omega_{2}$ is caused by the
decrease of the plume’s aspect ratio, rather than by a decreased roughness in
the case of a fractal object which the sheetlike plumes are not.
### 4.3 Distributions of geometric measures of sheetlike plumes
Figure 15: PDFs of (a) normalized area $4A/\pi D^{2}$, (b) normalized
perimeter $P/D$, (c) shape complexity $\Omega_{2}$, (d) normalized length
$l/D$, (e) normalized width $w/\delta_{th}$, and (f) aspect ratio
$\gamma_{pl}^{rod}$ sheetlike plumes, all obtained at $Ra=3.0\times 10^{9}$.
Solid curves are fittings of log-normal function to the respective data.
In a previous study, Zhou _et al._ [29] found that the area of sheetlike
plumes obeys log-normal statistics. Here, we further investigate distributions
of other geometric measures of sheetlike plumes. Figures 15(a)-(f) show the
measured PDFs of normalized geometric measures, i.e. $4A/\pi D^{2}$, $P/D$,
$\Omega_{2}$, $l/D$, $w/\delta_{th}$, and $\gamma_{pl}^{sheet}$, of sheetlike
plumes. It is seen that all these quantities have a same distribution, i.e.
the log-normal distribution. The same distributions were also found for all of
these quantities at other $Ra$ investigated in our experiments. Together with
the log-normal distributions found for mushroomlike plumes [21], these
findings suggest that the log-normal distribution is universal for thermal
plumes and log-normal statistics may be used to model them, at least in
turbulent RBC. In addition, this log-normal statistics of thermal plumes is
different from that of passive scalars measured in the same system, which is
found to obey a log-Poisson statistics [40].
### 4.4 Three-dimensional structures of sheetlike plumes
Figure 16: (a) Three-dimensional sheetlike/rodlike thermal plumes obtained
near the top plate by tomographic reconstruction technique ($Ra=2.0\times
10^{9}$, $\Gamma=1$). (b) An enlarged region of (a), showing the morphological
transformation of rodlike plumes into mushroomlike ones through
convolution/spiraling. (c) An individual rodlike plume.
Finally, we study the 3D structures of sheetlike plumes. Results obtained in
section 3.2 have suggested that the vertical extent of what is called
sheetlike plumes is only of the order of the thermal boundary layer thickness,
and hence not large enough to form a sheet. To see this more clearly , we used
a tomographic reconstruction technique to construct the 3D image of thermal
plumes from sequences of 2D images acquired near the top plate of the cell. To
achieve this, the convection cell was placed on a translational stage and, as
the cell traverses continuously at a speed of $\sim 1cm/sec$, a series of
photographs of TLC microspheres were recorded by a Nikon D3 camera (3CCD, with
a resolution of $2397\times 1591$ pixels) operating at 11 frames/s. As the
speed of the the horizontal motion of the sheetlike plumes are about
$0.4cm/sec$, the sequences of the horizontal slices of the plumes may be
regarded as taken at approximately the same time. In post-experiment analysis,
2D horizontal cuts of cold plumes were extracted from the images taken in each
run and a MATLAB script was used to reconstruct 3D thermal plumes from these
extracted 2D cuts. Figure 16(a) shows an example of the reconstruction of 3D
thermal plumes near the top plate. From the figure, one can see how mushroom-
like plumes are formed by the convolution (or spiraling) of sheetlike plumes.
Figure 16(b) shows this process more clearly, which is an enlarged region of
the image. Another noteworthy feature is that most plumes have a one-
dimensional structure, rather than a sheetlike shape. This geometric feature
may be illustrated more clearly by figure 16(c), which shows the
reconstruction of an enlarged individual 3D plume. It is seen that both the
height and the width of this plume extend only to a few millimeters, while its
length extends to a few centimeters. This is the most direct evidence thus far
that shows thermal plumes near the conducting plates are only one-dimensional
structures with horizontal length being much larger than their horizontal
width and vertical extent, which has been suggested previously [30].
Therefore, we may hereafter call thermal structures near the conducting plates
rod-like plumes. Future investigations will be focused on the geometric
properties of 3D structures of both rod-like and mushroom-like thermal plumes.
## 5 Conclusions
In this paper, we have presented a detailed experimental study of the
temperature and velocity fields in turbulent Rayleigh-Bénard convection using
the thermochromic-liquid-crystal technique. The number statistics and
geometric properties of sheetlike/rodlike thermal plumes were investigated.
Major findings are summarized as follows:
(i) When observed from above, the previous-called sheetlike thermal plumes
near the top plate seem to be only one-dimensional structures and should be
called rodlike plumes hereafter. These plumes evolve morphologically, i.e.
convolute or spiral, to mushroomlike plumes. The width ($\ell$) of the region
near the plates within which these rodlike plumes exist and evolve
morphologically is only associated with the thermal boundary thickness (i.e.
$\ell\sim$ several $\delta_{th}$).
(ii) The numbers of sheetlike/rodlike and mushroomlike plumes,
$N_{pl}^{sheet}$ and $N_{pl}^{mush}$, are found to both scale as $Ra^{0.3}$.
This finding suggests that the total amount of heat flow is dominated only by
the number of thermal (rodlike or mushroomlike) plumes and the normalized mean
heat content carried by each type of plumes is approximately independent on
$Ra$.
(iii) As the turbulent intensity is increased, large-sized sheetlike/rodlike
plumes are more easily to be fragmented into smaller ones and hence their
aspect ratios decrease.
(iv) Although sheetlike/rodlike plumes do not look like fractal objects, ,
power-law relations can be used to characterize the relationship between their
perimeter and area and between their size and shape complexity.
(v) Geometric measures of thermal plumes are found to have a universal
distribution, i.e. the log-normal distribution. Thus, log-normal statistics
may be used to model thermal plumes. In contrast, for the mixing of passive
scalars measured in the same system, it is found that the same quantities obey
log-Poisson statistics [40].
(vi) The Ra-dependency of the plume area density ($f_{r}\sim
Ra^{0.23}$)suggests that the plume number increase with $Ra$ ($N\sim
Ra^{0.3}$) can be largely attributed to the increased emission of plumes (as
$Ra$ is increased), rather than as a result of fragmentation of plumes.
## Acknowledgments
We gratefully acknowledge support of this work by the Research Grants Council
of Hong Kong SAR (Nos. CUHK403806 and 403807) (K.Q.X.) and by the Natural
Science Foundation of Shanghai (No. 09ZR1411200), “Chen Guang” project (No.
09CG41), and RFDP of Ministry of Education of China (No. 20093108120007)
(Q.Z.).
## References
## References
* [1] Siggia E D 1994 High Rayleigh number convection Annu. Rev. Fluid Mech. 26 137-168
* [2] Ahlers G, Grossmann S and Lohse D 2002 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection Rev. Mod. Phys. 81 503-537
* [3] Lohse D and Xia K Q 2010 Small scale properties of turbulent Rayleigh-Bénard convection Annu. Rev. Fluid Mech. 42 335-364
* [4] Morgan W J 1968 Rises trenches great faults and crustal blocks J. Geophys. Res. 73 1959
* [5] Jellinek A M and Manga M 2004 Links between long-lived hot spots, mantle plumes, D”, and plate tectonics Rev. Geophys. 42 RG3002
* [6] Rieutord M and Zahn J P 1995 Turbulent plumes in stellar convective envelopes Astron. Astrophys. 296 127-138
* [7] Shang X D, Qiu X L, Tong P and Xia K Q 2003 Measured local heat transport in turbulent Rayleigh-Bénard convection Phys. Rev. Lett. 90 074501
* [8] Shang X D, Qiu X L, Tong P and Xia K Q 2004 Measurements of the local convective heat flux in turbulent Rayleigh-Bénard convection Phys. Rev. E 70 026308
* [9] Gasteuil Y, Shew W L, Gibert M, Chilla F, Castaing B and Pinton J F 2007 Lagrangian temperature, velocity, and local heat flux measurement in Rayleigh-Bénard convection Phys. Rev. Lett. 99 234302
* [10] Grossmann S and Lohse D 2004 Fluctuations in turbulent Rayleigh-Bénard convection: The role of plumes Phys. Fluid 16 4462-4472
* [11] Moses E, Zocchi G and Libchaber A 1993 An experimental study of laminar plumes J. Fluid Mech. 251 581-601
* [12] Kaminski E and Jaupart C 2003 Laminar starting plumes in high-Prandtl-number fluids J. Fluid Mech. 478 287-298
* [13] Zocchi G, Moses E and Libchaber A 1990 Coherent structures in turbulent convection: an experimental study Physica (Amsterdam) A166 387 C407
* [14] Moses E, Zocchi G, Procaccia I and Libchaber A 1991 The Dynamics and Interaction of Laminar Thermal Plumes Europhys. Lett. 14 55-60
* [15] Gluckman B J, Willaime H and Gollub J P 1993 Geometry of isothermal and isoconcentration on surfaces in thermal turbulence Phys. Fluids A 5 647-661
* [16] Ciliberto S, Cioni S and Laroche C 1996 Large-scale flow properties of turbulent thermal convection Phys. Rev. E 54 R5901-5904
* [17] Zhang J, Childress S and Libchaber A 1997 Non-Boussinesq effect: Thermal convection with broken symmetry Phys. Fluids 9 1034-1042
* [18] Du Y B and Tong P 1998 Enhanced heat transport in turbulent convection over a rough surface Phys. Rev. Lett. 81 987-990
* [19] Qiu X L and Tong P 2001 Onset of convection in turbulent Rayleigh-Bénard convection Phys. Rev. Lett. 87 094501
* [20] Xi H D, Lam S and Xia K Q 2004 From laminar plumes to organized flows: the onset of large-scale circulation in turbulent thermal convection J. Fluid Mech. 503 47-56
* [21] Zhou S Q and Xia K Q 2002 Plume statistics in thermal turbulence: mixing of an active Scalar Phys. Rev. Lett. 89 184502
* [22] Zhou S Q and Xia K Q 2009 Statistical characterization of the thermal structures in turbulent Rayleigh-Bénard to be published
* [23] Tanaka H and Miyata H 1980 Turbulent natural convection in a horizontal water layer heated from below Int. J. Heat Mass Transfer 23 1273-1281
* [24] Vorobie P and Ecke R 2002 Turbulent rotating convection: an experimental study J. Fluid Mech. 458 191-218
* [25] Funfschilling D and Ahlers G 2004 Plume motion and large-scale circulation in a cylindrical Rayleigh-Bénard cell Phys. Rev. Lett. 92 194502
* [26] Haramina T and Tilgner A 2004 Coherent structures in boundary layers of Rayleigh-Bénard convection Phys. Rev. E 69 056306
* [27] Puthenveettil B A and Arakeri J H 2005 Plume structure in high-Rayleigh-number convection J. Fluid Mech. 542 217-249
* [28] Puthenveettil B A and Arakeri J H 2005 The multifractal nature of plume structure in high-Rayleigh-number convection J. Fluid Mech. 526 245-256
* [29] Zhou Q, Sun C and Xia K Q 2007 Morphological evolution of thermal plumes in turbulent Rayleigh-Bénard convection Phys. Rev. Lett. 98 074501
* [30] Funfschilling D, Brown E and Ahlers G 2008 Torsional oscillations of the large-scale circulation in turbulent Rayleigh-Béenard convection J. Fluid Mech. 607 119-139
* [31] Puthenveettil B A and Arakeri J H 2008 Convection due to an unstable density difference across a permeable membrane J. Fluid Mech. 609 139-170
* [32] Shishkina O and Wagner C 2008 Analysis of sheet-like plumes in turbulent Rayleigh-Bènard convection J. Fluid Mech. 599 383-404
* [33] Xi H D, Zhou Q and Xia K Q 2006 Azimuthal motion of the mean wind in turbulent thermal convection Phys. Rev. E 73 056312
* [34] Lui S L 1997 Experimental investigation of the temperature field in turbulent convection M.Phil. Thesis The Chinese University of Hong Kong
* [35] Lui S L and Xia K Q 1998 Spatial structure of the thermal boundary layer in turbulent convection Phys. Rev. E 57 5494-5503
* [36] Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomae S, Wu X Z, Zaleski S and Zanetti G 1989 Scaling of hard thermal turbulence in Rayleigh-Bénard convection J. Fluid Mech. 204 1-30
* [37] Xia K Q, Lam S and Zhou S Q 2002 Heat-flux measurement in high-Prandtl-number turbulent Rayleigh-Bénard convection Phys. Rev. Lett. 88 064501
* [38] Nikolaenko A and Ahlers G 2003 Nusselt number measurements for turbulent Rayleigh-Bénard convection Phys. Rev. Lett. 91 084501
* [39] Catrakis H J and Dimotakis P E 1998 Shape complexity in turbulence Phys. Rev. Lett. 80 968-971
* [40] Zhou Q and Xia K Q 2010 Mixing evolution and geometric properties of passive scalar field in high-Schmidt-number buoyancy-driven turbulence to be published
* [41] Mandelbrot B B 1982 The Fractal Geometry of Nature (Freeman, NY, 1982)
* [42] Lovejoy S 1982 Area-perimeter relation for rain and cloud areas Science 216 185-187
* [43] Sreenivasan K R 1991 Fractals and multifractals in fluid turbulence Annu. Rev. Fluid Mech. 23 539-600.
|
arxiv-papers
| 2010-02-11T05:25:02 |
2024-09-04T02:49:08.339112
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan Zhou and Ke-Qing Xia",
"submitter": "Quan Zhou",
"url": "https://arxiv.org/abs/1002.2275"
}
|
1002.2277
|
# Improved orbit predictions using two-line elements
Creon Levit NASA Ames Research Center, Moffett Field, MS202-3, CA 94035, USA
Corresponding author creon.levit@nasa.gov William Marshall NASA Ames Research
Center and Universities Space Research Association, Moffett Field, MS202-3, CA
94035, USA
###### Abstract
The density of orbital space debris constitutes an increasing environmental
challenge. There are three ways to alleviate the problem: debris mitigation,
debris removal and collision avoidance. This paper addresses collision
avoidance, by describing a method that contributes to achieving a requisite
increase in orbit prediction accuracy. Batch least-squares differential
correction is applied to the publicly available two-line element (TLE) catalog
of space objects. Using a high-precision numerical propagator, we fit an orbit
to state vectors derived from successive TLEs. We then propagate the fitted
orbit further forward in time. These predictions are compared to precision
ephemeris data derived from the International Laser Ranging Service (ILRS) for
several satellites, including objects in the congested sun-synchronous orbital
region. The method leads to a predicted range error that increases at a
typical rate of 100 meters per day, approximately a 10-fold improvement over
TLE’s propagated with their associated analytic propagator (SGP4).
Corresponding improvements for debris trajectories could potentially provide
initial conjunction analysis sufficiently accurate for an operationally viable
collision avoidance system.
We discuss additional optimization and the computational requirements for
applying all-on-all conjunction analysis to the whole TLE catalog, present and
near future. Finally, we outline a scheme for debris-debris collision
avoidance that may become practicable given these developments.
###### keywords:
Space debris , Conjunction analysis , Orbit determination
## 1 Introduction
Collisions in orbit pose a threat to spacecraft, to astronauts and to the
global commons of near-earth space. Several collisions have already occurred
between spacecraft and debris, while the Iridium 33/Cosmos 2251 collision of
January 2009 represented the first documented satellite-satellite collision.
Unfortunately, the cumulative number of collisions thus far is consistent with
the prescient predictions of a runaway chain reaction (Kessler & Cour-Palais,
1978).
Reducing debris from current and future space missions (debris mitigation) is
an important measure. However, even the unrealistic case of “no new launches”
is inadequate to curb runaway growth (Liou & Johnson, 2008). There are two
remaining ways to alleviate the problem: debris removal and collision
avoidance. For the latter, one needs prediction precision to be sufficiently
accurate so as not to imply an unwieldy number of collision avoidance
maneuvers. In addition, one needs a means of doing debris-debris collision
avoidance since applying collision avoidance only to the subset of
conjunctions involving maneuverable spacecraft does not suffice to curb the
growth. This paper describes a method to increase the orbit prediction
accuracy based on publicly available TLEs.
Many satellite owner operations have inadequate (if any) processes for
conjunction assessment and collision avoidance, since they would have to
screen their asset(s) against all other space objects. The only source of
knowledge at their disposal for the majority of other objects is the publicly
available two-line element (TLE) sets. But predictions based on TLEs using the
associated analytic propagator (SGP4) are not sufficiently accurate to warrant
maneuvering to avoid potential collisions: they imply an unacceptably large
number of potential collisions per space object, each of which has very low
probability. The problem is similar for debris-debris conjunctions except then
both objects, not just one, are subject to these imprecisions.
To address this TLE/SGP4 accuracy problem, we investigated several methods to
improve the propagation errors for non-maneuvering orbital objects whilst
using only TLEs as input data. The following research target was set:
“increase the predictive accuracy for orbital objects, using only historical
TLE data, such that it enables operational conjunction assessment for
collision avoidance”.
In the following sections, we describe one approach to this target, assess its
accuracy, and discuss the requirements for extension to the entire space
object catalog. Finally, we propose a new method of debris-debris collision
avoidance enabled by long-term high-accuracy conjunction assessment.
## 2 Method: TLE orbit fitting and propagation
ESA uses TLEs from the publicly available catalog to initially screen their
sun-synchronous orbit (SSO) spacecraft ERS2 and ENVISAT for conjunctions
(Flohrer, Krag, & Klinkrad, 2009). Telemetry from their operational spacecraft
provide precision orbital ephemeredes (POEs) for those spacecraft. For
screening against all other potentially conjuncting objects, only TLEs are
used. Flohrer, Krag, & Klinkrad (2008) describe a method to estimate error
covariances of TLEs in order to quantify collision probability assessments.
Their method provided inspiration for the present work. Here we extend and
adapt their approach: based solely on the object’s historical TLEs, we improve
the accuracy of the object’s predicted position, as opposed to quantifying the
accuracy of the object’s SGP4 propagation errors.
Our method is essentially to use TLE data as “pseudo-observations” and to fit
an orbit to these pseudo-observations using a high-precision special
perturbations propagator and traditional batch least-squares differential
correction. The fitted orbit is then propagated into the future using the same
high-precision orbit propagator. The prediction accuracy is assessed by
comparison with precision orbital ephemeris (POE) data from the International
Laser Ranging Service (Pearlman, Degnan, & Bosworth, 2002).
For each object we wish to analyze we choose a time window with two sections:
a fitting period and a subsequent prediction period. The fitting period is
initially set to ten days, typical of the period over which U.S. Space
Surveillance Network observations are fitted when generating TLEs for LEO
objects (Danielson et al., 2000). Section $4$ details a more principled
approach to determining the fitting period. The prediction period is 30 days.
For a given object, all TLEs with epochs within its entire window are
obtained. However, only TLEs with epochs within the fitting period are used
for the fitting process; those in the prediction period are only used for
validation and test.
Using all TLEs from the fitting period and interpolating using SGP4, we
generate a series of order $100$ state vectors (“pseudo-observations”) equally
spaced in time within the fitting period. We initialize the differential
correction with a state vector derived from the first TLE in the fitting
period, and then propagate that using a high precision propagator. We apply
batch least-squares differential correction to minimize the RMS error in the
radial, in track and cross track (RIC) components of relative positions
between the fitted orbit and the pseudo-observations. The trajectory obtained
from the converged differential corrector is our fit.
The high precision propagator incorporates a $60\times 60$ EGM2008 gravity
field including solid Earth tides, point masses for the solar and lunar
gravity fields and a NRL-MSISE $1990$ atmospheric drag model. We assume
constant values for solar F$10.7$ and Ap equal to their averages over the
fitting period. Solar radiation pressure is modelled using a simple the
biconic approximation with Earth and Moon as eclipsing bodies.
## 3 Results: Prediction Accuracy
Figure 1: Typical position errors with respect to POE (“truth”) using our
method (red), and using SGP4 (black) for Stella (upper panels) and Etalon-2
(lower panels). The prediction period is $0<t\leq 30$ days. The fitting period
is $-10\leq t\leq 0$ days for Stella and $-90\leq t\leq 0$ days for
Etalon-$2$. All fitting uses only publicly available TLEs. Truth data are only
used for plotting. Selection of the length of the fitting period is discussed
in Section 4. Also shown are the updated TLEs (no prediction) in blue.
We applied our method to four non-manoeuvring spacecraft for which POE data
were readily available (Ries, 2009): Stella, Starlette, Ajisai and Etalon-$2$.
The summary of basic orbital and physical properties for these spacecraft are
shown in Table $1$. They were picked in part for their range of orbital
altitudes.
Table 1: Key Parameters of Geodesic Spacecraft Used Satellite | Perigee | Eccentricity | Period | Inclination | Mass | Diameter
---|---|---|---|---|---|---
| (km) | | (min) | (deg) | (kg) | (m)
Stella | 800 | 0.0206 | 98.6 | 101 | 48 | 0.24
Starlette | 812 | 0.0206 | 104 | 49.8 | 47 | 0.24
Ajisai | 1490 | 0.0010 | 116 | 50.0 | 685 | 2.15
Etalon 2 | 19120 | 0.0007 | 675 | 65.5 | 1415 | 1.29
Figure $1$ shows examples for two of these spacecraft – the highest and lowest
altitude – comparing our method’s predictions to those of SGP4. The TLEs for
these objects exhibit quite different behaviours. Nevertheless, in both cases
the TLEs propagated with SGP4 (black) depart rapidly from the true position of
the satellite whereas our numerically fitted and numerically propagated orbits
(red) maintain higher accuracy, particularly over the long term.
Figure 2: “Box and whisker” plots of range errors for predictions using our
fit method (red), and using SGP4 (black) for four satellites. Each panel
summarizes $50$ runs corresponding to $50$ random starting dates in $2004$.
The $x$-axis is prediction time in days. The $y$-axis is prediction error in
km. The lower and upper whiskers extending from each box bound the minimum and
maximum prediction errors respectively. The lower and upper box edges bound
the $25^{th}$ and $75^{th}$ percentiles, respectively. The glyphs inside the
boxes mark the median prediction error. All errors are plotted with respect to
“truth” (i.e. POEs obtained from the ILRS). Truth data are only used for
plotting. All fitting uses only publicly available TLEs. Figure 3: Additional
summaries of range errors for predictions using our method (red), and using
SGP4 (black) for four satellites. On the LHS, just our method is plotted for a
full $30$ days and on the RHS both our method and the SGP4 propagated TLEs are
shown but just for the first $10$ days ($5$ in the case of Starlette since the
TLE was diverging rapidly). Each box and whisker plot summarizes $50$ runs
corresponding to $50$ random starting dates in 1994. The $x$-axis is
prediction time in days. The $y$-axis is prediction error in km. All errors
are plotted with respect to “truth” (i.e. POE obtained from the ILRS). Truth
data are only used for plotting. All fitting uses only publicly available
TLEs.
Precision orbit ephemeredes (“truth data”) define the $x$-axis for plotting,
but truth data are not used in the fitting process. Nor are they used for
tuning hyperparameters (e.g. length of fitting interval, see Section $4$).
Improvements in prediction accuracy are typically seen in all three axis
(RIC), especially more than day into the future. For predictions of less than
one day the errors appear to be dominated by TLE bias (with respect to truth)
which our method cannot remove.
For each of the four satellites, we performed $50$ runs of our method using
different starting dates distributed uniformly in $2004$. Summary statistics
of these runs using “box-and-whisker” plots appear in Figures $2$ and $3$. The
TLE+SGP4 errors grow more rapidly in every case. The typical prediction errors
for our method are $3$km at $30$ days out, corresponding to $\sim 100$ m/day
prediction error growth. The worst case is Stella where the median error grows
$<150$ m/day and the best is Etalon at $<30$ m/day. The maximum error for our
method over all four satellites is $<300$ m/day. These compare to typical
growth of $1,500$ m/day for TLE’s propagated with SGP4 for these objects. The
ratio of median prediction error using our method vs. SGP4 at $30$ days range
from $3$ for Etalon-$2$ to $50$ for Starlette, and averages $15$.
Thus for these satellites, our method exhibits approximately one order of
magnitude improvement in prediction accuracy over TLEs propagated with SGP4.
Since the improvement is in all three directions (RIC) the resultant decrease
in position covariance ellipsoid volume is likely to result in $2$ to $3$
orders of magnitude reduction in false positive conjunctions.
Since the TLEs for these objects have an instantaneous range bias from truth
of $0.8\pm 0.3$ km the error growth of approximately $1.5$ km/day means that
only after approximately one day can one detect the benefits of our method.
Our fitting method has a similar initial range bias to TLEs but error growth
of only $100$ m/day. These errors are similar to those published in the open
literature evaluating the high accuracy special perturbations catalog(s)
maintained by the US Space Command (Neal, Coffey, & Knowles, 1997; Coffey et
al., 1998; Boers et al., 2000). Table 2 compares the accuracy of various
tracking data and prediction methods.
Table 2: Accuracies of Different Sensors and Prediction Methodologies. Method | Prediction | Accuracy | Accuracy
---|---|---|---
| (days) | (m) | (m/day)
Laser Ranging (ILRS “truth”)111Pearlman, Degnan, & Bosworth (2002) | 0 | 0.1 | N.A.
Fence (raw direction cosines)222Hayden (1962); Gilbreath (1997); Schumacher et al. (2001) | 0 | 10 | N.A.
High Accuracy catalog + SP333Neal, Coffey, & Knowles (1997); Coffey et al. (1998); Boers et al. (2000) | 10 | 500-2000 | 50-200
TLEs + SGP4444Boyce (2004); Chan & Navarro (2001); Flohrer, Krag, & Klinkrad (2009); Kelso (2007); Muldoon & Elkaim (2008); Snow & Kaya (1999); Wang, Liu, & Zhang (2009) | 10 | 1,000-30,000 | 100-3,000
TLEs + new scheme | 10 | 500-2,000 | 50-200
## 4 Fitting Debris objects
Figure 4: TLEs from beyond the fitting period can be used instead of
precision ephemeris (POE) data when optimizing the length of the fitting
period. Left: prediction error with respect to future TLEs (“‘proxy”) vs.
fitting period. Right: prediction error with respect to POE data (“truth”) vs.
fitting period. Each box and whisker plot summarises $50$ runs corresponding
to $50$ random starting dates in $1994$.
The majority of tracked space objects (i.e debris) do not have precision
orbital ephemerides – they are not tracked by the ILRS and they transmit
nothing. Many are, however, tracked by the United States Space Surveillance
Network (SSN). Unfortunately, the only publicly available source of data
derived from these SSN observations are, at present, the TLEs accessible at
space-track.org (and archived e.g. at celestrak.com).
While there is currently no publicly available source of high-accuracy
trajectory for debris objects, we believe the method described above could be
applied to refine the predictions of debris objects’ future positions based
only upon comparatively inaccurate TLEs, facilitating conjunction assessment
and collision avoidance. However, there are several issues which complicate
the application of our procedure to debris:
The first set of issues arise when TLEs are the only source of data - we have
no “truth” in order to perform validation and test. This is solved by the use
of TLEs from beyond the fitting period as a proxy for truth data.
Our initial manual experiments suggested that different fitting periods were
optimal for different orbit categories. These results are consistent with
Alfriend et al. (2002); Danielson et al. (2000). We made a more principled
investigation, utilizing cross-validation, to optimize the length of the
fitting interval on a per-object basis using only TLEs for both fitting and
validation. Figure 4 shows how a fitting period of $10$ days was determined to
be optimal for satellite Ajisai. The important point shown in the figure is
that a $10$ day fitting period is optimal both when validating predictions
with truth data and when validating predictions with future TLEs. And the $10$
day fitting period is optimal whether predicting $5$, $10$ or $30$ days into
the future. Thus it appears “future” TLEs can be used as a proxy for truth
data when optimizing the length of the fitting period on a per object basis.
The second set of issues is that for debris objects one must also solve for
area to mass ratio (drag) and validate this as well using only future TLEs. We
are currently performing these experiments.
## 5 Outlook
### 5.1 Methodological Improvements
Further accuracy improvements could come from a principled analysis of the
number of fitting points and their weighting as a function of location within
the fitting period, the removal of outlier data (i.e. filtering) and other
techniques from statistical orbit determination, treating TLEs as
observations. Also, recent work Legendre et al. (2008); Muldoon & Elkaim
(2008) suggests that there may be simple global transformations of TLE data
that can improve TLE/SGP$4$ accuracy. These systematic corrections might also
be further refined on the basis of orbital parameters, resulting in a look-up-
table used to correct TLEs as a function of their orbital regimes.
In addition, the scheme could be tested on actual known conjunctions in order
to analyse its efficiency for conjunction assessment. These issues will be the
subject of a future paper.
### 5.2 Computational Scaling
As an initial assessment of the computational requirements needed to perform
conjunction analysis for all space objects in the publicly available catalog,
a simple conjunction analysis system was parallelized111 This work was
performed by Chris Henze of the NASA Advanced Supercomputing Division at NASA
Ames Research Center. Details will be forthcoming in a future publication. on
the Pleiades supercomputer at NASA Ames Research Center. Using approximately
$200$ CPU cores, checking all objects against all objects for conjunctions $7$
days into the future takes about a minute. The conjunctions involving
spacecraft precisely match the results from SOCRATES (Kelso & Alfano, 2005).
The same system takes about $40$ minutes to perform 7 days of all-on-all
conjunction assessment of a “simulated S-band fence” catalog (obtained from
the NASA Orbital Debris Office) containing “pseudo-TLEs” for approximately
$2.5$ million objects of size $\geq 2$cm. This demonstrates the feasibility of
doing all-on-all conjunction assessment on the present and expected near-
future catalogs. Scaling the fitting method discussed above to the future
catalog will be the subject of a later paper.
### 5.3 A New Scheme for Debris-Debris collision avoidance
An along track $\Delta V$ of $1$ cm/s provides of order 1km displacement per
day. Since error growth for our method is $~{}100$m per day, such small
manoeuvres might suffice for collision avoidance. For manoeuvres this small
one could use radiation pressure to impart the necessary momentum. To verify
the conjecture, we increased the flux of radiation by a factor of 10x the
solar radiation constant for 10 minutes on a typical debris object (area to
mass ratio $0.1$ m2/kg) in SSO. We then propagated that for a further 5 days.
The displacement was $>0.1$ km/day along track compared to a reference object
that was not so irradiated. Two cumulative such events, if one ensured
appropriate geometry, would thus lead to the necessary along-track
displacement.222This idea was first conceived in discussions with Dr. Rüdiger
Jehn from ESA, July 2009.
A factor of 10x the solar rational constant could conceivably be applied from
the ground. For example, a 10kW class laser attached to a 1m class telescope
with appropriate tracking, could, (considering diffraction limits only)
provide the necessary radiation pressure for a debris object of area $<$1m2 in
SSO. One would need to perform a detailed engineering analysis of the
feasibility of such a device, particularly the viability of maintaining small
beam divergence and tracking.
Debris-debris collision avoidance could eliminate the need for an active
debris removal program. The results of Liou & Johnson (2009) imply, to first
order, that performing debris-debris collision avoidance for 5 additional
carefully selected objects per annum would curb debris field growth.
Furthermore, any additional debris-debris collision avoidance would start
reducing the net debris density, eventually to below a critical level, because
debris creating collisions are being nulled whereas atmospheric drag continues
to cause objects re-entry. Thus this method could provide not just a stop gap
measure, but a permanent solution.
## 6 Conclusions
This fitting and propagation method, based solely on TLEs, could potentially
provide initial conjunction analysis sufficiently accurate for an
operationally viable collision avoidance system. If similar improvements are
possible for debris objects, as we expect, then it could be used for all-on-
all conjunction assessment. However, to ensure that the scheme works in an
operational setting, it would be necessary to have additional data: the so-
called “uncorrelated objects” (a.k.a. “analysts set’) which account for
approximately 30 percent of potential conjunctions in LEO (Neuman, 2008).
Further, we show that all-on-all conjunction assessment is possible with
moderate computer infrastructure, even with the large increase in size of the
catalog of tracked objects that is expected in the next few years.
Finally, given accurate predictions, we claim debris-debris collision
avoidance may be possible by externally inducing small manoeuvres using
radiation pressure from $10$ kW class power density ground-based lasers. If
feasible, this could negate the need for a large scale debris removal program.
## References
* Alfriend et al. (2002) K. Alfriend, S. Paik, V. Boikov, Z. Khutorovsky, & A. Testov. Comparison of the Russian & US Algorithms for Catalog Maintenance for Geosynchronous Satellites. Adv. Astronaut. Sci., 112(2):1157–1176, 2002.
* Boers et al. (2000) J. Boers, S. Coffey, W. Barnds, D. Johns, M. Davis, & J. Seago. Accuracy assessment of the naval space command special perturbations cataloging system. Spaceflight Mechanics 2000, volume 105 of Adv. Astronaut. Sci., pages 1291–1304, 2000.
* Boyce (2004) W. H. Boyce. Examination of NORAD TLE accuracy using the iridium constellation. Spaceflight Mechanics 2004, volume 119 of Adv. Astronaut. Sci., pages 2133–2142, 2004.
* Chan & Navarro (2001) J. Chan & D. Navarro. Comparison of NORAD two-line elements with INTELSAT orbital elements. Proceedings of the Third European Conference on Space Debris, volume SP-473, pages 771–780, 2001.
* Coffey et al. (1998) S. Coffey, H. Neal, C. Visel, & P. Conolty. Demonstration of a special-perturbations-based catalog in the naval space command system. Spaceflight mechanics 1998, volume 99 of Adv. Astronaut. Sci., pages 227–248, 1998.
* Danielson et al. (2000) D. Danielson, D. Canright, D. Perini, & P. Schumacher. The naval space command automatic differential correction process. Astrodynamics 1999, volume 103 of Adv. Astronaut. Sci., pages 991–1008, 2000.
* Flohrer, Krag, & Klinkrad (2009) T. Flohrer, H. Krag, & H. Klinkrad. ESA’s process for the identification & assessment of high-risk conjunction events. Adv. Space Res., 44(3):355–363, 2009.
* Flohrer, Krag, & Klinkrad (2008) T. Flohrer, H. Krag, & H. Klinkrad. Assessment & categorization of TLE orbit errors for the US SSN catalog. In Advanced Maui Optical & Space Surveillance Technologies Conference, page 513–524, 2008.
* Gilbreath (1997) G. Gilbreath. Calibrating the naval space surveillance fence using satellite laser ranging. Astrodynamics 1997, volume 97 of Adv. Astronaut. Sci., pages 403–416, 1997.
* Hayden (1962) L. Hayden. Optical calibration of the US Naval Space Surveillance System. Technical Report NRL 5741, U.S. Naval Research Laboratory, 1962.
* Kelso & Alfano (2005) T. S. Kelso & S. Alfano. Satellite orbital conjunction reports assessing threatening encounters in space (SOCRATES). Proceedings of the 15th AAS/AIAA Space Flight Mechanics Conference. Paper AAS 05-124, 2005.
* Kelso (2007) T. Kelso. Validation of SGP4 & IS-GPS-200 against GPS precision ephemerides (AAS 07-127). In 17th AAS/AIAA Space Flight Mechanics Conference, volume 127 of Adv. Astronaut. Sci., page 427, 2007.
* Kessler & Cour-Palais (1978) D. Kessler and B. Cour-Palais. Collision frequency of artificial satellites: The creation of a debris belt. J. of Geophys. Res., 83(A6):2637–2646, 1978.
* Legendre et al. (2008) P. Legendre, R. Garmier, G. Prat, B. Revelin, and S. Delavault. Improvement of the Two-Line Element Accuracy Assessment Based on a Mixture of Gaussian Laws Adv. Astronaut. Sci., 129(3):2189–2207, 2008.
* Liou & Johnson (2008) J.-C. Liou & N. Johnson. Instability of the present LEO satellite populations. Adv. Space Res., 41:1046–1053, 2008.
* Liou & Johnson (2009) J.-C. Liou & N. Johnson. A sensitivity study of the effectiveness of active debris removal in leo. Acta Astronaut., 64:236–243, 2009.
* Muldoon & Elkaim (2008) A. Muldoon & G. Elkaim. Improved orbit estimation using GPS measurements for conjunction analysis. In Inst. of Navigation Global Navigation Satellite Systems Meeting, pages 786–794, 2008.
* Neal, Coffey, & Knowles (1997) H. Neal, S. Coffey, & S. Knowles. Maintaining the space object catalog with special perturbations. Astrodynamics 1997, Adv. Astronaut. Sci., pages 1349–1360, 1997.
* Neuman (2008) L. Newman. NASA Goddard Space Flight Center. Personal communication, 2009.
* Pearlman, Degnan, & Bosworth (2002) M. Pearlman, J. Degnan, & J. Bosworth. The International Laser Ranging Service. Adv. Space Res., 30(2):135–143, 2002.
* Ries (2009) J. Ries. Precision Orbital Ephemerides (POEs) for satellites Stella, Starlette, Ajisai & Etalon-2 were obtained from John Ries at the Center for Space Research, University of Texas, Austin TX.
* Schumacher et al. (2001) P. Schumacher, G. Gilbreath, M. Davis, & E. Lydick. Precision of satellite laser ranging calibration of the naval space surveillance system. J. Guid. Control Dynam., 24(5):925–932, 2001.
* Snow & Kaya (1999) D. Snow & D. Kaya. Element set prediction accuracy assessment. In Astrodynamics 1999, volume 103 of Adv. Astronaut. Sci., pages 1937–1958, 1999.
* Wang, Liu, & Zhang (2009) R. Wang, J. Liu, & Q. Zhang. Propagation errors analysis of TLE data. Adv. Space Res., 43(7):1065–1069, 2009.
|
arxiv-papers
| 2010-02-11T05:31:04 |
2024-09-04T02:49:08.345010
|
{
"license": "Public Domain",
"authors": "Creon Levit, William Marshall",
"submitter": "William Marshall",
"url": "https://arxiv.org/abs/1002.2277"
}
|
1002.2302
|
# GUP and Correction to BTH
Sumit Ghosh Rabin Banerjee, Sumit Ghosh
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: rabin@bose.res.inE-mail: sumit.ghosh@bose.res.in
# Generalised Uncertainty Principle, Remnant Mass
and
Singularity Problem in Black Hole Thermodynamics
Sumit Ghosh Rabin Banerjee, Sumit Ghosh
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: rabin@bose.res.inE-mail: sumit.ghosh@bose.res.in
###### Abstract
We have derived a new Generalised Uncertainty Principle (GUP) based on certain
general assumptions. This GUP is consistent with predictions from string
theory. It is then used to study Schwarzschild black hole thermodynamics.
Corrections to the mass-temperature relation, area law and heat capacity are
obtained. We find that the evaporation process stops at a particular mass,
referred as the remnant mass. This is instrumental in bypassing the well known
singularity problem that occurs in a semiclassical approach.
## 1 Introduction
The introduction of gravity into quantum field theory brings an observer
independent minimum length scale in the picture [1]. A minimal length also
occurs in string theory [2], non commutative geometry [3] or can be obtained
from gedanken experiment [4]. This minimal length is expected to be close or
equal to Planck length ($L_{P}$). The manifestation of the inclusion of a
minimal length in theories has been observed from different perspectives - the
generalised uncertainty principle (GUP), modified dispersion relation (MDR),
deformed special relativity (DSR), to name a few. The modification or
deformation affect the well known semiclassical laws of black hole
thermodynamics [5, 6, 7, 8]. For instance, the black hole entropy is no longer
proportional to the horizon area [9, 10]. Another interesting result is the
existence of a remnant mass of a black hole. The existence of a remnant mass
of a black hole is verified by different approaches - using a generalised
uncertainty principle [11] or analysing the tunnelling probability [12].
In this paper we study the modifications to the laws of Schwarzschild black
hole thermodynamics by starting from a new GUP which is derived from certain
basic assumptions. The consequences of these modifications are investigated in
details. We show the existence of a critical (ans also singular) mass for the
black hole below (at) which the thermodynamic entities become complex (ill
defined). However both the critical and singular mass are less than another
mass - the remnant mass. Our analysis reveals that the black hole evaporation
does not lead to a singularity. This process terminates at a finite mass which
we call the remnant mass. Since, as already stated, both the critical and
singular masses are less than the remnant mass, the problematic situations are
avoided.
Section 2 gives the derivation of the GUP. Corrections induced by this GUP to
the black hole thermodynamics are given in section 3. The connection of
remnant mass with the singularity problem is discussed in section 4. Section 5
contains the discussions.
## 2 A Generalised Uncertainty Principle (GUP)
A particle with energy close to the Planck energy $E_{P}$ will disturb the
space time significantly at least upto a length of the order of the Planck
length. It is very natural to take the metric to be a function of the particle
energy111This is the effect of back reaction [13]. [14] . One can find the
explicit dependence by solving corresponding Einstein equation where the right
hand side is given by the energy momentum tensor of the particle. If we assume
that the a particle field is a linear superposition of plane wave solutions
($\thicksim e^{ik^{\mu}x_{\mu}}$), then one can easily guess that on
quantisation the particle momentum$(p)$ and energy$(E)$ may be non linear in
wave vector$(k)$ and angular frequency$(\omega)$ [14, 15] . In general we may
write
$\displaystyle\it{k^{\mu}=f(p^{\mu})}$ (1)
It is easy to show that both $k^{\mu}$ and $p^{\mu}$ can transform like a
Lorentz vector only for special types of function $f$. The standard form is
the obvious $p^{\mu}=\hbar k^{\mu}$ and a more general form is
$p^{\mu}=\phi\left(k^{\nu}k_{\nu}\right)k^{\mu}$ where
$\phi\left(k^{\nu}k_{\nu}\right)$ is a scalar function of the invariant
$\left(k^{\nu}k_{\nu}\right)$; the more general form is clearly equivalent to
generalising Planck’s constant to a function. For simplicity in this paper we
will forego Lorentz invariance and consider the following relations [16],
$\displaystyle\it{k=f(p)\hskip 28.45274pt\omega=f(E)}$ (2)
The function $f$ satisfies certain properties [14, 17] :
1. 1.
The function $\left(f\right)$ and its inverse $\left(f^{-1}\right)$ have to be
an odd function to preserve parity.
2. 2.
For small momentum/energy ($E<<E_{P}$) the function should be chosen to
satisfy the relationship $p=\hbar k$ .
3. 3.
We assume the existence of a minimum length, identified as the Planck length
$(L_{P})$ [1, 14, 17] that cannot be resolved. So the wave vector $k=f(p)$
should have an upper bound $\dfrac{2\pi}{L_{P}}$. Since the wave vector
$k=f(p)$ shows a saturation with respect to the momentum $p$, the momentum
$p=f^{-1}(k)$ will be a monotonically increasing function of $k$.
We also assume that the commutation relations
$\displaystyle[x,k]=i\;\;\;,\;\;\;[x,p(k)]=i\dfrac{\partial p}{\partial k}$
(3)
hold which lead to a uncertainty relation [18]
$\displaystyle\Delta p\Delta
x\geqslant|\left\langle\dfrac{1}{2}[x,p]\right\rangle|\;=\;\dfrac{1}{2}|\left\langle\dfrac{\partial
p}{\partial k}\right\rangle|$ (4)
Observe that we are not using the field theory commutator between the field
and its conjugate momentum. Rather our analysis is based on the algebra (3)
which is plausible.
The properties of the function $f(p)$, enlisted below (2) cannot be satisfied
by a finite order polynomial. A possible choice is
$k=f(p)=\dfrac{1}{L_{P}}\sum_{i=0}^{\infty}a_{i}(-1)^{i}{\left(\dfrac{L_{P}\;p}{\hbar}\right)}^{2i+1}$
(5)
Only odd powers of $p$ appear in the polynomial ensuring that $f(p)$ is odd in
$p$ (property 1). The coefficients $\\{a_{i}\\}$ are all positive with
$a_{0}=1$ (to satisfy $p=\hbar k$ at small energy(property 2)). The factor
$(-1)^{i}$ ensures saturation (property 3). The third property further implies
that for $p\rightarrow\infty,\hskip
14.22636ptk\rightarrow\dfrac{2\pi}{L_{P}}$, i.e.
$\displaystyle\sum_{i=0}^{\infty}a_{i}(-1)^{i}{\left(\dfrac{L_{P}\;p}{\hbar}\right)}^{2i+1}$
$\displaystyle\longrightarrow\;2\pi$ (6)
From (5) we get
$\dfrac{\partial k}{\partial
p}=\dfrac{1}{\hbar}\sum_{i=0}^{\infty}a_{i}(2i+1)(-1)^{i}\left(\dfrac{pL_{P}}{\hbar}\right)^{2i}$
(7)
Inverting this we obtain
$\dfrac{\partial p}{\partial
k}=\hbar\sum_{i=0}^{\infty}a^{\prime}_{i}\;\left(\dfrac{pL_{P}}{\hbar}\right)^{2i}$
(8)
where the new coefficients of expansions $\\{a^{\prime}_{i}\\}$ are functions
of $\\{a_{i}\\}$. It is very easy to show that the first coefficient of this
inverted series will be inverse of $a_{0}$, i.e. 1.
Hence the GUP following from (4) takes the form,
$\Delta x\Delta
p\geqslant\left\langle\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\left(\dfrac{L_{P}p}{\hbar}\right)^{2i}\right\rangle\geqslant\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\left(\dfrac{L_{P}}{\hbar}\right)^{2i}\left((\Delta
p)^{2}+\left\langle p\right\rangle^{2}\right)^{i}$ (9)
where we have used $\left\langle p^{2i}\right\rangle\geqslant\left\langle
p^{2}\right\rangle^{i}$. For minimum position uncertainty we put $\left\langle
p\right\rangle=0$ and our GUP becomes
$\Delta x\Delta
p\geqslant\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\left(\dfrac{L_{P}\Delta
p}{\hbar}\right)^{2i}$ (10)
Note that all $a^{\prime}_{i}$’s are positive. If only the first two terms in
(10) are considered we reproduce the GUP predicted by string theory [4, 19].
## 3 Thermodynamics of Schwarzschild black hole with corrections
The object of this section is to use the GUP (10) to evaluate different
thermodynamic entities of a Schwarzschild black hole and thereby find
relations among them.
Let us consider a Schwarzschild black hole with mass M. Let a pair (particle-
antiparticle) production occur near the event horizon. For simplicity we
consider the particles to be massless222For massive particle the expression
for temperature (11) will be modified.. The particle with negative energy
falls inside the horizon and that with positive energy escapes outside the
horizon and observed by some observer at infinity. The momentum of the emitted
particle$(p)$, which also characterises its temperature $(T)$ 333For
simplicity we consider the emitted spectrum to be thermal., is of the order of
its momentum uncertainty $(\Delta p)$ [11]. Consequently
$\displaystyle T=\dfrac{\Delta pc}{k_{B}}$ (11)
For thermodynamic equilibrium, the temperature of the particle gets identified
with the temperature of the black hole itself.
It is now possible to relate this temperature with the mass (M) of the black
hole by recasting the GUP (10) in terms of T and M. In that case the GUP has
to be saturated
$\Delta p\Delta
x={\epsilon}_{1}\dfrac{\hbar}{2}\sum_{i=0}^{\infty}a^{\prime}_{i}\;\left(\dfrac{\Delta
pL_{P}}{\hbar}\right)^{2i}$ (12)
where the new dimensionless parameter ${\epsilon}_{1}$ is a scale factor
saturating the uncertainty relation. We can later adjust it by calibrating
with some known result. We add that the product of $\Delta x$ and $\Delta p$
may be arbitrarily large but we assume that the lower limit can be achieved.
Near the horizon of a black hole the position uncertainty of a particle will
be of the order of the Schwarzschild radius of the black hole [6, 11],
$\displaystyle\Delta x\;=\;{\epsilon}_{2}\dfrac{2GM}{c^{2}}$ (13)
The new dimensionless parameter ${\epsilon}_{2}$ is introduced as a scale
factor and will be calibrated soon.
Substituting the values of $\Delta p$ (11) and $\Delta x$ (13) in (10), the
GUP is recast as
$\displaystyle
M\;=\;\epsilon\dfrac{M_{P}}{4}\sum_{i=0}^{\infty}a^{\prime}_{i}\;{\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)}^{2i-1}$
(14)
(where we have used the relations
$\epsilon=\dfrac{{\epsilon}_{1}}{{\epsilon}_{2}},~{}M_{P}=\dfrac{L_{P}c^{2}}{G}$
and $\dfrac{c\hbar}{L_{P}}=M_{P}c^{2}~{}~{},~{}M_{P}$ being the Planck mass.)
In the absence of correction due to quantum gravity effects only
$a^{\prime}_{0}=1$ will survive and we should reproduce the semi classical
result. In this approximation (14) reduces to
$M=\epsilon\dfrac{M_{P}^{2}c^{2}}{4k_{B}T}$ (15)
This will fix $\epsilon$. Comparison with the standard semi classical Hawking
temperature [9] $\left(T_{H}=\dfrac{M_{P}^{2}c^{2}}{8\pi Mk_{B}}\right)$
yields $\epsilon=\dfrac{1}{2\pi}$.
So the mass temperature relationship is
$\displaystyle
M\;=\;\dfrac{M_{P}}{8\pi}\sum_{i=0}^{\infty}a^{\prime}_{i}\;{\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)}^{2i-1}$
(16)
The heat capacity of the black hole , by definition, is given by
$\displaystyle C\;=\;c^{2}\dfrac{dM}{dT}$ (17)
Therefore from (16) we find that
$\displaystyle
C\;=\;\dfrac{k_{B}}{8\pi}\sum_{i=0}^{\infty}a^{\prime}_{i}\;(2i-1)\;\left(\dfrac{k_{B}T}{M_{P}\;c^{2}}\right)^{2i-2}$
(18)
The nature of the heat capacity will become more illuminating if we express it
in terms of particle energy $E=k_{B}T$ and Planck energy $E_{P}=M_{P}c^{2}$.
Then
$C\;=\;\dfrac{k_{B}}{8\pi}\sum_{i=0}^{\infty}a^{\prime}_{i}\;(2i-1)\;\left(\dfrac{E}{E_{P}}\right)^{2i-2}$
(19)
For $E<<E_{P}$ the first term will predominate, and since it is with a
negative signature the heat capacity will also be negative in this region. The
heat capacity increases monotonically as $E\rightarrow E_{P}$. There will be a
point at which the heat capacity vanishes. We consider the corresponding
temperature to be the maximum temperature attainable by a black hole during
evaporation. The process stops thereafter.
So a Schwarzschild black hole with a finite mass and temperature, by radiation
process, loses its mass and in turn its temperature increases. This state
corresponds to a negative heat capacity. Then it attains a temperature at
which $\dfrac{dM}{dT}$ becomes zero (zero heat capacity), i.e there will be no
further change of black hole mass with its temperature. The radiation process
ends here with a finite remnant mass with a finite temperature.
One can also determine the black hole entropy(S) in a similar way. According
to the first law of black hole thermodynamics it is given by
$S\;=\;\int\dfrac{c^{2}dM}{T}$ (20)
For technical simplification this definition is expressed in terms of the heat
capacity (17). Then exploiting (18 ) and carrying out the above integration we
finally obtain
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int\dfrac{CdT}{T}$ (21)
$\displaystyle=$
$\displaystyle\dfrac{k_{B}}{16\pi}\left[\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}\;+a^{\prime}_{1}\ln\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\;+\sum_{i=2}^{\infty}a^{\prime}_{i}\;\dfrac{(2i-1)}{(i-1)}\;\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2(i-1)}\right]$
If we want to express the heat capacity and the entropy in terms of the mass
we have to obtain an expression for $T^{2}$ in terms of M. We can do this by
squaring (16)
$\displaystyle\left(\dfrac{8\pi M}{M_{P}}\right)^{2}$ $\displaystyle=$
$\displaystyle\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}+2a^{\prime}_{1}+\left({a^{\prime}_{1}}^{2}+2a^{\prime}_{2}\right)\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}+2\left(a^{\prime}_{1}a^{\prime}_{2}+a^{\prime}_{3}\right)\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{4}+\ldots$
Then considering a finite number of terms, dictated by the order of the
approximation, we can obtain an expression for
$\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ in terms of M by inverting
(LABEL:mt ).
### 3.1 First order correction
We will next discuss the effect of first order correction. Consequently we can
neglect the contribution of $\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ and
higher order terms in (LABEL:mt). Then we get
$\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\;=\;\dfrac{1}{\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}}$ (23)
The critical mass below which the temperature becomes a complex quantity is
given by
$M_{cr}=\dfrac{\sqrt{2a^{\prime}_{1}}}{8\pi}M_{P}$ (24)
We will soon show that the evaporation process terminates with a mass greater
than this.
Substituting (23 ) in (21) we get
$\displaystyle\dfrac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}-\dfrac{a^{\prime}_{1}}{16\pi}ln\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)-\dfrac{a^{\prime}_{1}}{16\pi}ln(16\pi)$
(25)
where $S_{BH}=k_{B}\dfrac{4\pi M^{2}}{M_{P}^{2}}$ is the Bekenstein-Hawking
entropy.
We now obtain the area theorem from equation (25). This theorem will appear
more tractable if we introduce a new variable $A^{\prime}$ (reduced area)
defined as
$\displaystyle
A^{\prime}\;=\;16\pi\dfrac{G^{2}M^{2}}{c^{4}}-\dfrac{2a^{\prime}_{1}}{4\pi}\dfrac{G^{2}M_{P}^{2}}{c^{4}}\;=\;A-\dfrac{2a^{\prime}_{1}}{4\pi}L_{P}^{2}$
(26)
where A is the usual area of the horizon.
In terms of the reduced area the expression for entropy takes a familiar form
$\displaystyle\dfrac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle\dfrac{A^{\prime}}{4L_{P}^{2}}-\dfrac{a^{\prime}_{1}}{16\pi}\ln\left(\dfrac{A^{\prime}}{4L_{P}^{2}}\right)-\dfrac{a^{\prime}_{1}}{16\pi}\ln(16\pi)$
(27)
This is the area theorem in presence of the GUP (10) upto the first order
correction. The usual Bekenstein-Hawking semiclassical area law is reproduced
for $a^{\prime}_{1}=0$.
The important feature of (27) is that entropy is explicitly expressed as a
function of the reduced area and not the actual area [5, 6]. It has a
singularity at zero reduced area which corresponds to a singular mass given by
$M_{sing}=\dfrac{\sqrt{2a^{\prime}_{1}}}{8\pi}M_{P}$ (28)
We will subsequently prove that the reduced area is always positive in
presence of quantum gravity effect and the singularity is thereby avoided
during the evaporation process. Observe that, to this order, the critical mass
(24) and the singular mass (28) are identical.
The variation of temperature and entropy with mass for different values of
$a^{\prime}_{1}$ is shown in $fig~{}1,2$. The interesting fact about these
curves is their termination at some finite mass for non zero $a^{\prime}_{1}$.
We will explain the reason in the next section.
### 3.2 Second order correction
In this subsection the various thermodynamic variables are computed upto
second order. This implies that terms upto
$\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ in (LABEL:mt) are retained. With
a simple rearrangement one can easily obtain
$\displaystyle\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ $\displaystyle=$
$\displaystyle\dfrac{\left[\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1})\right]\pm\sqrt{\left[\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1})\right]^{2}-4({a^{\prime}_{1}}^{2}+2a^{\prime}_{2})}}{2\left({a^{\prime}_{1}}^{2}+2a^{\prime}_{2}\right)}$
(29)
Only the (-) sign is acceptable from the $(\pm)$ part, because the (+) sign
will not produce the semi classical result if we put
$a^{\prime}_{1}=a^{\prime}_{2}=0$.
At a first glance, one observes that the first order expression for
temperature (23) cannot be retrieved simply by putting $a^{\prime}_{2}=0$.
Instead, one has to put ${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}=0$, because both
${a^{\prime}_{1}}^{2}$ and $2a^{\prime}_{2}$ bear the signature of the second
order approximation (see (LABEL:mt)). To make this retrieval simpler we will
rearrange (29) using binomial expansion.
$\displaystyle\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}}\left[1+\dfrac{\left({a^{\prime}_{1}}^{2}+2a^{\prime}_{2}\right)}{\left[\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}\right]^{2}}+\ldots\right]$
From this expression it is clear that the first order expression for
temperature can be obtained from the second order expression by putting
${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}=0$. The critical mass is found by
equating the discriminant of (29) to zero. It is given by
$\displaystyle\left(\dfrac{8\pi
M_{cr}}{M_{P}}\right)^{2}=2a^{\prime}_{1}+2\sqrt{{a^{\prime}_{1}}^{2}+2a^{\prime}_{2}}$
(31)
which is greater than (24 ) with first order correction (since
$a^{\prime}_{1},a^{\prime}_{2}$ are all positive).
We conclude this section by computing the corrections (upto second order) to
the entropy and the area law. The corrected entropy follows from (21)
$\displaystyle S$ $\displaystyle=$
$\displaystyle\dfrac{k_{B}}{16\pi}\left[\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}\;+a^{\prime}_{1}\ln\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\;+3a^{\prime}_{2}\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\right]$
(32)
Substituting (LABEL:mt2e) in (32) we get
$\displaystyle\dfrac{S}{k_{B}}$ $\displaystyle=$
$\displaystyle\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)-\dfrac{a^{\prime}_{1}}{16\pi}ln\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)+\sum_{j=0}^{\infty}c_{j}(a^{\prime}_{1},a^{\prime}_{2})\left(\dfrac{S_{BH}}{k_{B}}-\dfrac{2a^{\prime}_{1}}{16\pi}\right)^{-j}-\dfrac{a^{\prime}_{1}}{16\pi}\ln(16\pi)$
where $S_{BH}=k_{B}\dfrac{4\pi M^{2}}{M_{P}^{2}}$ is the Bekenstein-Hawking
entropy and these new coefficients of expansion $c_{j}$ are functions of
$a^{\prime}_{1}$ and $a^{\prime}_{2}$. The coefficients $c_{j}$ will have an
explicit dependence on ${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}$, which follows
from (LABEL:mt2e). Hence we can get our first order result (25) by putting
${a^{\prime}_{1}}^{2}+2a^{\prime}_{2}=0$.
We can now obtain our area theorem in terms of the reduced area (26) from
(LABEL:s2),
$\dfrac{S}{k_{B}}\;=\;\dfrac{A^{\prime}}{4L_{P}^{2}}-\dfrac{a^{\prime}_{1}}{16\pi}\ln\left(\dfrac{A^{\prime}}{4L_{P}^{2}}\right)+\sum_{j=0}^{\infty}c_{j}(a^{\prime}_{1},a_{2})\left(\dfrac{A^{\prime}}{4L_{P}^{2}}\right)^{-j}-\dfrac{a^{\prime}_{1}}{16\pi}\ln(16\pi)$
(34)
This is the area theorem with second order correction. The expression looks
like the standard corrected area theorem [5, 6], with the role of the actual
area ($A$) being played by the reduced area ($A^{\prime}$).
Some comments are in order. The singularity is again at zero reduced area,
corresponding mass being given by (28). As shown in the next section, this
singular mass is also less than the remnant mass with second order correction.
## 4 Remnant mass and singularity problem
As discussed earlier here we show that the black hole evaporation terminates
at a finite mass which is greater than the either the critical mass
$M_{cr}$(24,31) or the singular mass (28). This demonstrates the internal
consistency of our calculation scheme. Consequently the usual singularity
problem whereby the temperature blows up, is avoided.
### 4.1 First order correction
Considering the first two terms in the series expansion for the heat capacity
(18), we obtain
$C\;=\;\dfrac{k_{B}}{8\pi}\left[-\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}+a^{\prime}_{1}\right]$
(35)
Substituting the value of $T^{2}$ from (23)
$C\;=\;\dfrac{k_{B}}{8\pi}\left[-\left(\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}\right)+a^{\prime}_{1}\right]$ (36)
The variation of heat capacity with mass for different values of
$a^{\prime}_{1}$ is shown in $fig~{}3$.
The collapse of the black hole is terminated when the heat capacity becomes
zero. The mass of the black hole now remains unchanged. This mass is called
the remnant mass. Its value is obtained by solving
$\dfrac{k_{B}}{8\pi}\left[-\left(\left(\dfrac{8\pi
M}{M_{P}}\right)^{2}-2a^{\prime}_{1}\right)+a^{\prime}_{1}\right]\;=\;0$ (37)
leading to,
$\displaystyle M_{rem}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{3a^{\prime}_{1}}}{8\pi}M_{P}$ (38)
Alternatively the value for remnant mass can also be obtained by minimising
the entropy (28),
$\dfrac{dS}{dM}\;=\;0$ (39)
and looking at the second derivative $\left(\dfrac{d^{2}S}{dM^{2}}>0\right)$.
The result (38) is reproduced.
The most important fact is that the value (38) is greater than the singular
mass (28) and also the critical mass (24). So we can say that the singularity
will be avoided during the evaporation process and the reduced area will be
positive. At the same time we also managed to avoid any possibility of dealing
with complex values for the thermodynamic entities
$\left(since,~{}M_{rem}>M_{cr}\right)$.
The remnant value of area, reduced area, temperature and entropy are now
expressed in terms of the coefficient $a^{\prime}_{1}$.
$\displaystyle A_{rem}$ $\displaystyle=$ $\displaystyle\dfrac{16\pi
G^{2}M_{rem}^{2}}{c^{4}}\;=\;\dfrac{3a^{\prime}_{1}}{64{\pi}^{2}}\dfrac{16\pi
G^{2}M_{P}^{2}}{c^{4}}$ (40) $\displaystyle A^{\prime}_{rem}$ $\displaystyle=$
$\displaystyle\dfrac{a^{\prime}_{1}L_{P}^{2}}{4\pi}\;=\;\dfrac{16\pi
G^{2}M_{rem}^{2}}{3c^{4}}=\dfrac{A_{min}}{3}$ (41) $\displaystyle T_{rem}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{a^{\prime}_{1}}}\dfrac{M_{P}c^{2}}{k_{B}}$ (42)
$\displaystyle S_{rem}$ $\displaystyle=$
$\displaystyle\dfrac{k_{B}}{16\pi}\left[a^{\prime}_{1}-a^{\prime}_{1}\ln\left(a^{\prime}_{1}\right)\right]$
(43)
The expressions for different thermodynamic entities including the remnant
mass involves only one parameter $a^{\prime}_{1}$ as a measure of quantum
gravity effect. If we put $a^{\prime}_{1}=0$ we get back all our semi
classical results and the remnant mass becomes zero. The final entropy and
heat capacity become zero while the temperature becomes infinite.
### 4.2 Second order correction
The heat capacity with the second order contribution is given by
$\displaystyle C$ $\displaystyle=$
$\displaystyle\dfrac{k_{B}}{8\pi}\left[-\left(\dfrac{M_{P}c^{2}}{k_{B}T}\right)^{2}+a^{\prime}_{1}+3a^{\prime}_{2}\left(\dfrac{k_{B}T}{M_{P}c^{2}}\right)^{2}\right]$
(44)
The expression for remnant mass can be found either from zero heat capacity
condition or by minimising the entropy. Adopting the first approach we will
compute the remnant mass here. Replacing the value of $T^{2}$ from (29) in
(44) and equating the r.h.s. to zero we obtain the remnant mass
$\left(\dfrac{8\pi
M_{rem}}{M_{P}}\right)^{2}=\dfrac{1}{6a^{\prime}_{2}}\left[-a^{\prime}_{1}\left({a^{\prime}_{1}}^{2}-13a^{\prime}_{2}\right)+\left({a^{\prime}_{1}}^{2}+5a^{\prime}_{2}\right)\sqrt{{a^{\prime}_{1}}^{2}+12a^{\prime}_{2}}\;\right]$
(45)
One can easily show that the r.h.s. is greater than $3a^{\prime}_{1}$ which is
the value (38) for $\left(\dfrac{8\pi M_{rem}}{M_{P}}\right)^{2}$ with first
order correction. The remnant mass is also greater than the critical mass (31)
for positive $a^{\prime}_{1}$ and $a^{\prime}_{2}$. The difference between
remnant mass and the critical mass for different values of $a^{\prime}_{1}$
and $a^{\prime}_{2}$ is shown in $fig.~{}4$ .
## 5 Discussion
The laws of black hole thermodynamics are known to be modified by the presence
of a generalised uncertainty principle (GUP) [5, 6, 7, 8].Here we have derived
a new GUP (based on the presence of a minimal length scale $(L_{P})$) which,
at the lowest orders, is also shown to be compatible with string theory
predictions. Using this GUP various aspects of Schwarzscild black hole
thermodynamics were examined.
Our calculations were performed upto two orders (in $L_{P}$) of corrections.
Corrected structures of mass-temperature relation, area theorem and heat
capacity were obtained. The usual semiclassical expressions were easily
derived.
An important consequence was that the black hole evaporation terminated at a
finite mass. This (remnant) mass was found to be greater than either the
critical mass (below which the thermodynamic variables become complex) or the
singular mass (where the thermodynamic variables become infinite).
Consequently the ill defined situations were bypassed. Also, contrary to
standard results [5, 6] using GUP our modified area law (27,34) is more
transparent when expressed in terms of reduced area defined in (26).
To put our results in a proper perspective let us compare with earlier
findings. A remnant mass was also found in [11] using stringy GUP and in [12]
employing notions of tunnelling. In the first case the remnant mass was given
by $M=M_{P}$ (which is consistent with our findings) and successfully avoided
the singularity. However, in contrast to our analysis, the calculations were
confined to the leading order only. Also, the remnant and the critical mass
became identical. Hence it was not possible to distinguish between the
termination of black hole evaporation and complexification of thermodynamic
variables. The calculation of [12], on the other hand, led to the result that
although there was a remnant mass, the singularity problem could not be
avoided.
## Acknowledgement
One of the authors (S.G.) thanks the Council of Scientific and Industrial
Research (C.S.I.R), Government of India, for financial support. We also thank
B. R. Majhi and S. K. Modak for discussions.
Figure 1: Temperature - mass curve with $a^{\prime}_{1}=1$ (blue),
$a^{\prime}_{1}=.75$ (red), $a^{\prime}_{1}=\dfrac{1}{3}$ (green) and
$a^{\prime}_{1}=0$ (black) [Semiclassical]
Figure 2: Entropy - mass curve with $a^{\prime}_{1}=1$ (blue),
$a^{\prime}_{1}=.75$ (red), $a^{\prime}_{1}=\dfrac{1}{3}$ (green) and
$a^{\prime}_{1}=0$ (black) [Semiclassical]
Figure 3: heat capacity - mass curve with $a^{\prime}_{1}=1$ (blue),
$a^{\prime}_{1}=.75$ (red), $a^{\prime}_{1}=\dfrac{1}{3}$ (green) and
$a^{\prime}_{1}=0$ (black) [Semiclassical]
Figure 4: difference between the remnant mass and the critical mass
$(M_{rem}-M_{cr})$ in unit of $M_{P}$ for different values of $a^{\prime}_{1}$
and $a^{\prime}_{2}$ shown from two different angles. The difference becomes
zero only at $a^{\prime}_{2}=0$. This point is not considered, since for
$a^{\prime}_{2}=0$, second order correction is not meaningful.
## References
* [1] G. Amelino-Camelia, Int.J.Mod.Phys. D11 (2002) (35-60) [arXiv:gr-qc/0012051]; Phys. Lett. B 510, 2001, (255-263) [arXiv:hep-th/0012238v1]
* [2] D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216, 1989, (41-47)
* [3] F. Girelli, E. R. Livine and D. Oriti, Nuclear Physics B 708, 2005, (411-433) [arXiv:gr-qc/0406100]
* [4] F. Scardigli, Phys. Lett. B 452, 1999, (39-44)
* [5] G. Amelino-Camelia, M. Arzano, Y. Ling and G. Mandanici, Class. Quant. Grav. 23 (2006), (2585-2606) [arXiv:gr-qc/0506110]
* [6] A. J. M. Medved, E. C. Vagenas, Phys. Rev. D 70, (124021), 2004
* [7] K. Nouicer, Phys. Lett. B 646, 2007, (63-71)
Y.M. Kim, Y.J. Park, Phys. Lett. B 655, 2007, (172-177)
* [8] A. Bina, S. Jalalzadeh, A. Moslehi, Phys. Rev. D 81, (023528), 2010 [arXiv:1001.0861]
* [9] S. W. Hawking, Nature (London) 248, 30 (1974); Commun. Math. Phys. 43, 199(1975)
* [10] J. D. Bekenstein, Phys. Rev. D 7 (8), 1973, (2333 - 2346)
* [11] R. J. Adler, P. Chen and D. I. Santiago, Gen. Rel. Grav. 33, 2001, (2101-2108) [arXiv:gr-qc/0106080]
* [12] L. Xiang, Phys. Lett. B 647, 2007, (207-210)
* [13] C. O. Loust$\acute{o}$, N. S$\acute{a}$nchez, Phys. Lett. B 4, 1988, (411-414)
* [14] S. Hossenfelder, Phys. Rev. D 73 (105013), 2006 [arXiv:hep-th/0603032]
* [15] Jo$\tilde{a}$o Magueijo, Phys. Rev. D 73 (124020), 2006
* [16] S. Hossenfelder, Class. Quant. Grav. 23, 2006, (1815-1821) [arXiv:hep-th/0510245]
* [17] S. Hossenfelder, Class. Quant. Grav. 25 (038003), 2008 [arXiv:0712.2811]
* [18] See for example David J. Griffiths, Introduction to quantum mechanics,(1994, Prentice Hall), Section 3.4
* [19] M. Maggiore, Phys. Lett. B 304, 1993, (65-69)
|
arxiv-papers
| 2010-02-11T08:50:16 |
2024-09-04T02:49:08.349652
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rabin Banerjee, Sumit Ghosh",
"submitter": "Sumit Ghosh",
"url": "https://arxiv.org/abs/1002.2302"
}
|
1002.2339
|
# R-parity violating resonant stop production at the Large Hadron Collider
Nishita Desaia and Biswarup Mukhopadhyayaa,b
aRegional Centre for Accelerator-based Particle Physics,
Harish-Chandra Research Institute
Chhatnag Road, Jhunsi,
Allahabad - 211 019, India
bIndian Association for Cultivation of Science,
Raja SC Mullick Road,
Jadavpur, Kolkata - 700032 , India
Email nishita@hri.res.in biswarup@hri.res.in
###### Abstract:
We have investigated the resonant production of a stop at the Large Hadron
Collider, driven by baryon number violating interactions in supersymmetry. We
work in the framework of minimal supergravity models with the lightest
neutralino being the lightest supersymmetric particle which decays within the
detector. We look at various dilepton and trilepton final states, with or
without b-tags. A detailed background simulation is performed, and all
possible decay modes of the lighter stop are taken into account. We find that
higher stop masses are sometimes easier to probe, through the decay of the
stop into the third or fourth neutralino and their subsequent cascades. We
also comment on the detectability of such signals during the 7 TeV run, where,
as expected, only relatively light stops can be probed. Our conclusion is that
the resonant process may be probed, at both 10 and 14 TeV, with the R-parity
violating coupling $\lambda^{\prime\prime}_{312}$ as low as 0.05, for a stop
mass of about 1 TeV. The possibility of distinguishing between resonant stop
production and pair-production is also discussed.
MSSM, supersymmetry, R-parity violation
††preprint: HRI-RECAPP-2010-002
## 1 Introduction
The current structure of the standard model (SM), with gauge invariance and
renormalisability built in, implies automatic lepton and baryon number
conservation. This is no longer true in the supersymmetric (SUSY) extension of
the SM [1, 2], where scalars carrying baryon or lepton number are present.
Thus the superpotential of the minimal SUSY standard model (MSSM), namely
$\displaystyle\mathcal{W}_{MSSM}=h^{d}_{ij}Q_{i}D^{c}_{j}H_{d}+h^{u}_{ij}Q_{i}U^{c}_{j}H_{u}+h^{l}_{ij}L_{i}E^{c}_{j}H_{d}+\mu
H_{u}H_{d}$ (1)
can in principle be augmented to include
$\displaystyle\mathcal{W}_{RPV}=\mu_{i}L_{i}H_{u}+\lambda_{ijk}L_{i}L_{j}E^{c}_{k}+\lambda^{\prime}_{ijk}L_{i}Q_{j}D^{c}_{k}+\lambda^{\prime\prime}_{ijk}U^{c}_{i}D^{c}_{j}D^{c}_{k}$
(2)
which contain terms that are gauge invariant and renormalisable but explicitly
violate lepton or baryon number. Here, L(E) is an $SU(2)$ doublet (singlet)
lepton superfield and Q (U,D) is (are) an $SU(2)$ doublet (singlet) quark
superfield(s). $H_{u}$ and $H_{d}$ are the two Higgs doublet superfields,
$\mu$ is the Higgsino parameter and $(i,j,k)$ are flavour indices. Each term
in equation (2) violates R-parity, defined as $R=(-1)^{3(B-L)-2S}$ (where B is
baryon number, L is lepton number and S is spin), against which all SM
particles are even whereas all superpartners are odd. The consequence of
violating R-parity is that superpartners need not be produced in pairs
anymore, and that the lightest superparticle (LSP) can now decay. The
strongest argument for studying R-parity violation is that it does not arise
as an essential symmetry of MSSM. However, the requirement of suppressing
proton decay prompts one to allow only one of B and L to be violated at a
time.
The collider phenomenology in the absence of R-parity may be very different
from that of the usual R-parity conserving MSSM. In particular, if the
R-parity violating (RPV) couplings are large enough, the LSP will decay within
the detector and one no longer has missing-$E_{T}$ as a convenient
discriminator. Although studies have taken place on such signals, closer looks
at them are often quite relevant in the wake of the Large Hadron Collider
(LHC). In particular, it is crucial to know the consequences of broken
R-parity in the production of sparticles. Here we perform a detailed
simulation in the context of the LHC, highlighting one possible consequence of
the B-violating term(s), namely, the resonant production of a squark—in this
case, the stop.
Many of the RPV couplings have been indirectly constrained from various decay
processes, including rare and flavour-violating decays and violation of weak
universality. The constraints derived are of two general kinds—those on
individual RPV couplings, assuming the existence of a single RPV term; and
those on the products of couplings when at least two terms are present, which
contribute to some (usually rare) process. The constraints obtained so far are
well-listed in the literature[3].
The L-violating terms are relatively well-studied, partly because of their
potential role in generating neutrino masses and are constrained by indirect
limits. In comparison, the baryon-number violating coupling are relatively
unconstrained. $\lambda^{\prime\prime}_{112,113}$ are constrained from double
nucleon decay and neutron-antineutron oscillations[4, 5, 6]. The rest of the
couplings are constrained only by the requirement that they remain
perturbative till the GUT scale. Limits on $\lambda^{\prime\prime}_{3ij}$ type
of couplings due to the ratio of Z-boson decay widths for hadronic versus
leptonic final states have been calculated for a stop mass of 100 GeV [7].
However, the results do not restrict the couplings for high stop masses of
concern here. The coupling $\lambda^{\prime\prime}_{3jk}$ is thus practically
unconstrained for large stop masses. It is also known that mixing in the quark
sector causes generation of couplings of different flavour structures and can
therefore be constrained by data from flavour changing neutral
currents(FCNC)[8]. Such effects arising from mixing in the quark and squark
sector can affect the contribution of R-parity violation to physical process
and alter the limits[9]. However these effects are model dependent and have
not been taken into account here.
It has been already noticed that such large values of
$\lambda^{\prime\prime}$-type couplings as are still allowed, not only cause
the LSP to decay, but also lead to resonant production of squarks via quark
fusion at the LHC. The rate of such fusion can in fact far exceed that of the
canonically studied squark-pair production. One would therefore like to know
how detectable the resonant process is at the LHC. Furthermore, one needs to
know the search limits in different phases of the LHC, and how best to handle
the backgrounds, both from the SM and the R-conserving SUSY processes. These
are some of the questions addressed in this paper.
Single stop production, mostly in the context of the Tevatron, was studied in
detail in[10, 11]. A full one-loop production cross section can be found
in[12]. A study of SUSY with the LSP decaying through baryon-number violating
couplings and therefore giving no missing energy was done in [13]. Further
studies on determining the flavour structure of baryon number violating
couplings and possible mass reconstruction following specific decay chains can
be found in[14, 15]. There have also been recent studies on possible LSPs[16]
and identification of R-parity violating decays of the LSP using jet
substructure methods[17]. A recent study on identification of stop-pair
production via top-tagging using jet-substructure can be found in [18].
We find, however, that the earlier studies on resonant stop production are
inadequate in the context of the LHC. We improve upon them in the following
respects:
* •
In the work done for the Tevatron, the sparticle masses were required to be
less than 500 GeV to be within reach. Thus, the gluino was also required to be
much lighter than a TeV to avoid large radiative corrections to squark masses.
This implied that in the constrained MSSM (cMSSM) [19] scenario, the LSP,
assumed to be the lightest neutralino ($\tilde{\chi}_{1}^{0}$), had to have
mass less than 100 GeV. If we allow only the term proportional to
$\lambda^{\prime\prime}_{3ij}$, the only three-body decay of
$\tilde{\chi}_{1}^{0}$ is
$\tilde{\chi}_{1}^{0}\rightarrow\bar{t}\bar{d}_{i}\bar{d}_{j}(tds)$. Since the
neutralino is much lighter than the top, it can decay only via a 4-body decay
and therefore has long lifetime and decays outside the detector for all
allowed values of $\lambda^{\prime\prime}_{3ij}$[20]. Thus, one still has the
canonical missing-$E_{T}$ signature. This was one of the main assumptions
in[11]. However, if the stop mass is beyond the Tevatron reach but within the
reach of the LHC, we may indeed have lightest-neutralino mass high enough to
allow decay within the detector.
* •
We focus on this richer and more challenging scenario within the framework of
minimal supergravity (mSUGRA) models. We investigate the LHC reach for
detection of the lightest stop assuming that the lightest neutralino is the
LSP which decays within the detector and the stop is heavy enough to be beyond
the reach of Tevatron.
* •
For a light stop, the only available R-parity conserving decay modes are into
the lightest neutralino($\tilde{\chi}_{1}^{0}$) and lighter
chargino($\tilde{\chi}_{1}^{+}$) i.e. $\tilde{t}\rightarrow
t\tilde{\chi}_{1}^{0},b\tilde{\chi}_{1}^{+}$. For stop mass near a TeV, the
decay modes into higher neutralinos and the heavier chargino may open up,
leading to different final states. We have found that this drastically
improves the detectability of the signature over the SM backgrounds.
* •
We have taken into account all the potential backgrounds at the LHC, including
those from $t{\bar{t}}+jets,Wt{\bar{t}}+jets,Zt{\bar{t}}+jets$, which pose
little problem at the Tevatron. A detailed investigation towards reducing
these backgrounds has been reported in the present study.
* •
In identifying signals of stop decay, it is often helpful to tag b quarks.
This is especially important since the reconstruction of energetic top quarks
in multi-top final states is difficult and has a low efficiency. However, the
b-jets produced from stop decay can be quite hard, especially when the stop is
heavy and one is looking at the decay in the $b\tilde{\chi}_{1}^{+}$ channel.
It is not clear that the b-tagging efficiency is appreciable at the LHC for
b-jets with $p_{T}\mathrel{\mathchoice{\lower
1.72218pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
1.72218pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
1.72218pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
1.72218pt\vbox{\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}$
100 GeV. With this is mind, we have performed a conservative analysis with a
b-jet has zero detection efficiency unless its $p_{T}$ lies in the range 50 -
100 GeV[21].
We present our results for centre of mass energies of 7, 10 and 14 TeV at the
LHC. In section 2, we discuss the rates for resonant stop production, the
different decays of the stop and our choice of benchmark points to account for
all of them. In section 3, we present a detailed description of cuts required
to isolate the signal and section 4 contains the numerical results from our
simulation. We also comment on the distinguishability of such a signal from
dilepton signals coming from R-parity conserving MSSM. It is possible to have
LSPs other than $\tilde{\chi}_{1}^{0}$ when R-parity is violated. We comment
on the possibility of detection in such cases in section 5 and summarise and
conclude in section 6.
## 2 Resonant stop production and decays
### 2.1 Stop production
The resonant stop production process depends on B-violating couplings
proportional to $\lambda^{\prime\prime}_{3ij}$, and also on fraction of the
right-chiral eigenstate (${\tilde{t}}_{R}$) in the mass eigenstate concerned.
We concentrate on the production of $\tilde{t}_{1}$ since the lighter stop
eigenstate usually has a higher fraction of ${\tilde{t}}_{R}$. The resonant
production cross section is given by
$\displaystyle\sigma_{\tilde{t}_{1}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\sin^{2}\theta_{\tilde{t}}}{3m^{2}_{\tilde{t}}}\times$
(3) $\displaystyle\sum_{i,j}{|\lambda^{\prime\prime}_{3ij}|^{2}\int
dx_{1}dx_{2}[f_{i}(x_{1})f_{j}(x_{2})+f_{i}(x_{2})f_{j}(x_{1})]\delta(1-\frac{m_{\tilde{t}_{1}}}{\sqrt{\hat{s}}})}$
where $\sin\theta_{\tilde{t}}$ is the amplitude of finding a $\tilde{t}_{R}$
in $\tilde{t}_{1}$, $f_{i}$ is the proton parton distribution function for a
parton of species $i$ and $x_{(1,2)}$ are the momentum fractions carried by
the respective partons. Out of the three possible $\lambda^{\prime\prime}$
couplings, contributions via $\lambda^{\prime\prime}_{313}$ and
$\lambda^{\prime\prime}_{323}$ are suppressed due to the small fraction of b
quarks in the proton. We therefore look at the production of top (anti) squark
through the fusion of the d and s (anti)quarks, via the coupling
$\lambda^{\prime\prime}_{312}$. Since the actual cross section for production
of the lightest stop depends on the mixing angle via
$\sin^{2}\theta_{\tilde{t}}$, it is useful to define the cross section in
terms of an effective coupling
$\lambda^{\prime\prime}_{eff}=\sin\theta_{\tilde{t}}\lambda^{\prime\prime}_{312}$.
$\displaystyle\sigma_{\tilde{t}_{1}}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{3m^{2}_{\tilde{t}_{1}}}|\lambda^{\prime\prime}_{eff}|^{2}\times$
$\displaystyle 2\int
dx_{1}dx_{2}[f_{d}(x_{1})f_{s}(x_{2})+f_{d}(x_{2})f_{s}(x_{1})$
$\displaystyle+f_{\bar{d}}(x_{1})f_{\bar{s}}(x_{2})+f_{\bar{d}}(x_{2})f_{\bar{s}}(x_{1})]\delta(1-\frac{m_{\tilde{t}_{1}}}{\sqrt{\hat{s}}})$
The production cross section at the LHC with centre-of-mass energies of 7, 10
and 14 TeV is given in Figure 1. As an illustration, we have chosen the value
$\lambda^{\prime\prime}_{eff}=0.2$ which is consistent with the existing limit
on $\lambda^{\prime\prime}_{312}$. In general, both $\tilde{t}_{1}$ and
$\tilde{t}_{2}$ will be produced. However, due to larger mass and smaller
fraction of $\tilde{t}_{R}$ , ${\tilde{t}_{2}}$ is rarely produced. For
comparison, we also present the ${\tilde{t}_{1}}$-pair production cross-
section via strong interaction. For $m_{\tilde{t}_{1}}>500$ GeV, the resonant
production dominates over pair-production for
$\lambda^{\prime\prime}_{eff}>0.01$ at 14 TeV. Resonant production can
therefore hold the key to heavy stop signals if baryon number is violated.
Figure 1: Production cross section at the LHC for $\sqrt{s}=7,10$ and $14$ TeV
with $\lambda^{\prime\prime}_{eff}=0.2$. The corresponding cross sections for
R-conserving pair production are also shown.
At next-to-leading order, the production cross section at $\sqrt{s}=14$ TeV is
modified by a k-factor of about 1.4[12]. The uncertainty due to
renormalisation and factorisation scales at lowest order is about 10% and
drops to 5% when NLO corrections are taken into account.
### 2.2 Stop decays and choice of benchmark points
Figure 2: Lighter stop decay branching fractions in different modes for
$\tan\beta=5$, $A_{0}=-1500$ (top left) ; $\tan\beta=40$, $A_{0}=-1500$ (top
right) and $\tan\beta=10$, $A_{0}=0$ (bottom).
$\lambda^{\prime\prime}_{312}=0.2$, $\mu>0$ and $m_{1/2}=450$ GeV in all
cases.
We wish to make our conclusions apply broadly to a general SUSY scenario and
to include all possible final states arising from stop decay. However, the
multitude of free parameters in the MSSM often encourages one to look for some
organising principle. A common practice in this regard is to embed SUSY in
high-scale breaking scheme. Following this practice, we have based our
calculation on the minimal supergravity (mSUGRA) model[22], mainly for
illustrating our claims in a less cumbersome manner. The high scale parameters
in this model are: $m_{0}$, the unified scalar parameter, $m_{1/2}$, the
unified gaugino parameter, $sign(\mu)$, where $\mu$ is the Higgsino mass
parameter, $A_{0}$, the unified trilinear coupling and $\tan\beta$, the ratio
of the two Higgs vacuum expectation values.
Although the production cross section of the stop depends only on the mass and
mixing angle of the stop, any strategy developed for seeing the ensuing
signals has to take note of the decay channels. We have tried to make our
analysis comprehensive by including all possible decay chains of the stop.
Thus we have included decays into $t\tilde{\chi}_{i}^{0}$,
$b\tilde{\chi}_{i}^{\pm}$, $t\tilde{g}$ and $ds$, of whom the first three are
R-conserving decays while the last one is R-violating. The charginos,
neutralinos or the gluino produced out of stop-decay have their usual cascades
until the LSP (here $\chi_{1}^{0}$, the lightest neutralino) is reached. The
$\chi_{1}^{0}$ thereafter undergoes three-body RPV decays driven by
$\lambda^{\prime\prime}_{312}$, to give rise to final states consisting
leptons and jets of various multiplicities.
We observe that for the same values of ($m_{0},m_{1/2}$), the mass and
branching fractions of the stop may vary drastically with different values of
($\tan\beta,A_{0}$). We shall choose $\mu>0$ for all the benchmark points as
it is favoured by the constraint from the muon anomalous magnetic moment[23].
Since we explicitly want to study the situation in which the neutralino decays
within the detector, the only available decay mode is
$\tilde{\chi}_{1}^{0}\rightarrow tds(\bar{t}\bar{d}\bar{s})$. We therefore
require that the neutralino mass be greater than the top mass to allow for a
three-body decay. We fix $m_{1/2}=450$ GeV which gives
$M_{\tilde{\chi}_{1}^{0}}\sim 180$ GeV. We also choose the high scale value of
$\lambda^{\prime\prime}_{312}\sim 0.065$ such that it gives a value of $0.2$
at the electroweak scale.
Figure 2 shows the branching fractions into various final states for three
different choices of $(\tan\beta,A_{0})$, namely, $(5,-1500)$, $(40,-1500)$
and $(10,0)$ for different stop masses, obtained by varying $m_{0}$. We notice
that, for low $m_{0}$, the dominant decay mode is $b\chi_{2}^{+}$ in the third
case of Figure 2, while it is $t\chi_{1}^{0}$ in the first two cases. We also
notice that the decays into higher neutralinos and charginos open up earlier
for $\tan\beta=$ 40 and compared to $\tan\beta=$ 5.
The Tevatron reach for single stop production is about $450$ GeV. We therefore
start with a benchmark point with stop mass of 500 GeV, just beyond this reach
(Point A). The major decay channels in this case are
$t\tilde{\chi}_{1}^{0},b\tilde{\chi}_{1}^{+}$. A stop mass of a TeV at the
electroweak scale may be obtained by various configurations in the high-scale
parameter space. However, from the above plots, one expects its decays to
change significantly with different parameters. Our objective is to determine
whether signal of resonant production of a stop of mass near a TeV can be
probed irrespective of what the high-scale parameters are. For this, we fix
$M_{\tilde{t}_{1}}\sim 1$ TeV. We first look at the case with $A_{0}=-1500$.
We construct two benchmark points with $\tan\beta=$ 5(Point B) and 40(Point C)
which correspond to the opposite ends of the allowed range in $\tan\beta$. We
see that for a stop mass of 1 TeV, the decays into the higgsino-like
$\tilde{\chi}_{2}^{+}$ and $\tilde{\chi}_{3}^{0}$ become dominant goes to high
$\tan\beta$.
Similarly, we also look at a point with $A_{0}=0$ $\tan\beta=10$ (Point D). In
this case, we find that the Higgsino channels open up fairly early and the
dominant decay is $b\tilde{\chi}_{2}^{+}$ followed by $t\tilde{\chi}_{3}^{0}$.
As we shall see in the next section, this plays a crucial role in enhancing
multi-lepton signals of a resonantly produced stop. Finally, since the decay
into a top and a gluino does not open up until much higher stop masses, we
also construct one point in which the stop decays dominantly into $t\tilde{g}$
(Point E).
Points A, B and E correspond to the same value of
$(\tan\beta,A_{0})=(5,-1500)$ and therefore provide a description of how the
signal changes when only $m_{0}$ is varied. This choice of parameters also
corresponds to the most conservative case in terms of signal since the decay
modes into the higher gauginos does not open for a large region in the
parameter space. We will therefore use these points to obtain limits on
$\lambda^{\prime\prime}_{eff}$.
We have tabulated the parameters and significant decay modes in Table 1. The
benchmark points were generated with RPV renormalisation group running of
couplings and masses using SOFTSUSY 3.0.2[24] and the RPV decays were
calculated with the ISAWIG interface to Isajet[25].
Point | $(m_{0},\tan\beta,A_{0})$ | $M_{\tilde{t}_{1}}$ | $\sin^{2}\theta_{\tilde{t}}$ | Dominant decay modes
---|---|---|---|---
A | (600,5,-1500) | 508 | 0.88 | $t\tilde{\chi}_{1}^{0}$ (0.35); $b\tilde{\chi}_{1}^{+}$ (0.19); $\bar{d}\bar{s}$ (0.48)
B | (1650,5,-1500) | 1002 | 0.97 | $t\tilde{\chi}_{1}^{0}$ (0.48);$t\tilde{\chi}_{2}^{0}$ (0.04); $b\tilde{\chi}_{1}^{+}$ (0.10); $\bar{d}\bar{s}$ (0.38)
C | (1570,40,-1500) | 1002 | 0.95 | $t\tilde{\chi}_{1}^{0}$ (0.35);$t\tilde{\chi}_{2}^{0}$ (0.05);
| | | | $b\tilde{\chi}_{1}^{+}$ (0.12); $b\tilde{\chi}_{2}^{+}$ (0.21); $\bar{d}\bar{s}$ (0.27)
D | (1250,10,0) | 1008 | 0.97 | $t\tilde{\chi}_{1}^{0}$ (0.13); $t\tilde{\chi}_{2}^{0}$ (0.04); $t\tilde{\chi}_{3}^{0}$ (0.20); $t\tilde{\chi}_{4}^{0}$ (0.13);
| | | | $b\tilde{\chi}_{1}^{+}$ (0.08); $b\tilde{\chi}_{2}^{+}$ (0.33); $\bar{d}\bar{s}$ (0.10)
E | (2450,5,-1500) | 1404 | 0.99 | $t\tilde{g}$ (0.39); $t\tilde{\chi}_{1}^{0}$ (0.15); $t\tilde{\chi}_{2}^{0}$ (0.02); $t\tilde{\chi}_{3}^{0}$ (0.08);
| | | | $t\tilde{\chi}_{4}^{0}$ (0.07); $b\tilde{\chi}_{1}^{+}$ (0.02); $b\tilde{\chi}_{2}^{+}$ (0.17); $\bar{d}\bar{s}$ (0.11)
Table 1: Benchmark points and the dominant decay modes of the lighter stop.
$\lambda^{\prime\prime}_{312}=0.2$, $\mu>0$ and $m_{1/2}=450$ GeV for all
benchmark points.
The decay width of the stop in the R-parity violating channel $ds$ depends
only on $\lambda^{\prime\prime}_{eff}$ and the stop mass. Therefore, the
branching ratio into this channel for same values of
$\lambda^{\prime\prime}_{eff}$ and stop mass depends only on the decay widths
of the other channels open at the same time. For the benchmarks under
consideration, $\tilde{\chi}_{1,2}^{0}$ and $\tilde{\chi}_{1}^{\pm}$ have
large gaugino fractions whereas $\tilde{\chi}_{3,4}^{0}$ and
$\tilde{\chi}_{2}^{\pm}$ have large higgsino fractions. The large top mass
means that stop coupling to higgsino-like chargino and neutralinos is large.
Thus as soon as these decays become kinematically allowed, they quickly
dominate over the decays into gaugino-like chargino and neutralinos. This can
be seen for points B, C and D which have nearly identical stop masses and
$\sin^{2}\theta_{\tilde{t}}$ ($\lambda^{\prime\prime}_{312}=0.2$ at
electroweak scale for all points). Large $\tan\beta$ opens up the
$b\tilde{\chi}_{2}^{+}$ mode early in point C as compared to point B and makes
the branching fraction into $ds$ for point C much lower. For point D, the
branching fraction into higgsino-like chargino and neutralinos is larger than
$60\%$ and the RPV decay fraction is only about $10\%$. The
$\tilde{t}-t-\tilde{g}$ coupling comes from strong interactions and therefore
the $t\tilde{g}$ channel dominates whenever it becomes kinematically allowed
(as in point E).
## 3 Event generation and selection
### 3.1 Event generation
Signal events have been generated using HERWIG 6.510[26], and jets have been
formed using anti-$k_{T}$ algorithm[27] from FastJet 2.4.1. SM backgrounds
have been calculated using Alpgen 2.13[28] showered through Pythia [29] with
MLM matching. We have used CTEQ6L1 parton distribution functions[30]. The
renormalisation and factorisation scales have been set at the lighter stop
mass ($M_{\tilde{t}_{1}}$) for signal, while the default option in ALPGEN has
been used for the backgrounds.
In R-parity conserving MSSM, the production of two heavy superparticles
requires a large centre-of-mass energy at the parton level. This allows us to
further suppress the SM background by applying cuts on global variables like
the “effective mass” ($M_{eff}$). Since we no longer have a large
missing-$E_{T}$ and the energy scale of the resonant production process is not
very high, the SM background cannot be suppressed so easily. We therefore
concentrate on leptonic signals with or without b-tags to identify the signal
over the background.
### 3.2 Event selection
Decay of the lighter stop in this scenario can lead to a variety of final
states. Out of them, we have chosen the following ones:
* •
Same-sign dileptons: $SSD$
* •
Same-sign dileptons with one b-tagged jet: $SSD+b$
* •
Trileptons: $3l$
We do not consider the RPV dijet channel as a viable signature due to the
enormous background from QCD processes. Similarly, we also omit opposite-sign
dileptons due to large backgrounds from Drell-Yan, $W^{+}W^{-}$, $t\bar{t}$
etc.
We have imposed the following identification requirements on leptons and jets:
* •
Leptons: A lepton ($l$) is considered isolated if (a) It is well separated
from each jet ($j$): $\Delta R_{lj}>0.4$, (b) The total hadronic deposit
within $\Delta R<0.35$ is less than $10$ GeV. We consider only those leptons
which fall within $|\eta|<2.5$ with $p_{T}>10$ GeV. Here, $\Delta
R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$ where $\eta$ is the pseudo-rapidity
and $\phi$ is the azimuthal angle.
* •
Jets: Jets have been formed using the anti-$k_{T}$ algorithm with parameter
$R=0.7$. We only retain jets with $p_{T}>20$ GeV and $|\eta|<2.5$.
* •
B-tagged jets: A jet is b-tagged with probability of $0.5$ if a b-hadron with
$50<p_{T}<100$ GeV lies within a cone of $0.7$ from the jet axis. We have set
the identification efficiency to be zero outside this window, in order to make
our estimates conservative.
We also apply the following extra cuts on various final states to enhance the
signal over background:
* •
Cut 1: Lepton-$p_{T}$: We demand that the $p_{T}$ of the leptons be greater
than $(40,30)$ GeV for dilepton and $(30,30,20)$ for trilepton channels. This
cut removes the background from semileptonic decays of b quarks. It strongly
suppresses the $b\bar{b}+jets$, $Wb\bar{b}+jets$ and $t\bar{t}+jets$
background in the $SSD$-channel coming from semileptonic b-decays.
* •
Cut 2: Missing $E_{T}$: At least one lepton in the signal always comes from
the decay of a W boson and is accompanied by a neutrino. We demand a
missing-$E_{T}$ greater than $30$ GeV from all events. This helps in reducing
the probability of jets faking leptons. Missing-$E_{T}$ has been defined as
$|\vec{p}_{T,visible}|$.
* •
Cut 3: Jet $p_{T}$: We demand that the number of jets, $n_{j}\geq 2$ with
$p_{T}(j_{1})>100$ GeV and $p_{t}(j_{2})>50.0$ GeV for $SSD$ and $SSD+b$. This
cut is useful when high stop mass is very high and the production cross
section is very low.
* •
Cut 4: Dilepton invariant mass: We also apply a cut on dilepton invariant mass
($M_{l1,l2}$) around Z-mass window ($|M_{l_{1},l_{2}}-M_{Z}|<15.0$ GeV) for
opposite sign dileptons of same flavour in trilepton events. This serves to
suppress contribution from $Zb\bar{b}+jets$ and $WZ+jets$ background to
trileptons.
Due to the Majorana nature of neutralinos, $\lambda^{\prime\prime}$-type
interactions result in equal rates for $tds$ and $\bar{t}\bar{d}\bar{s}$-type
final states. Therefore, the most promising signals are those involving same-
sign dileptons (SSD). This not only applies to $\tilde{\chi}_{1}^{0}$ but also
to the higher neutralinos produced in stop decay, whose cascades can give rise
to W’s. SSD have previously been used extensively for studying signals of
supersymmetry [31, 32] . The most copious backgrounds to SSD processes come
from the processes $t\bar{t}$ and $Wb\bar{b}$ due to one lepton from $W$ and
another from semileptonic decays of the b-quark. There is also a potentially
large contribution from $b\bar{b}$ due to $B^{0}-\bar{B}^{0}$ oscillations
along with semileptonic decays of both B-mesons. The effect of oscillations is
simulated in the Pythia program. The $p_{T}$-cuts on leptons have been
selected to minimise the background from heavy flavour decays [33]. We find
that after the isolation and $p_{T}$ cuts on leptons, $Wb\bar{b}$ and
$b\bar{b}$ cross sections fall to sub-femtobarn levels.
We simulate the $t\bar{t}+jets$ background up to two jets. The trilepton
channel has another source of backgrounds in $WZ+jets$; however, we have
checked and found them to be negligibly small after applying all the cuts. We
also generate $Wt\bar{t}+jets$ and $Zt\bar{t}+jets$ up to one jet.
It should be mentioned here that the dilepton and trilepton final states can
also arise in the same scenario from the pair-production of superparticles.
These include, for example, pair production of gluinos and electroweak
production of chargino-neutralino pairs. Such contributions have been
explicitly shown in the plots in section 4.
We also expect that the $p_{T}$ distribution of the $\tilde{t}_{1}$ becomes
significantly harder if the NLO corrections are taken into account[12]. Our
cuts on leptons have been designed to cut off the background from semileptonic
b-decays by requiring the $p_{T}$ to be about half the mass of the $W$.
Therefore, if only the lepton cuts are used, we do not expect a large change
in the efficiency of the cuts quoted in the next section.
## 4 Results
We present results using $\lambda^{\prime\prime}_{312}=0.2$; the predictions
for other values of this coupling can be obtained through scaling arguments.
If $\lambda^{\prime\prime}_{312}$ is scaled by a factor of $n$ then the
production cross section as well as the decay width of the RPV channel scale
by $n^{2}$. All other decay widths remain unchanged. If $f$ is the branching
fraction of $\tilde{t}\rightarrow\bar{d}\bar{s}$ before scaling, then the
signal rates in any other channel are scaled by a factor of
$\frac{R_{new}}{R_{old}}=\frac{n^{2}}{(n^{2}-1)f+1}$ (5)
### 4.1 Limits at $\sqrt{s}=$14, 10 TeV
The numerical results for various signals corresponding to the five benchmark
points for LHC running energy $\sqrt{s}=14,10$ TeV are presented in Tables
2(for the $SSD$ channel), 3 (for $SSD+b$) and 4 (for $3l$).
$SSD$ | | 14 TeV | | | | 10 TeV | |
---|---|---|---|---|---|---|---|---
Point | Cut 0 | Cut 1 | Cut 2 | Cut 3 | Cut 0 | Cut 1 | Cut 2 | Cut 3
A | 884.8 | 496.8 | 459.4 | 41.0 | 540.1 | 312.7 | 287.0 | 15.1
B | 64.7 | 43.7 | 41.4 | 19.3 | 30.6 | 21.0 | 19.8 | 9.6
C | 83.0 | 51.5 | 49.2 | 25.8 | 40.1 | 25.6 | 24.6 | 12.5
D | 145.4 | 71.9 | 68.9 | 41.1 | 65.1 | 32.3 | 31.0 | 19.0
E | 29.8 | 16.5 | 15.9 | 13.6 | 10.7 | 5.8 | 5.6 | 4.6
$t\bar{t}+nj$ | 687.9 | 26.3 | 24.7 | 10.0 | 307.0 | 8.7 | 7.0 | 3.6
$Wt\bar{t}+nj$ | 17.0 | 9.2 | 8.7 | 5.2 | 7.6 | 3.9 | 3.7 | 2.0
$Zt\bar{t}+nj$ | 12.7 | 6.7 | 6.7 | 4.1 | 4.9 | 2.3 | 2.2 | 1.4
Total | 717.6 | 42.2 | 40.1 | 19.3 | 319.5 | 14.9 | 12.9 | 7.0
Table 2: Effect of cuts on signal and SM background cross sections (in $fb$) in the $SSD$ channel at $\sqrt{s}=14,10$ TeV. Cut 0 refers to all events passing the identification cuts. Cuts 1-3 are described in the text. The numbers corresponding to best significance ($s/\sqrt{b}$) of the signal ($s$) with respect to the background ($b$) are highlighted in bold. $SSD+b$ | | 14 TeV | | | | 10 TeV | |
---|---|---|---|---|---|---|---|---
Point | Cut 0 | Cut 1 | Cut 2 | Cut 3 | Cut 0 | Cut 1 | Cut 2 | Cut 3
A | 243.2 | 134.7 | 121.2 | 14.1 | 158.7 | 67.5 | 61.8 | 6.6
B | 13.8 | 9.6 | 9.2 | 4.7 | 8.3 | 6.0 | 5.6 | 2.8
C | 25.3 | 15.2 | 14.6 | 8.0 | 12.7 | 7.8 | 7.4 | 4.1
D | 47.9 | 23.9 | 23.0 | 15.2 | 21.2 | 9.9 | 9.5 | 6.3
E | 11.3 | 6.2 | 6.0 | 5.3 | 4.1 | 2.2 | 2.1 | 1.8
$t\bar{t}+nj$ | 173.0 | 7.6 | 7.1 | 2.3 | 80.9 | 4.2 | 1.4 | 1.3
$Wt\bar{t}+nj$ | 6.7 | 0.8 | 0.8 | 0.6 | 4.0 | 2.1 | 1.9 | 1.3
$Zt\bar{t}+nj$ | 5.6 | 2.6 | 2.6 | 1.8 | 2.3 | 1.1 | 1.1 | 0.7
Total | 185.3 | 11.0 | 10.5 | 4.7 | 87.2 | 7.4 | 4.4 | 3.3
Table 3: Same as Table 2, but for the $SSD+b$ channel. $3l$ | | | 14 TeV | | | | 10 TeV |
---|---|---|---|---|---|---|---|---
Point | Cut 0 | Cut 1 | Cut 2 | Cut 4 | Cut 0 | Cut 1 | Cut 2 | Cut 4
A | 49.1 | 2.8 | 2.8 | 0.0 | 18.0 | 1.1 | 1.1 | 0.0
B | 2.0 | 0.6 | 0.6 | 0.1 | 1.1 | 0.2 | 0.2 | 0.0
C | 13.7 | 9.1 | 9.1 | 1.1 | 6.5 | 3.7 | 3.7 | 0.6
D | 48.2 | 29.6 | 29.0 | 8.6 | 24.1 | 14.5 | 14.1 | 4.5
E | 9.3 | 5.7 | 5.5 | 3.0 | 3.4 | 2.1 | 2.1 | 1.2
$t\bar{t}+nj$ | 2.1 | 0.0 | 0.0 | 0.0 | 1.8 | 0.0 | 0.0 | 0.0
$Wt\bar{t}+nj$ | 4.1 | 2.5 | 2.4 | 1.0 | 2.2 | 1.4 | 1.4 | 0.6
$Zt\bar{t}+nj$ | 30.8 | 20.7 | 19.7 | 2.7 | 11.3 | 7.3 | 4.7 | 1.1
Total | 37.0 | 23.2 | 22.1 | 3.7 | 15.3 | 8.7 | 6.1 | 1.7
Table 4: Effect of cuts on signal and SM background cross sections (in $fb$)
in the trilepton ($3l$) channel at $\sqrt{s}=14,10$ TeV. Cut 0 refers to all
events passing the identification cuts. Cuts 1, 2 and 4 are described in the
text. Cut 4 is necessary to eliminate background from $WZ+jets$.
We can make the following observations from the numerical results:
* •
The final states $SSD$ and $SSD+b$ consistently have substantial event rates
at both 14 and 10 TeV. Furthermore, the simultaneous observation of excesses
in the SSD and SSD +b channel can serve as definite pointer to the production
of a third generation squark.
* •
For point A, which is just above the Tevatron reach, we can achieve more than
$5\sigma$ significance in the $SSD$ channel with just 100 pb-1 data at both 14
and 10 TeV. For point E, which has $M_{\tilde{t}}=1500$ GeV, we can reach
$3\sigma$ with 1(3) $fb^{-1}$ and $5\sigma$ with 3(9) $fb^{-1}$ at 14(10) TeV.
Therefore, we can conclude that the entire range from 500-1500 GeV can be
successfully probed at the LHC for $\lambda^{\prime\prime}_{312}=0.2$.
* •
Stops decaying into higher neutralinos and charginos make the total rates
distinctly better. This is governed by Higgsino couplings and is therefore
most prominent for high $\tan\beta$ and low $A_{0}$. This effect is evident
from the large event rates for point D. We can successfully probe this point
in the SSD channel at $5\sigma$ with less than 1 $fb^{-1}$ data at both 10 and
14 TeV runs.
* •
The trilepton final state occurs when the stop can decay into $\chi_{2}^{+}$,
$\chi_{3,4}^{0}$ or $\tilde{g}$. Therefore, points A and B show almost no
signal and Point D has the largest signal in this channel. This advantage is
largely lost for benchmark point E due to the kinematic suppression in the
stop production process.
* •
Reach for the LHC: Assuming the conservative case of
($\tan\beta=5,~{}A_{0}=0$), with 10 $fb^{-1}$ luminosity, one can rule out
$\lambda_{eff}^{{}^{\prime\prime}}$ greater than 0.007–0.045 (0.007–0.062) for
stop masses between 500 and 1500 GeV at 95 % CL at $\sqrt{s}=$ 14 (10) TeV. A
$5\sigma$ discovery can be made in the same mass range for
$\lambda^{\prime\prime}_{eff}$ greater than 0.012–0.084 (0.012–0.12). However,
we observe that the reach in stop mass does not decrease monotonously with
stop mass. The opening of new decay channels can improve detection
considerably. The statements about minimum value of
$\lambda_{eff}^{{}^{\prime\prime}}$ that can be probed are therefore dependent
on the particular decays of the stop. We therefore tabulate the minimum values
of $\lambda_{eff}^{{}^{\prime\prime}}$ for each benchmark point at 10
$fb^{-1}$ for both 10 and 14 TeV in Table 5.
Point | | 14 TeV | | | 10 TeV |
---|---|---|---|---|---|---
| 95% CL | $3\sigma$ | $5\sigma$ | 95 % CL | $3\sigma$ | $5\sigma$
A | 0.007 | 0.009 | 0.012 | 0.007 | 0.009 | 0.012
B | 0.027 | 0.037 | 0.052 | 0.029 | 0.041 | 0.059
C | 0.026 | 0.035 | 0.048 | 0.028 | 0.038 | 0.052
D | 0.024 | 0.032 | 0.042 | 0.027 | 0.036 | 0.047
E | 0.045 | 0.062 | 0.084 | 0.062 | 0.087 | 0.12
Table 5: Values of minimum $\lambda_{eff}^{\prime\prime}$ that can be ruled
out at 95% CL, probed at $3\sigma$ or $5\sigma$ with 10 $fb^{-1}$ of data at
$\sqrt{s}=10,14$ for each of the benchmark points. The significance used is
$s/\sqrt{b}$ where $s$ is the signal and $b$ is the background. $s/b>0.2$ in
all cases.
In Figure 3, we present the effective mass distributions in $SSD$ channel for
all the benchmark points. Effective mass is defined as
$M_{eff}=\sum_{jets}|\vec{p}_{T}|+\sum_{leptons}|\vec{p}_{T}|+E_{T}/~{}~{}$
(6)
The contributions from resonant stop production is superposed in the figures
on the SM backgrounds and also RPC superparticle production processes. The RPC
contributions are much smaller and therefore do not provide a serious
background to our signals.
Figure 3: $M_{eff}$ distributions at $\sqrt{s}=14$ TeV. “SM” is the
contribution to the background from Standard Model processes. “SUSY” refers to
the contribution from R-conserving production processes. The inset in each
figure contains the distribution for the signal alone.
### 4.2 Observability at the early run of 7 TeV
The initial LHC run at $\sqrt{s}=$ 7 TeV will collect up to
$1~{}\mathrm{fb}^{-1}$ data. It will be difficult to observe RPV production of
a 1 TeV stop at this energy. However, we can make useful comments for lower
stop masses by looking at the $SSD$ channel. We therefore look two benchmark
points with low stop masses: the first is the ‘Point A’ described earlier and
the second is similar to ‘Point D’ (with $\tan\beta=10$ and $A_{0}=0$). Since
the $t\bar{t}$ backgrounds are much smaller at 7 TeV, we relax the jet-$p_{T}$
cuts. The high scale parameters, stop mass at electroweak scale and cut-flow
table for signal as well as background are given in Table 6.
We conclude that we can rule out up to $\lambda^{\prime\prime}_{eff}=0.025$
for a stop mass of 500 GeV at 7 TeV with $1~{}\mathrm{fb}^{-1}$ data and a
$5\sigma$ discovery can be made at stop mass 500 GeV for
$\lambda^{\prime\prime}_{eff}\geq 0.043$. For the case $\tan\beta=10$ and
$A_{0}=0$, the lowest possible theoretically allowed stop mass (with
$m_{1/2}=450$ GeV) is 775 GeV and we can rule out up to
$\lambda^{\prime\prime}_{eff}=0.054$ with $1~{}\mathrm{fb}^{-1}$ data.
Point | $(\tan\beta,A_{0},m_{0})$ | $M_{\tilde{t}_{1}}$ | Cut 0 | Cut 1 | Cut 2
---|---|---|---|---|---
$A$ | $(5,-1500,600)$ | 508 | 283.3 | 158.6 | 147.5
$D^{\prime}$ | $(10,0,100)$ | 775 | 70.0 | 33.9 | 32.0
$t\bar{t}+nj$ | | | 116.0 | 3.7 | 3.5
$Wt\bar{t}+nj$ | | | 4.3 | 2.3 | 2.1
$Zt\bar{t}+nj$ | | | 1.8 | 0.9 | 0.8
Total | | | 122.1 | 6.9 | 6.4
Table 6: The benchmark points for studying RPV stop production and the effect
of cuts on signal and SM background cross sections (in $fb$) in the $SSD$
channel at $\sqrt{s}=7$ TeV. Cut 0 refers to all events passing the
identification cuts. All other cuts are described in the text, we do not apply
Cut 3.
### 4.3 Differentiating from R-conserving signals
We now address the question whether the signals we suggest can be faked by an
R-parity conserving scenario in some other region(s) of the parameter space.
One possible way that our signal may be mimicked is if a point in the mSUGRA
parameter space (without RPV) gives similar kinematic distributions to any our
benchmark points. More specifically, one may have a peak in the same region
for the variable $M_{eff}$, defined in equation 6.
For each of our benchmark points A-D, an $M_{eff}$ peak in the same region
requires the strongly interacting sparticles to have masses in the range
already ruled out by the Tevatron data[34]. In particular, they require the
gluino mass $M_{\tilde{g}}<390$ GeV. Thus the question of faking arises only
for benchmark point E, which represents the highest mass where the signals
rates are appreciable.
We generate such a point (Point RC) with the parameters $m_{0}=300$,
$m_{1/2}=180$, $A_{0}=0$, $\tan\beta=10$, $\mu>0$ and the resultant sparticle
masses for coloured particles are $M_{\tilde{g}}=465$, $M_{\tilde{q}}\sim 500$
GeV. The $M_{eff}$ distributions for point E and point RC is shown in Figure
4. We present the following results at 14 TeV as an illustration.
Distributions at 10 TeV are almost identical.
The missing $E_{T}$ distribution is also not a good discriminator under such
circumstances, as can be seen from Figure 4. This is because the neutrinos
that contribute to missing-$E_{T}$ in the RPV case are highly boosted due to
the large masses of the particles produced in the initial hard scattering.
Thus, the $E_{T}$ spectrum is actually harder for the RPV case even though the
RPC case has a stable massive LSP. However, as the resultant spectrum is quite
light, the RPC production cross section ($\sim$ 40 pb) is about two orders of
magnitude greater than the RPV case with $\lambda^{\prime\prime}_{312}=0.2$
($\sim$480 fb). Consequently, the rate of the SSD signals, for example, are
much higher in the R-conserving scenario ($\sim$ 34 fb) as compared to those
from point E ($\sim$ 14 fb). For values of $\lambda^{\prime\prime}_{312}<0.2$,
we can therefore make a reliable distinction simply based on the number of
events expected in the SSD channel.
Another possible discriminator is the charge asymmetry. In the $SSD$ channel,
one can look at the ratio of negative to positive $SSD$
$\frac{N^{--}}{N^{++}}$. The fraction of $ds\rightarrow\tilde{t}_{1}^{*}$ is
more than the charge conjugate process
$\bar{d}\bar{s}\rightarrow\tilde{t}_{1}$ due to the difference in parton
distributions of $d$ and $\bar{d}$ in the proton. Therefore, one expects extra
negative sign leptons than positive ones. Whereas in the RPC case, since most
of the $SSD$ contribution comes from $\tilde{g}\tilde{g}$ production, we do
not expect a large asymmetry. In our illustration, we see that this ratio is
2.7 (1.4) for the RPV (RPC) case.
Figure 4: The normalised effective mass ($M_{eff}$) and missing energy
($E_{T}/~{}~{}$) distributions in the $SSD$ channel for Point E and an Point
RC. Point RC has been generated using
$m_{0}=300,m_{1/2}=180;A_{0}=0,tan\beta=10$ and $\mu>0$. The gluino mass is
465 GeV.
## 5 Non-$\tilde{\chi}_{1}^{0}$ LSPs
For RPV models, the restriction of having an uncharged LSP no longer exits. A
significant region of the mSUGRA parameter space with low $m_{0}$ corresponds
to a stau ($\tilde{\tau}$) LSP. With only $\lambda^{\prime\prime}_{312}$-type
couplings present, the stau can only decay via off-shell
$\tilde{\chi}_{1}^{0}$ and $\tilde{t}$ propagators into the four body decay
($\tilde{\tau}\rightarrow\tau tds$) if its mass, $m_{\tilde{\tau}}>m_{top}$ or
via the five body decay ($\tilde{\tau}\rightarrow\tau bWds$) if
$m_{\tilde{\tau}}<m_{top}$ where the top propagator is also off-shell. The
four-body decays of the stau in lepton-number violating scenarios was
calculated in [35]. Since the intermediate $\tilde{\chi}_{1}^{0}$ is of
Majorana character, we can always have one lepton of either sign from LSP
decay via the $W$ from an on-shell or off-shell top. Thus, for various types
of $\tilde{t}$ decays, the following situations may arise:
* •
For decays of $t\tilde{\chi}_{i}^{0}$-type, we can still have same-sign
dileptons with one lepton from top decay and the other from the decay of the
LSP.
* •
For decays of type $b\tilde{\chi}_{i}^{\pm}$ with $\chi_{i}^{\pm}\rightarrow
W^{\pm}\tilde{\chi}_{1}^{0}+X$, we have
$\tilde{\chi}_{1}^{0}\rightarrow\tau\tilde{\tau}$ and the SSD come from
$W^{\pm}$ and LSP decay respectively.
* •
For decays of the type $\tilde{t}\rightarrow b\tilde{\chi}_{i}^{+}$ with
$\tilde{\chi}_{i}^{+}\rightarrow\nu_{\tau}\tilde{\tau}+X$, we may still get
SSD from leptonic decay of the $\tau$ in the $\tilde{\tau}$ decay. If
$\tau$-identification is used, final states of the type same-sign
$(\tau+e/\mu)$ may be considered.
* •
Since the stau has to decay via four- or five-body processes, it is possible
that the lifetime of the $\tilde{\tau}$ is large and it is stable over the
length scale of the detector. In this case, it will leave a charged track like
a muon and one can look at same-sign leptons with this “muon” as one of the
leptons. It is also possible that the lifetime is large but the stau still
decays within the detector. In this case, a displaced vertex can be observed
in the detector.
We leave the detailed simulation of all scenarios of resonant stop production
with stau LSP to a future study.
Another possibility that arises with a large $\lambda^{\prime\prime}_{312}$ is
of having a stop LSP. In this case however, the decay will be almost entirely
via the RPV $\bar{d}\bar{s}$ di-jet channel. The overwhelmingly large dijet
backgrounds at the LHC would most likely make this situation unobservable.
## 6 Summary and Conclusion
We have performed a detailed analysis of resonant stop production at the LHC,
both for the 10 and 14 TeV runs, for values of the baryon number violating
coupling $\lambda^{\prime\prime}_{312}$ an order of magnitude below the
current experimental limit. Benchmark points have been chosen for this
purpose, which start just beyond the reach of the Tevatron and end close at
the LHC search limit. We find that the same-sign dilepton final states, both
with and without a tagged b, are most helpful in identifying the signal. The
trilepton signals can also be sometimes useful, especially when decays of the
resonant into higher neutralinos, the heavier chargino or the gluino open up.
At 14(10) TeV, we can probe stop masses up to 1500 GeV and values of
$\lambda^{\prime\prime}_{eff}$ down to 0.05(0.06) depending on the combination
of various SUSY parameters. For cases of stop mass below a TeV, the effective
mass distributions can enable us to distinguish between the resonant process
and contributions from R-parity conserving SUSY processes. For higher stop
masses, one has to rely on cross sections or the charge asymmetry.
We have used a particular B-violating coupling, namely,
$\lambda^{\prime\prime}_{312}$. One can also have resonant stop production
driven by $\lambda^{\prime\prime}_{313}$ and $\lambda^{\prime\prime}_{323}$.
In either of these cases, one expects a larger abundance of b quarks in the
final state. However, there is a suppression in the production rates due to
the b-distribution function in the proton.
In conclusion, resonant stop production is a potentially interesting channel
to look for SUSY in its baryon-number violating incarnation. Values of the
B-violating coupling(s) more than an order below the current experimental
limits can be definitely probed at the LHC, both at 10 and 14 TeV. If such
interactions really exist, our suggested strategy can not only yield
detectable event rates but also point towards resonant production as opposed
to pair-production of SUSY particles.
## Acknowledgements
We would like to thank Satyaki Bhattacharya, Gobinda Majumdar, Bruce Mellado
and Satyanarayan Mukhopadhyay for helpful discussions and comments. ND thanks
the Indian Association for the Cultivation of Science for hospitality while
part of this work was carried out. This work was partially supported by
funding available from the Department of Atomic Energy, Government of India
for the Regional Centre for Accelerator-based Particle Physics, Harish-Chandra
Research Institute. Computational work for this study was partially carried
out at the cluster computing facility of Harish-Chandra Research Institute
(http://cluster.mri.ernet.in).
## References
* [1] H. P. Nilles, Supersymmetry, Supergravity and Particle Physics, Phys. Rept. 110 (1984) 1–162.
* [2] G. L. Kane, (ed. ), Perspectives on supersymmetry. World Scientific, Singapore, 1998.
* [3] R. Barbier et al., R-parity violating supersymmetry, Phys. Rept. 420 (2005) 1–202, [hep-ph/0406039].
* [4] F. Zwirner, Observable Delta B=2 Transitions Without Nucleon Decay in a Minimal Supersymmetric Extension of the Standard Model, Phys. Lett. B132 (1983) 103–106.
* [5] H. K. Dreiner and G. G. Ross, R-parity violation at hadron colliders, Nucl. Phys. B365 (1991) 597–613.
* [6] R. Barbieri and A. Masiero, Supersymmetric Models with Low-Energy Baryon Number Violation, Nucl. Phys. B267 (1986) 679.
* [7] G. Bhattacharyya, D. Choudhury, and K. Sridhar, New LEP bounds on $B$ violating scalar couplings: R-parity violating supersymmetry or diquarks, Phys. Lett. B355 (1995) 193–198, [hep-ph/9504314].
* [8] K. Agashe and M. Graesser, R-parity violation in flavor changing neutral current processes and top quark decays, Phys. Rev. D54 (1996) 4445–4452, [hep-ph/9510439].
* [9] B. C. Allanach, A. Dedes, and H. K. Dreiner, Bounds on R-parity violating couplings at the weak scale and at the GUT scale, Phys. Rev. D60 (1999) 075014, [hep-ph/9906209].
* [10] E. L. Berger, B. W. Harris, and Z. Sullivan, Single-top-squark production via R-parity-violating supersymmetric couplings in hadron collisions, Phys. Rev. Lett. 83 (1999) 4472–4475, [hep-ph/9903549].
* [11] E. L. Berger, B. W. Harris, and Z. Sullivan, Direct probes of R-parity violating supersymmetric couplings via single top squark production, Phys. Rev. D63 (2001) 115001, [hep-ph/0012184].
* [12] T. Plehn, Single stop production at hadron colliders, Phys. Lett. B488 (2000) 359–366, [hep-ph/0006182].
* [13] H. Baer, C. Kao, and X. Tata, Impact of R-parity violation on supersymmetry searches at the tevatron, Phys. Rev. D51 (1995) 2180–2186, [hep-ph/9410283].
* [14] B. C. Allanach et al., Measuring supersymmetric particle masses at the LHC in scenarios with baryon-number R-parity violating couplings, JHEP 03 (2001) 048, [hep-ph/0102173].
* [15] B. C. Allanach, A. J. Barr, M. A. Parker, P. Richardson, and B. R. Webber, Extracting the flavour structure of a baryon-number R- parity violating coupling at the LHC, JHEP 09 (2001) 021, [hep-ph/0106304].
* [16] H. K. Dreiner and S. Grab, All Possible Lightest Supersymmetric Particles in R-Parity Violating mSUGRA, Phys. Lett. B679 (2009) 45–50, [arXiv:0811.0200].
* [17] J. M. Butterworth, J. R. Ellis, A. R. Raklev, and G. P. Salam, Discovering baryon-number violating neutralino decays at the LHC, Phys. Rev. Lett. 103 (2009) 241803, [arXiv:0906.0728].
* [18] T. Plehn, M. Spannowsky, M. Takeuchi, and D. Zerwas, Stop Reconstruction with Tagged Tops, [arXiv:1006.2833].
* [19] G. L. Kane, C. F. Kolda, L. Roszkowski and J. D. Wells, Study of constrained minimal supersymmetry, Phys. Rev. D49 (1994) 6173-6210. [hep-ph/9310291].
* [20] R parity Working Group Collaboration, B. Allanach et al., Searching for R parity violation at Run II of the Tevatron, hep-ph/9906224.
* [21] The ATLAS Collaboration, G. Aad et al., Expected Performance of the ATLAS Experiment - Detector, Trigger and Physics, arXiv:0901.0512.
* [22] P. Nath, R. L. Arnowitt, and A. H. Chamseddine, Applied N=1 Supergravity. World Scientific, Singapore, 1984.
* [23] U. Chattopadhyay and P. Nath, Upper limits on sparticle masses from g-2 and the possibility for discovery of SUSY at colliders and in dark matter searches, Phys. Rev. Lett. 86 (2001) 5854–5857, [hep-ph/0102157].
* [24] B. C. Allanach and M. A. Bernhardt, Including R-parity violation in the numerical computation of the spectrum of the minimal supersymmetric standard model: SOFTSUSY3.0, Comput. Phys. Commun. 181 (2010) 232–245, [arXiv:0903.1805].
* [25] F. E. Paige, S. D. Protopopescu, H. Baer, and X. Tata, ISAJET 7.69: A Monte Carlo event generator for p p, anti-p p, and e+ e- reactions, hep-ph/0312045.
* [26] G. Corcella et al., HERWIG 6.5 release note, hep-ph/0210213.
* [27] M. Cacciari, G. P. Salam, and G. Soyez, The anti-$k_{t}$ jet clustering algorithm, JHEP 04 (2008) 063, [arXiv:0802.1189].
* [28] M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau, and A. D. Polosa, ALPGEN, a generator for hard multiparton processes in hadronic collisions, JHEP 07 (2003) 001, [hep-ph/0206293].
* [29] T. Sjostrand, S. Mrenna, and P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026, [hep-ph/0603175].
* [30] J. Pumplin et al., New generation of parton distributions with uncertainties from global QCD analysis, JHEP 07 (2002) 012, [hep-ph/0201195].
* [31] R. M. Barnett, J. F. Gunion, and H. E. Haber, Discovering supersymmetry with like sign dileptons, Phys. Lett. B315 (1993) 349–354, [hep-ph/9306204].
* [32] H. K. Dreiner, M. Guchait, D. P. Roy, Like sign dilepton signature for gluino production at CERN LHC with or without R conservation, Phys. Rev. D49 (1994) 3270-3282. [hep-ph/9310291].
* [33] Z. Sullivan and E. L. Berger, Isolated leptons from heavy flavor decays: Theory and data, Phys. Rev. D82 (2010) 014001, [arXiv:1003.4997].
* [34] Particle Data Group Collaboration, C. Amsler et al., Review of particle physics, Phys. Lett. B667 (2008) 1.
* [35] H. K. Dreiner, S. Grab, and M. K. Trenkel, Stau LSP Phenomenology: Two versus Four-Body Decay Modes. Example: Resonant Single Slepton Production at the LHC, Phys. Rev. D79 (2009) 016002, [arXiv:0808.3079].
|
arxiv-papers
| 2010-02-11T13:22:12 |
2024-09-04T02:49:08.357137
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nishita Desai and Biswarup Mukhopadhyaya",
"submitter": "Nishita Desai",
"url": "https://arxiv.org/abs/1002.2339"
}
|
1002.2401
|
# The Progress of Solar Cycle 24 at High Latitudes
Richard C. Altrock
###### Abstract
The “extended” solar cycle 24 began in 1999 near 70∘ latitude, similarly to
cycle 23 in 1989 and cycle 22 in 1979. The extended cycle is manifested by
persistent Fe XIV coronal emission appearing near 70∘ latitude and slowly
migrating towards the equator, merging with the latitudes of sunspots and
active regions (the “butterfly diagram”) after several years. Cycle 24 began
its migration at a rate 40% slower than the previous two solar cycles, thus
indicating the possibility of a peculiar cycle. However, the onset of the
“Rush to the Poles” of polar crown prominences and their associated coronal
emission, which has been a precursor to solar maximum in recent cycles (cf.
Altrock 2003), has just been identified in the northern hemisphere.
Peculiarly, this “Rush” is leisurely, at only 50% of the rate in the previous
two cycles. The properties of the current “Rush to the Poles” yields an
estimate of 2013 or 2014 for solar maximum.
Air Force Research Laboratory, NSO/SP, PO Box 62, Sunspot, NM 88349, USA
## 1\. Introduction
Altrock (1997) and earlier authors (cf. Wilson et al. 1988) discussed the
high-latitude “extended” solar cycle seen in the Fe XIV corona prior to the
appearance of sunspots and active regions at lower latitudes. For example,
persistent coronal emission appeared near 70∘ latitude in 1979 and 1989 and
slowly migrated towards the equator, merging with the latitudes of sunspots
and active regions after several years. Wilson et al. (1988) discussed other
observational parameters that have similar properties, and this was updated by
Altrock, Howe and Ulrich (2008) for torsional oscillations.
Altrock (2007) showed that the high-latitude coronal emission was situated
above the high-latitude neutral line of the large-scale photospheric magnetic
field seen in Wilcox Solar Observatory synoptic maps, thus implying a
connection with the solar dynamo.
Altrock (2003) discussed coronal emission features seen in Fe XIV which, prior
to solar maximum in cycles 21 - 23, appeared above 50∘ latitude and began to
move towards the poles at a rate of 8 to 11 ∘$yr^{-1}$. This motion was
maintained for a period of 3 or 4 years, at which time the emission features
disappeared near the poles. This phenomenon has been referred to as the “Rush
to the Poles” (RttP). It was first identified in solar-crown prominences, and
it was first observed in the corona by Waldmeier (1964). Altrock concluded
that (i) the maximum of solar activity, as defined by the smoothed sunspot
number, occurred 1.5 $\pm$ 0.2 yr before the extrapolated linear fit to the
RttP reached the poles, and (ii) the RttP could be used to predict the date of
solar-cycle maximum up to three years prior to its occurrence. He stated that,
“For solar cycle 24, a prediction of the date of solar maximum can be made
when the RttP becomes apparent, approximately eleven years after its cycle-23
onset on 1997.58, or 2008 - 2009\. When that occurs, the average slope for
cycles 21 - 23, 9.38 $\pm$ 1.71 ∘$yr^{-1}$, can be used to predict the arrival
date of the RttP at the poles, and then the average lag [time between solar
maximum and the date the extrapolated linear fit to the RttP reached the
poles], 1.52 $\pm$ 0.20 yr, can be used to predict the date of solar maximum
…”
## 2\. Observations
Figure 1.: Sample polar plot of Fe XIV intensity at 1.15 Ro at solar minimum.
Intensity is zero at outer circle.
Observations of the Fe XIV 530.3 nm solar corona have been attempted three to
seven times a week since 1973 with the photoelectric coronal photometer and
40-cm coronagraph at the John W. Evans Solar Facility of the National Solar
Observatory at Sacramento Peak (Fisher 1973 and 1974; Smartt 1982). The
photometer automatically removes the highly-variable sky background. Scans at
0.15 solar radii ($R_{\odot}$) above the limb every 3∘ in position angle show
coronal features overlying active regions, prominences, large-scale magnetic
field boundaries, etc. Observations near solar minimum continue to show
coronal emission overlying high-latitude neutral lines even when there are no
active regions at the limb. Figure 1 shows a sample solar-minimum scan. Note
(i) a lack of low-latitude active region emission and (ii) emission occurring
at higher latitudes.
Figure 2.: Annual northern plus southern hemisphere averages of the number of
Fe XIV intensity maxima from 1973 through 2009. The “Rush to the Poles” around
2000 is indicated, as well as the extended solar cycle 24, beginning in
approximately 1999.
## 3\. Procedure
As discussed in Altrock (1997), the daily scans of the corona in Fe XIV at
1.15 $R_{\odot}$ are examined to determine the location in latitude of local
intensity maxima, and each maximum is plotted on a synoptic map of latitude
vs. time. Altrock (1997) Figure 3 shows such a synoptic map from 1973 to 1996.
Note that nowhere in this analysis is the value of the intensity used, and
that allows tracking of very faint features. To clarify the solar-cycle
behavior of the intensity maxima, the number of points at each latitude in the
synoptic map is averaged over a given time interval. This process allows us to
correct the figure for days of missing data, which is an important step in
order to correctly interpret the data. Figure 2 shows annual averages of the
number of intensity maxima, also averaged over the north and south
hemispheres.
## 4\. Discussion
In Figure 2 we can clearly see the nature of extended solar cycles and RttP
over the last 30+ years. Extended solar cycles begin near 70∘ latitude and end
near the equator about 18 years later, as can be seen in cycles 22 and 23.
Note that cycle 24 began similarly to cycles 22 and 23; however, it has been
migrating equatorward more slowly. The rates for cycles 22 - 24 have been
-5.3, -4.7 and -3.1 ∘$yr^{-1}$, respectively. Most recently, emission took a
sudden jump down to around 30∘ , and there is an indication that the RttP
could be developing. This suggests to use a higher-resolution (if noisier)
graphic.
Figure 3.: Seven-rotation (approximately semiannual) averages of the number
of Fe XIV emission maxima from 1973 through 2009. Note the cycle 24 “Rush to
the Poles” indicated by the label and linear fit in the upper right corner.
Figure 3 shows the data with a temporal average of 189 days, or seven 27-day
rotation periods (approximately semiannual). In addition we now examine both
hemispheres independently. Here we see that the extended cycle 24 (the
equatorward-moving emission) appears to have recently split into two branches,
most easily seen in the northern hemisphere. This has occurred previously,
notably in the northern hemisphere in approximately 1991, after which the
lower latitude branch eventually disappeared. So it is difficult to say to
what the current split may lead.
The more interesting development is the appearance in 2005 of the RttP in the
northern hemisphere, marked by a label, “Rush to the Poles”, and a linear fit,
both seen in the upper right-hand corner of the graph. No such feature is yet
evident in the southern hemisphere, which only may mean it is not yet visible
in these noisy data or that it is delayed. In any case, we can use the
northern hemisphere data as an indicator of when solar maximum will occur. The
current RttP rate is estimated to be 4.6∘$yr^{-1}$ (recall 9.4 $\pm$ 1.7
∘$yr^{-1}$ average in the previous three cycles [Altrock 2003]). This 50%
lower rate makes the earlier suggestion to use the previous higher rate to
estimate the time of cycle maximum invalid (see discussion in Introduction).
At the current rate, the extrapolated RttP will reach the north pole at
2016.3. If we apply the previously-determined 1.5 $\pm$ 0.2 yr offset between
solar maximum and arrival at the poles (see Introduction), this would imply
solar maximum at 2014.8 $\pm$ 0.2. However, using that offset could be
somewhat dubious, considering the slow “rush” this cycle.
A method that is possibly more reliable is to use the property that solar
maximum occurs when the center line of the RttP reaches a critical latitude.
In the previous three cycles this latitude was 76∘, 74∘ and 78∘, for an
average of 76∘ $\pm$ 2∘ [this can be determined from the figures in Altrock
(2003)]. At the current rate, this will occur at 2013.3 $\pm$ 0.5. If the RttP
rate increases, solar maximum would be earlier, although there is no reason to
believe that this will occur.
Thus, the two methods using the coronal “rush to the poles” result in
predictions for solar maximum at 2013.3 $\pm$ 0.5 and 2014.8 $\pm$ 0.5, or
2013-2014.
Nothing in this analysis yields the sunspot number to be expected at solar
maximum.
## 5\. Conclusions
The location of Fe XIV intensity maxima in time-latitude space displays an
18-year progression from near 70∘ to the equator, which has been referred to
as the “extended” solar cycle. Cycle 24 emission began proceeding towards the
equator similarly to previous cycles, although at a 40% slower rate. In
addition, in 2009 the northern hemisphere “Rush to the Poles” became evident
and is proceeding at a 50% slower rate than in recent cycles. Both of these
facts indicate that cycle 24 is peculiar. Analysis of the “Rush to the Poles”
indicates that solar maximum will occur in 2013 or 2014, but there is no
indication of the strength of the maximum. There is at this time no
confirmation of this prediction from the southern hemisphere.
### Acknowledgments.
The observations used herein are the result of a cooperative program of the
Air Force Research Laboratory and the National Solar Observatory. I am
grateful for the assistance of NSO personnel, especially John Cornett, Timothy
Henry, Lou Gilliam and Wayne Jones, for observing and data-reduction and
analysis services and maintenance of the Evans Solar Facility and its
instrumentation.
## References
* Altrock (1997) Altrock, Richard C. 1997, Solar Phys., 170, 411
* Altrock (2003) Altrock, Richard C. 2003, Solar Phys., 216, 343
* Altrock (2007) Altrock, Richard C. 2007, Eos, Trans. Am. Geophys. Union, 88(52), Fall Meeting Suppl., Abstract SH53A-1052
* Altrock et al. (2008) Altrock, R. C., Howe, R., & Ulrich, R. 2008, in Astronomical Society of the Pacific Conference Series Vol. 383, Subsurface and Atmospheric Influences on Solar Activity, ed. R. Howe, R. W. Komm, K. S. Balasubramaniam, and G. J. Petrie, (Astronomical Society of the Pacific), 335
* Fisher (1973) Fisher, R. R. 1973, A Photoelectric Photometer for the Fe XIV Solar Corona, AFCRL-TR-73-0696, 15 pp.
* Fisher (1974) Fisher, R. R. 1974, Solar Phys., 36, 343
* Smartt (1982) Smartt, R. N. 1982, in SPIE 331, Instrumentation in Astronomy IV, 442
* Waldmeier (1964) Waldmeier, M.: 1964, ZAp, 59, 205
* Wilson et al. (1988) Wilson, P. R., Altrock, R. C., Harvey, K. L., Martin, S. F., and Snodgrass, H. B. 1988, Nature, 333, 748
|
arxiv-papers
| 2010-02-11T19:40:45 |
2024-09-04T02:49:08.363667
|
{
"license": "Public Domain",
"authors": "Richard C. Altrock (Air Force Research Laboratory, NSO/SP, Sunspot,\n NM, USA)",
"submitter": "Richard Altrock",
"url": "https://arxiv.org/abs/1002.2401"
}
|
1002.2463
|
# Young’s integral inequality with upper and lower bounds
Douglas R. Anderson Concordia College, Department of Mathematics and Computer
Science, Moorhead, MN 56562 USA andersod@cord.edu
http://www.cord.edu/faculty/andersod/ , Steven Noren Concordia College,
Moorhead, MN 56562 USA srnoren@cord.edu and Brent Perreault Concordia
College, Moorhead, MN 56562 USA bmperrea@cord.edu
###### Abstract.
Young’s integral inequality is reformulated with upper and lower bounds for
the remainder. The new inequalities improve Young’s integral inequality on all
time scales, such that the case where equality holds becomes particularly
transparent in this new presentation. The corresponding results for difference
equations are given, and several examples are included. We extend these
results to piecewise-monotone functions as well.
###### Key words and phrases:
Young’s inequality, monotone functions, Pochhammer lower factorial, difference
equations, time scales.
###### 2000 Mathematics Subject Classification:
26D15, 39A12, 34N05
## 1\. introduction
In 1912, Young [13] presented the following highly intuitive integral
inequality, namely that any real-valued continuous function
$f:[0,\infty)\rightarrow[0,\infty)$ satisfying $f(0)=0$ with $f$ strictly
increasing on $[0,\infty)$ satisfies
$ab\leq\int_{0}^{a}f(t)dt+\int_{0}^{b}f^{-1}(y)dy$ (1.1)
for any $a,b\in[0,\infty)$, with equality if and only if $b=f(a)$. A useful
consequence of this theorem is Young’s inequality,
$ab\leq\frac{a^{p}}{p}+\frac{b^{q}}{q},\qquad\frac{1}{p}+\frac{1}{q}=1,$
with equality if and only if $a^{p}=b^{q}$, a fact derived from (1.1) by
taking $f(t)=t^{p-1}$ and $q=\frac{p}{p-1}$. Hardy, Littlewood, and Pólya
included Young’s inequality in their classic book [4], but there was no
analytic proof until Diaz and Metcalf [3] supplied one in 1970. Tolsted [11]
showed how to derive Cauchy, Hölder, and Minkowski inequalities in a
straightforward way from (1.1). For many other applications and extensions of
Young’s inequality, see Mitrinović, Pečarić, and Fink [10]. For the purposes
of this paper we recall some results that consider upper bounds for the
integrals in (1.1). Merkle [8] established the inequality
$\int_{0}^{a}f(t)dt+\int_{0}^{b}f^{-1}(y)dy\leq\max\\{af(a),bf^{-1}(b)\\},$
which has been improved and reformulated recently by Minguzzi [9] to the
inequality
$0\leq\int_{\alpha_{1}}^{a}f(t)dt+\int_{\beta_{1}}^{b}f^{-1}(y)dy-
ab+\alpha_{1}\beta_{1}\leq\left(f^{-1}(b)-a\right)\left(b-f(a)\right),$ (1.2)
where the hypotheses of Young’s integral inequality hold, except that
$f(\alpha_{1})=\beta_{1}$ has replaced $f(0)=0$.
One might wonder if there is a discrete version of (1.1) in the form of a
summation inequality, or more generally a time-scale version of (1.1), where a
time scale, introduced by Hilger [5], is any nonempty closed set of real
numbers. Wong, Yeh, Yu, and Hong [12] presented a version of Young’s
inequality on time scales $\mathbb{T}$ in the following form. Using the
standard notation [2] of the left jump operator $\rho$ given by
$\rho(t):=\sup\\{s\in\mathbb{T}:s<t\\}$, the right jump operator $\sigma$
given by $\sigma(t)=\inf\\{s\in\mathbb{T}:s>t\\}$, the compositions
$f\circ\rho$ and $f\circ\sigma$ denoted by $f^{\rho}$ and $f^{\sigma}$,
respectively, the graininess functions defined by $\mu(t)=\sigma(t)-t$ and
$\nu(t)=t-\rho(t)$, and the delta and nabla derivatives of $f$ at
$t\in\mathbb{T}$, denoted $f^{\Delta}(t)$ and $f^{\nabla}(t)$, respectively,
(provided they exist) are given by
$f^{\Delta}(t):=\lim_{s\rightarrow
t}\frac{f^{\sigma}(t)-f(s)}{\sigma(t)-s},\qquad
f^{\nabla}(t):=\lim_{s\rightarrow t}\frac{f^{\rho}(t)-f(s)}{\rho(t)-s},$
we have the following result.
###### Theorem 1.1 (Wong, Yeh, Yu, and Hong).
Let $f$ be right-dense continuous on $[0,c]_{\mathbb{T}}:=[0,c]\cap\mathbb{T}$
for $c>0$, strictly increasing, with $f(0)=0$. Then for
$a\in[0,c]_{\mathbb{T}}$ and $b\in[0,f(c)]_{\mathbb{T}}$ the inequality
$ab\leq\int_{0}^{a}f^{\sigma}(t)\Delta
t+\int_{0}^{b}(f^{-1})^{\sigma}(y)\Delta y$
holds.
If $\mathbb{T}=\mathbb{Z}$ and $f(t)=t$, then Theorem 1.1 says that
$ab\leq\sum_{t=0}^{a-1}(t+1)+\sum_{y=0}^{b-1}(y+1)=\frac{1}{2}\left(a(a+1)+b(b+1)\right)$
(1.3)
holds. Note that an if and only if clause concerning an actual equality is
missing in the formulation in Theorem 1.1, with equality impossible in the
simple example (1.3) except for the trivial case $a=0=b$. This omission was
rectified in [1] via the following theorem.
###### Theorem 1.2 (Anderson).
Let $\mathbb{T}$ be any time scale (unbounded above) with $0\in\mathbb{T}$.
Further, suppose that $f:[0,\infty)_{\mathbb{T}}\rightarrow\mathbb{R}$ is a
real-valued function satisfying
1. (i)
$f(0)=0$;
2. (ii)
$f$ is continuous on $[0,\infty)_{\mathbb{T}}$, right-dense continuous at $0$;
3. (iii)
$f$ is strictly increasing on $[0,\infty)_{\mathbb{T}}$ such that
$f(\mathbb{T})$ is also a time scale.
Then for any $a\in[0,\infty)_{\mathbb{T}}$ and $b\in[0,\infty)\cap
f(\mathbb{T})$, we have
$ab\leq\frac{1}{2}\int_{0}^{a}\left[f(t)+f(\sigma(t))\right]\Delta
t+\frac{1}{2}\int_{0}^{b}\left[f^{-1}(y)+f^{-1}(\sigma(y))\right]\Delta y,$
with equality if and only if $b=f(a)$.
Motivated by [9], in this paper we extend (1.2) to the general time scales
setting while, in the process, simplifying and extending Theorem 1.2 as well.
As these results on time scales will include new results in difference
equations as an important corollary, we will illustrate our new inequalities
using discrete examples with $\mathbb{T}=\mathbb{Z}$.
## 2\. Theorem Formulation
We begin this section by introducing a new and improved version of Theorem 1.2
to facilitate the subsequent results. For any time scale $\mathbb{T}$, we have
the following result.
###### Theorem 2.1 (Young’s Inequality on Time Scales).
Let $\mathbb{T}$ be any time scale with $\alpha_{1}\in\mathbb{T}$ and
$\sup\mathbb{T}=\infty$. Further, suppose that
$f:[\alpha_{1},\infty)_{\mathbb{T}}\rightarrow\mathbb{R}$ is a real-valued
function satisfying
1. (i)
$f(\alpha_{1})=\beta_{1}$;
2. (ii)
$f$ is continuous on $[\alpha_{1},\infty)_{\mathbb{T}}$, right-dense
continuous at $\alpha_{1}$;
3. (iii)
$f$ is strictly increasing on $[\alpha_{1},\infty)_{\mathbb{T}}$ such that
$\widetilde{\mathbb{T}}:=f(\mathbb{T})$ is also a time scale.
Then for any $a\in\left[\alpha_{1},\infty\right)_{\mathbb{T}}$ and
$b\in\left[\beta_{1},\infty\right)_{\widetilde{\mathbb{T}}}$, we have
$ab\leq\int_{\alpha_{1}}^{a}f(t)\Delta
t+\int_{\beta_{1}}^{b}f^{-1}(y)\widetilde{\nabla}y+\alpha_{1}\beta_{1},$ (2.1)
with equality if and only if $b\in\left\\{f^{\rho}(a),f(a)\right\\}$ for fixed
$a$, or with equality if and only if
$a\in\left\\{f^{-1}(b),\sigma(f^{-1}(b))\right\\}$ for fixed $b$. The
inequality in (2.1) is reversed if $f$ is strictly decreasing.
###### Proof.
The proof is modeled after the one given on $\mathbb{R}$ in [3]. Note that $f$
is delta integrable and $f^{-1}$ is nabla integrable by the continuity
assumption in (ii). For simplicity, define
$F(a,b):=\int_{\alpha_{1}}^{a}f(t)\Delta
t+\int_{\beta_{1}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\alpha_{1}\beta_{1}.$
(2.2)
Then, the inequality to be shown is just $F(a,b)\geq 0$.
(I). We will first show that
$F(a,b)\geq F(a,f(a))\quad\text{for}\quad
a\in[\alpha_{1},\infty)_{\mathbb{T}}\quad\text{and}\quad
b\in[\beta_{1},\infty)_{\widetilde{\mathbb{T}}},$
with equality if and only if $b\in\left\\{f^{\rho}(a),f(a)\right\\}$. For any
such $a$ and $b$ we have
$F(a,b)-F(a,f(a))=\int_{f(a)}^{b}\left[f^{-1}(y)-a\right]\widetilde{\nabla}y.$
(2.3)
Clearly if $b=f(a)$ then the integrals are empty, and if $b=f^{\rho}(a)$ then
$F(a,f^{\rho}(a))-F(a,f(a))=\int^{f(a)}_{f^{\rho}(a)}\left[a-f^{-1}(y)\right]\widetilde{\nabla}y=[f(a)-f^{\rho}(a)][a-f^{-1}(f(a))]=0.$
Otherwise, since $f^{-1}(y)$ is continuous and strictly increasing for
$y\in\widetilde{\mathbb{T}}$, the integrals in (2.3) are strictly positive for
$b<f^{\rho}(a)$ and $b>f(a)$.
(II). We will next show that $F(a,f(a))=F(a,f^{\rho}(a))=0$. For brevity, put
$\varphi(a)=F(a,f(a))$, that is
$\varphi(a):=\int_{\alpha_{1}}^{a}f(t)\Delta
t+\int_{\beta_{1}}^{f(a)}f^{-1}(y)\widetilde{\nabla}y-af(a)+\alpha_{1}\beta_{1}.$
First, assume $a$ is a right-scattered point. Then
$\displaystyle\varphi^{\sigma}(a)-\varphi(a)$ $\displaystyle=$
$\displaystyle\int_{a}^{\sigma(a)}f(t)\Delta
t+\int_{f(a)}^{f^{\sigma}(a)}f^{-1}(y)\widetilde{\nabla}y-\sigma(a)f^{\sigma}(a)+af(a)$
$\displaystyle=$
$\displaystyle[\sigma(a)-a]f(a)+\left[f^{\sigma}(a)-f(a)\right]f^{-1}(f^{\sigma}(a))-\sigma(a)f^{\sigma}(a)+af(a)=0.$
Therefore, if $a$ is a right-scattered point, then $\varphi^{\Delta}(a)=0$.
Next, assume $a$ is a right-dense point. Let
$\\{a_{n}\\}_{n\in\mathbb{N}}\subset[a,\infty)_{\mathbb{T}}$ be a decreasing
sequence converging to $a$. Then
$\displaystyle\varphi(a_{n})-\varphi(a)$ $\displaystyle=$
$\displaystyle\int_{a}^{a_{n}}f(t)\Delta
t+\int_{f(a)}^{f(a_{n})}f^{-1}(y)\widetilde{\nabla}y-a_{n}f(a_{n})+af(a)$
$\displaystyle\geq$
$\displaystyle(a_{n}-a)f(a)+[f(a_{n})-f(a)]a-a_{n}f(a_{n})+af(a)$
$\displaystyle=$ $\displaystyle(a_{n}-a)\left[f(a)-f(a_{n})\right],$
since the functions $f$ and $f^{-1}$ are strictly increasing. Similarly,
$\displaystyle\varphi(a_{n})-\varphi(a)$ $\displaystyle\leq$
$\displaystyle(a_{n}-a)f(a_{n})+[f(a_{n})-f(a)]a_{n}-a_{n}f(a_{n})+af(a)$
$\displaystyle=$ $\displaystyle(a_{n}-a)\left[f(a_{n})-f(a)\right].$
Therefore,
$0=\lim_{n\rightarrow\infty}\left[f(a)-f(a_{n})\right]\leq\lim_{n\rightarrow\infty}\frac{\varphi(a_{n})-\varphi(a)}{a_{n}-a}\leq\lim_{n\rightarrow\infty}\left[f(a_{n})-f(a)\right]=0.$
It follows that $\varphi^{\Delta}(a)$ exists, and $\varphi^{\Delta}(a)=0$ for
right-dense $a$ as well. In other words, in either case,
$\varphi^{\Delta}(a)=0$ for $a\in[\alpha_{1},\infty)_{\mathbb{T}}$. As
$\varphi(\alpha_{1})=0$, by the uniqueness theorem for initial value problems
we have that $\varphi(a)=0$ for all $a\in[\alpha_{1},\infty)_{\mathbb{T}}$.
From earlier we know that $F(a,f^{\rho}(a))=F(a,f(a))$. Thus as an overall
result, we have that
$F(a,b)\geq F(a,f(a))=0,$
with equality if and only if $b=f(a)$ or $b=f^{\rho}(a)$, as claimed. The case
with $a\in\left\\{f^{-1}(b),\sigma(f^{-1}(b))\right\\}$ for fixed $b$ is
similar and thus omitted. If $f$ is strictly decreasing, it is straightforward
to see that the inequality in (2.1) is reversed; the details are left to the
reader. ∎
We now focus on establishing an upper bound for Young’s integral. Before we
state and prove our main theorem, we need an auxiliary result via the
following lemma.
###### Lemma 2.2.
Let $f$ satisfy the hypotheses of Theorem 2.1, and let $F(a,b)$ be given as in
(2.2). For every $a,\alpha\in\mathbb{T}$ and
$b,\beta\in\widetilde{\mathbb{T}}$ we have
$F(a,b)+F(\alpha,\beta)\geq-(\alpha-a)(\beta-b),$ (2.4)
where equality holds if and only if
$\alpha\in\left\\{f^{-1}(b),\sigma(f^{-1}(b))\right\\}$ and
$\beta\in\left\\{f^{\rho}(a),f(a)\right\\}$.
###### Proof.
Fix $a\in\mathbb{T}$ and $b\in\widetilde{\mathbb{T}}$. By Young’s integral
inequality on time scales (Theorem 2.1) we have
$\int_{\alpha_{1}}^{a}f(t)\Delta
t+\int_{\beta_{1}}^{\beta}f^{-1}(y)\widetilde{\nabla}y+\alpha_{1}\beta_{1}\geq
a\beta,\quad\text{and}$ (2.5) $\int_{\alpha_{1}}^{\alpha}f(t)\Delta
t+\int_{\beta_{1}}^{b}f^{-1}(y)\widetilde{\nabla}y+\alpha_{1}\beta_{1}\geq\alpha
b,$ (2.6)
with equality if and only if $\beta\in\left\\{f^{\rho}(a),f(a)\right\\}$ and
$\alpha\in\left\\{f^{-1}(b),\sigma(f^{-1}(b))\right\\}$, respectively. By
rearranging it follows that
$\displaystyle\left(\int_{\alpha_{1}}^{a}f(t)\Delta
t+\int_{\beta_{1}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\alpha_{1}\beta_{1}\right)+\left(\int_{\alpha_{1}}^{\alpha}f(t)\Delta
t+\int_{\beta_{1}}^{\beta}f^{-1}(y)\widetilde{\nabla}y-\alpha\beta+\alpha_{1}\beta_{1}\right)$
$\displaystyle=\left(\int_{\alpha_{1}}^{a}f(t)\Delta
t+\int_{\beta_{1}}^{\beta}f^{-1}(y)\widetilde{\nabla}y+\alpha_{1}\beta_{1}\right)+\left(\int_{\alpha_{1}}^{\alpha}f(t)\Delta
t+\int_{\beta_{1}}^{b}f^{-1}(y)\widetilde{\nabla}y+\alpha_{1}\beta_{1}\right)-ab-\alpha\beta$
$\displaystyle\geq a\beta+\alpha
b-ab-\alpha\beta=-\left(\alpha-a\right)\left(\beta-b\right).$
Note that equality holds here if and only if it holds in (2.5) and (2.6),
videlicet if and only if
$\alpha\in\left\\{f^{-1}(b),\sigma(f^{-1}(b))\right\\}$ and
$\beta\in\left\\{f^{\rho}(a),f(a)\right\\}$. ∎
###### Theorem 2.3.
Let $\mathbb{T}$ be any time scale and
$f:[\alpha_{1},\alpha_{2}]_{\mathbb{T}}\rightarrow[\beta_{1},\beta_{2}]_{\widetilde{\mathbb{T}}}$
be a continuous strictly increasing function such that
$\widetilde{\mathbb{T}}=f(\mathbb{T})$ is also a time scale. Then for every
$a,\widehat{a}\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$ and
$b,\widehat{b}\in[\beta_{1},\beta_{2}]_{\widetilde{\mathbb{T}}}$ we have
$\left(f^{-1}(\widehat{b})-\widehat{a}\right)\left(f^{\rho}(\widehat{a})-\widehat{b}\right)\leq\int_{\widehat{a}}^{a}f(t)\Delta
t+\int_{\widehat{b}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\widehat{a}\widehat{b}\leq\left(f^{-1}(b)-a\right)\left(b-f^{\rho}(a)\right),$
(2.7)
where the equalities hold if and only if
$\widehat{b}\in\left\\{f^{\rho}(\widehat{a}),f(\widehat{a})\right\\}$ and
$b\in\left\\{f^{\rho}(a),f(a)\right\\}$. The inequalities are reversed if $f$
is strictly decreasing.
###### Proof.
Considering $F$ as in (2.2), and (2.4) with $\alpha=f^{-1}(b)$ and
$\beta=f^{\rho}(a)$ we have the equality
$F(a,b)+F\left(f^{-1}(b),f^{\rho}(a)\right)=-\left(f^{-1}(b)-a\right)\left(f^{\rho}(a)-b\right).$
As $f^{-1}(b)\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$ and
$f^{\rho}(a)\in[\beta_{1},\beta_{2}]_{\widetilde{\mathbb{T}}}$, via Young’s
integral inequality on time scales (Theorem 2.1 above) we see that
$F\left(f^{-1}(b),f^{\rho}(a)\right)\geq 0$. Consequently we have that
$0\leq F(a,b)\leq-\left(f^{-1}(b)-a\right)\left(f^{\rho}(a)-b\right);$ (2.8)
note that equality holds if and only if
$b\in\left\\{f^{\rho}(a),f(a)\right\\}$. Thus for any
$\widehat{a}\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$ and
$\widehat{b}\in[\beta_{1},\beta_{2}]_{\widetilde{\mathbb{T}}}$ we have from
(2.8) that
$0\leq-\left(f^{-1}(\widehat{b})-\widehat{a}\right)\left(f^{\rho}(\widehat{a})-\widehat{b}\right)-F(\widehat{a},\widehat{b}),$
(2.9)
with equality if and only if
$\widehat{b}\in\left\\{f^{\rho}(\widehat{a}),f(\widehat{a})\right\\}$.
Combining inequalities (2.8) and (2.9) we get
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
F(a,b)-\left(f^{-1}(\widehat{b})-\widehat{a}\right)\left(f^{\rho}(\widehat{a})-\widehat{b}\right)-F(\widehat{a},\widehat{b})$
$\displaystyle\leq$
$\displaystyle-\left(f^{-1}(b)-a\right)\left(f^{\rho}(a)-b\right)-\left(f^{-1}(\widehat{b})-\widehat{a}\right)\left(f^{\rho}(\widehat{a})-\widehat{b}\right)-F(\widehat{a},\widehat{b}).$
This can be rewritten as (2.7). If $f$ is strictly decreasing the proof is
similar and thus omitted. ∎
###### Remark 2.4.
In [9, Theorem 1.1] the author assumes $f(\widehat{a})=\widehat{b}$ $($see
(1.2) above$)$, so that the inequality in (2.7) is new for
$\mathbb{T}=\mathbb{R}$, as well as for $\mathbb{T}=\mathbb{Z}$ and general
time scales.
###### Remark 2.5.
Due to Lemma 2.2 and the first line of the proof of Theorem 2.3, there are
three other inequalities we could write in place of (2.7), namely
$\displaystyle\left(f^{-1}(\widehat{b})-\widehat{a}\right)\left(f(\widehat{a})-\widehat{b}\right)$
$\displaystyle\leq$ $\displaystyle\int_{\widehat{a}}^{a}f(t)\Delta
t+\int_{\widehat{b}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\widehat{a}\widehat{b}\leq\left(f^{-1}(b)-a\right)\left(b-f(a)\right),$
$\displaystyle\left(\sigma(f^{-1}(\widehat{b}))-\widehat{a}\right)\left(f^{\rho}(\widehat{a})-\widehat{b}\right)$
$\displaystyle\leq$ $\displaystyle\int_{\widehat{a}}^{a}f(t)\Delta
t+\int_{\widehat{b}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\widehat{a}\widehat{b}\leq\left(\sigma(f^{-1}(b))-a\right)\left(b-f^{\rho}(a)\right),$
$\displaystyle\left(\sigma(f^{-1}(\widehat{b}))-\widehat{a}\right)\left(f(\widehat{a})-\widehat{b}\right)$
$\displaystyle\leq$ $\displaystyle\int_{\widehat{a}}^{a}f(t)\Delta
t+\int_{\widehat{b}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\widehat{a}\widehat{b}\leq\left(\sigma(f^{-1}(b))-a\right)\left(b-f(a)\right).$
If we focus on just the upper bounds, for $a>\widehat{a}$, $b>\widehat{b}$,
and $b\geq f(a)$ we have the least of these upper bounds, leading to
$\int_{\widehat{a}}^{a}f(t)\Delta
t+\int_{\widehat{b}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\widehat{a}\widehat{b}\leq\left(f^{-1}(b)-a\right)\left(b-f(a)\right),$
whereas for $a>\widehat{a}$, $b>\widehat{b}$, and $b\leq f^{\rho}(a)$ we have
$\int_{\widehat{a}}^{a}f(t)\Delta
t+\int_{\widehat{b}}^{b}f^{-1}(y)\widetilde{\nabla}y-ab+\widehat{a}\widehat{b}\leq\left(\sigma(f^{-1}(b))-a\right)\left(b-f^{\rho}(a)\right).$
###### Corollary 2.6.
Pick $a_{i}\in\mathbb{T}$ with $a_{i}<a_{i+1}$. Let
$f_{1}:[\rho(a_{1}),a_{2}]_{\mathbb{T}}\rightarrow\mathbb{R}$ and
$f_{i}:[a_{i},a_{i+1}]_{\mathbb{T}}\rightarrow\mathbb{R}$ be continuous
strictly monotone functions for $i=2,\cdots,m$. Assume
$f:\mathbb{T}\rightarrow\mathbb{R}$ is continuous, where
$f(t):=f_{i}(t)\quad\text{and}\quad f^{-1}(y):=f_{i}^{-1}(y)$
for $t\in[a_{i},a_{i+1}]_{\mathbb{T}}$ and
$y\in[\min\\{f(a_{i}),f(a_{i+1})\\},\max\\{f(a_{i}),f(a_{i+1})\\}]_{f(\mathbb{T})}$,
respectively, and $i=1,2,\cdots,m$, that is to say
$f_{i}(a_{i+1})=f_{i+1}(a_{i+1})$ for $i=1,2,\cdots,m-1$. Set
$\displaystyle I_{f}:=\int_{a_{1}}^{a_{m+1}}f(t)\Delta
t+\int_{b_{1}}^{b_{m+1}}f^{-1}(y)\widetilde{\nabla}y-a_{m+1}b_{m+1}+a_{1}b_{1},\quad\text{and}$
$\displaystyle
K_{i}:=\left(a_{i}-f^{-1}(b_{i})\right)\left(f^{\rho}(a_{i})-b_{i}\right),\quad
i\in\\{1,m+1\\},$
where $f^{-1}(b_{1})\in[a_{1},a_{2}]_{\mathbb{T}}$ and
$f^{-1}(b_{m+1})\in[a_{m},a_{m+1}]_{\mathbb{T}}$. Then we have the following.
1. (i)
If $f_{1}$ and $f_{m}$ are both strictly increasing, then
$-K_{1}\leq I_{f}\leq K_{m+1}.$
The inequalities are reversed if $f_{1}$ and $f_{m}$ are both strictly
decreasing.
2. (ii)
If $f_{1}$ is strictly increasing and $f_{m}$ is strictly decreasing, then
$K_{m+1}-K_{1}\leq I_{f}\leq 0.$
The inequalities are reversed if $f_{1}$ is strictly decreasing and $f_{m}$ is
strictly increasing.
In all cases, the equalities hold if and only if
$b_{1}\in\left\\{f^{\rho}(a_{1}),f(a_{1})\right\\}$ and
$b_{m+1}\in\left\\{f^{\rho}(a_{m+1}),f(a_{m+1})\right\\}$.
###### Proof.
We will only prove the first part of (i), as the other parts follow in a
similar manner from Theorem 2.3. Assume $f_{1}$ and $f_{m}$ are both strictly
increasing. By Theorem 2.3 we have the inequalities
$\left(f^{-1}(b_{1})-a_{1}\right)\left(f^{\rho}(a_{1})-b_{1}\right)\leq\int_{a_{1}}^{a_{2}}f(t)\Delta
t+\int_{b_{1}}^{f(a_{2})}f^{-1}(y)\widetilde{\nabla}y-a_{2}f(a_{2})+a_{1}b_{1}\leq
0,$
the equalities
$0=\int_{a_{i}}^{a_{i+1}}f(t)\Delta
t+\int_{f(a_{i})}^{f(a_{i+1})}f^{-1}(y)\widetilde{\nabla}y-a_{i+1}f(a_{i+1})+a_{i}f(a_{i})=0$
for $i=2,\cdots,m-1$, and the inequalities
$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\int_{a_{m}}^{a_{m+1}}f(t)\Delta
t+\int_{f(a_{m})}^{b_{m+1}}f^{-1}(y)\widetilde{\nabla}y-a_{m+1}b_{m+1}+a_{m}f(a_{m})$
$\displaystyle\leq$
$\displaystyle\left(a_{m+1}-f^{-1}(b_{m+1})\right)\left(f^{\rho}(a_{m+1})-b_{m+1}\right),$
where equalities hold in the first line if and only if
$b_{1}\in\left\\{f^{\rho}(a_{1}),f(a_{1})\right\\}$, and equalities hold in
the third if and only if
$b_{m+1}\in\left\\{f^{\rho}(a_{m+1}),f(a_{m+1})\right\\}$. If we add these
expressions together, we obtain $-K_{1}\leq I_{f}\leq K_{m+1}$. This completes
the proof. ∎
###### Remark 2.7.
Corollary 2.6 for continuous piecewise-monotone functions is new even for
$\mathbb{T}=\mathbb{R}$, as well as for $\mathbb{T}=\mathbb{Z}$ and general
time scales. In the next result, Theorem 2.8, we extend the original Young
result for continuous functions to piecewise-continuous piecewise-monotone
functions on $\mathbb{R}$.
###### Theorem 2.8.
Let $a_{i}\in\mathbb{R}$ with $a_{i}<a_{i+1}$, and let
$f_{i}:[a_{i},a_{i+1}]\rightarrow\mathbb{R}$ be a continuous strictly monotone
function for $i=1,2,\cdots,m$. Let $f:[a_{1},a_{m+1}]\rightarrow\mathbb{R}$ be
the piecewise-continuous function given by
$f(x):=f_{i}(x)\quad\text{and}\quad f^{-1}(y):=f_{i}^{-1}(y)$
for $x\in(a_{i},a_{i+1})$ and
$y\in\left(\min\\{f(a_{i}),f(a_{i+1})\\},\max\\{f(a_{i}),f(a_{i+1})\\}\right)$,
respectively, for $i=1,\cdots,m$, with $f(a_{1})=f_{1}(a_{1})$ and
$f(a_{m+1})=f_{m}(a_{m+1})$. Set
$\displaystyle
F(b_{1},b_{m+1}):=\int_{a_{1}}^{a_{m+1}}f(x)dx+\int_{b_{1}}^{b_{m+1}}f^{-1}(y)dy-
a_{m+1}b_{m+1}+a_{1}b_{1}+\sum_{i=2}^{m}a_{i}\left[f_{i}(a_{i})-f_{i-1}(a_{i})\right],$
$\displaystyle
K_{i}:=\left(a_{i}-f^{-1}(b_{i})\right)\left(f(a_{i})-b_{i}\right),\quad
i\in\\{1,m+1\\},$
where $f^{-1}(b_{1})\in[a_{1},a_{2}]$ and $f^{-1}(b_{m+1})\in[a_{m},a_{m+1}]$.
Then we have the following.
1. (i)
If $f_{1}$ and $f_{m}$ are both strictly increasing, then
$-K_{1}\leq F(b_{1},b_{m+1})\leq K_{m+1}.$
The inequalities are reversed if $f_{1}$ and $f_{m}$ are both strictly
decreasing.
2. (ii)
If $f_{1}$ is strictly increasing and $f_{m}$ is strictly decreasing, then
$K_{m+1}-K_{1}\leq F(b_{1},b_{m+1})\leq 0.$
The inequalities are reversed if $f_{1}$ is strictly decreasing and $f_{m}$ is
strictly increasing.
In all cases, the equalities hold if and only if $b_{1}=f(a_{1})$ and
$b_{m+1}=f(a_{m+1})$.
###### Proof.
Apply Theorem 2.3 on $\mathbb{T}=\mathbb{R}$ to the pieces $f_{i}$ on
$[a_{i},a_{i+1}]$, using the appropriate inequalities for $f_{1}$ and $f_{m}$,
and equalities for $f_{i}$, $i=2,\cdots,m-1$. Then add up these expressions to
get the result. ∎
###### Theorem 2.9.
For some delta differentiable function $g$, let $f=g^{\Delta}$ satisfy the
hypotheses of Theorem 2.3, where
$f^{-1}=(g^{\Delta})^{-1}=(g^{*})^{\widetilde{\nabla}}$ on
$\widetilde{\mathbb{T}}=f(\mathbb{T})$. [If $\mathbb{T}=\mathbb{R}$ the
function $g^{*}$ is known as the Legendre transform of $g$.] If we pick
$\widehat{a}\widehat{b}=g(\widehat{a})+g^{*}(\widehat{b})$, then by Theorem
2.3 we have
$\displaystyle\left((g^{*})^{\widetilde{\nabla}}(\widehat{b})-\widehat{a}\right)\left(g^{\nabla}(\widehat{a})-\widehat{b}\right)\leq
g(a)+g^{*}(b)-ab\leq\left((g^{*})^{\widetilde{\nabla}}(b)-a\right)\left(b-g^{\nabla}(a)\right),$
where the equalities hold if and only if
$\widehat{b}\in\left\\{g^{\nabla}(\widehat{a}),g^{\Delta}(\widehat{a})\right\\}$
and $b\in\left\\{g^{\nabla}(a),g^{\Delta}(a)\right\\}$.
In the following theorem we reconsider Theorem 2.3 above. This allows us to
get a Young-type integral inequality without having to find $f^{-1}$.
###### Theorem 2.10.
Let the hypotheses of Theorem 2.3 hold. Then for any
$a,\alpha,\widehat{a},\widehat{\alpha}\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$
we have
$\displaystyle\left(\widehat{\alpha}-\widehat{a}\right)\left(f^{\rho}(\widehat{a})-f(\widehat{\alpha})\right)$
$\displaystyle\leq$ $\displaystyle\int_{\widehat{a}}^{a}f(t)\Delta
t-\int_{\widehat{\alpha}}^{\alpha}f(t)\Delta
t+(\alpha-a)f(\alpha)+(\widehat{a}-\widehat{\alpha})f(\widehat{\alpha})$
(2.10) $\displaystyle\leq$
$\displaystyle\left(\alpha-a\right)\left(f(\alpha)-f^{\rho}(a)\right),$
where the equalities hold if and only if
$\widehat{\alpha}\in\\{\rho(\widehat{a}),\widehat{a}\\}$ and
$\alpha\in\\{\rho(a),a\\}$.
###### Proof.
By Theorem 2.3 with $\widehat{a}=\widehat{\alpha}$,
$\widehat{b}=f(\widehat{\alpha})$, $a=\alpha$ and $b=f(\alpha)$ we have
$\int_{f(\widehat{\alpha})}^{f(\alpha)}f^{-1}(y)\widetilde{\nabla}y=\alpha
f(\alpha)-\widehat{\alpha}f(\widehat{\alpha})-\int_{\widehat{\alpha}}^{\alpha}f(t)\Delta
t$ (2.11)
for any $\alpha,\widehat{\alpha}\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$.
Since $\alpha,\widehat{\alpha}\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$ are
arbitrary, we substitute (2.11) into (2.7) to obtain (2.10). ∎
###### Example 2.11.
Consider the generalized polynomial functions $h_{n}(t,s)$ on time scales
defined recursively in the following way [2, Section 1.6]:
$h_{0}(t,s)\equiv 1,\quad
h_{k+1}(t,s)=\int_{s}^{t}h_{k}(\tau,s)\Delta\tau,\quad t,s\in\mathbb{T},\quad
k\in\mathbb{N}_{0}.$
If we take $f(t)=h_{n}(t,\widehat{\alpha})$ for any $n\in\mathbb{N}$, then by
Theorem 2.10 we have
$\displaystyle\left(\widehat{\alpha}-\widehat{a}\right)h_{n}(\rho(\widehat{a}),\widehat{\alpha})$
$\displaystyle\leq$ $\displaystyle
h_{n+1}(a,\widehat{a})-h_{n+1}(\alpha,\widehat{\alpha})+(\alpha-a)h_{n}(\alpha,\widehat{\alpha})$
(2.12) $\displaystyle\leq$
$\displaystyle\left(\alpha-a\right)\left[h_{n}(\alpha,\widehat{\alpha})-h_{n}(\rho(a),\widehat{\alpha})\right],$
for any
$a,\widehat{a},\alpha,\widehat{\alpha}\in[\alpha_{1},\alpha_{2}]_{\mathbb{T}}$,
where the equalities hold if and only if
$\widehat{\alpha}\in\\{\rho(\widehat{a}),\widehat{a}\\}$ and
$\alpha\in\\{\rho(a),a\\}$.
## 3\. results for difference equations
In this section we concentrate on the discrete case. For
$\mathbb{T}=\mathbb{Z}$ we have the following new discrete results, which are
corollaries of the theorems above. Recall that
$[\alpha_{1},\alpha_{2}]_{\mathbb{Z}}=\\{\alpha_{1},\alpha_{1}+1,\alpha_{1}+2,\cdots,\alpha_{2}-1,\alpha_{2}\\}$.
The first two theorems are direct translations to $\mathbb{T}=\mathbb{Z}$ of
Theorem 2.3 and Theorem 2.10, respectively.
###### Theorem 3.1.
Let
$f:[\alpha_{1},\alpha_{2}]_{\mathbb{Z}}\rightarrow[\beta_{1},\beta_{2}]_{\widetilde{\mathbb{Z}}}$
be a strictly increasing function, where
$\widetilde{\mathbb{Z}}=f(\mathbb{Z})$. Then for every
$a,\widehat{a}\in[\alpha_{1},\alpha_{2}]_{\mathbb{Z}}$ and
$b,\widehat{b}\in[\beta_{1},\beta_{2}]_{\widetilde{\mathbb{Z}}}$ we have
$\displaystyle\left(f^{-1}(\widehat{b})-\widehat{a}\right)\left(f(\widehat{a}-1)-\widehat{b}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{t=\widehat{a}}^{a-1}f(t)+\sum_{y\in(\widehat{b},b]\cap\widetilde{\mathbb{Z}}}f^{-1}(y)\widetilde{\mu}(y)-ab+\widehat{a}\widehat{b}$
(3.1) $\displaystyle\leq$
$\displaystyle\left(f^{-1}(b)-a\right)\left(b-f(a-1)\right),$
where the equalities hold if and only if
$\widehat{b}\in\left\\{f(\widehat{a}-1),f(\widehat{a})\right\\}$ and
$b\in\left\\{f(a-1),f(a)\right\\}$.
###### Theorem 3.2.
Let $f:\mathbb{Z}\rightarrow\mathbb{R}$ be a strictly increasing function.
Then for any integers $a,\widehat{a},\alpha,\widehat{\alpha}$ we have
$\displaystyle\left(\widehat{\alpha}-\widehat{a}\right)\left[f(\widehat{a}-1)-f(\widehat{\alpha})\right]$
$\displaystyle\leq$
$\displaystyle\sum_{t=\widehat{a}}^{a-1}f(t)-\sum_{t=\widehat{\alpha}}^{\alpha-1}f(t)+(\alpha-a)f(\alpha)+(\widehat{a}-\widehat{\alpha})f(\widehat{\alpha})$
(3.2) $\displaystyle\leq$
$\displaystyle\left(\alpha-a\right)\left[f(\alpha)-f(a-1)\right],$
where the equalities hold if and only if
$\widehat{\alpha}\in\\{\widehat{a}-1,\widehat{a}\\}$ and
$\alpha\in\\{a-1,a\\}$.
###### Example 3.3.
Consider the Pochhammer lower factorial function
$f_{k}(t)=t^{\underline{k}}:=t(t-1)\cdots(t-k+1),\quad t,k\in\mathbb{Z},$
also known as $t$ to the $k$ falling [6], or the falling factorial power
function [7]. It is clear that $f_{k}$ is strictly increasing on the integer
interval $[k-1,\infty)_{\mathbb{Z}}$. By Theorem 3.2 we have
$\displaystyle(a-\alpha)f_{k}(\alpha)\leq\frac{1}{k+1}\left[f_{k+1}(a)-f_{k+1}(\alpha)\right]\leq(a-\alpha)f_{k}(a-1)$
(3.3)
for $a,\alpha\in\\{k-1,k,k+1,\cdots\\}$, where the equalities hold if and only
if $\alpha\in\\{a-1,a\\}$.
###### Example 3.4.
For any real $B>1$ and any integers $a\geq\alpha\in\mathbb{Z}$ we have
$B^{\alpha}\leq\frac{B^{a}-B^{\alpha}}{(a-\alpha)(B-1)}\leq B^{a-1},$
where the equalities hold if and only if $\alpha\in\\{a-1,a\\}$.
###### Example 3.5.
Consider Theorem 2.9. Let $\mathbb{T}=\mathbb{Z}$ and $f(t)=B^{t}$ for $B>1$.
Then $\widetilde{\mathbb{T}}=B^{\mathbb{Z}}$, and we have
$g(t)=\frac{B^{t}-B^{\alpha}}{B-1}$, $f^{-1}(y)=\log_{B}y$, and
$g^{*}(y)=y\log_{B}y-\frac{y}{B-1}+\beta\left(\alpha+\frac{1}{B-1}-\log_{B}\beta\right)$.
It is easy to check that $f^{-1}=(g^{*})^{\widetilde{\nabla}}$ on
$\widetilde{\mathbb{T}}$ and $\alpha\beta=g(\alpha)+g^{*}(\beta)$. Therefore
we have the inequalities
$\displaystyle\left(\log_{B}\beta-\alpha\right)\left(B^{\alpha-1}-\beta\right)$
$\displaystyle\leq$
$\displaystyle\frac{B^{a}-B^{\alpha}}{B-1}+b\log_{B}b-\frac{b}{B-1}+\beta\left(\alpha+\frac{1}{B-1}-\log_{B}\beta\right)-ab$
$\displaystyle\leq$
$\displaystyle\left(\log_{B}b-a\right)\left(b-B^{a-1}\right),$
where the equalities hold if and only if
$\beta\in\left\\{B^{\alpha-1},B^{\alpha}\right\\}$ and
$b\in\left\\{B^{a-1},B^{a}\right\\}$.
###### Example 3.6.
Let $f(t)=\sin\left[\frac{\pi t}{2k}\right]$ for $k\in\mathbb{N}$. Then $f$ is
strictly increasing on $[-k,k]_{\mathbb{Z}}$, so that for any
$a\geq\alpha\in[-k,k]_{\mathbb{Z}}$ we have by Theorem 3.2 that
$\sin\left[\frac{\alpha\pi}{2k}\right]\leq\frac{1}{2(a-\alpha)}\left(\cos\left[\frac{(2\alpha-1)\pi}{4k}\right]-\cos\left[\frac{(2a-1)\pi}{4k}\right]\right)\csc\left[\frac{\pi}{4k}\right]\leq\sin\left[\frac{(a-1)\pi}{2k}\right],$
with equalities if and only if $\alpha\in\\{a-1,a\\}$.
###### Example 3.7.
Let $f(t)=\binom{t}{k}$ for $t,k\in\mathbb{N}$ with $t\geq k$. Then $f$ is
strictly increasing on $[k,\infty)_{\mathbb{Z}}$, whereby for any
$a\geq\alpha\in[k,\infty)_{\mathbb{Z}}$ we have
$\binom{\alpha}{k}\leq\frac{(a-k)\binom{a}{k}-(\alpha-k)\binom{\alpha}{k}}{(a-\alpha)(k+1)}\leq\binom{a-1}{k},$
with equalities if and only if $\alpha\in\\{a-1,a\\}$.
## References
* [1] D. R. Anderson, Young’s integral inequality on time scales revisited, _J. Inequalities in Pure Appl. Math._ , Volume 8 (2007), Issue 3, Article 64, 5 pages.
* [2] M. Bohner and A. Peterson, _Dynamic Equations on Time Scales: An Introduction with Applications_ , Birkhäuser, Boston (2001).
* [3] J. B. Diaz and F. T. Metcalf, An analytic proof of Young’s inequality, _American Math. Monthly_ , 77 (1970) 603–609.
* [4] G. H. Hardy, J. E. Littlewood, and G. Pólya, _Inequalities_ , Cambridge, 1934.
* [5] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, _Results Math._ 18 (1990), no. 1–2, 18–56.
* [6] W. G. Kelley and A. C. Peterson, _Difference Equations: An Introduction with Applications_ , 2e, Harcourt/Academic Press, San Diego, 2001.
* [7] V. Lampret, Approximating real Pochhammer products: a comparison with powers, _Cent. Eur. J. Math._ , 7(3) (2009) 493–505.
* [8] M. Merkle, A contribution to Young’s inequality, _Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz._ , 461–497 (1974), 265–267.
* [9] E. Minguzzi, An equivalent form of Young’s inequality with upper bound, _Appl. Anal. Discrete Math._ 2 (2008) 213–216.
* [10] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, _Classical and New Inequalities in Analysis_ , Kluwer Academic Publishers, Dordrecht, 1993.
* [11] E. Tolsted, An elementary derivation of the Cauchy, Holder, and Minkowski inequalities from Young’s inequality, _Math. Mag._ , 37 (1964), 2–12.
* [12] F. H. Wong, C. C. Yeh, S. L. Yu, and C. H. Hong, Young’s inequality and related results on time scales, _Appl. Math. Letters_ , 18 (2005) 983–988.
* [13] W. H. Young, On classes of summable functions and their Fourier series, _Proc. Royal Soc._ , Series (A), 87 (1912) 225–229.
|
arxiv-papers
| 2010-02-12T02:38:05 |
2024-09-04T02:49:08.368492
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson, Steven Noren, and Brent Perreault",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1002.2463"
}
|
1002.2500
|
Wiman and Arima theorems for quasiregular mappings
O. Martio, V.M. Miklyukov, and M. Vuorinen
File: arima9b.tex, 2009-12-21, printed: 2024-8-27, 15.40
Abstract. Generalizations of the theorems of Wiman and of Arima on entire
functions are proved for spatial quasiregular mappings.
2000 Mathematics Subject Classification. Primary 30 C 65, 35 J 70. Secondary
58 J 05
Key words and phrases. Quasiregular mapping, elliptic PDE
## 1 Main results
It follows from the Ahlfors theorem that an entire holomorphic function $f$ of
order $\rho$ has no more than $[2\rho]$ distinct asymptotic curves where $[r]$
stands for the largest integer $\leq r$. This theorem does not give any
information if $\rho<1/2$. This case is covered by two theorems: if an entire
holomorphic function $f$ has order $\rho<{1/2}$ then
$\lim\sup_{r\to\infty}\min_{|z|=r}|f(z)|=\infty.$ (Wiman [22]) and if $f$ is
an entire holomorphic function of order $\rho>0$ and $l$ is a number
satisfying the conditions $0<l\leq 2\pi,$ $l<{{\pi}\over{\rho}},$ then there
exists a sequence of circular arcs $\\{|z|=r_{k},\;\theta_{k}\leq\arg
z\leq\theta_{k}+l\\},$ $r_{k}\to\infty,$ $0\leq\theta_{k}<2\pi,$ along which
$|f(z)|$ tends to $\infty$ uniformly with respect to $\arg z$ (Arima [1]).
Below we prove generalizations of these theorems for quasiregular mappings for
$n\geq 2$. The next two theorems are generalizations of the theorems of Wiman
and of Arima for quasiregular mappings on manifolds.
1.1. Theorem. Let ${\cal M},{\cal N}$ be $n$-dimensional noncompact Riemannian
manifolds without boundary. Assume that $h:{\cal M}\to(0,\infty)$ is a special
exhaustion function of the manifold ${\cal M}$ and $u$ is a nonnegative growth
function on the manifold ${\cal N}$, which is a subsolution of an equation
(3.3) with the structure conditions (3.1), (3.2) and the structure constants
$p=n$, $\nu_{1}$, $\nu_{2}$.
Let $f:{\cal M}\to{\cal N}$ be a non-constant quasiregular mapping. Suppose
that the manifold ${\cal M}$ is such that
$\int\limits^{\infty}\lambda_{n}(\Sigma_{h}(t);1)dt=\infty.$ (1.2)
If now
$\liminf_{\tau\to\infty}\max_{h(m)=\tau}u(f(m))\exp\Bigl{\\{}-C\int\limits^{\tau}\lambda_{n}(\Sigma_{h}(t);1)dt\Bigr{\\}}=0$
(1.3)
then
$\limsup_{\tau\to\infty}\min_{h(m)=\tau}u(f(m))=\infty.$
Here
$C=\Bigl{(}n-1+n\Bigl{(}\bigl{(}{\nu_{2}\over\nu_{1}}\bigr{)}^{2}K^{2}(f)-1\Bigr{)}^{1/2}\Bigr{)}^{-1}$
is a constant, $K(f)$ is the maximal dilatation of $f$, $\Sigma_{h}(t)$ is a
$h$-sphere in the manifold ${\cal M}$, $\lambda_{n}(U)$ is a fundamental
frequency of an open subset $U\subset\Sigma_{h}(t)$, and
$\lambda_{n}(\Sigma_{h}(t);1)=\inf\lambda_{n}(U)$ where the infimum is taken
over all open sets $U\subset\Sigma_{h}(t)$ with $U\neq\Sigma_{h}(t)$. (See
Sections 4 and 6.)
1.4. Theorem. Let ${\cal M},{\cal N}$ be $n$-dimensional noncompact Riemannian
manifolds without boundary. Assume that $h:{\cal M}\to(0,\infty)$ is a special
exhaustion function of the manifold ${\cal M}$ and $u$ is a nonnegative growth
function on the manifold ${\cal N}$, which is a subsolution of an equation
(3.3) with the structure conditions (3.1), (3.2) and the structure constants
$p=n$, $\nu_{1}$, $\nu_{2}$.
Let $f:{\cal M}\to{\cal N}$ be a quasiregular mapping and
$M(\tau)=\max_{\Sigma_{h}(\tau)}u(f(m))$. If for some $\gamma>0$ the mapping
$f$ satisfies the condition
$\liminf_{\tau\to\infty}M(\tau+1)\exp\Bigl{\\{}-\gamma\int\limits^{\tau}\lambda_{n}(\Sigma_{h}(t);1)\,dt\Bigr{\\}}=0,$
(1.5)
then for each $k=1,2,\ldots$ there exists an $h$-sphere $\Sigma_{h}(t_{k})$
and an open set $U\subset\Sigma_{h}(t_{k})$, for which
$u(f)|_{U}\geq k\quad\hbox{and}\quad\lambda_{n}(U)<{n\gamma\over
C}\,\lambda_{n}(\Sigma_{h}(t_{k});1).$ (1.6)
The proofs of these results are based upon Phragmén-Lindelöf’s and Ahlfors
theorems for differential forms of ${\cal WT}$–classes obtained in [16].
For $n$-harmonic functions on abstract cones similar theorems were obtained in
[15].
Our notation is as in [4] and [16]. We assume that the results of [16] are
known to the reader and we only recall some results on qr-mappings.
## 2 Quasiregular mappings
Let ${\cal M}$ and ${\cal N}$ be Riemannian manifolds of dimension $n$. A
continuous mapping $F:{\cal M}\to{\cal N}$ of the class $W_{n,{\rm
loc}}^{1}({\cal M})$ is called a quasiregular mapping if $F$ satisfies
$|F^{\prime}(m)|^{n}\leq KJ_{F}(m)$ (2.1)
almost everywhere on ${\cal M}$. Here $F^{\prime}(m):T_{m}({\cal M})\to
T_{F(m)}({\cal N})$ is the formal derivative of $F(m)$, further,
$|F^{\prime}(m)|=\max_{|h|=1}|F^{\prime}(m)h|$. We denote by $J_{F}(m)$ the
Jacobian of $F$ at the point $m\in{\cal M}$, i.e. the determinant of
$F^{\prime}(m)$.
The best constant $K\geq 1$ in the inequality (2.1) is called the outer
dilatation of $F$ and denoted by $K_{O}(F)$. If $F$ is quasiregular then the
least constant $K\geq 1$ for which we have
$J_{F}(m)\leq Kl(F^{\prime}(m))^{n}$ (2.2)
almost everywhere on ${\cal M}$ is called the inner dilatation and denoted by
$K_{I}(F)$. Here
$l(F^{\prime}(m))=\min_{|h|=1}|F^{\prime}(m)h|.$
The quantity
$K(F)=\max\\{K_{O}(F),K_{I}(F)\\}$
is called the maximal dilatation of $F$ and if $K(F)\leq K$ then the mapping
$F$ is called $K$-quasiregular.
If $F:{\cal M}\to{\cal N}$ is a quasiregular homeomorphism then the mapping
$F$ is called quasiconformal. In this case the inverse mapping $F^{-1}$ is
also quasiconformal in the domain $F({\cal M})\subset{\cal N}$ and
$K(F^{-1})=K(F)$.
Let ${\cal A}$ and ${\cal B}$ be Riemannian manifolds of dimensions $\dim{\cal
A}=k$, $\dim{\cal B}=n-k$, $1\leq k<n$, and with scalar products
$\langle\,,\rangle_{A}$, $\langle\,,\rangle_{B}$, respectively. The Cartesian
product ${\cal N}={\cal A}\times{\cal B}$ has the natural structure of a
Riemannian manifold with the scalar product
$\langle\,,\rangle=\langle\,,\rangle_{{\cal A}}+\langle\,,\rangle_{{\cal B}}.$
We denote by $\pi:{\cal A}\times{\cal B}\to{\cal A}$ and $\eta:{\cal
A}\times{\cal B}\to{\cal B}$ the natural projections of the manifold ${\cal
N}$ onto submanifolds.
If $w_{{\cal A}}$ and $w_{{\cal B}}$ are volume forms on ${\cal A}$ and ${\cal
B}$, respectively, then the differential form $w_{{\cal N}}=\pi^{*}w_{{\cal
A}}\wedge\eta^{*}w_{{\cal B}}$ is a volume form on ${\cal N}$.
2.3. Theorem[4]. Let $F:{\cal M}\to{\cal N}$ be a quasiregular mapping and let
$f=\pi\circ F:{\cal M}\to{\cal A}$. Then the differential form $f^{*}w_{{\cal
A}}$ is of the class ${\cal W}{\cal T}_{2}$ on ${\cal M}$ with the structure
constants $p=n/k$, ${\nu}_{1}={\nu}_{1}(n,k,K_{O})$ and
${\nu}_{2}={\nu}_{2}(n,k,K_{O})$.
2.4. Remark. The structure constants can be chosen to be
${\nu}_{1}^{-1}=(k+{n-k\over\overline{c}^{2}})^{-n/2}n^{n/2}\,K_{O},\quad{\nu}_{2}^{-1}=\underline{c}^{n-k}\,,$
where $\overline{c}=\overline{c}(k,n,K_{O})$ and
$\underline{c}=\underline{c}(k,n,K_{O})$ are, respectively, the greatest and
smallest positive roots of the equation
$(k\xi^{2}+(n-k))^{n/2}-n^{n/2}\,K_{O}\,\xi^{k}=0.$ (2.5)
## 3 Domains of growth
Let $D\subset{\bf C}$ be an unbounded domain and let $w=f(z)$ be a holomorphic
function continuous on the closure $\overline{D}$. The Phragmén–Lindelöf
principle [18] traditionally refers to the alternatives of the following type:
$\alpha)$ If ${\,\rm Re}f(z)\leq 1$ everywhere on $\partial D$, then either
${\,\rm Re}f(z)$ grows with a certain rate as $z\to\infty$, or ${\,\rm
Re}f(z)\leq 1$ for all $z\in D$;
$\beta)$ If $|f(z)|\leq 1$ on $\partial D$, then either $|f(z)|$ grows with a
certain rate as $|z|\to\infty$ or $|f(z)|\leq 1$ for all $z\in D$.
Here the rate of growth of the quantities ${\,\rm Re}f(z)$ and $|f(z)|$
depends on the ”width” of the domain $D$ near infinity.
It is not difficult to prove that these conditions are equivalent with the
following conditions:
$\alpha_{1})$ If ${\,\rm Re}f(z)=1$ on $\partial D$ and ${\,\rm Re}f(z)\geq 1$
in $D$, then either ${\,\rm Re}f(z)$ grows with a certain rate as $z\to\infty$
or $f\equiv{\rm const}$;
$\beta_{1})$ If $|f(z)|=1$ on $\partial D$ and $|f(z)|\geq 1$ in $D$ then
either $|f(z)|$ grows with a certain rate as $z\to\infty$ or $f\equiv{\rm
const}$.
Let $D$ be an unbounded domain in ${\bf R}^{n}$ and let
$f=(f_{1},f_{2},\ldots,f_{n}):D\to{\bf R}^{n},$ be a quasiregular mapping. We
assume that $f\in C^{0}(\overline{D})$. It is natural to consider the
Phragmén–Lindelöf alternative under the following assumptions:
$a)$ $f_{1}(x)|_{\partial D}=1$ and $f_{1}(x)\geq 1$ everywhere in $D$,
$b)$ $\sum\limits_{i=1}^{p}f_{i}^{2}(x)|_{\partial D}=1$ and
$\sum\limits_{i=1}^{p}f_{i}^{2}(x)\geq 1$ on $D$, $1<p<n$,
$c)$ $|f(x)|=1$ on $\partial D$ and $|f(x)|\geq 1$ on $D$.
Several formulations of the Phragmén–Lindelöf theorem under various
assumptions can be found in [17], [21], [6], [14], [13]. However, these
results are mainly of qualitative character. Here we give a new approach to
Phragmén–Lindelöf type theorems for quasiregular mappings, based on
isoperimetry, that leads to almost sharp results. Our approach can be used to
prove Phragmén–Lindelöf type results for quasiregular mappings of Riemannian
manifolds.
Let ${\cal N}$ be an $n$-dimensional noncompact Riemannian $C^{2}$-manifold
with piecewise smooth boundary $\partial{\cal N}$ (possibly empty). A function
$u\in C^{0}(\overline{{\cal N}})\cap W_{n,\rm loc}^{1}({\cal N})$ is called a
growth function with ${\cal N}$ as a domain of growth if (i) $u\geq 1,$ (ii)
$u|\partial{\cal N}=1$ if $\partial{\cal N}\neq\emptyset,$ and
$\sup_{y\in{\cal N}}u(y)=+\infty.$
We consider a quasiregular mapping $f:{\cal M}\to{\cal N}$, $f\in C^{0}({\cal
M}\cup\partial M)$, where ${\cal M}$ is a noncompact Riemannian
$C^{2}$-manifold, $\dim{\cal M}=n$ and $\partial{\cal M}\neq\emptyset$. We
assume that $f(\partial{\cal M})\subset\partial{\cal N}$. In what follows we
mean by the Phragmén–Lindelöf principle an alternative of the form: either the
function $u(f(m))$ has a certain rate of growth in ${\cal M}$ or $f(m)\equiv
const$.
By choosing the domain of growth ${\cal N}$ and the growth function $u$ in a
special way we can obtain several formulations of Phragmén–Lindelöf theorems
for quasiregular mappings. In view of the examples in [17], the best results
are obtained if an $n$-harmonic function is chosen as a growth function. In
the case a) the domain of growth is ${\cal N}=\\{y=(y_{1},\ldots,y_{n})\in{\bf
R}^{n}:y_{1}\geq 0\\}$ and as the function of growth it is natural to choose
$u(y)=y_{1}+1$; in the case b) the domain ${\cal N}$ is the set
$\\{y=(y_{1},\ldots,y_{n})\in{\bf R}^{n}:\sum_{i=1}^{p}y_{i}^{2}\geq 1\\}$,
$1<p<n$, and $u(y)=(\sum_{i=1}^{p}y_{i}^{2})^{(n-p)/(2(n-1))}$; in the case c)
the domain of growth is ${\cal N}=\\{y\in{\bf R}^{n}:|y|>1\\}$ and
$u(y)=\log|y|+1$.
In the general case we shall consider growth functions which are $A$-solutions
of elliptic equations [8]. Namely, let ${\cal M}$ be a Riemannian manifold and
let
$A:T({\cal M})\to T({\cal M})$
be a mapping defined a.e. on the tangent bundle $T({\cal M}).$ Suppose that
for a.e. $m\in{\cal M}$ the mapping $A$ is continuous on the fiber $T_{m},$
i.e. for a.e. $m\in{\cal M}$ the function $A(m,\cdot):T_{m}\to T_{m}$ is
defined and continuous; the mapping $m\mapsto A_{m}(X)$ is measurable for all
measurable vector fields $X$ (see [8]).
Suppose that for a.e. $m\in{\cal M}$ and for all $\xi\in T_{m}$ the
inequalities
$\nu_{1}\,|\xi|^{p}\leq\langle\xi,A(m,\xi)\rangle,$ (3.1)
and
$|A(m,\xi)|\leq\nu_{2}\,|\xi|^{p-1}$ (3.2)
hold with $p>1$ and for some constants $\nu_{1},\nu_{2}>0$. It is clear that
we have $\nu_{1}\leq\nu_{2}$.
We consider the equation
${\rm div}\,\,A(m,\nabla f)=0.$ (3.3)
Solutions to (3.3) are understood in the weak sense, that is, $A$-solutions
are $W_{p,loc}^{1}$-functions satisfying the integral identity
$\int\limits_{{\cal M}}\langle\nabla\theta,A(m,\nabla f)\rangle*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}_{{\cal M}}=0$ (3.4)
for all $\theta\in W_{p}^{1}({\cal M})$ with compact support in ${\cal M}$.
A function $f$ in $W_{p,loc}^{1}({\cal M})$ is a $A$-subsolution of (3.3) in
${\cal M}$ if
${\rm div}\,\,A(m,\nabla f)\geq 0$ (3.5)
weakly in ${\cal M}$, i.e.
$\int\limits_{{\cal M}}\langle\nabla\theta,A(m,\nabla f)\rangle*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}_{{\cal M}}\leq 0$ (3.6)
whenever $\theta\in W^{1}_{p}({\cal M})$, is nonnegative with compact support
in ${\cal M}$.
A basic example of such an equation is the $p$-Laplace equation
${\rm div}\,(|\nabla f|^{p-2}\nabla f)=0.$ (3.7)
## 4 Exhaustion functions
Below we introduce exhaustion and special exhaustion functions on Riemannian
manifolds and give illustrating examples.
4.1. Exhaustion functions of boundary sets . Let $h:{\cal M}\to(0,h_{0})$,
$0<h_{0}\leq\infty$, be a locally Lipschitz function such that
${\rm ess}\,\inf_{Q}|\nabla h|>0\quad\forall\quad Q\subset\subset{\cal M}\,.$
(4.2)
For arbitrary $t\in(0,h_{0})$ we denote by
$B_{h}(t)=\\{m\in{\cal M}:h(m)<t\\},\quad\Sigma_{h}(t)=\\{m\in{\cal
M}:h(m)=t\\}$
the $h$-balls and $h$-spheres, respectively.
Let $h:{\cal M}\to{\bf R}$ be a locally Lipschitz function such that there
exists a compact $K\subset{\cal M}$ with $|\nabla h(x)|>0$ for a.e. $m\in{\cal
M}\setminus K$. We say that the function $h$ is an exhaustion function for a
boundary set $\Xi$ of ${\cal M}$ if for an arbitrary sequence of points
$m_{k}\in{\cal M}$, $k=1,2,\ldots$ the function $h(m_{k})\to h_{0}$ if and
only if $m_{k}\to\xi$.
It is easy to see that this requirement is satisfied if and only if for an
arbitrary increasing sequence $t_{1}<t_{2}<\ldots<h_{0}$ the sequence of the
open sets $V_{k}=\\{m\in{\cal M}:h(m)>t_{k}\\}$ is a chain, defining a
boundary set $\xi$. Thus the function $h$ exhausts the boundary set $\xi$ in
the traditional sense of the word.
The function $h:{\cal M}\to(0,h_{0})$ is called the exhaustion function of the
manifold ${\cal M}$ if the following two conditions are satisfied
(i) for all $t\in(0,h_{0})$ the $h$–ball $\overline{B_{h}(t)}$ is compact;
(ii) for every sequence $t_{1}<t_{2}<\ldots<h_{0}$ with
$\lim\nolimits_{k\to\infty}t_{k}=h_{0}$, the sequence of $h$-balls
$\\{B_{h}(t_{k})\\}$ generates an exhaustion of ${\cal M}$, i.e.
$B_{h}(t_{1})\subset B_{h}(t_{2})\subset\ldots\subset
B_{h}(t_{k})\subset\ldots\quad\mbox{and}\quad\cup_{k}B_{h}(t_{k})={\cal M}.$
4.3. Example. Let ${\cal M}$ be a Riemannian manifold. We set
$h(m)=\mbox{dist}(m,m_{0})$ where $m_{0}\in{\cal M}$ is a fixed point. Because
$|\nabla h(m)|=1$ almost everywhere on ${\cal M}$, the function $h$ defines an
exhaustion function of the manifold ${\cal M}$.
4.4. Special exhaustion functions . Let ${\cal M}$ be a noncompact Riemannian
manifold with the boundary $\partial{\cal M}$ (possibly empty). Let $A$
satisfy (3.1) and (3.2) and let $h:{\cal M}\to(0,h_{0})$ be an exhaustion
function, satisfying the following additional conditions:
$a_{1})$ there is $h^{\prime}>0$ such that $h^{-1}((0,h^{\prime}))$ is compact
and $h$ is a solution of (3.3) in the open set $K=h^{-1}((h^{\prime},h_{0}));$
$a_{2})$ for a.e. $t_{1},\;t_{2}\in(h^{\prime},h_{0})$, $t_{1}<t_{2}$,
$\int\limits_{\Sigma_{h}(t_{2})}\langle{{\nabla h}\over{|\nabla
h|}},A(x,\nabla h)\rangle\,d{\cal
H}^{n-1}=\int\limits_{\Sigma_{h}(t_{1})}\langle{{\nabla h}\over{|\nabla
h|}},A(x,\nabla h)\rangle\,d{\cal H}^{n-1}.$
Here $d{\cal H}^{n-1}$ is the element of the $(n-1)-$dimensional Hausdorff
measure on $\Sigma_{h}.$ Exhaustion functions with these properties will be
called the special exhaustion functions of ${\cal M}$ with respect to $A$. In
most cases the mapping $A$ will be the $p-$Laplace operator (3.7) and, unless
otherwise stated, $A$ is the $p$-Laplace operator.
Since the unit vector $\nu={{\nabla h}/{|\nabla h|}}$ is orthogonal to the
$h$–sphere $\Sigma_{h}$, the condition $a_{2})$ means that the flux of the
vector field $A(m,\nabla h)$ through $h$–spheres $\Sigma_{h}(t)$ is constant.
In the following we consider domains $D$ in ${\bf R}^{n}$ as manifolds ${\cal
M}$. However, the boundaries $\partial D$ of $D$ are allowed to be rather
irregular. To handle this situation we introduce $(A,h)$-transversality
property for ${\cal M}$.
Let $h:{{\cal M}}\to(0,h_{0})$ be a $C^{2}$-exhaustion function. We say that
${\cal M}$ satisfies the $(A,h)$-transversality property if for a.e.
$t_{1},t_{2},$ $h<t_{1}<t_{2}<h_{0}$, and for every $\varepsilon>0$ there
exists an open set
$G=G_{\varepsilon}(t_{1},t_{2})\subset
B_{h}(t_{2})\setminus\overline{B}_{h}(t_{1})$
with piecewise regular boundary such that
${\cal H}^{n-1}\left(\Sigma_{h}(t_{1})\cap\Sigma_{h}(t_{2})\setminus\partial
G\right)<\varepsilon\,,$ (4.5) ${\cal
H}^{n}\left(\left(B_{h}(t_{2})\setminus\overline{B}_{h}(t_{1})\right)\setminus
G\right)<\varepsilon,$ (4.6) $\langle A(m,\nabla h(m),v)\rangle=0$ (4.7)
where $v$ is the unit inner normal to $\partial G$.
We say that ${\cal M}$ satisfies the $h$-transversality condition if ${\cal
M}$ satisfies the $(A,h)$-transversality condition for the $p$-Laplace
operator $A(m,\xi)=|\xi|^{p-2}\xi$. In this case (4.7) reduces to
$\langle\nabla h(m),v\rangle=0\,.$ (4.8)
4.9. Example. Let $D$ be a bounded domain in ${\bf R}^{2}$ and let
${{\cal M}}=\\{(x_{1},x_{2},x_{3})\in{\bf R}^{3}:(x_{1},x_{2})\in
D,x_{3}>0\\}$
be a cylinder with base $D$. The function $h:(0,\infty)\to{\bf R}$,
$h(x)=x_{3}$, is an exhaustion function for ${\cal M}$. Since every domain $D$
in ${\bf R}^{2}$ can be approximated by smooth domains $D^{\prime}$ from
inside, it is easy to see that for $0<t_{1}<t_{2}<\infty$ the domain
$G=D^{\prime}\times(t_{1},t_{2})$ can be used as an approximating domain
$G_{\varepsilon}(t_{1},t_{2})$. Note that the transversality condition (4.7)
is automatically satisfied for the $p$-Laplace operator
$A(m,\xi)=|\xi|^{p-2}\xi$
4.10. Lemma. Suppose that an exhaustion function $h\in C^{2}({{\cal
M}}\setminus K)$ satisfies the equation (3.3) in ${{\cal M}}\setminus K$ and
that the function $A(m,\xi)$ is continuously differentiable. If ${\cal M}$
satisfies the $(A,h)$-transversality condition, then $h$ is a special
exhaustion function on the manifold ${\cal M}$.
Proof. It suffices to show $a_{2})$. Let $h^{\prime}<t_{1}<t_{2}<h_{0}$ and
$\varepsilon>0$. Choose an open set $G$ as in the definition of the
$(A,h)$-transversality condition. $|A(m,\nabla h(m))|\leq M<\infty$ for every
$m\in{\cal M}$, and (4.5) - (4.7) together with the Gauss formula imply for
a.e. $t_{1},t_{2}$
$\begin{array}[]{ll}&\left|\displaystyle\int\limits_{\Sigma_{h}(t_{2})}\displaystyle\langle{{\nabla
h}\over{|\nabla h|}},A(m,\nabla h)\displaystyle\rangle d{\cal
H}^{n-1}-\displaystyle\int\limits_{\Sigma_{h}(t_{1})}\displaystyle\langle{{\nabla
h}\over{|\nabla h|}},A(m,\nabla h)\displaystyle\rangle d{\cal
H}^{n-1}\right|\leq\\\ \\\ &\leq\left|\displaystyle\int\limits_{\partial
G\cup\Sigma_{h}(t_{2})}\displaystyle\langle{{\nabla h}\over{|\nabla
h|}},A(m,\nabla h)\displaystyle\rangle d{\cal
H}^{n-1}-\displaystyle\int\limits_{\partial
G\cup\Sigma_{h}(t_{1})}\displaystyle\langle{{\nabla h}\over{|\nabla
h|}},A(m,\nabla h)\displaystyle\rangle d{\cal H}^{n-1}\right|+\varepsilon
M=\\\ \\\ &=\left|\displaystyle\int\limits_{\partial
G}\displaystyle\langle{{\nabla h}\over{|\nabla h|}},A(m,\nabla
h)\displaystyle\rangle d{\cal H}^{n-1}\right|+\varepsilon
M=\left|\displaystyle\int\limits_{\partial G}\displaystyle\langle v,A(m,\nabla
h)\displaystyle\rangle d{\cal H}^{n-1}\right|+\varepsilon M=\\\ \\\
&=\left|\displaystyle\int\limits_{G}\displaystyle{\rm div}\,A(m,\nabla
h)d{\cal H}^{n}\right|+\varepsilon M=\varepsilon M\,.\\\ \\\ \end{array}$
Since $\varepsilon>0$ is arbitrary, $a_{2})$ follows. $\Box$
4.11. Example. Fix $1\leq n\leq p$. Let $x_{1},x_{2},\ldots,x_{n}$ be an
orthonormal system of coordinates in ${\bf R}^{n},$ $1\leq n<p$. Let
$D\subset{\bf R}^{n}$ be an unbounded domain with piecewise smooth boundary
and let ${\cal B}$ be an $(p-n)$-dimensional compact Riemannian manifold with
or without boundary. We consider the manifold ${\cal M}=D\times{\cal B}$.
We denote by $x\in D$, $b\in{\cal B}$, and $(x,b)\in{\cal M}$ the points of
the corresponding manifolds. Let $\pi:D\times{\cal B}\to D$ and
$\eta:D\times{\cal B}\to{\cal B}$ be the natural projections of the manifold
${\cal M}$.
Assume now that the function $h$ is a function on the domain $D$ satisfying
the conditions $b_{1})$, $b_{2})$ and the equation (3.7). We consider the
function $h^{*}=h\circ\pi:{\cal M}\to(0,\infty)$.
We have
$\nabla h^{*}=\nabla(h\circ\pi)=(\nabla_{x}h)\circ\pi$
and
${\rm div}\,(|\nabla h^{*}|^{p-2}\nabla h^{*})={\rm
div}\,\bigl{(}|\nabla(h\circ\pi)|^{p-2}\nabla(h\circ\pi)\bigr{)}=$ $={\rm
div}\,\bigl{(}|\nabla_{x}h|^{p-2}\circ\pi(\nabla_{x}h)\circ\pi\bigr{)}=\Bigl{(}\sum_{i=1}^{n}{\partial\over\partial
x_{i}}\bigl{(}|\nabla_{x}h|^{p-2}{\partial h\over\partial
x_{i}}\bigr{)}\Bigr{)}\circ\pi.$
Because $h$ is a special exhaustion function of $D$ we have
${\rm div}\,(|\nabla h^{*}|^{p-2}\nabla h^{*})=0.$
Let $(x,b)\in\partial{\cal M}$ be an arbitrary point where the boundary
$\partial{\cal M}$ has a tangent hyperplane and let $\nu$ be a unit normal
vector to $\partial{\cal M}$.
If $x\in\partial D$, then $\nu=\nu_{1}+\nu_{2}$ where the vector
$\nu_{1}\in{\bf R}^{k}$ is orthogonal to $\partial D$ and $\nu_{2}$ is a
vector from $T_{b}({\cal B})$. Thus
$\langle\nabla
h^{*},\nu\rangle=\langle(\nabla_{x}h)\circ\pi,\nu_{1}\rangle=0,$
because $h$ is a special exhaustion function on $D$ and satisfies the property
$b_{2})$ on $\partial D$. If $b\in\partial{\cal B}$, then the vector $\nu$ is
orthogonal to $\partial{\cal B}\times{\bf R}^{n}$ and
$\langle\nabla h^{*},\nu\rangle=\langle(\nabla_{x}h)\circ\pi,\nu\rangle=0,$
because the vector $(\nabla_{x}h)\circ\pi$ is parallel to ${\bf R}^{n}$.
The other requirements for a special exhaustion function for the manifold
${\cal M}$ are easy to verify.
Therefore, the function
$h^{*}=h^{*}(x,b)=h\circ\pi:{\cal M}\to(0,\infty)$ (4.12)
is a special exhaustion function on the manifold ${\cal M}=D\times{\cal B}$.
4.13. Example. We fix an integer $k$, $1\leq k\leq n,$ and set
$d_{k}(x)=\Bigl{(}\sum\limits_{i=1}^{k}x_{i}^{2}\Bigr{)}^{1/2}\,.$
It is easy to see that $|\nabla d_{k}(x)|=1$ everywhere in ${\bf
R}^{n}\setminus\Sigma_{0}$ where $\Sigma_{0}=\\{x\in{\bf
R}^{n}:d_{k}(x)=0\\}$. We shall call the set
$B_{k}(t)=\\{x\in{\bf R}^{n}:d_{k}(x)<t\\}$
a $k$-ball and the set
$\Sigma_{k}(t)=\\{x\in{\bf R}^{n}:d_{k}(x)=t\\}$
a $k$-sphere in ${\bf R}^{n}$.
We shall say that an unbounded domain $D\subset{\bf R}^{n}$ is $k$-admissible
if for each $t>\inf_{x\in D}d_{k}(x)$ the set $D\cap B_{k}(t)$ has compact
closure.
It is clear that every unbounded domain $D\subset{\bf R}^{n}$ is
$n$-admissible. In the general case the domain $D$ is $k$-admissible if and
only if the function $d_{k}(x)$ is an exhaustion function of $D$. It is not
difficult to see that if a domain $D\subset{\bf R}^{n}$ is $k$-admissible,
then it is $l$-admissible for all $k<l<n$.
Fix $1\leq k<n$. Let $\Delta$ be a bounded domain in the $(n-k)$-plane
$x_{1}=\ldots=x_{k}=0$ and let
$D=\\{x=(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{n})\in{\bf
R}^{n}:(x_{k+1},\ldots,x_{n})\in\Delta\\}\,.$
The domain $D$ is $k$-admissible. The $k$-spheres $\Sigma_{k}(t)$ are
orthogonal to the boundary $\partial D$ and therefore $\langle\nabla
d_{k},\nu\rangle=0$ everywhere on the boundary. The function
$h(x)=\cases{\log d_{k}(x),&$p=k$,\cr d_{k}^{(p-k)/(p-1)}(x),&$p\neq k$,\cr}$
satisfies (3.3). By Lemma 4 the function $h$ is a special exhaustion function
of the domain $D$. Therefore the domain $D$ has $p$-parabolic type for $p\geq
k$ and $p$-hyperbolic type for $p<k$.
4.14. Example. Fix $1\leq k<n$. Let $\Delta$ be a bounded domain in the plane
$x_{1}=\ldots=x_{k}=0$ with a (piecewise) smooth boundary and let
$D=\\{x=(x_{1},\ldots,x_{n})\in{\bf
R}^{n}:(x_{k+1},\ldots,x_{n})\in\Delta\\}={\bf R}^{n-k}\times\Delta$ (4.15)
be the cylinder domain with base $\Delta.$
The domain $D$ is $k$-admissible. The $k$-spheres $\Sigma_{k}(t)$ are
orthogonal to the boundary $\partial D$ and therefore $\langle\nabla
d_{k},\nu\rangle=0$ everywhere on the boundary, where $d_{k}$ is as in Example
4.
Let $h=\phi(d_{k})$ where $\phi$ is a $C^{2}-$function with $\phi^{\prime}\geq
0$. We have $\nabla h=\phi^{\prime}\;\nabla d_{k}$ and since $|\nabla
d_{k}|=1$, we obtain
$\sum_{i=1}^{n}{{\partial}\over{\partial x_{i}}}\Bigl{(}|\nabla
h|^{n-2}\;{{\partial h}\over{\partial
x_{i}}}\Bigr{)}=\sum_{i=1}^{k}{{\partial}\over{\partial
x_{i}}}\Bigl{(}(\phi^{\prime})^{n-1}\;{{\partial d_{k}}\over{\partial
x_{i}}}\Bigr{)}$
$=(n-1)\;(\phi^{\prime})^{n-2}\;\phi^{\prime\prime}+{{k-1}\over{d_{k}}}\;(\phi^{\prime})^{n-1}.$
From the equation
$(n-1)\;\phi^{\prime\prime}+{{k-1}\over{d_{k}}}\;\phi^{\prime}=0$
we conclude that the function
$h(x)=\bigl{(}d_{k}(x)\bigr{)}^{{n-k}\over{n-1}}$ (4.16)
satisfies the equation (3.7) in $D\setminus K$ and thus it is a special
exhaustion function of the domain $D.$
4.17. Example. Let $(r,\theta)$, where $r\geq 0$, $\theta\in S^{n-1}(1)$, be
the spherical coordinates in ${{\bf R}}^{n}$. Let $U\subset S^{n-1}(1)$,
$\partial U\neq\emptyset,$ be an arbitrary domain with a piecewise smooth
boundary on the unit sphere $S^{n-1}(1)$. We fix $0\leq r_{1}<\infty$ and
consider the domain
$D=\\{(r,\theta)\in{{\bf R}}^{n}:r_{1}<r<\infty,\;\theta\in U\\}.$ (4.18)
As above it is easy to verify that the given domain is $n$–admissible and the
function
$h(|x|)=\log{{|x|}\over{r_{1}}}$ (4.19)
is a special exhaustion function of the domain $D$ for $p=n$.
4.20. Example. Let ${\cal A}$ be a compact Riemannian manifold, $\dim{\cal
A}=k,$ with piecewise smooth boundary or without boundary. We consider the
Cartesian product ${\cal M}={\cal A}\times{\bf R}^{n}$, $n\geq 1$. We denote
by $a\in{\cal A}$, $x\in{\bf R}^{n}$ and $(a,x)\in{\cal M}$ the points of the
corresponding spaces. It is easy to see that the function
$h(a,x)=\cases{\log|x|,&$p=n$,\cr|x|^{p-n\over p-1},&$p\neq n$,\cr}$
is a special exhaustion function for the manifold ${\cal M}$. Therefore, for
$p\geq n$ the given manifold has $p$-parabolic type and for $p<n$
$p$-hyperbolic type.
4.21. Example. Let $(r,\theta)$, where $r\geq 0$, $\theta\in S^{n-1}(1)$, be
the spherical coordinates in ${\bf R}^{n}$. Let $U\subset S^{n-1}(1)$ be an
arbitrary domain on the unit sphere $S^{n-1}(1)$. We fix $0\leq
r_{1}<r_{2}<\infty$ and consider the domain
$D=\\{(r,\theta)\in{\bf R}^{n}:r_{1}<r<r_{2},\;\theta\in U\\}$
with the metric
$ds^{2}_{\cal M}=\alpha^{2}(r)dr^{2}+\beta^{2}(r)dl_{\theta}^{2},$ (4.22)
where $\alpha(r),\,\beta(r)>0$ are $C^{0}$-functions on $[r_{1},r_{2})$ and
$dl_{\theta}$ is an element of length on $S^{n-1}(1)$.
The manifold ${\cal M}=(D,ds^{2}_{\cal M})$ is a warped Riemannian product. In
the case $\alpha(r)\equiv 1$, $\beta(r)=1$, and $U=S^{n-1}$ the manifold
${\cal M}$ is isometric to a cylinder in ${\bf R}^{n+1}$. In the case
$\alpha(r)\equiv 1$, $\beta(r)=r$, $U=S^{n-1}$ the manifold ${\cal M}$ is a
spherical annulus in ${\bf R}^{n}$.
The volume element in the metric (4.22) is given by the expression
$d\sigma_{\cal M}=\alpha(r)\,\beta^{n-1}(r)\,dr\,dS^{n-1}(1).$
If $\phi(r,\theta)\in C^{1}(D)$, then the length of the gradient $\nabla\phi$
in ${\cal M}$ takes the form
$|\nabla\phi|^{2}={1\over\alpha^{2}}(\phi^{\prime}_{r})^{2}+{1\over\beta^{2}}|\nabla_{\theta}\phi|^{2},$
where $\nabla_{\theta}\phi$ is the gradient in the metric of the unit sphere
$S^{n-1}(1)$.
For the special exhaustion function $h(r,\theta)\equiv h(r)$ the equation
(3.7) reduces to the following form
${d\over
dr}\left(\Bigl{(}{1\over\alpha(r)}\Bigr{)}^{p-1}\bigl{(}h^{\prime}_{r}(r)\bigr{)}^{p-1}\beta^{n-1}(r)\right)=0.$
Solutions of this equation are the functions
$h(r)=C_{1}\int\limits_{r_{1}}^{r}{\alpha(t)\over\beta^{n-1\over
p-1}(t)}\,dt+C_{2}$
where $C_{1}$ and $C_{2}$ are constants.
Because the function $h$ satisfies obviously the boundary condition $a)_{2}$
as well as the other conditions of (4), we see that under the assumption
$\int\limits^{r_{2}}{\alpha(t)\over\beta^{n-1\over p-1}(t)}\,dt=\infty$ (4.23)
the function
$h(r)=\int\limits_{r_{1}}^{r}{\alpha(t)\over\beta^{n-1\over p-1}(t)}\,dt$
(4.24)
is a special exhaustion function on the manifold ${\cal M}$.
4.25. Theorem. Let $h:{\cal M}\to(0,h_{0})$ be a special exhaustion function
of a boundary set $\xi$ of the manifold ${\cal M}$. Then
(i) if $h_{0}=\infty$, the set $\xi$ has $p$-parabolic type,
(ii) if $h_{0}<\infty$, the set $\xi$ has $p$-hyperbolic type.
Proof. Choose $0<t_{1}<t_{2}<h_{0}$ such that $K\subset B_{h}(t_{1})$. We need
to estimate the $p$-capacity of the condenser $(B_{h}(t_{1}),{\cal M}\setminus
B_{h}(t_{2});{\cal M})$. We have
$\hbox{\rm cap}_{p}(\overline{B}_{h}(t_{1}),{\cal M}\setminus
B_{h}(t_{2});{\cal M})={J\over(t_{2}-t_{1})^{p-1}}$ (4.26)
where
$J=\int\limits_{\Sigma_{h}(t)}|\nabla h|^{p-1}d{\cal H}_{{\cal M}}^{n-1}$
is a quantity independent of $t>h(K)=\sup\\{h(m):m\in K\\}$. Indeed, for the
variational problem [16, (2.9)] we choose the function $\varphi_{0}$,
$\varphi_{0}(m)=0$ for $m\in B_{h}(t_{1})$,
$\varphi_{0}(m)={h(m)-t_{1}\over t_{2}-t_{1}},\ m\in B_{h}(t_{2})\setminus
B_{h}(t_{1})$
and $\varphi_{0}(m)=1$ for $m\in{\cal M}\setminus B_{h}(t_{2})$. Using the
Kronrod–Federer formula [3, Theorem 3.2.22], we get
$\begin{array}[]{ll}\hbox{\rm cap}_{p}(B_{h}(t_{1}),{\cal M}\setminus
B_{h}(t_{2});{\cal M})&\leq\displaystyle\int\limits_{{\cal
M}}|\nabla\varphi_{0}|^{p}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}_{{\cal
M}}\leq\\\ \\\
&\leq{1\over(t_{2}-t_{1})^{p}}\displaystyle\int\limits_{t_{1}<h(m)<t_{2}}|\nabla
h(m)|^{p}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}_{{\cal M}}=\\\ \\\
&=\displaystyle\int\limits_{t_{1}}^{t_{2}}dt\displaystyle\int\limits_{\Sigma_{h}(t)}|\nabla
h(m)|^{p-1}d{\cal H}_{{\cal M}}^{n-1}.\\\ \\\ \end{array}$
Because the special exhaustion function satisfies the equation (3.7) and the
boundary condition $a)_{2}$, one obtains for arbitrary $\tau_{1},\tau_{2}$,
$h(K)<\tau_{1}<\tau_{2}<h_{0}$
$\int\limits_{\Sigma_{h}(t_{2})}|\nabla h|^{p-1}d{\cal H}_{{\cal
M}}^{n-1}-\int\limits_{\Sigma_{h}(t_{1})}|\nabla h|^{p-1}d{\cal H}_{{\cal
M}}^{n-1}=$ $=\int\limits_{\Sigma_{h}(t_{2})}|\nabla h|^{p-2}\langle\nabla
h,\nu\rangle d{\cal H}_{{\cal M}}^{n-1}-\int\limits_{\Sigma_{h}(t_{1})}|\nabla
h|^{p-2}\langle\nabla h,\nu\rangle d{\cal H}_{{\cal M}}^{n-1}=$
$=\int\limits_{t_{1}<h(m)<t_{2}}{\rm div}\,_{{\cal M}}(|\nabla h|^{p-2}\nabla
h)*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}_{{\cal M}}=0.$
Thus we have established the inequality
$\hbox{\rm cap}_{p}(B_{h}(t_{1}),{\cal M}\setminus B_{h}(t_{2});{\cal
M})\leq{J\over(t_{2}-t_{1})^{p-1}}\,.$
By the conditions, imposed on the special exhaustion function, the function
$\varphi_{0}$ is an extremal in the variational problem [16, (2.9)]. Such an
extremal is unique and therefore the preceding inequality holds as an
equality. This conclusion proves the equation (4.26).
If $h_{0}=\infty$, then letting $t_{2}\to\infty$ in (4.26) we conclude the
parabolicity of the type of $\xi$. Let $h_{0}<\infty$. Consider an exhaustion
$\\{{\cal U}_{k}\\}$ and choose $t_{0}>0$ such that the $h$-ball
$B_{h}(t_{0})$ contains the compact set $K$.
Set $t_{k}=\sup\nolimits_{m\in\partial{\cal U}_{k}}h(m)$. Then for
$t_{k}>t_{0}$ we have
$\hbox{\rm cap}_{p}(\overline{U}_{k_{0}},{\cal U}_{k};{\cal M})\geq\hbox{\rm
cap}_{p}(B_{h}(t_{0}),B_{h}(t_{k});{\cal M})=J/(t_{k}-t_{0})^{p-1}\,,$
and hence
$\liminf_{k\to\infty}\hbox{\rm cap}_{p}(\overline{U}_{k_{0}},{\cal
U}_{k};{\cal M})\geq J/(h_{0}-t_{0})^{p-1}>0,$
and the boundary set $\xi$ has $p$-hyperbolic type. $\Box$
## 5 Wiman theorem
Now we will prove Theorem 1.
5.1. Fundamental frequency. Let $U\subset\Sigma_{h}(\tau)$ be an open set. We
need further the following quantity
$\lambda_{p}(U)=\inf{\left(\displaystyle\int\limits_{U}|\nabla
h|^{-1}|\nabla_{2}\varphi|^{p}d{\cal H}_{{\cal
M}}^{n-1}\right)^{1/p}\over\left(\displaystyle\int\limits_{U}|\nabla
h|^{p-1}|\varphi|^{p}d{\cal H}_{{\cal M}}^{n-1}\right)^{1/p}}$ (5.2)
where the infimum is taken over all functions $\varphi\in W_{p}^{1}(U)$111By
the definition, $\varphi$ is a $W^{1}_{p}$-function on an open set $U$, if $f$
belongs to this class on every component of $U$. with ${\rm
supp}\,\varphi\subset U$. Here $\nabla_{2}\varphi$ is the gradient of
$\varphi$ on the surface $\Sigma_{h}(\tau)$.
In the case $|\nabla h|\equiv 1$ this quantity is well–known and can be
interpreted, in particular, as the best constant in the Poincaré inequality.
Following [19] we shall call this quantity the fundamental frequency of the
rigidly supported membrane $U$.
Observe a useful property of the fundamental frequency.
5.3. Lemma. Let $U\subset\Sigma_{h}(\tau)$ be an open set and let $U_{i}$ be
the components of $U$, $i=1,2,\ldots$. Then
$\lambda_{p}(U)=\inf_{i}\lambda_{p}(U_{i}).$
Proof. To prove this property we fix arbitrary functions $\varphi_{i}$ with
${\rm supp}\,\varphi_{i}\subset U_{i}$. Set $\varphi(m)=\varphi_{i}(m)$ for
$m\in U_{i}$ and $\varphi=0$ for $U\setminus(\cup_{i}U_{i})$. Hence
$\lambda^{p}_{p}(U_{i})\int\limits_{U_{i}}|\nabla
h|^{p-1}\,|\varphi_{i}|^{p}d{\cal H}^{n-1}\leq\int\limits_{U_{i}}|\nabla
h|^{-1}\,|\nabla_{2}\varphi_{i}|^{p}d{\cal H}^{n-1}.$
Summation yields
$\left(\inf_{i}\lambda^{p}_{p}(U_{i})\right)\,\sum\limits_{i}\int\limits_{U_{i}}|\nabla
h|^{p-1}\,|\varphi_{i}|^{p}d{\cal
H}^{n-1}\leq\sum\limits_{i}\int\limits_{U_{i}}|\nabla
h|^{-1}\,|\nabla_{2}\varphi_{i}|^{p}d{\cal H}^{n-1}$
and we obtain
$\left(\inf_{i}\lambda^{p}_{p}(U_{i})\right)\,\int\limits_{U}|\nabla
h|^{p-1}\,|\varphi|^{p}d{\cal H}^{n-1}\leq\int\limits_{U}|\nabla
h|^{-1}\,|\nabla_{2}\varphi|^{p}d{\cal H}^{n-1}.$
This gives
$\inf_{i}\lambda_{p}(U_{i})\leq\lambda_{p}(U).$
The reverse inequality is evident. Indeed, if $U_{i}$ is a component of $U$,
then evidently
$\lambda_{p}(U)\leq\lambda_{p}(U_{i})$
and hence
$\lambda_{p}(U)\leq\inf\limits_{i}\lambda_{p}(U_{i}).\quad\Box$
We also need the following statement.
5.4. Lemma. Under the above assumptions for a.e. $\tau\in(0,h_{0})$ we have
$\varepsilon(\tau;{\cal F}_{B})\geq\lambda_{p}(\Sigma_{h}(\tau))/c,$ (5.5)
where $\lambda_{p}$ is the fundamental frequency of the membrane
$\Sigma_{h}(\tau)$ defined by formula (5.2) and
$c=c({\nu}_{1},{\nu}_{2},p)=\cases{c_{1}\quad&for $p\leq 2$\,,\cr
c_{2}\quad&for $p\geq 2$\,,\cr}$
where
$c_{1}=\sqrt{{\nu}_{2}^{2}-{\nu}_{1}^{2}}+2^{(2-p)/2}\,{\nu}_{1}p^{-1}(p-1)^{(p-1)/p}$
and
$c_{2}=\sqrt{{\nu}_{2}^{2}-{\nu}_{1}^{2}}+{\nu}_{1}\,{p-1\over p}.$
For the proof see Lemma 4.3 in [14].
We now use these estimates for proving Phragmén-Lindelöf type theorems for the
solutions of quasilinear equations on manifolds.
5.6. Theorem. Let $h:{\cal M}\to(0,\infty)$ be an exhaustion function. Suppose
that the manifold ${\cal M}$ satisfies the condition
$\int\limits^{\infty}\lambda_{p}(\Sigma_{h}(t))dt=\infty.$ (5.7)
Let $f$ be a continuous solution of the equation (3.3) with (3.1), (3.2) on
${\cal M}$ such that
$\limsup_{m\to m_{0}}f(m)\leq 0,\quad\hbox{for all}\;\,m_{0}\in\partial{\cal
M}\,.$ (5.8)
Then either $f(m)\leq 0$ everywhere on ${\cal M}$ or
$\qquad\liminf_{\tau\to\infty}\int\limits_{\tau<h(m)<\tau+1}|\nabla
h||f(m)||\nabla f(m)|^{p-1}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}}\exp\Bigl{\\{}-c_{3}\int\limits^{\tau}\lambda_{p}(\Sigma_{h}(t))dt\Bigr{\\}}>0,$
(5.9)
and
$\liminf_{\tau\to\infty}\int\limits_{\tau<h(m)<\tau+1}|\nabla
h(m)|^{p}|f(m)|^{p}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}}\exp\Bigl{\\{}-c_{3}\int\limits^{\tau}\lambda_{p}(\Sigma_{h}(t))dt\Bigr{\\}}>0.$
(5.10)
In particular, if $h$ is a special exhaustion function on ${\cal M}$, then
$\liminf_{\tau\to\infty}M(\tau+1)\exp\Bigl{\\{}-{c_{3}\over
p}\int\limits^{\tau}\lambda_{p}(\Sigma_{h}(\tau))dt\Bigr{\\}}>0.$ (5.11)
Here
$M(t)=\sup\limits_{m\in\Sigma_{h}(t)}|f(m)|$
and $c_{3}=\nu_{1}c^{-1}$ where $c$ is the constant of Lemma 5.
Proof. We assume that at some point $m_{1}\in{\rm int}\,{\cal M}$ we have
$f(m_{1})>0$. We consider the set
${\cal O}=\\{m\in{\cal M}:f(m)>f(m_{1})\\}.$
By Corollary [16, 4.57] the set ${\cal O}$ is noncompact.
The function $h$ is an exhaustion function on ${\cal O}$. Using the relation
[16, 6.74] for the function $f(m)-f(m_{1})$ on ${\cal O}$ we have
$\liminf_{\tau\to\infty}\int\limits_{{\cal O}(\tau)}|\nabla
h||f(m)-f(m_{1})||A(m,\nabla f)|*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}}\exp\Bigl{\\{}-\nu_{1}\int\limits_{\tau_{0}}^{\tau}\varepsilon(t;{\cal
F}_{{\cal O}})dt\Bigr{\\}}>0,$
where ${\cal O}(\tau)=\\{m\in{\cal O}:\tau<h(m)<\tau+1\\}$.
By Lemma 5 the following inequality holds
$\varepsilon(t;{\cal F}_{{\cal O}})\geq\lambda_{p}(\Sigma_{h}(t)\cap{\cal
O})/c.$
Because $\Sigma_{h}(t)\cap{\cal O}\subset\Sigma_{h}(t)$ it follows that
$\lambda_{p}(\Sigma_{h}(t)\cap{\cal O})\geq\lambda_{p}(\Sigma_{h}(t))$ and
hence
$\varepsilon(t;{\cal F}_{{\cal O}})\geq\lambda_{p}(\Sigma_{h}(t))/c\,.$
Thus using the requirement (3.2) for the equation (3.3), we arrive at the
estimate
$\liminf_{\tau\to\infty}\int\limits_{{\cal O}(\tau)}|\nabla
h(m)||f(m)-f(m_{1})||\nabla f(m)|^{p-1}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}}\exp\Bigl{\\{}-c_{3}\int^{\tau}\lambda_{p}(\Sigma_{h}(t))dt\Bigr{\\}}>0.$
Further we observe that from the condition $f(m)>f(m_{1})>0$ on ${\cal O}$ it
follows that
$\begin{array}[]{ll}&\displaystyle\int\limits_{{\cal O}(\tau)}|\nabla
h|\,|f(m)-f(m_{1})||\nabla f(m)|^{p-1}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}}=\\\ \\\ &=\displaystyle\int\limits_{{\cal O}(\tau)}f(m)\,|\nabla
h|\,|\nabla f(m)|^{p-1}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}}-f(m_{1})\int\limits_{{\cal O}(\tau)}|\nabla h|\,|\nabla
f(m)|^{p-1}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}\leq\\\ \\\
&\leq\displaystyle\int\limits_{\tau<h(m)<\tau+1}|\nabla h|\,|f(m)||\nabla
f(m)|^{p-1}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}.\\\ \\\ \end{array}$
From this relation we arrive at (5.9).
The proof of (5.10) is carried out exactly in the same way by means of the
inequality [16, 5.75].
In order to convince ourselves of the validity of (5.11) we observe that by
the maximum principle we have
$\int\limits_{\tau<h(m)<\tau+1}{|\nabla h(m)|^{p}|f(m)|^{p}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}}\leq
M^{p}(\tau+1)\int\limits_{\tau<h(m)<\tau+1}|\nabla h(m)|^{p}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}.$
But $h$ is a special exhaustion function and therefore by (4.26) we can write
$\int\limits_{\tau<h(m)<\tau+1}|\nabla h(m)|^{p}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}=J,$
where $J$ is a number independent of $\tau$.
The relation (5.10) implies then that (5.11) holds. $\Box$
5.12. Example. Let ${\cal A}$ be a compact Riemannian manifold with nonempty
piecewise smooth boundary, $\dim{\cal A}=k\geq 1$, and let ${\cal M}={\cal
A}\times{\bf R}^{n}$, $n\geq 1$. Choosing as a special exhaustion function of
${\cal M}$ the function $h(a,x)$, defined in Example 4 we have
$\Sigma_{h}(t)={\cal A}\times S^{n-1}(t).$
Then using the fact that $h(a,x)|_{\Sigma_{h}(t)}=t$ we find
$|\nabla h(a,x)|_{\Sigma_{h}(t)}=h^{\prime}(t)=\cases{\quad e^{-t}&for
$p=n$\cr\displaystyle{p-n\over p-1}\,t^{(1-n)/(p-n)}&for $p\neq n$.\cr}$
Therefore on the basis of (5.2) we get
$\lambda_{p}(\Sigma_{h}(t))={1\over
h^{\prime}(t)}\;\inf{\Bigl{(}\displaystyle\int\limits_{{\cal A}\times
S^{n-1}(t)}|\nabla_{2}\phi|^{p}d{\cal H}_{{\cal
M}}^{n-1}\Bigr{)}^{1/p}\over\Bigl{(}\displaystyle\int\limits_{{\cal
A}\times{\bf R}^{n}}|\phi|^{p}d{\cal H}_{{\cal M}}^{n-1}\Bigr{)}^{1/p}}\;.$
Computation yields
$\begin{array}[]{ll}|\nabla_{2}\phi(a,x)|^{2}&=|\nabla_{\cal
A}\phi(a,x)|^{2}+|\nabla_{S^{n-1}(t)}\phi(a,x)|^{2}=\\\ \\\ &=|\nabla_{\cal
A}\phi(a,x)|^{2}+{1\over
t^{2}}\Bigl{|}\nabla_{S^{n-1}(1)}\,\phi\Bigl{(}a,{x\over|x|}\Bigr{)}\Bigr{|}^{2}\\\
\\\ \end{array}$
and
$d{\cal H}_{{\cal M}}^{n-1}=d\sigma_{\cal A}\,dS^{n-1}(t),$
where $d\sigma_{\cal A}$ is an element of $k$-dimensional area on ${\cal A}$.
Therefore
$\begin{array}[]{ll}&\lambda_{p}(\Sigma_{h}(t))=\\\ \\\ &={1\over
h^{\prime}(t)}\,\inf{\Bigl{(}\displaystyle\int\limits_{\cal A}d\sigma_{\cal
A}\displaystyle\int\limits_{S^{n-1}(t)}(|\nabla_{\cal
A}\phi(a,x)|^{2}+|\nabla_{S^{n-1}(t)}\phi(a,x)|^{2})^{p/2}dS^{n-1}(t)\Bigr{)}^{1/p}\over\Bigr{(}\displaystyle\int\limits_{\cal
A}d\sigma_{\cal
A}\int\limits_{S^{n-1}(t)}\phi^{p}(a,x)dS^{n-1}(t)\Bigr{)}^{1/p}}=\\\ \\\
&={1\over h^{\prime}(t)}\inf{\Bigl{(}\displaystyle\int\limits_{\cal
A}d\sigma_{\cal A}\\!\displaystyle\int\limits_{S^{n-1}(1)}\\!(|\nabla_{\cal
A}\,\phi(a,{x\over|x|})|^{2}+{1\over
t^{2}}|\nabla_{S^{n-1}(t)}\phi(a,{x\over|x|})|^{2})^{p/2}dS^{n-1}(1)\Bigr{)}^{1/p}\over\Bigl{(}\displaystyle\int\limits_{\cal
A}d\sigma_{\cal
A}\int\limits_{S^{n-1}(1)}\phi^{p}(a,{x\over|x|})\,dS^{n-1}(1)\Bigr{)}^{1/p}}\\\
\\\ \end{array}$
and we obtain
$\begin{array}[]{ll}&\lambda_{p}(\Sigma_{h}(t))=\\\ \\\ &={1\over
h^{\prime}(t)}\inf{\Bigl{(}\displaystyle\int\limits_{\cal A}d\sigma_{\cal
A}\displaystyle\int\limits_{S^{n-1}(1)}(|\nabla_{\cal A}\psi|^{2}+{1\over
t^{2}}|\nabla_{S^{n-1}(1)}\psi|^{2})^{p/2}dS^{n-1}(1)\Bigr{)}^{1/p}\over\Bigl{(}\displaystyle\int\limits_{\cal
A}d\sigma_{\cal
A}\displaystyle\int\limits_{S^{n-1}(1)}\psi^{p}\,dS^{n-1}(1)\Bigr{)}^{1/p}},\\\
\\\ \end{array}$ (5.13)
where the infimum is taken over all functions $\psi=\psi(a,x)$ with
$\psi(a,x)\in W^{1}_{p}({\cal A}\times
S^{n-1}(1)),\quad\psi(a,x)|_{a\in\partial{\cal A}}=0,\quad\mbox{for
all}\;\;x\in S^{n-1}(1).$
In the particular case $n=1$ Theorem 5 has a particularly simple content. Here
$h(x)$ is a function of one variable, $\Sigma_{h}(t)={\cal A}\times S^{0}(t)$
is isometric to $\Sigma_{h}(1)$. Therefore $h^{\prime}(t)\equiv 1$ and by
(5.13) we have
$\lambda_{p}(\Sigma_{h}(t))\equiv\lambda_{p}(\Sigma_{h}(1))\equiv\lambda_{p}({\cal
A})\quad\mbox{for all}\;\;t\in R^{1}.$ (5.14)
In the same way (5.11) can be written in the form
$\liminf_{t\to\infty}\max_{|x|=t}|f(a,x)|\,\exp\Bigl{\\{}-{{c_{3}}\over
p}\,\lambda_{n}({\cal A})\Bigr{\\}}>0.$ (5.15)
Let $n\geq 2$. We do not know of examples where the quantity (5.13) had been
exactly computed. Some idea about the rate of growth of the quantity $M(\tau)$
in the Phragmén–Lindelöf alternative can be obtained from the following
arguments. Simplifying the numerator of (5.13) by ignoring the second summand
we get the estimate
$\lambda_{p}(\Sigma_{h}(t))\geq{1\over
h^{\prime}(t)}\inf_{\psi}{\Bigl{(}\displaystyle\int\limits_{\cal
A}d\sigma_{\cal A}\displaystyle\int\limits_{S^{n-1}(1)}|\nabla_{\cal
A}\psi(a,x)|^{p}\,dS^{n-1}(1)\Bigr{)}^{1/p}\over\Bigl{(}\displaystyle\int\limits_{\cal
A}d\sigma_{\cal
A}\displaystyle\int\limits_{S^{n-1}(1)}\psi^{p}(a,x)\,dS^{n-1}(1)\Bigr{)}^{1/p}}.$
For each fixed $x\in S^{n-1}(1)$ the function $\psi(a,x)$ is finite on ${\cal
A}$, because from the definition of the fundamental frequency it follows that
$\Bigl{(}\int\limits_{\cal A}|\nabla_{\cal A}\psi(a,x)|^{p}\,d\sigma_{\cal
A}\Bigr{)}^{1/p}\geq\lambda_{p}({\cal A})\Bigl{(}\int\limits_{\cal
A}\psi^{p}(a,x)\,d\sigma_{\cal A}\Bigr{)}^{1/p}.$
From this we get
$\lambda_{p}(\Sigma_{h}(t))\geq{1\over h^{\prime}(t)}\lambda_{p}({\cal A}).$
(5.16)
Thus
$\begin{array}[]{ll}\displaystyle\int\limits_{\tau_{0}}^{\tau}\lambda_{p}(\Sigma_{h}(r))\,dr&\geq\displaystyle\int\limits_{\tau_{0}}^{\tau}\lambda_{p}({\cal
A}){dh(r)\over h^{\prime}(r)}\quad=\lambda_{p}({\cal
A})\displaystyle\int\limits_{\tau_{0}}^{\tau}r^{\prime}(h)\,dh=\\\ \\\
&=\lambda_{p}({\cal A})(r(\tau)-r(\tau_{0})).\\\ \\\ \end{array}$
Here $r(h)$ is the inverse function of $h(r)$. Because
$\max_{h(|x|)=\tau}|f(a,x)|\,\exp\Bigl{\\{}-{c_{3}\over p}\lambda_{p}({\cal
A})\,r(\tau)\Bigr{\\}}=\max_{|x|=r(\tau)}|f(a,x)|\,\exp\Bigl{\\{}-{c_{3}\over
p}\lambda_{p}({\cal A})\,r(\tau)\Bigr{\\}},$
the relation (5.11) can be written in the form (5.15).
5.17. Example. Let $U\subset S^{n-1}$ be an arbitrary domain with nonempty
boundary. We consider a warped Riemannian product ${\cal
M}=(r_{1},r_{2})\times U$ equipped with the metric (4.22) of the domain $D$.
We now analyze Theorem 5 in this case.
The function $h(r)$, given by the equation (4.24) under the requirement (4.23)
is a special exhaustion function on ${\cal M}$. We compute the quantity
$\lambda_{p}(\Sigma_{h}(\tau))$ as follows
$\begin{array}[]{ll}&\left.|\nabla
h(|x|)\right|_{\Sigma_{h}(\tau)}=h^{\prime}(r(\tau))=\alpha(r(\tau))/\beta^{n-1}(r(\tau)),\\\
\\\
&\left|\nabla_{2}\phi\right|_{\Sigma_{h}(\tau)}=|\nabla_{S^{n-1}(1)}\phi|/\beta(r(\tau))\\\
\\\ \end{array}$
and
$d{\cal H}_{{\cal M}}^{n-1}=\beta^{n-1}(r(\tau))dS^{n-1}(1),\quad
r(\tau)=h^{-1}(\tau).$
Therefore, observing that
${1\over h^{\prime}(r(\tau))}=r^{\prime}(\tau),$
we have
$\begin{array}[]{ll}\lambda_{p}(\Sigma_{h}(\tau))&={1\over
h^{\prime}(r(\tau))}\;\inf_{\phi}{\Bigl{(}\displaystyle\int\limits_{\Sigma_{h}(\tau)}|\nabla_{2}\phi|^{p}d{\cal
H}_{{\cal
M}}^{n-1}\Bigr{)}^{1/p}\over\Bigl{(}\displaystyle\int\limits_{\Sigma_{h}(\tau)}\phi^{p}d{\cal
H}_{{\cal M}}^{n-1})^{1/p}}=\\\ \\\
&={r^{\prime}(\tau)\over\beta(r(\tau))}\;\inf{\Bigl{(}\displaystyle\int\limits_{U}|\nabla_{S^{n-1}(1)}\phi|^{p}dS^{n-1}(1)\Bigr{)}^{1/p}\over\Bigl{(}\displaystyle\int\limits_{U}\phi^{p}dS^{n-1}(1)\Bigr{)}^{1/p}}.\\\
\\\ \end{array}$
Thus
$\lambda_{p}(\Sigma_{h}(\tau))={r^{\prime}(\tau)\over\beta(r(\tau))}\;\lambda_{p}(U).$
(5.18)
Further we get
$\int\limits_{\tau_{0}}^{\tau}\lambda_{h}(\Sigma_{h}(\tau))\,d\tau=\lambda_{p}(U)\int\limits_{r(\tau_{0})}^{r(\tau)}{dr\over\beta(r)}$
and
$\max_{h(|x|)=\tau}|f(x)|\,\exp\Bigl{\\{}-{c_{3}\over
p}\lambda_{p}(U)\int\limits^{r(\tau)}{dr\over\beta(r)}\Bigr{\\}}=\max_{|x|=r(\tau)}|f(x)|\,\exp\Bigl{\\{}-{c_{3}\over
p}\lambda_{p}(U)\int\limits^{r(\tau)}{dr\over\beta(r)}\Bigr{\\}}.$
Thus the relation (5.11) attains the form
$\liminf_{r\to\infty}\max_{|x|=r}|f(x)|\,\exp\Bigl{\\{}-{c_{3}\over
p}\lambda_{p}(U)\int\limits^{r}{dr\over\beta(r)}\Bigr{\\}}>0.$ (5.19)
5.20. Proof of Theorem 1. We assume that
$\limsup_{\tau\to\infty}\min_{m\in\Sigma_{h}(\tau)}u(f(m))=K<\infty.$
Consider the set
${\cal O}=\\{m\in{\cal X}:u(f(m))>qK\\},\ q<1.$
It is clear that for a suitable choice of $q$ the set ${\cal O}$ is not empty.
By assumptions the function $u$ satisfies (3.3) with (3.1), (3.2) and
structure constants $p=n$, $\nu_{1}$, $\nu_{2}$. Since $f$ is quasiregular, by
Lemma 14.38 of [8] the function $u(f(m))$ is a subsolution of another equation
of the form (3.3) with structure constants $\nu^{\prime}_{1}=\nu_{1}/K_{O}$,
$\nu_{2}^{\prime}=\nu_{2}K_{I}$ where $K_{O}$, $K_{I}$ are outer and inner
dilatations of $f$. In view of the maximum principle for subsolutions the set
${\cal O}$ does not have relatively compact components. Without restricting
generality we may assume that ${\cal O}$ is connected. Because for
sufficiently large $\tau$ the condition
${\cal O}\cap\Sigma_{h}(\tau)\neq\emptyset$
holds, we see that
$\lambda_{n}({\cal
O}\cap\Sigma_{h}(\tau))\geq\lambda_{n}(\Sigma_{h}(\tau);1).$
Therefore the condition (1.2) on the manifold ${\cal X}$ implies the following
property
$\int\limits^{\infty}\lambda_{n}({\cal O}\cap\Sigma_{h}(\tau))d\tau=\infty.$
Observing that
$\max_{m\in\Sigma_{h}(\tau)}u(f(m))\geq\max_{m\in\Sigma_{h}(\tau)\cap{\cal
O}}u(f(m)),$
we see that by (1.3)
$\liminf_{\tau\to\infty}\max_{\Sigma_{h}(\tau)\cap{\cal
O}}u(f(m))\exp\Bigl{\\{}-C\int\limits^{\tau}\lambda_{n}({\cal
O}\cap\Sigma_{h}(t))dt\Bigr{\\}}=0$
with the constant $C$ of Theorem 1.
It is easy to see that $C=c_{3}/n$. Using (5.11) with $p=n$ for the function
$u(f(m))$ in the domain ${\cal O}$ we see that $u(f(m))\equiv qK$ on ${\cal
O}$. This contradicts with the definition of the domain ${\cal O}$. $\Box$
5.21. Example. As the first corollary we shall now prove a generalization of
Wiman’s theorem for the case of quasiregular mappings $f:{\cal M}\to{\bf
R}^{n}$ where ${\cal M}$ is a warped Riemannian product.
For $0\leq r_{1}<r_{2}\leq\infty$ let
$D=\\{m=(r,\theta)\in{\bf R}^{n}:r_{1}<r<r_{2},\theta\in S^{n-1}(1)\\}$
be a ring domain in ${\bf R}^{n}$ and let ${\cal M}=(r_{1},r_{2})\times
S^{n-1}(1)$ be an $n$-dimensional Riemannian manifold on $D$ with the metric
$ds_{{\cal M}}^{2}=\alpha^{2}(r)dr^{2}+\beta^{2}(r)dl_{n-1}^{2},$
where $\alpha(r),\;\beta(r)>0$ are continuously differentiable on
$[r_{1},r_{2})$ and $dl_{n-1}$ is an element of length on $S^{n-1}(1)$.
As we have proved in Example 4, under condition (4.23), the function
$h(r)=\int\limits_{r_{1}}^{r}{\alpha(t)\over\beta(t)}dt$
is a special exhaustion function on ${\cal M}$.
Let $f:{\cal M}\to{\bf R}^{n}$ be a quasiregular mapping. We set
$u(y)=\log^{+}|y|$. This function is a subsolution of the equation (3.3) with
$p=n$ and also satisfies all the other requirements imposed on a growth
function.
We find
$\lambda_{n}(S^{n-1}(\tau);1)={1\over\beta(r(\tau))}\lambda_{n}(S^{n-1}(1);1)$
and further
$\lambda_{n}(\Sigma_{h}(\tau);1)={\lambda_{n}(S^{n-1}(1);1)\over\beta(r(\tau))\;h^{\prime}(r(\tau))}.$
Therefore the requirement (1.2) on the manifold will be fullfilled, if
$\int\limits^{r_{2}}{dr\over\beta(r)}=\infty$ (5.22)
holds.
Because
$\begin{array}[]{rcl}&\displaystyle\max_{\Sigma_{h}(\tau)=\tau}\log^{+}|f(r,\theta)|\;\exp\Bigl{\\{}-C\int\limits^{\tau}\lambda_{n}(\Sigma_{h}(t);1)\;dt\Bigr{\\}}\leq\\\
&\displaystyle\leq\max_{r=h^{-1}(\tau)}\log^{+}|f(r,\theta)|\;\exp\Bigl{\\{}-C\,\lambda_{n}(S^{n-1}(1);1)\int\limits^{h^{-1}(\tau)}{dr\over\beta(r)}\Bigr{\\}},\end{array}$
(5.23)
we see that, in view of (1.3), it suffices that
$\liminf_{\tau\to
r_{2}}\max_{\Sigma_{h}(\tau)}\;\log^{+}|f(r,\theta)|\;\exp\Bigl{\\{}-C\,\lambda_{n}(S^{n-1}(1);1)\int\limits^{\tau}{dt\over\beta(t)}\Bigr{\\}}=0.$
(5.24)
In this way we get
5.25. Corollary. Let $f:{\cal M}\to{\bf R}^{n}$ be a non-constant
quasiregular mapping from the warped Riemannian product ${\cal
M}=(r_{1},r_{2})\times S^{n-1}(1)$ and $h$ a special exhaustion function of
${\cal M}$. If the manifold ${\cal M}$ has property (5.22) and the mapping $f$
has property (5.24), then
$\limsup_{\tau\to r_{2}}\min_{\Sigma_{h}(\tau)}|f(r,\theta)|=\infty.$
5.26. Example. Suppose that under the assumptions of Example 5 we have (in
addition) $r_{1}=0$, $r_{2}=\infty$, and the functions
$\alpha(r)=\beta(r)\equiv 1$, that is, ${\cal M}=(0,\infty)\times S^{n-1}(1)$
with the metric $ds^{2}_{{\cal M}}=dr^{2}+dl^{2}_{n-1}$ is an $n$-dimensional
half–cylinder. As the special exhaustion function of the manifold ${\cal M}$
we can take $h\equiv r$. The condition (5.22) is obviously fullfilled for the
manifold.
The condition (5.24) for the mapping $f$ attains the form
$\liminf_{r\to\infty}\max_{\theta\in
S^{n-1}(1)}\;\log^{+}|f(r,\theta)|\,e^{-C\lambda_{n}(S^{n-1}(1);1)r}=0.$
(5.27)
5.28. Corollary. If ${\cal M}=(0,\infty)\times S^{n-1}(1)$ is a half–cylinder
and $f:{\cal M}\to{\bf R}^{n}$ is a non-constant quasiregular mapping
satisfying (5.27), then
$\limsup_{r\to\infty}\min_{\theta\in S^{n-1}(1)}|f(r,\theta)|=\infty.$
We assume that in Example 5 the quantities $r_{1}=0$, $r_{2}=\infty$, and the
functions $\alpha(r)\equiv 1$, $\beta(r)=r$, that is, the manifold is ${\bf
R}^{n}$. As the special exhaustion function we choose $h=\log|x|$. This
function satisfies (3.5) with $p=n$ and $\nu_{1}=\nu_{2}=1$. The condition
(5.22) for the manifold is obviously fullfilled.
The condition (5.27) attains the form
$\liminf_{r\to\infty}\max_{|x|=r}\;\log^{+}|f(x)|\,r^{-C^{\prime}\lambda_{n}(S^{n-1}(1);1)}=0,$
(5.29)
where
$C^{\prime}=\left(n-1+n\left(K^{2}(f)-1\right)^{1/2}\right)^{-1}.$
We have
5.30. Corollary. Let $f:{\bf R}^{n}\to{\bf R}^{n}$ be a non-constant
quasiregular mapping satisfying (5.29). Then
$\limsup_{r\to\infty}\min_{|x|=r}|f(x)|=\infty.$
## 6 Asymptotic tracts and their sizes
Wiman’s theorem for the quasiregular mappings $f:{\bf R}^{n}\to{\bf R}^{n}$
asserts the existence of a sequence of spheres $S^{n-1}(r_{k})$,
$r_{k}\to\infty$, along which the mapping $f(x)$ tends to $\infty$. It is
possible to further strengthen the theorem and to specify the sizes of the
sets along which such a convergence takes place. For the formulation of this
result it is convenient to use the language of asymptotic tracts discussed by
MacLane [11].
6.1. Tracts. Let $D$ be a domain in the complex plane $C$ and let $f$ be a
holomorphic function on $D$. A collection of domains $\\{{\cal D}(s):s>0\\}$
is called an asymptotic tract of $f$ if
a) each of the sets ${\cal D}(s)$ is a component of the set
$\\{z\in D:|f(z)|>s>0\\};$
b) for all $s_{2}>s_{1}>0$ we have ${\cal D}(s_{2})\subset{\cal D}(s_{1})$ and
$\cap_{s>0}\overline{\cal D}(s)=\emptyset$.
Two asymptotic tracts $\\{{\cal D}^{\prime}(s)\\}$ and $\\{{\cal D}"(s)\\}$
are considered to be different if for some $s>0$ we have ${\cal
D}^{\prime}(s)\cap{\cal D}"(s)=\emptyset$.
Below we shall extend this notion to quasiregular mappings $f:{\cal M}\to{\cal
N}$ of Riemannian manifolds. We study the existence of an asymptotic tract and
its size.
Let ${\cal M},{\cal N}$ be $n$-dimensional connected noncompact Riemannian
manifolds and let $u=u(y)$ be a growth function on ${\cal N}$, which is a
positive subsolution of the equation (3.3) with structure constants $p=n$,
$\nu_{1}$, $\nu_{2}$.
A family $\\{{\cal M}(s)\\}$ is called an asymptotic tract of a quasiregular
mapping $f:{\cal M}\to{\cal N}$ if
a) each of the sets $\\{{\cal M}(s)\\}$ is a component of the set
$\\{m\in{\cal M}:u(f(m))>s>0\\};$
b) for all $s_{2}>s_{1}>0$ we have ${{\cal M}}(s_{2})\subset{{\cal M}}(s_{1})$
and $\cap_{s>0}\overline{{\cal M}}(s)=\emptyset$.
Let $f:{\cal M}\to{\bf R}^{n}$ be a quasiregular mapping having a point
$a\in{\bf R}^{n}$ as a Picard exceptional value, that is $f(m)\neq a$ and
$f(m)$ attains on ${\cal M}$ all values of $B(a,r)\setminus\\{a\\}$ for some
$r>0$.
The set $\\{\infty\\}\cup\\{a\\}$ has $n$-capacity zero in ${\bf R}^{n}$ and
there is a solution $g(y)$ in ${\bf R}^{n}\setminus\\{a\\}$ of the equation
(3.3) such that $g(y)\to\infty$ as $y\to a$ or $y\to\infty$ (cf. [8, Ch. 10,
polar sets]). As the growth function on ${\bf R}^{n}\setminus\\{a\\}$ we
choose the function $u(y)=\max(0,g(y))$. It is clear that this function is a
subsolution of the equation (3.3) in ${\bf R}^{n}\setminus\\{a\\}$.
The function $u(f(m))$ also is a subsolution of an equation of the form (3.3)
on ${\cal M}$. Because the mapping $f(m)$ attains all values in the punctured
ball $B(a,r)$, then among the components of the set
$\\{m\in{\cal M}:u(f(m))>s\\}$
there exists at least one ${\cal M}(s)$ having a nonempty intersection with
$f^{-1}(B(a,r))$. Then by the maximum principle for subsolutions such a
component cannot be relatively compact.
Letting $s\to\infty$ we find an asymptotic tract $\\{{\cal M}(s)\\}$, along
which a quasiregular mapping tends to a Picard exceptional value $a\in{\bf
R}^{n}$.
Because one can find in every asymptotic tract a curve $\Gamma$ along which
$u(f(m))\to\infty$, we obtain the following generalization of Iversen’s
theorem [9].
6.2. Theorem. Every Picard exceptional value of a quasiregular mapping
$f:{\cal M}\to{\bf R}^{n}$ is an asymptotic value.
The classical form of Iversen’s theorem asserts that if $f$ is an entire
holomorphic function of the plane, then there exists a curve $\Gamma$ tending
to infinity such that
$f(z)\to\infty\quad\mbox{as}\;\;z\to\infty\quad\mbox{on}\;\Gamma.$
We prove a generalization of this theorem for quasiregular mappings $f:{\cal
M}\to{\cal N}$ of Riemannian manifolds.
The following result holds.
6.3. Theorem. Let $f:{\cal M}\to{\cal N}$ be a non-constant quasiregular
mapping between $n$-dimensional noncompact Riemannian manifolds without
boundaries. If there exists a growth function $u$ on ${\cal N}$ which is a
positive subsolution of the equation (3.3) with $p=n$ and on ${\cal M}$ a
special exhaustion function, then the mapping $f$ has at least one asymptotic
tract and, in particular, at least one curve $\Gamma$ on ${\cal M}$ along
which $u(f(m))\to\infty$.
Proof. Let $h:{\cal M}\to(0,\infty)$ be a special exhaustion function of the
manifold ${\cal M}$. Set
$\liminf_{\tau\to\infty}\min_{h(m)=\tau}u(f(m))=K.$ (6.4)
If $K=\infty$, then $u(f(m))$ tends uniformly on ${\cal M}$ to $\infty$ for
$h(m)\to\infty$. The asymptotic tract $\\{{\cal M}(s)\\}$ generates mutual
inclusion of the components of the set $\\{m\in{\cal M}:h(m)>s\\}$.
Let $K<\infty$. For an arbitrary $s>K$ we consider the set
${\cal O}(s)=\\{m\in{\cal M}:u(f(m))>s\\}.$
Because $u(f(m))$ is a subsolution, the non-empty set ${\cal O}(s)$ does not
have relatively compact components. By a standard argument we choose for each
$s>K$, as ${\cal M}(s)$ a component of the set ${\cal O}(s)$ having property
b) of the definition of an asymptotic tract. We now easily complete the proof
for the theorem. $\Box$
6.5. Proof of Theorem 1. We fix a growth function $u$ and a spectial
exhaustion function $h$ as in Section 4. Let $f:{\cal M}\to{\cal N}$ be a non-
constant quasiregular mapping. We set
$M(\tau)=\max_{h(m)=\tau}u(f(m)).$
Let $K$ be the quantity defined in (6.4). The case $K=\infty$ is degenerate
and has no interest in the present case.
Suppose now that $K<\infty$. For $s>K$ we consider the set ${\cal M}(s)$,
defined in the proof of the preceding theorem. Define
$\tau_{0}=\tau_{0}(s)>\inf_{m\in{\cal M}(s)}h(m).$
Because $u(f(m))$ is a subsolution of an equation of the form (3.3) on ${\cal
M}$ by Theorem [16, 5.59] we have for an arbitrary $\tau>\tau_{0}$
$\int\limits_{B_{h}(\tau_{0})\cap{\cal M}(s)}|\nabla u(f(m))|^{n}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm
1}}\leq\exp\Bigl{\\{}-\nu_{1}\int\limits_{\tau_{0}}^{\tau}\varepsilon(t)\;dt\Bigr{\\}}\int\limits_{B_{h}(\tau)\cap{\cal
M}(s)}|\nabla u(f(m))|^{n}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}.$
Using the inequality (4.5) of [14] for the quantity $\varepsilon(t)$ we get
$\begin{array}[]{ll}&\displaystyle\int\limits_{B_{h}(\tau_{0})\cap{\cal
M}(s)}|\nabla u(f(m))|^{n}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}\leq\\\
\\\ &\leq\exp\Bigl{\\{}-{{\nu_{1}}\over
c}\displaystyle\int\limits_{\tau_{0}}^{\tau}\lambda_{n}(\Sigma_{h}(t)\cap{\cal
M}(s))\,dt\Bigr{\\}}\displaystyle\int\limits_{B_{h}(\tau)\cap{\cal
M}(s)}|\nabla u(f(m))|^{n}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm
1}},\end{array}$
where
$c=\sqrt{\nu_{2}^{-2}-\nu_{1}^{-2}}+{n-1\over n}\nu_{1}.$
By [16, 5.71] we have
$\begin{array}[]{rcl}\left({{\nu_{1}}\over{\nu_{2}}}\right)^{n}\displaystyle\int\limits_{B_{h}(\tau)}|\nabla
u(f(m))|^{n}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}&\leq&\displaystyle
n^{n}\int\limits_{B_{h}(\tau+1)\setminus B_{h}(\tau)}|\nabla
h|^{n}\,|u(f(m))|^{n}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}\leq\\\
&\leq&\displaystyle n^{n}\,M^{n}(\tau+1)\,V(\tau),\end{array}$ (6.6)
where
$V(\tau)=\int\limits_{B_{h}(\tau+1)\setminus B_{h}(\tau)}|\nabla_{{\cal
M}}h|^{n}*{\hbox{\rm 1}\hskip-3.80005pt\hbox{\rm 1}}.$
But $h$ is a special exhaustion function and as in the proof of (4.26) we get
$V(\tau)\leq J\equiv{\rm const}$
for all sufficiently large $\tau$. Hence
$\int\limits_{B_{h}(\tau)}|\nabla u(f(m))|^{n}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}\leq J\,M^{n}(\tau+1)$
and further
$\int\limits_{B_{h}(\tau_{0})\cap{\cal M}(s)}|\nabla u(f(m))|^{n}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}\leq
J\,M^{n}(\tau+1)\,\exp\Bigl{\\{}-C\int\limits_{\tau_{0}}^{\tau}\lambda_{n}(\Sigma_{h}(t)\cap{\cal
M}(s))\,dt\Bigr{\\}},$
where $C=\nu_{1}/c$ and $c$ is defined in Lemma 5.
Under these circumstances, from the condition (1.5) for the growth of
$M(\tau)$ it follows that for all $\varepsilon>0$ and for all sufficiently
large $\tau$ we have
$\int\limits_{B_{h}(\tau_{0})\cap{\cal M}(s)}|\nabla u(f(m))|^{n}*{\hbox{\rm
1}\hskip-3.80005pt\hbox{\rm 1}}\leq
J\,\varepsilon\,\exp\Bigl{\\{}\int\limits_{\tau_{0}}^{\tau}\bigl{(}n\gamma\lambda_{n}(\Sigma_{h}(t);1)-C\lambda_{n}(\Sigma_{h}(t)\cap{\cal
M}(s))\bigr{)}\,dt\Bigr{\\}}.$ (6.7)
If we assume that for all $\tau>\tau_{0}$
$\int\limits_{\tau_{0}}^{\tau}\bigl{(}n\gamma\lambda_{n}(\Sigma_{h}(t);1)-C\lambda_{n}(\Sigma_{h}(t)\cap{\cal
M}(s))\bigr{)}\,dt\leq 0,$
then because $\varepsilon>0$ was arbitrary, it would follow from (6.7) that
$|\nabla u(f(m))|\equiv 0$ on $B_{h}(\tau_{0})\cap{\cal M}(s)$ which is
impossible.
Hence there exists $\tau=\tau(K)>\tau_{0}(K)$ for which
$\lambda_{n}(\Sigma_{h}(\tau)\cap{\cal M}(s))<{n\gamma\over
C}\lambda_{n}(\Sigma_{h}(\tau);1).$ (6.8)
Letting $K\to\infty$ we see that $\tau_{0}\to\infty$. Using each time the
relation (6.7) we get Theorem 1. $\Box$
In the formulation of the theorem we used only a part of the information about
the sizes of the sets ${\cal M}(s)$ which is contained in (6.7). In
particular, the relation (6.7) to some extent characterizes also the linear
measure of those $t>\tau_{0}$ for which the intersection of the sets ${\cal
M}(s)$ with the $h$-spheres $\Sigma_{h}(t)$ is not too narrow.
We consider the case of warped Riemannian product ${\cal
M}=(r_{1},r_{2})\times S^{n-1}(1)$ with the metric $ds^{2}_{\cal M}$ described
in Example 5. Let $h$ be a special exhaustion function of the manifold ${\cal
M}$ of the type (4.24) with $p=n$, satisfying condition (4.23).
Here, as in Example 5,
$\displaystyle\lambda_{n}(\Sigma_{h}(\tau);1)=\displaystyle{\lambda_{n}(S^{n-1}(1);1)\over\beta(r(\tau))\,h^{\prime}(r(\tau))},\quad\lambda_{n}(U)=\displaystyle{\lambda_{n}(U^{*})\over\beta(r(\tau))\,h^{\prime}(r(\tau))},$
(6.9)
where $r(\tau)=h^{-1}(\tau)$ and $U^{*}\subset S^{n-1}(1)$ is the image of the
set $U$ under the similarity mapping
$x\mapsto{x\over\beta(r(\tau))}$
of ${\bf R}^{n}$.
Let $f:{\cal M}\to{\bf R}^{n}$ be a non-constant quasiregular mapping. We
choose as a growth function $u$ the function $u=\log^{+}|y|$. This function
satisfies (3.5) with $p=n$ and $\nu_{1}=\nu_{2}=1.$ The condition (1.5) can be
written as follows
$\liminf_{\tau\to
r_{2}}\max_{r=\tau}\;\log^{+}|f(r,\theta)|\,\exp\Bigl{\\{}-\gamma\lambda_{n}(S^{n-1}(1);1)\int\limits^{r}{dt\over\beta(t)}\Bigr{\\}}=0.$
(6.10)
Hence we obtain
6.11. Corollary. If a quasiregular mapping $f:{\cal M}\to{\bf R}^{n}$ has the
property (6.10) for some $\gamma>0$, then for each $k=1,2,\ldots$ there are
spheres $S^{n-1}(t_{k})$, $t_{k}\in(r_{1},r_{2})$, $t_{k}\to r_{2}$, and open
sets $U\subset S^{n-1}(t_{k})$ for which
$|f(m)|>k\;\;\hbox{for all}\;\;m\in
U\quad\hbox{and}\quad\lambda_{n}(U)<{n\gamma\over
C^{\prime}}\lambda_{n}(S^{n-1}(1);1),$
where as above
$C^{\prime}=\left(n-1+n\left(K^{2}(f)-1\right)^{1/2}\right)^{-1}.$
Corresponding estimates of the quantities $\lambda_{n}(U^{*})$ and
$\lambda_{n}(S^{n-1}(1);1)$ were given in [17] in terms of the
$(n-1)$-dimensional surface area and in terms of the best constant in the
embedding theorem of the Sobolev space $W^{1}_{n}$ into the space $C$ on open
subsets of the sphere. This last constant can be estimated without
difficulties in terms of the maximal radius of balls contained in the given
subset.
## References
* [1] K. Arima: On maximum modulus of integral functions.— J. Math. Soc. Japan, 1952, v. 5, 62–66.
* [2] V.A. Botvinnik, V.M. Miklyukov: Phragmén-Lindelöf’s theorems for $n$-dimensional mappings with bounded distortion. (Russian) – Sibirsk. Math. Zh., v. 21, n. 2, 1980, 232–235.
* [3] H. Federer: Geometric measure theory. – Die Grundlehren der math. Wiss. Vol. 153, Springer-Verlag, Berlin-Heidelberg-New York, 1969
* [4] D. Franke, O. Martio, V. M. Miklyukov, M. Vuorinen, and R. Wisk: Quasiregular mappings and $\cal{WT}$ -classes of differential forms on Riemannian manifolds. – Pacific J. Math., v. 202, n. 1, 2002, 73-92.
* [5] V.M. Gol’dstein and Yu.G. Reshetnyak: Introduction to the theory of functions with generalized derivatives and quasiconformal mappings. (Russian) – Izdat. “Nauka”, Moscow, 1983.
* [6] S. Granlund, P. Lindqvist, and O. Martio: Phragmén–Lindelöf’s and Lindelöf’s theorem. – Ark. Mat. 23 (1985), 103–128.
* [7] G.H. Hardy, J.E. Littlewood and G. Polya: Inequalities. – 1934.
* [8] J. Heinonen, T. Kilpeläinen, and O. Martio: Nonlinear potential theory of degenerate elliptic equations. — Clarendon Press, 1993\.
* [9] F. Iversen: Recherches sur les fonctions inverses des fonctions meromorphes. – Thesis Helsinki 1914.
* [10] V.M. Kesel’man: On Riemannian manifolds of $\alpha-$parabolic type. (Russian) – Izv. vuzov Mat. 4 (1984), 81–83.
* [11] G.R. MacLane: Asymptotic values of holomorphic functions. – Rice Univ. Studies 49 (1963), 1–83.
* [12] O. Martio, V. Miklyukov and M. Vuorinen: Phragmén – Lindelöf’s principle for quasiregular mappings and isoperimetry. (in Russian) – Dokl. Akad. Nauk v. 347 n. 3, ( 1996), 303–305.
* [13] O. Martio, V.M. Miklyukov, and M. Vuorinen: Morrey’s lemma on Riemannian manifolds, Collection of papers in memory of Martin Jurchescu, Rev. Roumaine Math. Pures Appl. 43 (1998), no. 1-2, 183–210.
* [14] O. Martio, V.M. Miklyukov, and M. Vuorinen: Critical points of $A-$solutions of quasilinear elliptic equations, Houston Math. J., v. 25, n. 3, 1999, p. 583-601.
* [15] O. Martio, V. Miklyukov, and M. Vuorinen: Generalized Wiman and Arima theorems for $n$-subharmonic functions on cones, J. Geom. Anal. 13 (2003), 605–630.
* [16] O. Martio, V. Miklyukov and M. Vuorinen: Ahlfors theorems for differential forms, Reports of the Dep. of Math., University of Helsinki, 2005.
* [17] V.M. Miklyukov: Asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion. (Russian) – Mat. Sb. 11 (1980), 42–66; English transl. Math. USSR Sb. v. 39, 37–60, 1981.
* [18] E. Phragmén and E. Lindelöf: Sur une extension d’un principe classique de l’analyse et sur quelques propriétés des fonctions monogenènes dans le voisinage d’un point singulier. – Acta Math. 31 (1908), 381-406.
* [19] G. Pôlya and G. Szegö: Isoperimetric inequalities in mathematical physics. – Princeton, Princeton University Press, 1951.
* [20] Yu.G. Reshetnyak: Spatial mappings with bounded distortion. (Russian) – Izdat. “Nauka”, Sibirsk. Otdelenie, Novosibirsk, 1982.
* [21] S. Rickman and M. Vuorinen: On the order of quasiregular mappings. – Ann. Acad. Sci. Fenn. Math. 7 (1982), 221–231.
* [22] A. Wiman: Sur une extension d’un théoréme de M. Hadamard. – Arkiv för Math., Astr. och Fys., 1905, v. 2, n. 14, p. 1-5.
Martio :
Department of Mathematics and Statistics
University of Helsinki
00014 Helsinki
FINLAND
Email: olli.martio@helsinki.fi
Miklyukov :
Department of Mathematics
Volgograd State University
2 Prodolnaya 30
Volgograd 400062
RUSSIA
E-mail:miklyuk@hotmail.com
Vuorinen :
Department of Mathematics
FIN-20014 University of Turku
FINLAND
E-mail: vuorinen@utu.fi
|
arxiv-papers
| 2010-02-12T09:22:43 |
2024-09-04T02:49:08.373748
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Olli Martio, Vladimir M. Miklyukov and Matti K. Vuorinen",
"submitter": "Matti Vuorinen",
"url": "https://arxiv.org/abs/1002.2500"
}
|
1002.2546
|
-title
11institutetext: Institute of Electrophysics & Radiation Technologies,
NAS of Ukraine, Kharkov, Ukraine 22institutetext: NSC Kharkov Institute of
Physics & Technology
NAS of Ukraine, Kharkov, Ukraine
# The method of unitary clothing transformations in the theory of
nucleon–nucleon scattering
I. Dubovyk 11 e.a.dubovyk@mail.ru A. Shebeko 22 shebeko@kipt.kharkov.ua
###### Abstract
The clothing procedure, put forward in quantum field theory (QFT) by Greenberg
and Schweber, is applied for the description of nucleon–nucleon ($N$–$N$)
scattering. We consider pseudoscalar ($\pi$ and $\eta$), vector ($\rho$ and
$\omega$) and scalar ($\delta$ and $\sigma$) meson fields interacting with
$1/2$ spin ($N$ and $\bar{N}$) fermion ones via the Yukawa–type couplings to
introduce trial interactions between ”bare” particles. The subsequent unitary
clothing transformations (UCTs) are found to express the total Hamiltonian
through new interaction operators that refer to particles with physical
(observable) properties, the so–called clothed particles. In this work, we are
focused upon the Hermitian and energy–independent operators for the clothed
nucleons, being built up in the second order in the coupling constants. The
corresponding analytic expressions in momentum space are compared with the
separate meson contributions to the one–boson–exchange potentials in the meson
theory of nuclear forces. In order to evaluate the $T$ matrix of the $N$–$N$
scattering we have used an equivalence theorem that enables us to operate in
the clothed particle representation (CPR) instead of the bare particle
representation (BPR) with its huge amount of virtual processes. We have
derived the Lippmann–Schwinger(LS)–type equation for the CPR elements of the
$T$–matrix for a given collision energy in the two–nucleon sector of the
Hilbert space $\mathcal{H}$ of hadronic states and elaborated a code for its
numerical solution in momentum space.
## 1 Introductory remarks
We know that there are a number of high precision, boson–exchange models of
the two nucleon force $V_{NN}$, such as Paris Lacom80 , Bonn MachHolElst87 ,
Nijmegen Stocks94 , Argonne WirStocksSchia95 , CD Bonn Mach01 potentials and
a fresh family of covariant one–boson–exchange (OBE) ones GrossStad08 . Note
also successful treatments based on chiral effective field theory
OrdoRayKolck94 ; EpelGloeMeiss00 , for a review see Epel05 .
In this talk, we would like to draw attention to the first application of
unitary clothing transformations SheShi01 ; KorCanShe07 in describing the
nucleon-nucleon ($N$–$N$) scattering. Recall that such transformations $W$,
being aimed at the inclusion of the so–called cloud or persistent effects,
make it possible the transition from the bare–particle representation (BPR) to
the clothed–particle representation (CPR) in the Hilbert space $\mathcal{H}$
of meson–nucleon states. In this way, a large amount of virtual processes
induced with the meson absorption/emission, the $N\overline{N}$–pair
annihilation/production and other cloud effects can be accumulated in the
creation (destruction) operators $\alpha_{c}$ for the ”clothed” (physical)
mesons and nucleons. Such a bootstrap reflects the most significant
distinction between the concepts of clothed and bare particles.
In the course of the clothing procedure all the generators of the Poincaré
group get one and the same sparse structure on $\mathcal{H}$ SheShi01 . Here
we will focus upon one of them, viz., the total Hamiltonian
$H=H_{F}(\alpha)+H_{I}(\alpha)\equiv H(\alpha)$ (1)
with
$H_{I}(\alpha)=V(\alpha)+\mbox{mass and vertex counterterms},$ (2)
where free part $H_{F}(\alpha)$ and interaction $V(\alpha)$ depend on creation
(destruction) operators $\alpha^{{\dagger}}(\alpha)$ in the BPR , i.e.,
referred to bare particles with physical masses KorCanShe07 , where they have
been introduced via the mass–changing Bogoliubov–type UTs. To be more
definite, let us consider fermions (nucleons and antinucleons) and bosons
($\pi$–, $\eta$–, $\rho$–, $\omega$–mesons, etc.) interacting via the Yukawa-
type couplings for scalar (s), pseudoscalar (ps) and vector (v) mesons (see,
e.g., MachHolElst87 ). Then, using a trick prompted by the derivation of Eq.
(7.5.22) in WeinbergBook1995 to eliminate in a proper way the vector–field
component $\varphi_{\rm{v}}^{0}$ , we have $V(\alpha)=V_{s}+V_{ps}+V_{\rm{v}}$
with
$V_{s}=g_{s}\int
d\vec{x}\,\bar{\psi}(\vec{x})\psi(\vec{x})\varphi_{s}(\vec{x})$ (3)
$V_{ps}=ig_{ps}\int
d\vec{x}\,\bar{\psi}(\vec{x})\gamma_{5}\psi(\vec{x})\varphi_{ps}(\vec{x})$ (4)
$V_{\rm{v}}=\int
d\vec{x}\,\left\\{g_{\rm{v}}\bar{\psi}(\vec{x})\gamma_{\mu}\psi(\vec{x})\varphi_{\rm{v}}^{\mu}(\vec{x})\phantom{\frac{f}{4}}\right.\\\
\left.+\frac{f_{\rm{v}}}{4m}\bar{\psi}(\vec{x})\sigma_{\mu\nu}\psi(\vec{x})\varphi_{\rm{v}}^{\mu\nu}(\vec{x})\right\\}\\\
+\int
d\vec{x}\,\left\\{\frac{g_{\rm{v}}^{2}}{2m_{\rm{v}}^{2}}\bar{\psi}(\vec{x})\gamma_{0}\psi(\vec{x})\bar{\psi}(\vec{x})\gamma_{0}\psi(\vec{x})\right.\\\
\left.+\frac{f_{\rm{v}}^{2}}{4m^{2}}\bar{\psi}(\vec{x})\sigma_{0i}\psi(\vec{x})\bar{\psi}(\vec{x})\sigma_{0i}\psi(\vec{x})\right\\},$
(5)
with the boson fields $\varphi_{b}$ and the fermion field $\psi$, where
$\varphi_{\rm{v}}^{\mu\nu}(\vec{x})=\partial^{\mu}\varphi_{\rm{v}}^{\nu}(\vec{x})-\partial^{\nu}\varphi_{\rm{v}}^{\mu}(\vec{x})$
the tensor of the vector field included. The mass (vertex) counterterms are
given by Eqs. (32)–(33) of Ref. KorCanShe07 (the difference $V_{0}(\alpha)$
\- $V(\alpha)$ where a primary interaction $V_{0}(\alpha)$ is derived from
$V(\alpha)$ replacing the ”physical” coupling constants by ”bare” ones).
The corresponding set $\alpha$ involves operators $a^{{\dagger}}(a)$ for the
bosons, $b^{{\dagger}}(b)$ for the nucleons and $d^{{\dagger}}(d)$ for the
antinucleons. In their terms, e.g., we have the free pion and fermion parts
$H_{F}(\alpha)=\int
d\vec{k}\,\omega_{\vec{k}}a^{\dagger}(\vec{k})a(\vec{k})\\\ +\int
d\vec{p}\,E_{\vec{p}}\sum\limits_{\mu}\left[b^{\dagger}(\vec{p},\mu)b(\vec{p},\mu)+d^{\dagger}(\vec{p},\mu)d(\vec{p},\mu)\right]$
(6)
and the primary trilinear interaction
$V(\alpha)\sim
ab^{{\dagger}}b+ab^{{\dagger}}d^{{\dagger}}+adb+add^{{\dagger}}+H.c.$ (7)
with the three-legs vertices. Here
$\omega_{\vec{k}}=\sqrt{m^{2}_{b}+\vec{k}^{2}}\,\,\,$($E_{\vec{p}}=\sqrt{m^{2}+\vec{p}^{2}}$)
the pion (nucleon) energy with physical mass $m_{b}(m)$, $\mu$ the fermion
polarization index.
We have tried to draw parallels with that field–theoretic background which has
been employed in boson–exchange models. First of all, we imply the approach by
the Bonn group MachHolElst87 ; Mach01 , where, following the idea by Schütte
Schutte , the authors started from the total Hamiltonian (in our notations),
$H=H_{F}(\alpha)+V(\alpha)$ (8)
with the boson-nucleon interaction
$V(\alpha)\sim ab^{{\dagger}}b+H.c.$ (9)
## 2 Analytic expressions for the quasipotentials in momentum space
As shown in SheShi01 , after eliminating the so-called bad terms111By
definition, they prevent the bare vacuum $\Omega_{0}$
($a|\Omega_{0}\rangle=b|\Omega_{0}\rangle=\ldots=0$) and the bare one–particle
states $|1bare\rangle\equiv a^{{\dagger}}|\Omega_{0}\rangle$
($b^{{\dagger}}|\Omega_{0}\rangle,\ldots$) to be $H$ eigenstates. from
$V(\alpha)$ the primary Hamiltonian $H(\alpha)$ can be represented in the
form,
$H(\alpha)=K_{F}(\alpha_{c})+K_{I}(\alpha_{c})\equiv K(\alpha_{c})$ (10)
The free part of the new decomposition is determined by
$K_{F}(\alpha_{c})=\int
d\vec{k}\,\omega_{\vec{k}}a_{c}^{\dagger}(\vec{k})a_{c}(\vec{k})\\\ +\int
d\vec{p}\,E_{\vec{p}}\sum\limits_{\mu}\left[b_{c}^{\dagger}(\vec{p},\mu)b_{c}(\vec{p},\mu)+d_{c}^{\dagger}(\vec{p},\mu)d_{c}(\vec{p},\mu)\right]$
(11)
while $K_{I}$ contains only interactions responsible for physical processes,
these quasipotentials between the clothed particles, e.g.,
$K_{I}^{(2)}(\alpha_{c})=K(NN\to NN)+K(\bar{N}\bar{N}\to\bar{N}\bar{N})\\\
+K(N\bar{N}\to N\bar{N})+K(bN\to bN)+K(b\bar{N}\to b\bar{N})\\\ +K(bb\to
N\bar{N})+K(N\bar{N}\to bb)$ (12)
In accordance with the clothing procedure developed in SheShi01 they obey the
following requirements:
i) The physical vacuum (the $H$ lowest eigenstate) must coincide with a new
no–particle state $\Omega$, i.e., the state that obeys the equations
$a_{c}(\vec{k})\left|\Omega\right\rangle=b_{c}(\vec{p},\mu)\left|\Omega\right\rangle=d_{c}(\vec{p},\mu)\left|\Omega\right\rangle=0,$
$\forall\,\,{\vec{k},\,\vec{p},\,\mu}\,\,\,\,\,\left(\left\langle\Omega|\Omega\right\rangle=1\right).$
(13)
ii) New one-clothed-particle states $|\vec{k}\rangle_{c}\equiv
a_{c}^{{\dagger}}(\vec{k})\Omega$ etc. are the eigenvectors both of $K_{F}$
and $K$,
$K(\alpha_{c})|\vec{k}\rangle_{c}=K_{F}(\alpha_{c})|\vec{k}\rangle_{c}=\omega_{k}|\vec{k}\rangle_{c}$
(14) $K_{I}(\alpha_{c})|\vec{k}\rangle_{c}=0$ (15)
iii) The spectrum of indices that enumerate the new operators must be the same
as that for the bare ones .
iv) The new operators $\alpha_{c}$ satisfy the same commutation rules as do
their bare counterparts $\alpha$, since the both sets are connected to each
other via the similarity transformation
$\alpha_{c}=W^{{\dagger}}\alpha W,$ (16)
with a unitary operator $W$ to be obtained as in SheShi01 .
It is important to realize that operator $K(\alpha_{c})$ is the same
Hamiltonian $H(\alpha)$. Accordingly [10,11] the $N$–$N$ interaction operator
in the CPR has the following structure:
$K(NN\rightarrow NN)=\sum\limits_{b}K_{b}(NN\rightarrow NN),$
$K_{b}(NN\rightarrow
NN)=\int\sum\limits_{\mu}d\vec{p}^{\prime}_{1}\,d\vec{p}^{\prime}_{2}\,d\vec{p}_{1}\,d\vec{p}_{2}\\\
\times
V_{b}(1^{\prime},2^{\prime};1,2)b_{c}^{\dagger}(1^{\prime})b_{c}^{\dagger}(2^{\prime})b_{c}(1)b_{c}(2),$
(17)
where the symbol $\sum\limits_{\mu}$ denotes the summation over nucleon spin
projections, $1=\\{\vec{p}_{1},\mu_{1}\\}$, etc.
For our evaluations of the c–number matrices $V_{b}$ we have employed some
experience from Refs. SheShi01 ; KorCanShe07 to get in the second order in
the coupling constants
$V_{b}(1^{\prime},2^{\prime};1,2)=\frac{1}{(2\pi)^{3}}\frac{m^{2}}{\sqrt{E_{\vec{p}^{\prime}_{1}}E_{\vec{p}^{\prime}_{2}}E_{\vec{p}_{1}}E_{\vec{p}_{2}}}}\\\
\times\delta\left(\vec{p}^{\prime}_{1}+\vec{p}^{\prime}_{2}-\vec{p}_{1}-\vec{p}_{2}\right)v_{b}(1^{\prime},2^{\prime};1,2),$
(18) $v_{s}(1^{\prime},2^{\prime};1,2)\\\
=-\frac{g_{s}^{2}}{2}\bar{u}(\vec{p}^{\prime}_{1})u(\vec{p}_{1})\frac{1}{(p_{1}-p^{\prime}_{1})^{2}-m_{s}^{2}}\bar{u}(\vec{p}^{\prime}_{2})u(\vec{p}_{2}),$
(19) $v_{ps}(1^{\prime},2^{\prime};1,2)\\\
=\frac{g_{ps}^{2}}{2}\bar{u}(\vec{p}^{\prime}_{1})\gamma_{5}u(\vec{p}_{1})\frac{1}{(p_{1}-p^{\prime}_{1})^{2}-m_{ps}^{2}}\bar{u}(\vec{p}^{\prime}_{2})\gamma_{5}u(\vec{p}_{2}),$
(20)
$v_{\rm{v}}(1^{\prime},2^{\prime};1,2)=\frac{1}{2}\frac{1}{(p^{\prime}_{1}-p_{1})^{2}-m_{\rm{v}}^{2}}\\\
\times\left[\bar{u}(\vec{p}^{\prime}_{1})\left\\{(g_{\rm{v}}+f_{\rm{v}})\gamma_{\nu}-\frac{f_{\rm{v}}}{2m}(p^{\prime}_{1}+p_{1})_{\nu}\right\\}u(\vec{p}_{1})\right.\\\
\times\bar{u}(\vec{p}^{\prime}_{2})\left\\{(g_{\rm{v}}+f_{\rm{v}})\gamma^{\nu}-\frac{f_{\rm{v}}}{2m}(p^{\prime}_{2}+p_{2})^{\nu}\right\\}u(\vec{p}_{2})\\\
-\bar{u}(\vec{p}^{\prime}_{1})\left\\{(g_{\rm{v}}+f_{\rm{v}})\gamma_{\nu}-\frac{f_{\rm{v}}}{2m}(p^{\prime}_{1}+p_{1})_{\nu}\right\\}u(\vec{p}_{1})\\\
\times\bar{u}(\vec{p}^{\prime}_{2})\frac{f_{\rm{v}}}{2m}\left\\{(\hat{p}_{1}^{\prime}+\hat{p}_{2}^{\prime}-\hat{p}_{1}-\hat{p}_{2})\gamma^{\nu}\right.\\\
\left.\phantom{\frac{f_{\rm{v}}}{2m}}\left.-(p^{\prime}_{1}+p^{\prime}_{2}-p_{1}-p_{2})^{\nu}\right\\}u(\vec{p}_{2})\right],$
(21)
where $m_{b}$ the mass of the clothed boson (its physical value) and
$\hat{q}=q_{\mu}\gamma^{\mu}$. In the framework of the isospin formalism one
needs to add the factor $\vec{\tau}(1)\vec{\tau}(2)$ in the corresponding
expressions.
At this point, our derivation of the vector-boson contribution (21) is to be
specifically commented. Actually, it is the case, where for a
Lorentz–invariant Lagrangian it is not necessarily to have ”… the interaction
Hamiltonian as the integral over space of a scalar interaction density; we
also need to add non–scalar terms to the interaction density …” (quoted from
p.292 of Ref. WeinbergBook1995 ). Let us recall that the density in question
has the property,
$U_{F}(\Lambda,a)\mathscr{H}(x)U_{F}^{-1}(\Lambda,a)=\mathscr{H}(\Lambda
x+a),$ (22)
where the operators $U_{F}(\Lambda,a)$ realize a unitary irreducible
representation of the Poincaré group in the Hilbert space of states for free
(non–interacting) fields.
By definition, the first clothing transformation $W^{(1)}=\exp[R^{(1)}]$
(${R^{(1)}}^{{\dagger}}=-R^{(1)}$) eliminates all interactions linear in the
coupling constants, viz.,
$V^{(1)}=V_{s}+V_{ps}+V_{\rm{v}}^{(1)},$
with
$V_{\rm{v}}^{(1)}=\int
d\vec{x}\left\\{g_{\rm{v}}\bar{\psi}(\vec{x})\gamma_{\mu}\psi(\vec{x})\varphi_{\rm{v}}^{\mu}(\vec{x})\phantom{\frac{f_{\rm{v}}}{4m}}\right.\\\
\left.+\frac{f_{\rm{v}}}{4m}\bar{\psi}(\vec{x})\sigma_{\mu\nu}\psi(\vec{x})\varphi_{\rm{v}}^{\mu\nu}(\vec{x})\right\\}.$
(23)
Following Ref.SheShi01 we have
$R^{(1)}=-i\lim\limits_{\varepsilon\rightarrow
0+}\int\limits_{0}^{\infty}V_{D}^{(1)}(t)e^{-\varepsilon t}dt$ (24)
if $m_{b}<2m$. Here
$V_{D}^{(1)}\equiv\exp[iH_{F}t]V^{(1)}\exp[-iH_{F}t]=\int\mathscr{H}^{(1)}(x)d\vec{x},$
where $\mathscr{H}^{(1)}(x)$ is the Lorentz scalar.
The corresponding interaction operator in the CPR (12) can be written as
$K_{I}^{(2)}(\alpha_{c})=\frac{1}{2}\left[R^{(1)}(\alpha_{c}),V^{(1)}(\alpha_{c})\right]+V^{(2)}(\alpha_{c}),$
(25)
where we have kept only the contributions of the second order in the coupling
constants, so
$V^{(2)}=\int
d\vec{x}\left\\{\frac{g_{\rm{v}}^{2}}{2m_{\rm{v}}^{2}}\bar{\psi}(x\vec{)}\gamma_{0}\psi(\vec{x})\bar{\psi}(\vec{x})\gamma_{0}\psi(\vec{x})\right.\\\
\left.+\frac{f_{\rm{v}}^{2}}{4m^{2}}\bar{\psi}(\vec{x})\sigma_{0i}\psi(\vec{x})\bar{\psi}(\vec{x})\sigma_{0i}\psi(\vec{x})\right\\}.$
(26)
We point out that all quantities in the r.h.s. of Eq.(25) depend on the new
creation(destruction) operators $\alpha_{c}$. In particular, it means that in
the standard Fourier expansions of the fields involved in the definitions of
$V^{(1)}$ and $V^{(2)}$ one should replace the set $\\{\alpha\\}$ by the set
$\\{\alpha_{c}\\}$. Thus, there is an essential distinction between
$V^{(1)}$($V^{(2)}$), on the one hand, and the first(second) integral in the
r.h.s. of Eq.(5), on the other hand.
For this exposition we do not intend to derive all interactions between the
clothed mesons and nucleons, allowed by formula (25). Our aim is more humble,
viz., to find in the r.h.s. of Eq.(25) terms of the type (17), responsible for
the $N$–$N$ interaction. Meanwhile, in case of the vector mesons we encounter
an interplay between the commutator $[R^{(1)}_{\rm{v}},V_{\rm{v}}^{(1)}]/2$
and the integral (26). Indeed, after a simple algebra we find
$\frac{1}{2}\left[R^{(1)},V^{(1)}\right]_{\rm{v}}(NN\rightarrow NN)\\\
=K_{\rm{v}}(NN\rightarrow NN)+K_{cont}(NN\rightarrow NN),$
where the first term has the structure of Eq.(17) with the coefficients by
(21). At the same time the second term $K_{cont}$ completely cancels the
non–scalar operator $V^{(2)}$. The latter may be associated with a contact
interaction since it does not contain any propagators (cf. the approach by the
Osaka group TamuraSato88 ), being expressed through the
$b^{{\dagger}}_{c}(b_{c})$. In other words, the first UCT enables us to remove
the non–invariant terms directly in the Hamiltonian. In our opinion, such a
cancellation, first discussed here, is a pleasant feature of the CPR.
Moreover, as it was shown in Ref.SheShi01 , for each boson included the
corresponding relativistic and properly symmetrized $N$–$N$ interaction, the
kernel of integral equations for the $N$–$N$ bound and scattering states, is
determined by
$\left\langle
b_{c}^{\dagger}(\vec{p}^{\prime}_{1})b_{c}^{\dagger}(\vec{p}^{\prime}_{2})\Omega\right|K_{b}(NN\rightarrow
NN)\left|b_{c}^{\dagger}(\vec{p}_{1})b_{c}^{\dagger}(\vec{p}_{2})\Omega\right\rangle\\\
=V_{b}^{dir}(1^{\prime},2^{\prime};1,2)-V_{b}^{exc}(1^{\prime},2^{\prime};1,2),$
(27)
where we have separated the so–called direct
$V_{b}^{dir}(1^{\prime},2^{\prime};1,2)=-V_{b}(1^{\prime},2^{\prime};1,2)-V_{b}(2^{\prime},1^{\prime};2,1)$
(28)
and exchange
$V_{b}^{exc}(1^{\prime},2^{\prime};1,2)=V_{b}^{dir}(2^{\prime},1^{\prime};1,2)$
(29)
terms. For example, the one–pion–exchange contribution can be divided into the
two parts:
$V_{\pi}^{dir}(1^{\prime},2^{\prime};1,2)=-\frac{g_{\pi}^{2}}{(2\pi)^{3}}\frac{m^{2}}{\sqrt{E_{\vec{p}^{\prime}_{1}}E_{\vec{p}^{\prime}_{2}}E_{\vec{p}_{1}}E_{\vec{p}_{2}}}}\\\
\times\delta\left(\vec{p}^{\prime}_{1}+\vec{p}^{\prime}_{2}-\vec{p}_{1}-\vec{p}_{2}\right)\bar{u}(\vec{p}^{\prime}_{1})\gamma_{5}u(\vec{p}_{1})\bar{u}(\vec{p}^{\prime}_{2})\gamma_{5}u(\vec{p}_{2})\\\
\times\frac{1}{2}\left\\{\frac{1}{(p_{1}-p^{\prime}_{1})^{2}-m_{\pi}^{2}}+\frac{1}{(p_{2}-p^{\prime}_{2})^{2}-m_{\pi}^{2}}\right\\}$
(30)
and
$V_{\pi}^{exc}(1^{\prime},2^{\prime};1,2)=-\frac{g_{\pi}^{2}}{(2\pi)^{3}}\frac{m^{2}}{\sqrt{E_{\vec{p}^{\prime}_{1}}E_{\vec{p}^{\prime}_{2}}E_{\vec{p}_{1}}E_{\vec{p}_{2}}}}\\\
\times\delta\left(\vec{p}^{\prime}_{1}+\vec{p}^{\prime}_{2}-\vec{p}_{1}-\vec{p}_{2}\right)\bar{u}(\vec{p}^{\prime}_{1})\gamma_{5}u(\vec{p}_{2})\bar{u}(\vec{p}^{\prime}_{2})\gamma_{5}u(\vec{p}_{1})\\\
\times\frac{1}{2}\left\\{\frac{1}{(p_{2}-p^{\prime}_{1})^{2}-m_{\pi}^{2}}+\frac{1}{(p_{1}-p^{\prime}_{2})^{2}-m_{\pi}^{2}}\right\\}$
(31)
to be depicted in Fig.1, where the dashed lines correspond to the following
Feynman–like ”propagators”:
$\frac{1}{2}\left\\{\frac{1}{(p_{1}-p^{\prime}_{1})^{2}-m_{\pi}^{2}}+\frac{1}{(p_{2}-p^{\prime}_{2})^{2}-m_{\pi}^{2}}\right\\}$
on the left panel and
$\frac{1}{2}\left\\{\frac{1}{(p_{2}-p^{\prime}_{1})^{2}-m_{\pi}^{2}}+\frac{1}{(p_{1}-p^{\prime}_{2})^{2}-m_{\pi}^{2}}\right\\}$
on the right panel. Other distinctive features of the result (27) have been
discussed in SheShi01 ; KorCanShe07 .
Figure 1: The Feynman–like diagrams for the direct and exchange contributions
in the r.h.s. of Eq.(27).
## 3 The field–theoretic description of the elastic N–N scattering
### 3.1 The $T$–matrix in the CPR
In order to evaluate the $N$–$N$ scattering amplitude for the collision energy
$E$ we will regard a field operator $T$ that meets the equation
$T(E+i0)=H_{I}+H_{I}(E+i0-H_{F})^{-1}T(E+i0)$ (32)
and whose matrix elements $\langle NN|T(E+i0)|NN\rangle$ on the energy shell
$E=E_{1}+E_{2}=E_{1}^{{}^{\prime}}+E_{2}^{{}^{\prime}}$ can be expressed
through the phase shifts and mixing parameters.
Unlike nonrelativistic quantum mechanics (NQM) in relativistic QFT the
interaction $H_{I}$ does not conserve the particle number, being the spring of
particle creation and destruction. The feature makes the problem of finding
the $N$–$N$ scattering matrix much more complicated than in the framework of
nonrelativistic approach.
Such a general field–theoretic consideration can be simplified with the help
of an equivalence theorem ShebekoFB17 according to which the $S$ matrix
elements in the Dirac (D) picture, viz.,
$S_{fi}\equiv\langle\alpha^{{\dagger}}...\Omega_{0}|S(\alpha)|\alpha^{{\dagger}}...\Omega_{0}\rangle$
(33)
are equal to the corresponding elements
$S_{fi}^{c}\equiv\langle\alpha_{c}^{{\dagger}}...\Omega|S(\alpha_{c})|\alpha_{c}^{{\dagger}}...\Omega\rangle$
(34)
of the $S$ matrix in the CPR once the UCTs
$W_{D}(t)=\exp(iK_{F}t)W\exp(-iK_{F}t)$
obey the condition
$W_{D}(\pm\infty)=1$ (35)
The $T$ operator in the CPR satisfies the equation
$T_{cloth}(E+i0)=K_{I}\\\ +K_{I}(E+i0-K_{F})^{-1}T_{cloth}(E+i0)$ (36)
and the matrix
$T_{fi}\equiv\langle f;b|T(E+i0)|i;b\rangle\\\ =\langle
f;c|T_{cloth}(E+i0)|i;c\rangle\equiv T_{fi}^{c},$ (37)
where $|;b\rangle$ ( $|;c\rangle$ ) are the $H_{F}$ ( $K_{F}$ ) eigenvectors,
may be evaluated relying upon properties of the new interaction
$K_{I}(\alpha_{c})$.
If in Eq.(36) we approximate $K_{I}$ by $K_{I}^{(2)}$, then initial task of
evaluating the CPR matrix elements can be reduced to solving the equation
$\langle 1^{\prime},2^{\prime}|T_{NN}(E)|1,2\rangle=\langle
1^{\prime},2^{\prime}|K_{NN}|1,2\rangle\\\ +\langle
1^{\prime},2^{\prime}|K_{NN}(E+i0-K_{F})^{-1}T_{NN}(E)|1,2\rangle.$ (38)
### 3.2 The $R$–matrix equation and its angular–momentum decomposition
For practical applications one prefers to work with the corresponding
$R$–matrix that meets the set of equations
$\left\langle{1^{\prime}2^{\prime}}\right|\bar{R}(E)\left|{12}\right\rangle=\left\langle{1^{\prime}2^{\prime}}\right|\bar{K}_{NN}\left|{12}\right\rangle\\\
+\int\limits_{34}\\!\\!\\!\\!\\!\\!\\!\\!\sum{\left\langle{1^{\prime}2^{\prime}}\right|\bar{K}_{NN}\left|{34}\right\rangle\frac{{\left\langle{34}\right|\bar{R}(E)\left|{12}\right\rangle}}{{E-E_{3}-E_{4}}}}$
(39)
with $\bar{R}(E)=R(E)/2$ and $\bar{K}_{NN}=K_{NN}/2$, where the operation
$\int\limits_{34}\\!\\!\\!\\!\\!\\!\sum$ means the summation over nucleon
polarizations and the $p.v.$ integration over nucleon momenta. The kernel of
Eq.(39) is
$\left\langle{1^{\prime}2^{\prime}}\left|\bar{K}_{NN}\right|{12}\right\rangle=\delta\left(\vec{p}^{\prime}_{1}+\vec{p}^{\prime}_{2}-\vec{p}_{1}-\vec{p}_{2}\right)\left\langle{1^{\prime}2^{\prime}}\left|\bar{V}\right|{12}\right\rangle\\\
\equiv\delta\left(\vec{p}^{\prime}_{1}+\vec{p}^{\prime}_{2}-\vec{p}_{1}-\vec{p}_{2}\right)\\\
\times\left\langle{\vec{p}^{\prime}_{1}\mu^{\prime}_{1}\tau^{\prime}_{1},\vec{p}^{\prime}_{2}\mu^{\prime}_{2}\tau^{\prime}_{2}\left|{\bar{V}}\right|\vec{p}_{1}\mu_{1}\tau_{1},\vec{p}_{2}\mu_{2}\tau_{2}}\right\rangle$
The subsequent calculations are essentially simplified in the center–of–mass
system (c.m.s) in which
$\left\langle\vec{p}^{\prime}\mu^{\prime}_{1}\mu^{\prime}_{2},\tau^{\prime}_{1}\tau^{\prime}_{2}\left|{\bar{R}(E)}\right|\vec{p}\mu_{1}\mu_{2},\tau_{1}\tau_{2}\right\rangle\\\
=\left\langle{\vec{p}^{\prime}\mu^{\prime}_{1}\mu^{\prime}_{2},\tau^{\prime}_{1}\tau^{\prime}_{2}\left|{\bar{V}}\right|\vec{p}\mu_{1}\mu_{2},\tau_{1}\tau_{2}}\right\rangle\\\
+\sum p.v.\int
d\vec{q}\left\langle\vec{p}^{\prime}\mu^{\prime}_{1}\mu^{\prime}_{2},\tau^{\prime}_{1}\tau^{\prime}_{2}\left|{\bar{V}}\right|\vec{q}\mu_{3}\mu_{4},\tau_{3}\tau_{4}\right\rangle\\\
\times\frac{\left\langle\vec{q}\mu_{3}\mu_{4},\tau_{3}\tau_{4}\left|{\bar{R}(E)}\right|\vec{p}\mu_{1}\mu_{2},\tau_{1}\tau_{2}\right\rangle}{E-2E_{\vec{q}}}$
(40)
Here the quantum numbers $\mu(\tau)$ are the individual spin (isospin)
projections.
Accordingly Eq. (27)
$\left\langle{1^{\prime}2^{\prime}}\right|{\bar{V}}\left|{12}\right\rangle=\frac{1}{2(2\pi)^{3}}\frac{m^{2}}{E_{\vec{p}^{\prime}}E_{\vec{p}}}\\\
\times\sum\limits_{b}[v_{b}^{dir}(1^{\prime},2^{\prime};1,2)-v_{b}^{exc}(1^{\prime},2^{\prime};2,1)]$
(41)
with
$v_{b}^{dir}(1^{\prime},2^{\prime};1,2)=-v_{b}(1^{\prime},2^{\prime};1,2)-v_{b}(2^{\prime},1^{\prime};2,1)$
(42)
and
$v_{b}^{exc}(1^{\prime},2^{\prime};2,1)=v_{b}^{dir}(2^{\prime},1^{\prime};1,2),$
where the separate boson contributions are determined by Eqs. (19)–(21) with
$\vec{p}_{1}=\vec{p}=-\vec{p}_{2}$ and
$\vec{p}^{\prime}_{1}=\vec{p}^{\prime}=-\vec{p}^{\prime}_{2}$.
Following a common practice we are interested in the angular–momentum
decomposition of Eq.(40) assuming a nonrelativistic analog of relativistic
partial wave expansions (see Werle66 and refs. therein) for two–particle
states. For example, the clothed two–nucleon state (the so–called two–nucleon
plane wave) can be represented as
$\left|\vec{p}\mu_{1}\mu_{2},\tau_{1}\tau_{2}\right\rangle=\sum\left(\tfrac{1}{2}\mu_{1}\tfrac{1}{2}\mu_{2}\left|SM_{S}\right.\right)\left(\tfrac{1}{2}\tau_{1}\tfrac{1}{2}\tau_{2}\left|TM_{T}\right.\right)\\\
\left(lm_{l}SM_{S}\left|JM_{J}\right.\right)Y_{lm_{l}}^{*}\left(\vec{p}/p\right)\left|pJ(lS)M_{J},TM_{T}\right\rangle,$
(43)
where $J$, $S$ and $T$ are, respectively, total angular momentum, spin and
isospin of the $NN$ pair, being the eigenvalues of the operators
$\vec{J}_{ferm}$, $\vec{S}_{ferm}$ and $\vec{T}_{ferm}$. Here
$\vec{J}_{ferm}=\vec{L}_{ferm}+\vec{S}_{ferm},$ (44)
where $\vec{L}_{ferm}$ ($\vec{S}_{ferm}$) the orbital (spin) momentum of the
fermion field,
$\vec{L}_{ferm}=\frac{i}{2}\sum\limits_{\mu}\int
d\vec{p}\,\,\vec{p}\times\left[\frac{\partial
b_{c}^{\dagger}(\vec{p}\mu)}{\partial\vec{p}}b_{c}(\vec{p}\mu)\right.\\\
-b_{c}^{\dagger}(\vec{p}\mu)\frac{\partial
b_{c}(\vec{p}\mu)}{\partial\vec{p}}+\frac{\partial
d_{c}^{\dagger}(\vec{p}\mu)}{\partial\vec{p}}d_{c}(\vec{p}\mu)\\\
\left.-d_{c}^{\dagger}(\vec{p}\mu)\frac{\partial
d_{c}(\vec{p}\mu)}{\partial\vec{p}}\right]$ (45)
and
$\vec{S}_{ferm}=\frac{1}{2}\sum\limits_{\mu\mu^{\prime}}\int
d\vec{p}\,\,\chi^{{\dagger}}_{\mu^{\prime}}\mathbf{\sigma}\chi_{\mu}\left\\{b_{c}^{\dagger}(\vec{p}\mu^{\prime})b_{c}(\vec{p}\mu)\right.\\\
\left.-d_{c}^{\dagger}(\vec{p}\mu^{\prime})d_{c}(\vec{p}\mu)\right\\},$ (46)
where $\chi_{\mu^{\prime}}(\chi_{\mu})$ are the Pauli spinors. For brevity, we
do not show the isospin operator $\vec{T}_{ferm}$.
The corresponding eigenvalue equations look as
${\vec{J}}^{\,2}_{ferm}\left|{pJ(lS)M_{J}}\right\rangle=J(J+1)\left|{pJ(lS)M_{J}}\right\rangle$
$J_{ferm}^{z}\left|{pJ(lS)M_{J}}\right\rangle=M_{J}\left|{pJ(lS)M_{J}}\right\rangle$
(47)
and
${\vec{S}}^{\,2}_{ferm}\left|{\vec{p}SM_{S}}\right\rangle=S(S+1)\left|{\vec{p}SM_{S}}\right\rangle$
$S_{ferm}^{z}\left|{\vec{p}SM_{S}}\right\rangle=M_{S}\left|{\vec{p}SM_{S}}\right\rangle$
(48)
Doing so, we have introduced the vectors222For a moment, the isospin quantum
numbers are suppressed.
$\left|{\vec{p}SM_{S}}\right\rangle=\sum{\left({\left.\frac{1}{2}\mu_{1}\frac{1}{2}\mu_{2}\right|SM_{S}}\right)}\left|{\vec{p}\mu_{1}\mu_{2}}\right\rangle$
(49)
and
$\left|{pJ(lS)M_{J}}\right\rangle\\\
=\int{d\hat{\vec{p}}}\,Y_{lm_{l}}\left(\vec{p}/p\right)\left|{\vec{p}SM_{S}}\right\rangle\left({lm_{l}SM_{S}\left|{JM_{J}}\right.}\right)$
(50)
A simple way of deriving Eqs.(47)–(48) is to use the transformation
$U_{F}^{c}(R)|\vec{p}SM_{S}\rangle=|R\vec{p}SM_{S}^{\prime}\rangle
D_{M_{S}^{\prime}M_{S}}^{(S)}(R)$ (51) $\forall\,R\,\in\,\mbox{the rotation
group}$
One should note that in our case the separable ansatz
$\left|{\vec{p}_{1}\vec{p}_{2}\mu_{1}\mu_{2}}\right\rangle=\left|{\vec{p}_{1}\mu_{1}}\right\rangle\left|{\vec{p}_{2}\mu_{2}}\right\rangle$
often exploited in relativistic quantum mechanics (RQM) (see, e.g., Werle66
and KeisPoly91 ) does not work. However, one can employ the similarity
transformation 333Sometimes it is convenient to handle the operators
$b_{c}^{{\dagger}}(p\mu)=\sqrt{p_{0}}b_{c}^{{\dagger}}(\vec{p}\mu)$ and their
adjoints $b_{c}(p\mu)$ that meet covariant relations
$\left\\{{b_{c}^{\dagger}(p^{\prime}\mu^{\prime}),b_{c}(p\mu)}\right\\}=p_{0}\delta(\vec{p}^{\prime}-\vec{p})\delta_{\mu^{\prime}\mu}$
$U_{F}^{c}\left({\Lambda,a}\right)b_{c}^{\dagger}\left({p\mu}\right)U_{F}^{c}{\dagger}\left({\Lambda,a}\right)\\\
=e^{i\Lambda p\cdot a}b_{c}^{\dagger}\left({\Lambda
p\mu^{\prime}}\right)D_{\mu^{\prime}\mu}^{(\frac{1}{2})}\left({W\left({\Lambda,p}\right)}\right)$
(52)
with the Wigner rotation $W(\Lambda,p)$ (e.g., for rotations $W(R,p)=R$ ) and
the property of the physical vacuum $\Omega$ to be invariant with respect to
unitary transformations $U_{F}^{c}$ in the CPR (some details can be found in a
separate paper).
The use of expansion (43) gives rise to the well known JST representation, in
which
$\left\langle
p^{\prime}J^{\prime}(l^{\prime}S^{\prime})M^{\prime}_{J},T^{\prime}M^{\prime}_{T}\left|\bar{R}(E)\\{\bar{V}\\}\right|pJ(lS)M_{J},TM_{T}\right\rangle\\\
=\bar{R}(E)\\{\bar{V}\\}_{l^{\prime}l}^{JST}(p^{\prime},p)\delta_{J^{\prime}J}\delta_{M^{\prime}_{J}M_{J}}\delta_{S^{\prime}S}\delta_{T^{\prime}T}\delta_{M^{\prime}_{T}M_{T}},$
(53)
so Eq.(40) reduces to the set of simple integral equations,
$\bar{R}_{l^{\prime}\,l}^{JST}(p^{\prime},p)=\bar{V}_{l^{\prime}\,l}^{JST}(p^{\prime},p)\\\
+\sum\limits_{l^{\prime\prime}}{p.v.}\int\limits_{0}^{\infty}\frac{q^{2}\,dq}{2(E_{p}-E_{q})}\bar{V}_{l^{\prime}\,l^{\prime\prime}}^{JST}(p^{\prime},q)\bar{R}_{l^{\prime\prime}{\rm{}}l}^{JST}(q,p)$
(54)
to be solved for each submatrix $\overline{R}^{JST}$ composed of the elements
$\bar{R}_{l^{\prime}l}^{JST}(p^{\prime},p)\equiv\bar{R}_{l^{\prime}l}^{JST}(p^{\prime},p;2E_{p}),$
(55)
where $E_{p}=\sqrt{\vec{p}^{2}+m^{2}}$ the collision energy in the c.m.s.. One
should note that in view of the charge independence assumed in this work one
has to solve two separate equations for isospin values $T=0$ and $T=1$.
## 4 Results of numerical calculations and their discussion
In the course of our computations we have used the so–called matrix inversion
method (MIM) (see BJ76 and refs. therein). Since we deal with the
relativistic dispersion law for the particle energies, the well known
substraction procedure within the MIM leads to equations
$R_{l^{\prime}{\rm{}}l}^{JST}(p^{\prime},p)=V_{l^{\prime}{\rm{}}l}^{JST}(p^{\prime},p)\\\
+\frac{1}{2}\sum\limits_{l^{\prime\prime}}\int\limits_{0}^{\infty}\frac{dq}{p^{2}-q^{2}}\left\\{q^{2}(E_{p}+E_{q})V_{l^{\prime}l^{\prime\prime}}^{JST}(p^{\prime},q)R_{l^{\prime\prime}l}^{JST}(q,p)\right.\\\
\left.-2p^{2}E_{p}V_{l^{\prime}l^{\prime\prime}}^{JST}(p^{\prime},p)R_{l^{\prime\prime}l}^{JST}(p,p)\right\\}.$
(56)
Table 1: The best–fit parameters for the two models. The third (fourth) column taken from Table A.1 Mach89 (obtained by solving Eqs.(56) with a least squares fitting to OBEP values in Table 2). All masses are in $MeV$, and $n_{b}=1$ except for $n_{\rho}=n_{\omega}=2.$ Meson | | Potential B | UCT
---|---|---|---
$\pi$ | $g^{2}_{\pi}/4\pi$ | 14.4 | 14.5
| $\Lambda_{\pi}$ | 1700 | 2200
| $m_{\pi}$ | 138.03 | 138.03
$\eta$ | $g^{2}_{\eta}/4\pi$ | 3 | 2.8534
| $\Lambda_{\eta}$ | 1500 | 1200
| $m_{\eta}$ | 548.8 | 548.8
$\rho$ | $g^{2}_{\rho}/4\pi$ | 0.9 | 1.3
| $\Lambda_{\rho}$ | 1850 | 1450
| $f_{\rho}/g_{\rho}$ | 6.1 | 5.85
| $m_{\rho}$ | 769 | 769
$\omega$ | $g^{2}_{\omega}/4\pi$ | 24.5 | 27
| $\Lambda_{\omega}$ | 1850 | 2035.59
| $m_{\omega}$ | 782.6 | 782.6
$\delta$ | $g^{2}_{\delta}/4\pi$ | 2.488 | 1.6947
| $\Lambda_{\delta}$ | 2000 | 2200
| $m_{\delta}$ | 983 | 983
$\sigma,\,T=0$ | $g^{2}_{\sigma}/4\pi$ | 18.3773 | 19.4434
| $\Lambda_{\sigma}$ | 2000 | 1538.13
| $m_{\sigma}$ | 720 | 717.7167
$\sigma,\,T=1$ | $g^{2}_{\sigma}/4\pi$ | 8.9437 | 10.8292
| $\Lambda_{\sigma}$ | 1900 | 2200
| $m_{\sigma}$ | 550 | 568.8612
To facilitate comparison with some derivations and calculations from Refs.
MachHolElst87 , Mach89 , we introduce the notation
$\left\langle{\vec{p}^{\prime}\,\mu^{\prime}_{1}\mu^{\prime}_{2}}\right|v^{UCT}_{b}\left|{\vec{p}\,\mu_{1}\mu_{2}}\right\rangle\\\
\equiv-F^{2}_{b}(p^{\prime},p)\left[v_{b}(1^{\prime},2^{\prime};1,2)+v_{b}(2^{\prime},1^{\prime};2,1)\right]$
for the regularized UCT quasipotentials in the c.m.s. As in Ref.MachHolElst87
, we put that invariants $F_{b}(p^{\prime},p)=F_{b}(\Lambda p^{\prime},\Lambda
p)$ have a phenomenological form,
$F_{b}(p^{\prime},p)=\left[\frac{\Lambda_{b}^{2}-m_{b}^{2}}{\Lambda_{b}^{2}-(p^{\prime}-p)^{2}}\right]^{n_{b}}\equiv
F_{b}[(p^{\prime}-p)^{2}]$
Doing so, we have
$\left\langle\vec{p}^{\,\prime}\,\mu^{\prime}_{1}\mu^{\prime}_{2}\right|v^{UCT}_{s}\left|\vec{p}\,\mu_{1}\mu_{2}\right\rangle\\\
=g_{s}^{2}\bar{u}(\vec{p}^{\,\prime})u(\vec{p})\frac{F^{2}_{s}[(p^{\prime}-p)^{2}]}{(p^{\prime}-p)^{2}-m_{s}^{2}}\bar{u}(-\vec{p}^{\,\prime})u(-\vec{p}),$
(57)
$\left\langle\vec{p}^{\,\prime}\,\mu^{\prime}_{1}\mu^{\prime}_{2}\right|v^{UCT}_{ps}\left|\vec{p}\,\mu_{1}\mu_{2}\right\rangle\\\
=-g_{ps}^{2}\bar{u}(\vec{p}^{\,\prime})\gamma_{5}u(\vec{p})\frac{F^{2}_{ps}[(p^{\prime}-p)^{2}]}{(p^{\prime}-p)^{2}-m_{ps}^{2}}\bar{u}(-\vec{p}^{\,\prime})\gamma_{5}u(-\vec{p})$
(58)
Table 2: Neutron–proton phase shifts (in degrees) for various laboratory energies (in MeV). The OBEP(OBEP∗)–rows taken from Table 5.2 Mach89 (calculated by solving Eqs.(61) with the model parameters from the third column in Table 1). The UCT∗(UCT)–rows calculated by solving Eqs.(56) with the parameters from the third (fourth) column in Table 1. As in MachHolElst87 , we have used the bar convention Stapp for the phase parameters. State | Potential | 25 | 50 | 100 | 150 | 200 | 300
---|---|---|---|---|---|---|---
| OBEP | 50.72 | 39.98 | 25.19 | 14.38 | 5.66 | -8.18
${{}^{1}}S_{0}$ | OBEP∗ | 50.71 | 39.98 | 25.19 | 14.37 | 5.66 | -8.18
| UCT∗ | 66.79 | 53.01 | 36.50 | 25.27 | 16.54 | 3.12
| UCT | 50.03 | 39.77 | 25.55 | 15.20 | 6.92 | -6.07
| OBEP | -7.21 | -11.15 | -16.31 | -20.21 | -23.47 | -28.70
${{}^{1}}P_{1}$ | OBEP∗ | -7.17 | -11.15 | -16.32 | -20.21 | -23.48 | -28.71
| UCT∗ | -7.40 | -11.70 | -17.73 | -22.63 | -26.98 | -34.54
| UCT | -7.15 | -10.95 | -15.62 | -18.90 | -21.49 | -25.41
| OBEP | 0.68 | 1.58 | 3.34 | 4.94 | 6.21 | 7.49
${{}^{1}}D_{2}$ | OBEP∗ | 0.68 | 1.58 | 3.34 | 4.94 | 6.21 | 7.49
| UCT∗ | 0.68 | 1.59 | 3.40 | 5.10 | 6.52 | 8.20
| UCT | 0.68 | 1.56 | 3.22 | 4.68 | 5.77 | 6.68
| OBEP | 9.34 | 12.24 | 9.80 | 4.57 | -1.02 | -11.48
${{}^{3}}P_{0}$ | OBEP∗ | 9.34 | 12.24 | 9.80 | 4.57 | -1.02 | -11.48
| UCT∗ | 9.48 | 12.53 | 10.32 | 5.27 | -0.15 | -10.27
| UCT | 9.30 | 12.16 | 9.81 | 4.73 | -0.68 | -10.76
| OBEP | -5.33 | -8.77 | -13.47 | -17.18 | -20.49 | -26.38
${{}^{3}}P_{1}$ | OBEP∗ | -5.33 | -8.77 | -13.47 | -17.18 | -20.48 | -26.38
| UCT∗ | -5.27 | -8.62 | -13.09 | -16.56 | -19.63 | -25.06
| UCT | -5.28 | -8.58 | -12.85 | -16.06 | -18.86 | -23.79
| OBEP | 3.88 | 9.29 | 17.67 | 22.57 | 24.94 | 25.36
${{}^{3}}D_{2}$ | OBEP∗ | 3.89 | 9.29 | 17.67 | 22.57 | 24.94 | 25.36
| UCT∗ | 3.86 | 9.15 | 17.12 | 21.51 | 23.47 | 23.48
| UCT | 3.89 | 9.25 | 17.31 | 21.77 | 23.75 | 23.61
| OBEP | 80.32 | 62.16 | 41.99 | 28.94 | 19.04 | 4.07
${{}^{3}}S_{1}$ | OBEP∗ | 80.31 | 62.15 | 41.98 | 28.93 | 19.03 | 4.06
| UCT∗ | 92.30 | 72.71 | 51.44 | 38.10 | 28.20 | 13.70
| UCT | 79.60 | 61.53 | 41.57 | 28.75 | 19.08 | 4.60
| OBEP | -2.99 | -6.86 | -12.98 | -17.28 | -20.28 | -23.72
${{}^{3}}D_{1}$ | OBEP∗ | -2.99 | -6.87 | -12.99 | -17.28 | -20.29 | -23.72
| UCT∗ | -2.74 | -6.43 | -12.36 | -16.54 | -19.47 | -22.78
| UCT | -3.00 | -6.90 | -13.12 | -17.66 | -21.11 | -26.03
| OBEP | 1.76 | 2.00 | 2.24 | 2.58 | 3.03 | 4.03
$\varepsilon_{1}$ | OBEP∗ | 1.76 | 2.00 | 2.24 | 2.58 | 3.03 | 4.03
| UCT∗ | 0.02 | -0.12 | -0.17 | 0.04 | 0.41 | 1.40
| UCT | 1.80 | 2.01 | 2.19 | 2.50 | 2.90 | 3.83
| OBEP | 2.62 | 6.14 | 11.73 | 14.99 | 16.65 | 17.40
${{}^{3}}P_{2}$ | OBEP∗ | 2.62 | 6.14 | 11.73 | 14.99 | 16.65 | 17.39
| UCT∗ | 2.80 | 6.61 | 12.71 | 16.28 | 18.10 | 18.91
| UCT | 2.57 | 6.00 | 11.32 | 14.18 | 15.37 | 15.07
| OBEP | 0.11 | 0.34 | 0.77 | 1.04 | 1.10 | 0.52
${{}^{3}}F_{2}$ | OBEP∗ | 0.11 | 0.34 | 0.77 | 1.04 | 1.10 | 0.52
| UCT∗ | 0.11 | 0.34 | 0.77 | 1.05 | 1.13 | 0.64
| UCT | 0.11 | 0.34 | 0.75 | 1.00 | 1.03 | 0.41
| OBEP | -0.86 | -1.82 | -2.84 | -3.05 | -2.85 | -2.02
$\varepsilon_{2}$ | OBEP∗ | -0.86 | -1.82 | -2.84 | -3.05 | -2.85 | -2.02
| UCT∗ | -0.87 | -1.83 | -2.82 | -2.99 | -2.75 | -1.88
| UCT | -0.86 | -1.83 | -2.84 | -3.05 | -2.89 | -2.18
Figure 2: Neutron-proton phase parameters for the uncoupled partial waves,
plotted versus the nucleon kinetic energy in the lab. system. Dashed[solid]
curves calculated with Potential B parameters (Table 1) by solving Eqs.
(56)[(61)]. Dotted represent the solutions of Eqs. (56) with UCT parameters
(Table 1). The rhombs show original OBEP results (see Table 2).
$\left\langle{\vec{p}^{\,\prime}\,\mu^{\prime}_{1}\mu^{\prime}_{2}}\right|v^{UCT}_{\rm{v}}\left|{\vec{p}\,\mu_{1}\mu_{2}}\right\rangle=-\frac{F^{2}_{\rm
v}[(p^{\prime}-p)^{2}]}{\left({p^{\prime}-p}\right)^{2}-m_{\rm{v}}^{2}}\\\
\times\left\\{\bar{u}(\vec{p}^{\,\prime})\left[\left(g_{\rm{v}}+f_{\rm{v}}\right)\gamma_{\nu}-\frac{f_{\rm{v}}}{2m}\left(p^{\prime}+p\right)_{\nu}\right.\right.\\\
\left.-\frac{f_{\rm{v}}}{2m}(E_{\vec{p}^{\prime}}-E_{\vec{p}})[\gamma_{0}\gamma_{\nu}-g_{0\nu}]\right]u\left(\vec{p}\right)\\\
\times\bar{u}\left(-\vec{p}^{\,\prime}\right)\left[\left({g_{\rm{v}}+f_{\rm{v}}}\right)\gamma^{\nu}-\frac{f_{\rm{v}}}{2m}\overline{\left(p^{\prime}+p\right)}^{\nu}\right.\\\
\left.-\frac{f_{\rm{v}}}{2m}(E_{\vec{p}^{\prime}}-E_{\vec{p}})[\gamma^{0}\gamma^{\nu}-g^{0\nu}]\right]u(-\vec{p})$
$-\frac{{f_{\rm
v}}^{2}}{4m^{2}}(E_{p^{\prime}}-E_{p})^{2}\bar{u}(\vec{p}^{\,\prime})[\gamma_{0}\gamma_{\nu}-g_{0\nu}]u(\vec{p})\\\
\left.\phantom{\frac{f_{\rm{v}}}{2m}}\times\bar{u}(-\vec{p}^{\,\prime})[\gamma^{0}\gamma^{\nu}-g^{0\nu}]u(-\vec{p})\right\\},$
(59)
where
$\overline{(p^{\prime}+p)}^{\nu}=(E_{\vec{p}^{\prime}}+E_{\vec{p}},-(\vec{p}^{\prime}+\vec{p}))$.
At first sight, such a regularization can be achieved via a simple
substitution $g_{b}\rightarrow g_{b}F_{b}(p^{\prime},p)$ with some cutoff
functions $F_{b}(p^{\prime},p)$ depending on the 4–momenta $p^{\prime}$ and
$p$. However, the principal moment is to satisfy the requirement (22) for the
Hamiltonian invariant under space inversion, time reversal and charge
conjugation. In this context, let us remind that the baryon–nucleon–nucleon
form factors are expressed through the matrix elements $\langle
p^{\prime}|j_{b}(0)|p\rangle$ of the corresponding baryon current density
$j_{b}(x)$ at $x=0$ between physical(clothed) one–nucleon states Gas . Such
matrix elements might be evaluated in terms of the cutoffs
$F_{b}(p^{\prime},p)$ using some idea from SheShi00 (cf. the clothed particle
representation of a current therein).
Figure 3: The same in Fig. 2 but for the coupled waves.
Replacing in equations (57)–(59)
$\frac{1}{(p^{\prime}-p)^{2}-m_{b}^{2}}\,F^{2}_{b}[(p^{\prime}-p)^{2}]$
by
$\frac{-1}{(\vec{p}^{\prime}-\vec{p})^{2}+m_{b}^{2}}\,F^{2}_{b}[-(\vec{p}^{\prime}-\vec{p})^{2}]$
and neglecting the tensor-tensor term
$\frac{{f_{\rm v}}^{2}}{4m^{2}}(E_{p^{\prime}}-E_{p})^{2}\\\
\times\bar{u}(\vec{p}^{\,\prime})[\gamma_{0}\gamma_{\nu}-g_{0\nu}]u(\vec{p})\bar{u}(-\vec{p}^{\,\prime})[\gamma^{0}\gamma^{\nu}-g^{0\nu}]u(-\vec{p})$
(60)
in (59), we obtain approximate expressions that with the common factor
$(2\pi)^{-3}m^{2}/E_{p^{\prime}}E_{p}$
instead of
$(2\pi)^{-3}m/\sqrt{E_{p^{\prime}}E_{p}}$
are equivalent to Eqs. (E.21)–(E.23) from MachHolElst87 . Such an equivalence
becomes coincidence if in our formulae instead of the canonical two-nucleon
basis $\left|{\vec{p}\,\mu_{1}\mu_{2}}\right\rangle$ one uses the helicity
basis as in MachHolElst87 .
In the context, we have considered the set of equations
${{}^{B}}R_{l^{\prime}{\rm{}}l}^{JST}(p^{\prime},p)={{}^{B}}V_{l^{\prime}{\rm{}}l}^{JST}(p^{\prime},p)\\\
+m\sum\limits_{l^{\prime\prime}}\int\limits_{0}^{\infty}\frac{dq}{p^{2}-q^{2}}\left\\{q^{2}\,{{}^{B}}V_{l^{\prime}l^{\prime\prime}}^{JST}(p^{\prime},q){{}^{B}}R_{l^{\prime\prime}l}^{JST}(q,p)\right.\\\
\left.-p^{2}\,{{}^{B}}V_{l^{\prime}l^{\prime\prime}}^{JST}(p^{\prime},p){{}^{B}}R_{l^{\prime\prime}l}^{JST}(p,p)\right\\},$
(61)
where the superscript $B$ refers to the partial matrix elements of the
potential $B$ defined in Mach89 with the just mentioned interchange of the
bases.
Our calculations of the $R$ matrices that meet the equations (56) and (61) are
twofold. On the one hand, we will check reliability of our numerical procedure
(in particular, its code). On the other hand, we would like to show
similarities and discrepancies between our results and those by the Bonn group
both on the energy shell and beyond it. These results are depicted in Figs.
2–3 and collected in Table 2.
As seen in Figs. 2–3, the most appreciable distinctions between the UCT and
OBEP curves take place for the phase shifts with the lowest $l-$values. As the
orbital angular momentum increases the difference between the solid and dashed
curves decreases. Such features may be explained if one takes into account
that the approximations under consideration affect mainly high–momentum
components of the UCT quasipotentials (their behavior at ”small” distances).
With the $l$–increase the influence of small distances is suppressed by the
centrifugal barrier repulsion.
Of course, it would be more instructive to compare the corresponding
half–off–energy–shell $R$–matrices (see definition (55)). Their
$p^{\prime}$–dependencies not shown here have been prepared for a separate
publication. They are necessary to know when calculating the $\psi^{(\pm)}$
scattering states for a two–nucleon system. In the context, one should
emphasize that hitherto we have explored the OBEP and UCT $R$–matrices in the
c.m.s., where the both approaches yield most close results. It is not the case
in those situations when the c.m.s. cannot be referred to everywhere (e.g., in
the reactions $NN\rightarrow\gamma NN$ and $\gamma d\rightarrow pn$). In this
respect our studies of the differences between UCT and OBE approaches are
under way.
## 5 Summary
The present work has been made to develop a consistent field–theoretical
approach in the theory of nucleon–nucleon scattering. It has been shown that
the method of UCT’s, based upon the notion of clothed particles, is proved to
be appropriate in achieving this purpose.
Using the unitary equivalence of the CPR to the BPR, we have seen how in the
approximation $K_{I}=K_{I}^{(2)}$ the extremely complicated scattering problem
in QFT can be reduced to the three–dimensional LS–type equation for the
$T$–matrix in momentum space.The equation kernel is given by the clothed
two–nucleon interaction of the class [2.2]. Such a conversation becomes
possible owing to the property of $K_{I}^{(2)}$ to leave the two–nucleon
sector and its separate subsectors to be invariant.
Special attention has been paid to the elimination of auxiliary field
components. We encounter such a necessity for interacting vector and fermion
fields when in accordance with the canonical formalism the interaction
Hamiltonian density embodies not only a scalar contribution but nonscalar
terms too. It has proved (at least, for the primary $\rho N$ and $\omega N$
couplings) that the UCT method allows us to remove such noncovariant terms
directly in the Hamiltonian. To what extent this result will take place in
higher orders in coupling constants it will be a subject of further
explorations.
## References
* (1) M. Lacombe et al., Phys. Rev. C21, (1980) 861
* (2) R. Machleidt, K. Holinde and C. Elster, Phys. Rep. 149, (1987) 1
* (3) V.G.J. Stocks et al., Phys. Rev. C49, (1994) 2950
* (4) R.B. Wiringa, V.G.J. Stocks and R. Schiavilla, Phys. Rev. C51, (1995) 38
* (5) R. Machleidt, Phys. Rev. C63, (2001) 024001
* (6) F. Gross and A. Stadler, Few Body Syst. 44, (2008) 295
* (7) C. Ordonez, L. Ray and U. van Kolck, Phys. Rev. Lett. 72, (1994) 1982
* (8) E. Epelbaum, W. Glöckle and U.-G. Meissner, Nucl. Phys. A671, (2000) 295
* (9) E. Epelbaum, Prog. Part. Nucl. Phys. 57, (2006) 654
* (10) A.V. Shebeko and M.I. Shirokov, Phys. Part. Nucl. 32, (2001) 31
* (11) V. Korda, L. Canton and A. Shebeko, Ann. Phys. 322, (2007) 736
* (12) S. Weinberg, The Quantum Theory of Fields, (University Press, Cambridge, 1995) Vol. 1
* (13) D.A. Schütte, Nucl. Phys. A221, (1974) 450
* (14) K. Tamura, T. Niva, T. Sato and H. Ohtsubo, Prog. Theor. Phys. 80, (1988) 138.
* (15) A.V. Shebeko, Nucl. Phys. A737, (2004) 252
* (16) J.Werle, Relativistic Theory of Reactions (PWN – Polish Scientific Publishers, Warszawa, 1966)
* (17) B.D.Keister, W.N.Polyzou, Adv. Nucl. Phys. 20, (1991) 266
* (18) G.E. Brown and A.D Jackson, Nucleon–Nucleon Interaction (Amsterdam: North–Holland Publ. Co. 1976)
* (19) H.Stapp et al., Phys. Rev. 105, (1957) 302
* (20) R. Machleidt, Adv. Nucl. Phys. 19, (1989) 189
* (21) S. Gasiorowicz, Elementary Particle Physics, (John Wiley & Sons, New York, 1966)
* (22) A.V. Shebeko and M.I. Shirokov, Prog. Part. Nucl. Phys. 44, (2000) 75.
|
arxiv-papers
| 2010-02-12T13:36:51 |
2024-09-04T02:49:08.381756
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. Dubovyk, A. Shebeko",
"submitter": "Alexander Shebeko",
"url": "https://arxiv.org/abs/1002.2546"
}
|
1002.2554
|
# Sum-product estimates for rational functions333MSC classification: 05E15,
11T23, 11B75, 14N10
Boris Bukh111B.Bukh@dpmms.cam.ac.uk. Centre for Mathematical Sciences,
Cambridge CB3 0WB, England and Churchill College, Cambridge CB3 0DS, England.
Jacob Tsimerman222jtsimerm@math.princeton.edu. Department of Mathematics,
Princeton University, Princeton, NJ 08544, USA
###### Abstract
We establish several sum-product estimates over finite fields that involve
polynomials and rational functions.
First, $\lvert f(A)+f(A)\rvert+\lvert AA\rvert$ is substantially larger than
$\lvert A\rvert$ for an arbitrary polynomial $f$ over $\mathbb{F}_{p}$.
Second, a characterization is given for the rational functions $f$ and $g$ for
which $\lvert f(A)+f(A)\rvert+\lvert g(A,A)\rvert$ can be as small as $\lvert
A\rvert$, for large $\lvert A\rvert$. Third, we show that under mild
conditions on $f$, $\lvert f(A,A)\rvert$ is substantially larger than $\lvert
A\rvert$, provided $\lvert A\rvert$ is large.
We also present a conjecture on what the general sum-product result should be.
## 1 Introduction and statement of the results
### Sum-product estimates
For a polynomial $f(x_{1},\dotsc,x_{k})$ and sets $A_{1},\dotsc,A_{k}$ define
$f(A_{1},\dotsc,A_{k})=\\{f(a_{1},\dotsc,a_{k}):a_{i}\in A_{i}\\}$. As mostly
we will be dealing with the case $A_{1}=\dotsb=A_{k}$, we write
$f(A)=f(A,\dotsc,A)$ for brevity. We also employ the notation $A+B=\\{a+b:a\in
A,\ b\in B\\}$ and $AB=\\{ab:a\in A,\ b\in B\\}$. The notation $X\ll Y$ means
$X\leq CY$ for some effective absolute constant $C$, whereas
$X\ll_{r,s,\dotsc}Y$ means $X\leq C(r,s,\dotsc)Y$ for some function $C$.
The sum-product theorem of Erdős and Szemerédi [ES83] states that if $A$ is a
finite set of real numbers, then either $A+A$ or $AA$ has at least $\lvert
A\rvert^{1+c}$ elements where $c>0$ is an absolute constant. After several
improvements, the current record, due to Solymosi [Sol08], is that the result
holds with $c=1/3-o(1)$, and it is conjectured that $c=1-o(1)$ is admissible.
In practice, most applicable are the sum-product estimates when $A$ is in a
finite field or a finite ring. Not only they have been used to tackle a wide
range of problems (see [Bou09] for a survey), but they are also more general,
for as was shown in [VWW08] the uniform sum-product estimates in
$\mathbb{F}_{p}$ imply the sum-product estimates over the complex numbers. The
first estimate in $\mathbb{F}_{p}$ was proved by Bourgain, Katz and Tao
[BKT04] for $\lvert A\rvert\geq p^{\delta}$ for arbitrarily small, but fixed
$\delta>0$. The restriction was subsequently removed by Bourgain and Konyagin
[BK03]. There was a rapid series of improvements, with the best known bounds
for $A\subset\mathbb{F}_{p}$ being
$\lvert A+A\rvert+\lvert AA\rvert\gg\begin{cases}\lvert
A\rvert^{12/11-o(1)},&\text{if }\lvert A\rvert\leq p^{1/2},\text{(see
\cite[cite]{[\@@bibref{}{rudnev_twelve}{}{}]})},\\\ \lvert
A\rvert^{13/12}(\lvert A\rvert/\sqrt{p})^{1/12-o(1)},&\text{if
}p^{1/2}\leq\lvert A\rvert\leq p^{35/68}\text{ (see
\cite[cite]{[\@@bibref{}{li_slight}{}{}]})},\\\ \lvert A\rvert(p/\lvert
A\rvert)^{1/11-o(1)},&\text{if }p^{35/68}\leq\lvert A\rvert\leq
p^{13/24}\text{ (see \cite[cite]{[\@@bibref{}{li_slight}{}{}]})},\\\ \lvert
A\rvert\cdot\lvert A\rvert/\sqrt{p},&\text{if }p^{13/24}\leq\lvert A\rvert\leq
p^{2/3}\text{ (see \cite[cite]{[\@@bibref{}{garaev_sharp}{}{}]})},\\\ \lvert
A\rvert(p/\lvert A\rvert)^{1/2},&\text{if }\lvert A\rvert>p^{2/3}\text{ (see
\cite[cite]{[\@@bibref{}{garaev_sharp}{}{}]})}.\end{cases}$ (1)
Of these results, Garaev’s estimate of $\lvert A\rvert(p/\lvert
A\rvert)^{1/2}$ for $\lvert A\rvert>p^{2/3}$ is notable in that it is the only
sharp bound. It is likely that $\lvert A+A\rvert+\lvert AA\rvert\gg\min(\lvert
A\rvert(p/\lvert A\rvert)^{1/2-o(1)},\lvert A\rvert^{2-o(1)})$.
Of use are also the statements that one of $A+A$ or $f(A)$ is substantially
larger than $A$, where $f$ is a rational function, that is possibly different
from $f(x,y)=xy$. For example, Bourgain [Bou05] showed that either $A+A$ or
$1/A+1/A$ is always large, and used this estimate to give new bounds on
certain bilinear Kloosterman sums. In application to a construction of
extractors, in the same paper Bourgain asked for sum-product estimates for
$f(x_{1},\dotsc,x_{k})=x_{1}^{t}+\dotsb+x_{k}^{t}$. The most general result of
the kind is due to Vu[Vu08], who generalized an earlier argument of Hart,
Iosevich and Solymosi[HIS07]. Call a polynomial $f(x,y)$ degenerate, if it is
a function of a linear form in $x$ and $y$. In [Vu08] it was shown that if $f$
is a bivariate non-degenerate polynomial of degree $d$, then
$\lvert A+A\rvert+\lvert f(A)\rvert\gg\begin{cases}\lvert A\rvert(\lvert
A\rvert/d^{2}\sqrt{p})^{1/2}d^{-1},&\text{if }p^{1/2}<\lvert A\rvert\leq
d^{4/5}p^{7/10},\\\ \lvert A\rvert(p/\lvert A\rvert)^{1/3}d^{-1/3},&\text{if
}\lvert A\rvert\geq d^{4/5}p^{7/10}.\end{cases}$ (2)
The same argument was used in [HLS09] to establish a version of (2) for
$\lvert A+B\rvert+\lvert f(A)\rvert$.
As nearly all applications of sum-product estimates in finite fields have
taken advantage of validity of the estimates for $A$ of very small size, it is
of interest to extend this result to $\lvert A\rvert<\sqrt{p}$. The result can
also be improved qualitatively because for some polynomials $f$ it is true
that $f(A)$ is much larger than $A$ no matter how large or small $A+A$ is. So,
for example, Bourgain [Bou05] showed that $x^{2}+xy$ and $x(y+a)$ for $a\neq
0$ are such polynomials (Bourgain actually showed that $x^{2}+xy$ grows even
if $x$ and $y$ range over different sets; see also [HH09] for a
generalization).
The sum-product estimate results are connected to the problem of giving good
upper bounds on the number points in a Cartesian product set, such as $A\times
A\times A$, that lie on a given variety. For example, the estimate (2) is
related to a bound on the the number of points on a surface, in the case $A$
has small additive doubling (see Lemma 20 for the explicit form). Some results
in this direction for the special case where $A$ is an interval have been
obtained by Fujiwara [Fuj88] and Schmidt [Sch86]. The proof of theorem 6 below
and conjecture at the end of the paper give additional links.
The goal of this paper is to communicate new sum-product estimates for
polynomial and rational functions. Our results are of two kinds: The first
kind are valid even for small sets (of size $|A|>p^{\epsilon}$ for every
$\epsilon>0$). The second kind extend Vu’s characterization to a more general
setting, but are valid only for large sets ($|A|>p^{c}$ for a fixed constant
$0<c<1$). We thus expect the estimates for the small sets to be more useful in
applications, whereas the large set results illuminate the general picture.
### Small sets.
The results in this section are stated only for sets of size $\lvert
A\rvert<\sqrt{p}$. Modification for large $\lvert A\rvert$ involve no
alterations in the fabric of the proofs, but would introduce much clutter.
Moreover, for large $\lvert A\rvert$, the large-set results are not only more
general, but yield sharper quantitative estimates. We did not optimize the
numeric constants that appear in the bounds below because the results are very
unlikely to be sharp for any value of the constants.
###### Theorem 1.
Let $f\in\mathbb{F}_{p}[X]$ be a polynomial of degree $d\geq 2$. Then for
every set $A\subset\mathbb{F}_{p}$ of size $\lvert A\rvert\leq\sqrt{p}$ we
have
$\lvert A+A\rvert+\lvert f(A)+f(A)\rvert\gg\lvert A\rvert^{1+\frac{1}{16\cdot
6^{d}}}.$
Note that by Ruzsa’s triangle inequality (Lemma 8 below) this implies that for
$\lvert A\rvert\leq\sqrt{p}$ and any polynomial $g$ of the form
$g(x,y)=x+f(y)$ with $\deg f=d\geq 2$ we have $\lvert g(A)\rvert\gg\lvert
A\rvert^{1+\frac{1}{32\cdot 6^{d}}}$, which is a generalization of [HLS09,
Theorem 3.1].
The next result is an extension of (2) to sets of any size for polynomials of
degree two.
###### Theorem 2.
There exists an absolute constant $c>0$ such that whenever
$f\in\mathbb{F}_{p}[X,Y]$ is a bivariate quadratic polynomial that is not of
the form $f=g(ax+by)$ for some univariate polynomial $g$, then for every
$A\subset\mathbb{F}_{p}$ of size $\lvert A\rvert\leq\sqrt{p}$ we have
$\lvert A+A\rvert+\lvert f(A)\rvert\gg\lvert A\rvert^{1+c}.$
Our final small-set result is another generalization of the sum-product
theorem itself:
###### Theorem 3.
Suppose $f=\sum_{i=1}^{k}a_{i}x^{d_{i}}\in\mathbb{F}_{p}[X]$ is a polynomial
with $k$ terms, and an integer $d\geq 2$ satisfies $d_{i}\leq d$ for all
$i=1,\dotsc,k$. Then for every positive integer $r$, and every set
$A\subset\mathbb{F}_{p}$ of size $p^{4/r}d^{40r}\leq\lvert
A\rvert\leq\sqrt{p}$ we have
$\lvert AA\rvert+\lvert f(A)+f(A)\rvert\gg\lvert A\rvert^{1+\varepsilon},$
where
$\varepsilon=(5000(r+k)^{2}\log_{2}d)^{-k}.$
The main appeal of this estimate is that the dependence on the degree of $f$
is merely logarithmic, which suggests that the exponents in all the sum-
product estimates should not depend on the degree. Further evidence that the
exponents in sum-product results should not depend on the degree is provided
by the sum-product estimates for large subsets $A\subset\mathbb{F}_{q}$, which
we present now.
### Large sets.
For a polynomial $f\in\mathbb{F}_{q}[X_{1},\dotsc,X_{n}]$ and sets
$A_{1},\dotsc,A_{n}\subset\mathbb{F}_{p}$ write $N(f;A_{1},\dotsc,A_{n})$ for
the number of solution to $f(x_{1},\dotsc,x_{n})=0$ in $x_{i}\in A_{i}$. The
commonly used case $N(f;A,\dotsc,A)$ will be abbreviated as $N(f;A)$. More
generally, if $V$ is a variety in $\mathbb{A}^{n}_{\mathbb{F}_{q}}$, then
$N(V;A_{1},\dotsc,A_{n})$ is the number of points of $V$ on
$A_{1}\times\dotsb\times A_{n}$. The principal result that generalizes (2) is
###### Theorem 4.
Let $f(x,y,z)$ be an irreducible polynomial of degree $d$ which is not of the
form $P(ax+by,z)$ or $P(x,y)$. Moreover, let $A,B\subset\mathbb{F}_{q}$.
Assume $d<q^{1/40}$. Then
$\lvert A+A\rvert+\frac{\lvert B\rvert\lvert
A\rvert^{4}}{N(f;A,A,B)^{2}}\gg\begin{cases}\lvert A\rvert(\lvert
A\rvert/\sqrt{q})^{1/2}d^{-1},&\text{if }q^{1/2}\leq\lvert A\rvert\leq
d^{4/5}q^{7/10},\\\ \lvert A\rvert(q/\lvert A\rvert)^{1/3}d^{-1},&\text{if
}\lvert A\rvert\geq d^{4/5}q^{7/10}.\end{cases}$
In particular, the inequality (2) holds (with slightly worse dependence on
$d$) as witnessed by setting $f(x,y,z)=g(x,y)-z$, $B=g(A)$ and noting that
$N(f;A,A,B)=\lvert A\rvert^{2}$.
The condition that $f$ be irreducible is purely for convenience, since
$\max(N(f_{1}),N(f_{2}))\leq N(f_{1}f_{2})\leq N(f_{1})+N(f_{2})$ holds for
reducible polynomials. On the other hand, the condition that $f$ is not of the
form $P(ax+by,z)$ is essential because if $f$ is of this form, then for
$A=\\{1,\dotsc,n\\}$ and an appropriate $B$ the result fails. Though the
theorem is formulated only for polynomials, the questions about growth of
rational function can be reduced to it, of which the following result is an
example.
###### Theorem 5.
Let $f(x)\in\mathbb{F}_{q}(x)$, $g(x,y)\in\mathbb{F}_{q}(x,y)$ be non-constant
rational functions of degree at most $d$, and assume $g(x,y)$ is not of the
form $G(af(x)+bf(y)+c),G(x)$, or $G(y)$ with $a,b,c\in\mathbb{F}_{q}$. Then if
$\lvert A\rvert\geq q^{\frac{1}{2}}$ and $d<q^{1/50}$, we have the estimate
$|f(A)+f(A)|+|g(A,A)|\gg\begin{cases}\lvert A\rvert(\lvert
A\rvert/\sqrt{q})^{1/2}d^{-2},&\text{if }q^{1/2}\leq\lvert A\rvert\leq
d^{8/5}q^{7/10},\\\ \lvert A\rvert(q/\lvert A\rvert)^{1/3}d^{-2},&\text{if
}\lvert A\rvert\geq d^{8/5}q^{7/10}.\end{cases}$
The main feature of all the results above is the abelian group structure
inherent in $A+A$ and $AA$. That structure permits us to use sumset
inequalities from the additive combinatorics, as well as the Fourier
transform. The following result shows that for most polynomials $f$, the set
$f(A)$ always grows even in the absence of any group structure.
###### Definition.
Let $f(x,y)\in\mathbb{K}[x,y]$ be a polynomial of degree $d$ in the
$x$-variable. Call $f(x,y)$ _monic_ in the $x$ variable if the coefficient of
the $x^{d}$ term is a non-zero constant of $\mathbb{K}$. That is,
$f(x,y)=cx^{d}+g(x,y),$
where $c\in\mathbb{K}\setminus\\{0\\}$, and $g(x,y)$ is of degree $\leq d-1$
in the $x$-variable.
###### Theorem 6.
Let $f(x,y)\in\mathbb{F}_{q}[x,y]$ be a polynomial of degree $d$ which is non-
composite, and is not of the form $g(x)+h(y)$ or $g(x)h(y)$. Suppose also that
$f(x,y)$ is monic in each variable. Then if $\lvert A\rvert,\lvert B\rvert\geq
q^{7/8}$,
$\lvert f(A,B)\rvert\gg_{d}\min(\sqrt[3]{q\lvert A\rvert\lvert B\rvert},\lvert
A\rvert^{3/4}\lvert B\rvert^{3/4}q^{-7/16}).$
This result is to be compared with the estimate of Elekes and Rónyai over the
real numbers:
###### Theorem 7 ([ER00], Theorem 2).
Let $f(x,y)\in\mathbb{R}(x,y)$ be a rational function of degree $d$ which is
not of the form $G(g(x)+h(y))$, $G(g(x)h(y))$ or
$G(\frac{g(x)+h(y)}{1-g(x)h(y)})$. Then there exists a constant $c=c(d)>0$
such that whenever $\lvert A\rvert=\lvert B\rvert=n$,
$\lvert f(A,B)\rvert\gg_{d}n^{1+c}.$
The statement appearing in [ER00] is quantitatively weaker, but the bound of
$n^{1+c}$ follows from the proof. Note that the case
$\frac{g(x)+h(y)}{1-g(x)h(y)}$ arises because $\mathbb{R}$ is not
algebraically closed. Indeed, $\frac{x+y}{1-xy}=G(F(x)F(y))$ where
$G(x)=\frac{x-1}{i\cdot(x+1)}$ and $F(x)=\frac{1+ix}{1-ix}$.
The rest of the paper is organized as follows. In sections 2 and 3 we gather
analytic and algebraic tools used in the paper. Theorems 1, 2 and 3 on small-
set estimates are proved in section 4. All the large-set results, apart from
Theorem 6 on $f(A,B)$, are proved in section 5, whereas Section 6 is devoted
to Theorem 6. It is followed by the proofs of the algebraic lemmas used
throughout the paper. The paper ends with several remarks and a conjecture.
## 2 Analytic tools
We shall need a number of tools from additive combinatorics that we collect
here.
###### Lemma 8 (Ruzsa’s triangle inequalities, [Ruz09], Theorems 1.8.1 and
1.8.7).
For every abelian group $G$ and every triple of sets $A,B,C\subset G$ we have
$\lvert A\pm C\rvert\lvert B\rvert\leq\lvert A\pm B\rvert\lvert B\pm C\rvert,$
where the result is valid for all eight possible choices of the signs.
Let $s*A=A+A+\dotsb+A$ where $A$ appears $s$ times as a summand.
###### Lemma 9 (Plünnecke’s inequality, [Ruz09], Theorem 1.1.1).
For every abelian group $G$ and every $A\subset G$ we have
$\lvert s*A-t*A\rvert/\lvert A\rvert\leq(\lvert A\pm A\rvert/\lvert
A\rvert)^{s+t},$
where the result is valid for either choice of the sign.
Let $\lambda\cdot A=\\{\lambda a:a\in A\\}$ be the $\lambda$-dilate of $A$.
The following result of the first author is used in the proof of Theorem 3 to
obtain the logarithmic dependence on the degree.
###### Lemma 10 ([Buk08], Theorem 3).
If $\Gamma$ is an abelian group and $A\subset\Gamma$ is a finite set
satisfying $\lvert A+A\rvert\leq K\lvert A\rvert$ or $\lvert A-A\rvert\leq
K\lvert A\rvert$, then $\lvert\lambda_{1}\cdot A+\dotsb+\lambda_{k}\cdot
A\rvert\leq K^{P}\lvert A\rvert$, where
$P=7+12\sum_{i=1}^{k}\log_{2}(1+\lvert\lambda_{i}\rvert).$
###### Lemma 11 (Szemerédi-Trotter theorem for $\mathbb{F}_{p}$, [BKT04]).
Let $\mathcal{P}$ and $\mathcal{L}$ be families of points and lines in
$\mathbb{F}_{p}^{2}$ of cardinality
$\lvert\mathcal{P}\rvert,\lvert\mathcal{L}\rvert\leq N\leq p^{2-\alpha}$ with
$\alpha>0$. Then we have
$\lvert\\{(p,l)\in\mathcal{P}\times\mathcal{L}:p\in l\\}\rvert\ll
N^{3/2-\varepsilon}$
for some $\varepsilon=\varepsilon(\alpha)>0$ that depends only on $\alpha$.
For sets $A,B$ in an abelian group, and a bipartite graph $G\subset A\times B$
we put $A+_{G}B=\\{a+b:(a,b)\in G\\}$ to denote their sumset along $G$.
Similarly, $A\cdot_{G}B=\\{ab:(a,b)\in G\\}$ will denote their productset
along $G$.
###### Lemma 12 (Balog-Szemerédi-Gowers theorem, Lemma 4.1 in [SSV05]).
Let $\Gamma$ be an abelian group, and $A,B\subset\Gamma$ be two $n$-element
sets. Suppose $G\subset A\times B$ is a bipartite graph with $n^{2}/K$ edges,
and $\lvert A+_{G}B\rvert\leq Cn$. Then one can find subsets
$A^{\prime}\subset A$ and $B^{\prime}\subset B$ such that $\lvert
A^{\prime}\rvert\geq n/(16K^{2})$, $\lvert B^{\prime}\rvert\geq n/(4K)$, and
$\lvert A^{\prime}+B^{\prime}\rvert\leq 2^{12}C^{3}K^{5}n$.
In the proofs of Theorems 4 and 5 we will use Fourier transform on
$\mathbb{F}_{q}^{n}$. For that we endow $\mathbb{F}_{q}^{n}$ with probability
measure, and its dual with the counting measure. Thus, the Fourier transform
is defined by $\hat{f}(\xi)=\tfrac{1}{q^{n}}\sum_{x}f(x)\exp(2\pi
ix\cdot\xi)$, Plancherel’s theorem asserts that
$\tfrac{1}{q^{n}}\sum_{x}f(x)\overline{g(x)}=\langle
f,g\rangle_{\mathbb{F}_{q}^{n}}=\langle\hat{f},\hat{g}\rangle_{\hat{\mathbb{F}}_{q}^{n}}=\sum_{\xi}\hat{f}(\xi)\overline{\hat{g}(\xi)},$
the convolutions are defined by $(f*g)(y)=\frac{1}{q^{n}}\sum_{x}f(x)g(y-x)$,
and satisfy $\widehat{f*g}=\hat{f}\hat{g}$.
## 3 Algebraic tools
Here we record several results in algebraic geometry that are repeatedly used
throughout the paper.
Throughout the paper $\mathbb{A}_{\mathbb{K}}^{n}$ denotes the $n$-dimensional
affine space over $\overline{\mathbb{K}}$, the algebraic closure of
$\mathbb{K}$. When the field is clear from the context, we write simply
$\mathbb{A}^{n}$. For a reducible variety $V\subset\mathbb{P}^{N}$, define the
_total degree_ $\deg(V)$ to be the sum of the degrees of all irreducible
components of $V$.
###### Lemma 13 (Generalized Bezout’s theorem, [Ful98], p. 223, Example
12.3.1).
Let $V_{1}$ and $V_{2}$ be two varieties in $\mathbb{P}^{N}$ and let $W$ be
their intersection. Then
$\deg(W)\leq\deg(V_{1})\deg(V_{2}).$
Much of this paper is about proving non-trivial upper bounds on the number of
points on varieties in Cartesian products. Both for comparison, and because we
need it several times in the proofs, we give an explicit “trivial bound”.
Recall that $N(V;A_{1},\dotsc,A_{n})$ stands for the number of points of $V$
on $A_{1}\times\dotsb\times A_{n}$. The following result is a generalization
of the well-known Schwartz–Zippel lemma to varieties.
###### Lemma 14.
Let $V$ be an $m$-dimensional variety of degree $d$ in $\mathbb{A}^{n}$. Let
$A_{1},\dotsc,A_{n}\subset\mathbb{A}^{1}$ be finite sets of the same size.
Then
$N(V;A_{1},\dotsc,A_{n})\leq d\lvert A_{1}\rvert^{m}.$
###### Proof.
The proof is by induction on $m$, the case $m=0$ being trivial. If $V$ is
reducible, then its degree is the sum of the degrees of its components. Thus,
we can assume that $V$ is irreducible. If $d=1$, then $V$ is a hyperplane, and
the lemma is immediate. If $d\geq 2$, then the irreducibility of $V$ implies
that for each $a\in A_{1}$ the hyperplane $H_{a}=\\{x_{1}=a\\}$ intersects $V$
in a variety of dimension $m-1$. Indeed, if $\dim V\cap H_{a}=m$, then $V\cap
H_{a}$ is a component of $H_{a}$ contradicting irreducibility. By Bezout’s
theorem (Lemma 13) the degree of $V\cap H_{a}$ is at most $d$, which by
induction implies that
$N(V;A_{1},\dotsc,A_{n})=\sum_{a\in A_{1}}N(V\cap
H_{a};A_{2},\dotsc,A_{n})\leq\lvert A_{1}\rvert\cdot d\lvert
A_{2}\rvert^{m-1}.\qed$
Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$. A
rational function $f(x,y)\in\mathbb{K}(x,y)$ is called _composite_ if there
exist rational functions $Q(u),r(x,y)$ with $Q(r(x,y))=f(x,y)$ and $\deg Q\geq
2$.
###### Lemma 15 (Bertini–Krull theorem, [Sch00], p. 221).
A polynomial $f(x,y)\in\mathbb{K}(x,y)$ of degree at most $p-1$ is composite
if and only if for a generic (cofinite) set $t\in\mathbb{K}$, the variety
$f(x,y)=t$ is reducible.
We also use the explicit bound on the Fourier transform over a curve due to
Bombieri:
###### Lemma 16 (Theorem 6, [Bom66]).
Let $P(x_{1},x_{2})$ be a polynomial of degree $d$ over $\mathbb{F}_{q}$
without linear factors, and define the set $S\subset\mathbb{F}_{q}^{2}$ to be
the zero set of $P$. If $\xi$ is any non-zero additive character of
$\mathbb{F}^{2}_{q}$, we have the following bound:
$\left|\sum_{\vec{x}\in S}\xi(\vec{x})\right|\leq 2d^{2}q^{\frac{1}{2}}.$
## 4 Small sets
We first prove that either $A+A$ or $f(A)+f(A)$ grows. The proof is inspired
by Weyl’s differencing method for estimating exponential sums. For $\deg f=2$
differencing reduces the problem to the standard sum-product estimates,
whereas for $\deg f\geq 3$, differencing lets us replace $f$ by a polynomial
of lower degree.
###### Proof of Theorem 1.
The proof is by induction on $k$. We could use theorem 2 as our base case, but
its proof is particularly simple when restricted to this special case, so we
give it here.
Suppose $f(x)=ax^{2}+bx+c$ is a polynomial of degree two, which is to say
$a\neq 0$. Assume that $\lvert A+A\rvert+\lvert
f(A)+f(A)\rvert\leq\Delta\lvert A\rvert$. Then by the triangle inequality
(Lemma 8) we have $\lvert f(A)-f(A)\rvert\leq\lvert f(A)+f(A)\rvert^{2}/\lvert
A\rvert\leq\Delta^{2}\lvert A\rvert$ and $\lvert A-A\rvert\leq\lvert
A+A\rvert^{2}/\lvert A\rvert\leq\Delta^{2}\lvert A\rvert$. As
$f(x)-f(y)=a(x-y)(x+y+b/a)$, it follows that there are at least $\lvert
A\rvert^{2}$ solutions to
$uw=z,\qquad\text{with }u\in A-A,\ w\in A+A+b/a,\ z\in(1/a)\cdot(f(A)-f(A)).$
By the Balog-Szemerédi-Gowers theorem (Lemma 12) applied to $A-A$ and
$A+A+b/a$ in the multiplicative group $\mathbb{F}_{p}^{*}$, we infer that
there are $A_{1}\subset A-A$ and $A_{2}\subset A+A+b/a$ of sizes $\lvert
A_{1}\rvert\geq\Delta^{2}\lvert A\rvert/(16\Delta^{8})$ and $\lvert
A_{2}\rvert\geq\Delta^{2}\lvert A\rvert/(4\Delta^{4})$ such that $\lvert
A_{1}A_{2}\rvert\leq 2^{12}(\Delta^{4})^{5}\Delta^{2}\lvert A\rvert$, and
hence
$\lvert A_{2}A_{2}\rvert\leq 2^{24}\Delta^{44}\lvert A\rvert\leq
2^{26}\Delta^{46}\lvert A_{2}\rvert$
by the triangle inequality. Since $A_{2}+A_{2}\subset A+A+A+A+2b/a$,
Plünnecke’s inequality (Lemma 9) implies
$\lvert A_{2}+A_{2}\rvert\leq\lvert A+A+A+A\rvert\leq\Delta^{4}\lvert
A\rvert\leq 4\Delta^{6}\lvert A_{2}\rvert.$
These inequalities contradict (1) unless $2^{26}\Delta^{46}\gg\lvert
A_{2}\rvert^{1/12}\geq(\lvert A\rvert/4\Delta^{2})^{1/12}$. Simple arithmetic
gives $\Delta\gg\lvert A\rvert^{1/554}$.
Hence, we know that whenever $\lvert A\rvert\leq\sqrt{p}$, and $f$ is
quadratic, we have $\lvert A+A\rvert+\lvert f(A)+f(A)\rvert\geq C\lvert
A\rvert^{1+1/554}$ for some constant $0<C\leq 1$. We shall prove that whenever
$\deg f=d\geq 3$, we have $\lvert A+A\rvert+\lvert f(A)+f(A)\rvert\geq C\lvert
A\rvert^{1+\frac{1}{16\cdot 6^{d}}}$ with the same constant $C$. Suppose that
$\deg f=d\geq 3$ and the claim has been proved for all polynomials of degree
$d-1$. Assume that $\lvert A+A\rvert+\lvert f(A)+f(A)\rvert\leq\Delta\lvert
A\rvert$. Let $t$ be any number having at least $\lvert A\rvert^{2}/\lvert
A-A\rvert\geq\Delta^{-2}\lvert A\rvert$ representations as $t=a_{1}-a_{2}$
with $a_{1},a_{2}\in A$. Define $A^{\prime}=\\{a\in A:a+t\in A\\}$ and
$g(x)=f(x+t)-f(x)$. From the choice of $t$ it follows that $\lvert
A^{\prime}\rvert\geq\Delta^{-2}\lvert A\rvert$. Plünnecke’s inequality (Lemma
9) tells us that
$\lvert g(A^{\prime})+g(A^{\prime})\rvert\leq\lvert
f(A)+f(A)-f(A)-f(A)\rvert\leq\lvert f(A)+f(A)\rvert^{4}/\lvert
A\rvert^{3}\leq\Delta^{4}\lvert A\rvert\leq\Delta^{6}\lvert A^{\prime}\rvert.$
We also have $\lvert A^{\prime}+A^{\prime}\rvert\leq\lvert
A+A\rvert\leq\Delta\lvert A\rvert\leq\Delta^{3}\lvert A^{\prime}\rvert$.
However, $g$ is a polynomial of degree $d-1$, and by the induction hypothesis
this implies $\Delta^{6}\geq C\lvert A\rvert^{\frac{1}{16\cdot 6^{d-1}}}$.
Since $C\leq 1$, we have $C^{1/6}\geq C$, and the induction step is complete.
∎
We note that the argument in [Bou05] for $x^{2}+xy$ does not seem to
generalize to an arbitrary quadratic polynomial, as that argument crucially
depends on $x^{2}+xy$ being linear in $y$, and so our proof of Theorem 2 is
again based on the idea of differencing.
###### Proof of Theorem 2.
Assume that $f(x,y)=ax^{2}+by^{2}+cxy+dx+ey$ is a non-degenerate quadratic
polynomial, and $\lvert A+A\rvert+\lvert f(A)\rvert\leq\Delta\lvert A\rvert$.
The Cauchy–Schwarz inequality implies that the equation
$ax_{1}^{2}+by_{1}^{2}+cx_{1}y_{1}+dx_{1}+ey_{1}=ax_{2}^{2}+by_{2}^{2}+cx_{2}y_{2}+dx_{2}+ey_{2},\qquad
x_{1},x_{2},y_{1},y_{2}\in A$
has at least $\Delta^{-1}\lvert A\rvert^{3}$ solutions. Changing the variables
to $v_{1}=x_{1}+x_{2}$, $v_{2}=y_{1}+y_{2}$, $u_{1}=x_{1}-x_{2}$,
$u_{2}=y_{1}-y_{2}$ we conclude that there are at least $\Delta^{-1}\lvert
A\rvert^{3}$ solutions to
$au_{1}v_{1}+bu_{2}v_{2}+\tfrac{1}{2}c(u_{2}v_{1}+u_{1}v_{2})+du_{1}+eu_{2}=0,\qquad
u_{1},u_{2}\in A-A,\ v_{1}\in A+A.$
Rewrite the equation as
$u_{1}(av_{1}+\tfrac{1}{2}cv_{2}+d)+u_{2}(bv_{2}+\tfrac{1}{2}cv_{1}+e)=0,\qquad
u_{1},u_{2}\in A-A,\ v_{1},v_{2}\in A+A.$ (3)
Consider
$g(v_{1},v_{2})=\frac{av_{1}+\tfrac{1}{2}cv_{2}+d}{bv_{2}+\tfrac{1}{2}cv_{1}+e}$.
Suppose $g$ is a constant function. Then $ab-\tfrac{1}{4}c^{2}=0$, which
implies that there are $s,t\in\mathbb{F}_{p^{2}}$ such that
$ax^{2}+by^{2}+cxy=(sx+ty)^{2}$. Hence
$g(v_{1},v_{2})=\frac{s^{2}v_{1}+stv_{2}+d}{t^{2}v_{2}+stv_{1}+e}$, and it is
evident that $dx+ey$ is a constant multiple of $sx+ty$, contradicting non-
degeneracy of $f$. We conclude that $g(v_{1},v_{2})$ cannot be a constant
function.
Without loss of generality we assume that $g$ depends non-trivially on
$v_{1}$. Call $v\in A+A$ _bad_ if $g(v_{1},v)$ is a constant function. As $v$
is bad only when the linear functions in the numerator and denominator of $g$
are proportional, there is at most one bad $v$, which we denote
$v_{\operatorname{bad}}$. When specialized to $v_{2}=v_{\operatorname{bad}}$
equation (3) takes the form $\alpha u_{1}+\beta u_{2}=0$, where $\alpha$ and
$\beta$ are constants, that are not simultaneously zero. Thus there are at
most $\lvert A-A\rvert\lvert A+A\rvert$ solutions to (3) with
$v_{2}=v_{\operatorname{bad}}$.
Choose a value for $v_{2}$ for which there are at least $N=\Delta^{-1}\lvert
A\rvert^{3}/\lvert A+A\rvert-\lvert A-A\rvert$ solutions to (3) and which is
not bad. Since we can assume that $\Delta<\lvert A\rvert^{3/4}/2$, it follows
that $N\geq\tfrac{1}{2}\Delta^{-1}\lvert A\rvert^{3}/\lvert
A+A\rvert\geq\tfrac{1}{2}\Delta^{-2}\lvert A\rvert^{2}$. Let $L_{u,v_{1}}$
denote the line in $\mathbb{F}_{p}^{2}$ with the equation
$(u_{1}-u)(av_{1}+\tfrac{1}{2}cv_{2}+d)+u_{2}(bv_{2}+\tfrac{1}{2}cv_{1}+e)=0.$
Since $g$ is not a constant function of $v_{1}$, the slope of $L_{u,v_{1}}$
uniquely determines $v_{1}$, from which we conclude that the family
$\mathcal{L}=\\{L_{u,v_{1}}:v_{1}\in A+A,\ u\in A-A\\}$
contains $\lvert A+A\rvert\lvert A-A\rvert$ distinct lines. Consider the point
set $\mathcal{P}=\bigl{(}(A-A)+(A-A)\bigr{)}\times(A-A)$. Each solution to (3)
yields $\lvert A-A\rvert$ incidences between $\mathcal{P}$ and $\mathcal{L}$,
one for each value of $u$. By Lemmas 8 and 9 we have
$\lvert\mathcal{P}\rvert\leq\Delta^{10}\lvert A\rvert^{2}$ and
$\lvert\mathcal{L}\rvert\leq\Delta^{3}\lvert A\rvert^{2}$. Szemerédi-Trotter
theorem (Lemma 11) implies that $\Delta^{-1}\lvert A-A\rvert
N\leq(\Delta^{10}\lvert A\rvert^{2})^{3/2-\epsilon}$. Since $\lvert
A-A\rvert\geq\lvert A\rvert$, we are done. ∎
The sum-product estimate for $\lvert AA\rvert+\lvert f(A)+f(A)\rvert$ is more
delicate than the results above. Similarly to the proof of Theorem 1, the aim
is to use the upper bounds on $f(A)+f(A)+\dotsb+f(A)$ to obtain an upper bound
on $g(A)+g(A)$ for some simpler polynomial $g$. However, now ‘simpler’ means
‘having fewer non-zero terms’, and one needs to add far more copies of $f(A)$
to obtain a simpler $g$. If $A$ was a product set itself, $A=BC$, then our aim
would be to find $b_{1},b_{2},\dotsc\in B$ so that the polynomial
$h(x)=f(b_{1}x)+f(b_{2}x)+\dotsb$ has fewer terms than $f$ has. If $\lvert
B\rvert\geq p^{\varepsilon}$ we can hope to use the pigeonhole principle to
find $h(x)$ and $h^{\prime}(x)$, in which one of the terms is the same. Then
the polynomial $h(x)-h^{\prime}(x)$ would have fewer terms. Unfortunately, it
might happen such $h(x)$ and $h^{\prime}(x)$ are always equal, in which case
$h(x)-h^{\prime}(x)=0$. However, as $BB$ is small, for each fixed $\lambda$ we
can find a large set $B^{\prime}\subset B$ and an element $g$ such that
$gb^{\lambda}\in B$ for all $b\in B^{\prime}$. Using $B^{\prime}$ in place of
$B$ then permits us to use not only the terms of the form $f(bx)$, but also of
the form $f(gb^{\lambda}x)$. As it turns out, that suffices to complete the
proof. Regrettably, $A$ is not necessarily a product set, and that requires us
to work with multiplications along a graph, introducing additional technical
complications.
The following lemma is used to find an analogue of $B^{\prime}$ in the sketch
above. Recall that $\lambda\cdot A=\\{\lambda a:a\in A\\}$.
###### Lemma 17.
Suppose $\lambda_{1},\dotsc,\lambda_{r}$ are non-zero integers, and $\Gamma$
is an abelian group. Furthermore, assume that $A\subset\Gamma$ satisfies
$\lvert A+A\rvert\leq K\lvert A\rvert$. Let
$P=43\sum_{i=1}^{r}\log_{2}(1+\lvert\lambda_{i}\rvert).$
Then there is a set $B\subset A$ of size $\lvert
B\rvert\geq\tfrac{1}{2}K^{-P}\lvert A\rvert$ and elements $g_{1},\dotsc,g_{r}$
such that for every $b\in B$ the set
$\\{a\in A:a+\lambda_{i}\cdot b+g_{i}\in A\text{ for all }i=1,\dotsc,r\\}$ (4)
has at least $\tfrac{1}{2}K^{-P}\lvert A\rvert$ elements.
###### Proof.
For given $g_{1},\dotsc,g_{r}\in\Gamma$ define
$S(g_{1},\dotsc,g_{r})=\sum_{b\in A}\\#\\{a\in A:a+\lambda_{i}\cdot b+g_{i}\in
A\text{ for all }i=1,\dotsc,r\\}.$
Summing over all $(g_{1},\dotsc,g_{r})\in\Gamma^{r}$ we obtain
$\displaystyle\sum_{g_{1},\dotsc,g_{r}}S(g_{1},\dotsc,g_{r})$
$\displaystyle=\sum_{a,b\in
A}\\#\\{(g_{1},\dotsc,g_{r})\in\Gamma^{r}:a+\lambda_{i}\cdot b+g_{i}\in
A\text{ for all }i=1,\dotsc,r\\}$ $\displaystyle=\sum_{a,b\in A}\lvert
A\rvert^{r}=\lvert A\rvert^{r+2}.$
Since $S(g_{1},\dotsc,g_{r})=0$ unless $g_{i}\in A-A-\lambda_{i}\cdot A$,
there is a way to choose $g_{1},\dotsc,g_{r}$ so that
$\displaystyle S(g_{1},\dotsc,g_{r})$ $\displaystyle\geq\lvert
A\rvert^{r+2}\prod_{i=1}^{r}\lvert A-A-\lambda_{i}\cdot A\rvert^{-1}$
$\displaystyle\geq\lvert
A\rvert^{2}K^{-\sum_{i=1}^{r}\bigl{(}7+12+12+12\log_{2}(1+\lvert\lambda_{i}\rvert)\bigr{)}}$
$\displaystyle\geq K^{-P}\lvert A\rvert^{2}$
by Lemma 10 and the inequality $31+12\log_{2}(1+\lambda)\leq
43\log_{2}(1+\lambda)$ valid for $\lambda\geq 1$. Having chosen
$g_{1},\dotsc,g_{r}$, define $B$ to be the set of all $b\in A$ for which the
set in (4) has at least $\tfrac{1}{2}K^{-P}\lvert A\rvert$ elements. Since the
elements $b\in A\setminus B$ contribute at most $\tfrac{1}{2}K^{-P}\lvert
A\rvert^{2}$ to $S(g_{1},\dotsc,g_{r})$, the lemma follows. ∎
Let $p_{t}(x_{1},\dotsc,x_{n})=x_{1}^{t}+\dotsb+x_{n}^{t}$ be the $t$’th power
sum polynomial.
###### Lemma 18.
Suppose $\mathbb{K}$ is a field, and
$0<t_{1}<t_{2}<\dotsb<t_{r}<\operatorname{char}\mathbb{K}$ are integers. Let
$w\colon\mathbb{A}_{\mathbb{K}}^{r}\to\mathbb{A}_{\mathbb{K}}^{r}$ be given by
$w(x_{1},\dotsc,x_{r})_{i}=p_{t_{i}}(x_{1},\dotsc,x_{r})$. Furthermore, assume
that there are sets $S_{1},\dotsc,S_{r}\subset\mathbb{K}$ of size $n$ each,
and $S=S_{1}\times\dotsb\times S_{r}$ is their product. Then there is a set
$S^{\prime}\subset S$ of size $\lvert S^{\prime}\rvert\geq\lvert
S\rvert-n^{r-1}\sum_{i}t_{i}$ such that for all $x\in S^{\prime}$ the number
of solutions to
$w(x)=w(y)\qquad\text{ with }y\in S^{\prime}$
is at most $\prod_{i}t_{i}$.
###### Proof.
The Jacobian determinant of $w$ is
$J(x)=\det(\partial p_{t_{i}}/\partial
x_{j})_{ij}=\det(t_{i}x_{j}^{t_{i}-1})_{ij}.$
The polynomial $J$ is of degree $\sum_{i}(t_{i}-1)$, and it is non-zero since
its degree in $x_{i}$ is $t_{r}-1<\operatorname{char}\mathbb{K}$. By Lemma 14
$J$ vanishes in at most $n^{r-1}\sum_{i}(t_{i}-1)$ points of $S$. Thus the set
$S^{\prime}=\\{x\in S:J(x)\neq 0\\}$ is of size $\lvert
S^{\prime}\rvert\geq\lvert S\rvert-n^{r-1}\sum_{i}t_{i}$.
For each $x\in S^{\prime}$ the variety $V_{x}=w^{-1}(w(x))$ is zero-
dimensional. The bound on the number of points of $V_{x}$ then follows from
Bezout’s theorem (Lemma 13). ∎
###### Proof of Theorem 3.
The proof is by induction on $k$. Suppose $k=1$ and $f=ax^{d}$. Let
$B=\\{x^{d}:x\in A\\}$. Then $\lvert B\rvert\geq\lvert A\rvert/d$. The
condition $\lvert AA\rvert+\lvert f(A)+f(A)\rvert\leq\Delta\lvert A\rvert$
implies $\lvert BB\rvert+\lvert B+B\rvert\leq d\Delta\lvert B\rvert$. Then (1)
implies $d\Delta\geq(\lvert A\rvert/d)^{1/12}$, which together with $\lvert
A\rvert\geq d^{20}$ establishes the base case.
Suppose $f=\sum_{i=1}^{k}a_{i}x^{d_{i}}$ and $k\geq 2$. If
$g=\gcd(d_{1},\dotsc,d_{k})\neq 1$, then upon replacing $A$ by $\\{a^{g}:a\in
A\\}$ and $f$ by $\sum a_{i}x^{d_{i}/g}$ the problem reduces to the case
$\gcd(d_{1},\dotsc,d_{k})=1$. Assume $\lvert AA\rvert+\lvert
f(A)+f(A)\rvert\leq\Delta\lvert A\rvert$. For $b\in\Gamma$ define
$A_{b}=\\{a\in A:ba\in A\\}$. Since $\gcd(d_{1},\dotsc,d_{k})=1$, there is a
pair of exponents $d_{i},d_{j}$ such that $d_{i}$ and $d_{j}$ are not a power
of the same integer. Without loss of generality we may assume that these two
exponents are $d_{1}$ and $d_{2}$. Let
$\lambda_{i}=d_{1}^{r-i}d_{2}^{i}\text{ for }i=1,\dotsc,r.$ (5)
Then by the choice of $d_{1}$ and $d_{2}$, all the $\lambda$’s are distinct.
Lemma 17 then yields a set $B\subset A$ of size $\lvert
B\rvert\geq\tfrac{1}{2}\Delta^{-P}\lvert A\rvert$ and elements
$g_{1},\dotsc,g_{r}$ such that if we define
$A_{b}=\\{a\in A:g_{i}b^{\lambda_{i}}a\in A\text{ for all }i=1,\dotsc,r\\},$
then for every $b\in B$
$\lvert A_{b}\rvert\geq\tfrac{1}{2}\Delta^{-P}\lvert A\rvert,$
where $P\leq 43r\log_{2}(1+d^{r})\leq 100r^{2}\log_{2}d$. Thus,
$\displaystyle\sum_{b_{1},\dotsc,b_{r}\in A}\lvert A_{b_{1}}\cap\dotsb\cap
A_{b_{r}}\rvert$ $\displaystyle=\sum_{a\in A}\bigl{(}\\#\\{b\in A:a\in
A_{b}\\}\bigr{)}^{r}$ $\displaystyle\geq\frac{(\sum_{a\in A}\bigl{(}\\#\\{b\in
A:a\in A_{b}\\})^{r}}{\lvert A\rvert^{r-1}}$
$\displaystyle\geq(\tfrac{1}{2}\Delta^{-P})^{2r}\lvert A\rvert^{r+1}.$
Define a graph $G$ as follows. Its vertex set is
$V(G)=\\{(b_{1},\dotsc,b_{r})\in B^{r}:\lvert A_{b_{1}}\cap\dotsb\cap
A_{b_{r}}\rvert\geq 2^{-2r-1}\Delta^{-2Pr}\lvert A\rvert\\},$
and the pair $(b_{1},\dotsc,b_{r})$, $(b_{r+1},\dotsc,b_{2r})$ is an edge of
$G$ if $\lvert A_{b_{1}}\cap\dotsb\cap A_{b_{2r}}\rvert\geq
2^{-4r-3}\Delta^{-4Pr}\lvert A\rvert$. Since $\lvert V(G)\rvert\lvert
A\rvert+\lvert B\rvert^{r}(2^{-2r-1}\Delta^{-2Pr}\lvert
A\rvert)\geq(\tfrac{1}{2}\Delta^{-P})^{2r}\lvert A\rvert^{r+1}$, the number of
vertices in $G$ is $\lvert V(G)\rvert\geq 2^{-2r-1}\Delta^{-2Pr}\lvert
A\rvert^{r}$. Moreover, the graph $G$ contains no independent set of size
$2^{2r+2}\Delta^{2Pr}$. Indeed, if
$\\{(b_{i,1},\dotsc,b_{i,r})\\}_{i=1}^{m}\subset V(G)$ is an independent set
of size $m$, then
$\displaystyle\lvert
A\rvert\geq\left\lvert\bigcup_{i=1}^{m}A_{b_{i,1}}\cap\dotsb\cap
A_{b_{i,r}}\right\rvert\geq m2^{-2r-1}\Delta^{-2Pr}\lvert
A\rvert-\binom{m}{2}2^{-4r-3}\Delta^{-4Pr}\lvert A\rvert,$
implying $m<2^{2r+2}\Delta^{2Pr}$.
For each $\mathbf{b}\in\mathbb{F}_{p}^{r}$ define
$\mathbf{b}^{\lambda}=(b_{1}^{\lambda},\dotsc,b_{r}^{\lambda})$. Let
$u(b)=(b^{d_{1}},\dotsc,b^{d_{k}})$, and for each
$\mathbf{b}=(b_{1},\dotsc,b_{r})\in V(G)$ define
$u(\mathbf{b})=u(b_{1})+\dotsb+u(b_{r})=(p_{d_{1}}(\mathbf{b}),p_{d_{2}}(\mathbf{b}),\dotsc,p_{d_{k}}(\mathbf{b}))$.
Now the argument breaks into two cases. In the first case, we will reduce the
problem to the case of a polynomial with at most $k-1$ terms, whereas in the
second case, we will show that a certain sumset associated to $V(G)$ is too
large.
Case 1: Suppose there is an index $i\in[r]$ and
$\mathbf{b}\mathbf{b}^{\prime}\in E(G)$ such that
$u(\mathbf{b}^{\lambda_{i}})\neq u(\mathbf{b}^{\prime\lambda_{i}})$, but
$u(\mathbf{b}^{\lambda_{i}})$ and $u(\mathbf{b}^{\prime\lambda_{i}})$ share a
coordinate, that is $p_{d_{j}}(\mathbf{b})=p_{d_{j}}(\mathbf{b}^{\prime})$ for
some $j$. Then
$\displaystyle g(x)$
$\displaystyle=f(g_{i}b_{1}^{\lambda_{i}}x)+\dotsb+f(g_{i}b_{r}^{\lambda_{i}}x)-f(g_{i}b_{1}^{\prime\lambda_{i}}x)+\dotsb+f(g_{i}b_{r}^{\prime\lambda_{i}}x)$
$\displaystyle=\sum_{j=1}^{k}a_{j}g_{j}^{d_{j}}(p_{d_{j}}(\mathbf{b}^{\lambda_{i}})-p_{d_{j}}(\mathbf{b}^{\prime\lambda_{i}}))x^{d_{j}}$
is a non-constant polynomial with one fewer term than $f$. Moreover, if we
define $A^{\prime}=A_{b_{1}}\cap\dotsb\cap A_{b_{r}}\cap
A_{b_{1}^{\prime}}\cap\dotsb\cap A_{b_{r}^{\prime}}$ then
$g(A^{\prime})+g(A^{\prime})\subset(2r)*f(A)-(2r)*f(A)$. By the definition of
the graph $G$,
$\lvert A^{\prime}\rvert\geq\frac{\lvert A\rvert}{2^{2r+3}\Delta^{2Pr}}.$
A simple, but tedious calculation shows that if $\Delta$ is small enough for
the conclusion of the theorem to fail, then $\lvert A\rvert\geq
d^{40r}p^{4/r}$ implies $\lvert A^{\prime}\rvert\geq d^{40(r+1)}p^{4/(r+1)}$,
permitting us to apply the induction hypothesis with $r$ and $k$ replaced by
$r+1$ and $k-1$ respectively. Then, since
$\lvert g(A^{\prime})+g(A^{\prime})\rvert\leq\Delta^{4r}\lvert
A\rvert=\Bigl{(}\Delta^{4r}\frac{\lvert A\rvert}{\lvert
A^{\prime}\rvert}\Bigr{)}\lvert A^{\prime}\rvert,$
the induction hypothesis implies that $\Delta^{4r}\frac{\lvert A\rvert}{\lvert
A^{\prime}\rvert}\geq\lvert A^{\prime}\rvert^{\varepsilon}$, where
$\varepsilon=(5000(r+k)^{2}\log_{2}d)^{k-1}$. Another tedious calculation
shows that this implies the desired lower bound on $\Delta$.
Case 2: Suppose that for all $i\in[r]$ and all edges
$\mathbf{b}\mathbf{b}^{\prime}\in E(G)$ either all coordinates of
$u(\mathbf{b}^{\lambda_{i}})$ and $u(\mathbf{b}^{\prime\lambda_{i}})$ are
equal, or all of them are distinct. In particular,
$p_{d_{1}}(\mathbf{b}^{\lambda_{i}})=p_{d_{1}}(\mathbf{b}^{\prime\lambda_{i}})$
and
$p_{d_{2}}(\mathbf{b}^{\lambda_{i}})=p_{d_{2}}(\mathbf{b}^{\prime\lambda_{i}})$
happen simultaneously. Let
$w(\mathbf{b})=(p_{d_{1}^{r}d_{2}}(\mathbf{b}),p_{d_{1}^{r-1}d_{2}^{2}}(\mathbf{b}),\dotsc,p_{d_{2}^{r+1}}(\mathbf{b})).$
Since
$p_{d_{1}}(\mathbf{b}^{\lambda_{i}})=p_{d_{1}^{r-i+1}d_{2}^{i}}(\mathbf{b})$
and
$p_{d_{2}}(\mathbf{b}^{\lambda_{i}})=p_{d_{1}^{r-i}d_{2}^{i+1}}(\mathbf{b})$,
it follows that if $\mathbf{b}\mathbf{b}^{\prime}\in E(G)$, then either
$w(\mathbf{b})=w(\mathbf{b}^{\prime})$ or $w(\mathbf{b})$ differs from
$w(\mathbf{b}^{\prime})$ in all coordinates.
By Lemma 18 there is a set $X\subset V(G)$ of size at least $\lvert
V(G)\rvert-\lvert A\rvert^{r-1}rd^{r+1}\geq\tfrac{1}{2}\lvert V(G)\rvert$ such
that $\lvert X\cap w^{-1}(w(\mathbf{b}))\rvert\leq d^{r^{2}+r}$ for all
$\mathbf{b}\in X$. Since $\lvert A\rvert\geq d^{40r}$, and $2^{r}\geq r$
either $\Delta$ is sufficiently large to stop the argument here, or we have
$\lvert X\rvert\geq\lvert V(G)\rvert-\lvert A\rvert^{r-1}rd^{r+1}\geq\lvert
A\rvert^{r-1}(2^{-2r-1}\Delta^{-2Pr}\lvert A\rvert-rd^{r+1})\geq\lvert
A\rvert^{r}2^{-2r-2}\Delta^{-2Pr}.$
By the pigeonhole principle there is an $h\in\mathbb{F}_{p}$ such that the
hyperplane $H=\\{x_{1}=h\\}\subset\mathbb{F}_{p}^{r}$ contains $w(\mathbf{b})$
for at least $\lvert X\rvert/p$ values of $\mathbf{b}\in X$. Pick
$X^{\prime}\subset X$ such that the points $\\{w(\mathbf{b})\\}_{\mathbf{b}\in
X^{\prime}}$ all lie in $H$ and are distinct, and such that
$\lvert X^{\prime}\rvert\geq\frac{1}{d^{r^{2}+r}}\cdot\frac{\lvert
X\rvert}{p}\geq\frac{2^{-2r-2}\Delta^{-2Pr}\lvert A\rvert^{r}}{pd^{r^{2}+r}}.$
Since such an $X^{\prime}$ is an independent set in $G$, we conclude that
$\lvert X^{\prime}\rvert\leq 2^{2r+2}\Delta^{2Pr}$, and the theorem follows. ∎
## 5 Large Sets
The results in this section require a more systematic use of the idea of
differencing appearing in the proof of Theorem 1, and used throughout the
previous section. The differencing is a special instance of a general strategy
that consists in repeatedly applying the Cauchy–Schwarz inequality to increase
the number of variables involved. Geometrically a single application of the
Cauchy–Schwarz corresponds to taking fiber products of two varieties. For
illustration, consider the problem of showing that $f(A)$ grows. Let
$g(x,y,z)=f(x,y)-z$. It suffices to show that $N(g;A,A,C)$ is much smaller
than $\lvert A\rvert^{2}$ for every set $C\subset\mathbb{F}_{q}$ of size
$\lvert C\rvert=\lvert A\rvert$. This is a problem of bounding the number of
points of a particular variety $V=\\{g(x,y,z)=0\\}$ on the Cartesian product
$A\times B\times C$. The Cauchy–Schwarz inequality tells us that the number of
points of $V^{\prime}=\\{g(x,y_{1},z_{1})=0,g(x,y_{2},z_{2})=0\\}$ on $A\times
B\times B\times C\times C$ is at least $N(g;A,B,C)^{2}/\lvert A\rvert$, and
thus it suffices to establish a non-trivial upper bound for the number of
points on $V^{\prime}$. The variety $V^{\prime}$ is the fiber product of $V$
with itself for the projection map on the first coordinate. However, in
general one can use less trivial fiber products.
The problem about sets, whether $f(A)$ is large, reduces to the problem about
counting points on the variety $\\{f(x,y)-z=0\\}$ on Cartesian products, but
to pass in the other direction the only tool currently available is the Balog-
Szemerédi-Gowers theorem (Lemma 12), which applies only for linear $f$. It is
because of this we are forced to work with varieties rather than sets. We note
however that the analogues of the sumset inequalities that were extensively
used in the previous section are easy to prove with Cauchy–Schwarz as above.
For instance, let $g(x,y,z)=x+y-z$, and $A=B=C$, then the lower bound for the
number of points on $V=\\{x+y-z=0\\}$ implies a lower bound for the number of
points on $V^{\prime}=\\{x+y_{1}-z_{1}=0,x+y_{2}-z_{2}=0\\}$, which upon
elimination of $x$ variable, yields a lower bound for the number of points on
$V^{\prime\prime}=\\{y_{1}+z_{2}-y_{2}-z_{1}=0\\}$. This relation corresponds
to the Plünnecke inequality for $A+A-A$. For general analogues of Ruzsa’s
triangle inequalities and Plünnecke’s inequality see [Raz07]. Another
interesting example of a systematic use of fiber products in additive
combinatorics is in [KT99].
To establish the estimates on $N(f;A,A,B)$ promised in theorem 4 we will use
the special case of $N(f;A,A,A)$ as a stepping-stone. The general result will
be deduced via an application of Cauchy–Schwarz as explained above.
###### Lemma 19.
Suppose $n\geq 2$ and $f(\vec{x})$ is an $n$-variable polynomial of degree $d$
with no linear factors, and $A\subset\mathbb{F}_{q}$. Then if $|A|\geq
q^{1/2}$ and $d<q^{1/5n}$, we have the estimate
$\lvert A+A\rvert+\frac{\lvert A\rvert^{n}}{N(f;A)}\gg_{n}\begin{cases}\lvert
A\rvert(\lvert A\rvert/\sqrt{q})^{1/2}d^{-1},&\text{if }q^{1/2}\leq\lvert
A\rvert\leq d^{4/5}q^{7/10},\\\ \lvert A\rvert(q/\lvert
A\rvert)^{1/3}d^{-1/3},&\text{if }\lvert A\rvert\geq
d^{4/5}q^{7/10}.\end{cases}$
###### Proof.
We will first establish the case $n=2$, and then reduce the general case to
it.
Case $n=2$: The proof is a standard Fourier-analytic argument. Namely, let
$B=A+A$, and $S\subset\mathbb{F}_{q}^{2}$ be the set of solutions to
$f(\vec{x})=0$, and let $C=S\cap A^{2}$ be the subset of the solutions with
coordinates in $A$. Since $C+(A\times A)\subset B\times B$ it follows that
$\langle\chi_{B\times
B}*\chi_{-A\times-A},\chi_{S}\rangle_{\mathbb{F}_{q}^{2}}\geq\frac{|C|\cdot|A|^{2}}{q^{4}}.$
Using Plancherel’s theorem, and evaluating the term $\xi=0$ separately, we
obtain
$\frac{|C|\cdot|A|^{2}}{q^{4}}\leq\langle\hat{\chi}_{B\times
B}\cdot\hat{\chi}_{-A\times-A},\hat{\chi}_{S}\rangle_{\hat{\mathbb{F}}_{q}^{2}}=\frac{\lvert
B\rvert^{2}\lvert A\rvert^{2}}{q^{5}}+\sum_{\xi\neq 0}\hat{\chi}_{B\times
B}(\xi)\hat{\chi}_{-A\times-A}(\xi)\hat{\chi}_{S}(-\xi).$
Lemma 14 gives the bound $\lvert S\rvert\leq dq$. Moreover, since $f$ has no
linear factors, by Lemma 16 $\hat{\chi}_{S}(\xi)\ll d^{2}q^{-3/2}$. Therefore,
by Cauchy–Schwarz and Parseval
$\displaystyle\frac{|C|\cdot|A|^{2}}{q^{4}}$ $\displaystyle\ll d\frac{\lvert
B\rvert^{2}\lvert
A\rvert^{2}}{q^{5}}+d^{2}q^{-3/2}\langle\lvert\hat{\chi}_{B\times
B}\rvert,\lvert\hat{\chi}_{-A\times-A}\rvert\rangle_{\hat{\mathbb{F}}_{q}^{2}}$
$\displaystyle\leq d\frac{\lvert B\rvert^{2}\lvert
A\rvert^{2}}{q^{5}}+d^{2}q^{-3/2}\lVert\hat{\chi}_{B\times
B}\rVert^{1/2}\lVert\hat{\chi}_{-A\times-A}\rVert^{1/2}$
$\displaystyle=d\frac{\lvert B\rvert^{2}\lvert
A\rvert^{2}}{q^{5}}+d^{2}\frac{\lvert B\rvert\lvert A\rvert}{q^{7/2}}.$
Since $\lvert C\rvert=N(f;A,A)$, we deduce that either $\frac{\lvert
A\rvert^{2}}{N(f;A)}\lvert B\rvert^{2}\geq\frac{1}{2d}q\lvert A\rvert^{2}$ or
$\lvert B\rvert\frac{\lvert A\rvert^{2}}{N(f;A)}\geq\frac{\lvert
A\rvert^{3}}{d^{2}q^{1/2}}$. In both cases, at least one of $\lvert
B\rvert=\lvert A+A\rvert$ and $\frac{\lvert A\rvert^{2}}{N(f;A)}$ is as large
as claimed, and the result follows.
Case $n>2$: Having established the $n=2$ case, we proceed by induction on $n$.
Note that we can suppose $f$ is irreducible. Now by Lemma 32, one can pick a
variable, say $x_{1}$, such that for all but $d^{n}(n-1)$ ‘bad’ elements
$c\in\mathbb{F}_{q}$ of $x_{1}$, the polynomial
$f_{c}=f(c,x_{2},x_{3},\dots,x_{n})$ has no linear factors. Then
$N(f;A)\leq d^{n}(n-1)|A|^{n-2}+\sum_{\textrm{`good' }c}N(f_{c};A).$
Now, if the first term on the RHS is bigger, we are done. Else, we have:
$\lvert A+A\rvert+\frac{\lvert A\rvert^{n}}{N(f,A)}\geq\lvert
A+A\rvert+\frac{\lvert A\rvert^{n}}{2\max_{\textrm{good c}}N(f_{c},A)},$
and we are done by induction. ∎
###### Lemma 20.
Let $W$ be an $m$-dimensional irreducible variety of degree $d\geq 2$ in
$\mathbb{A}_{\mathbb{F}_{q}}^{n}$, and $A\subset\mathbb{F}_{q}$. Then if
$|A|\geq q^{1/2}$ and $d<q^{1/5n}$, we have the estimate
$\lvert A+A\rvert+\frac{\lvert
A\rvert^{m+1}}{N(W;A)}\gg_{n}\begin{cases}\lvert A\rvert(\lvert
A\rvert/\sqrt{q})^{1/2}d^{-1},&\text{if }q^{1/2}\leq\lvert A\rvert\leq
d^{4/5}q^{7/10},\\\ \lvert A\rvert(q/\lvert A\rvert)^{1/3}d^{-1},&\text{if
}\lvert A\rvert\geq d^{4/5}q^{7/10}.\end{cases}$
###### Proof.
By Lemma 33, we can find a subset of $m+1$ coordinates
$y_{1},y_{2},..,y_{m+1}$ on which to project $W$ such that the image lies in
an $m$-dimensional variety $W^{\prime}$ of degree $d^{\prime}\geq 2$, which is
thus defined by an equation $f(\vec{y})=0$ for a polynomial $f$ of degree
$d^{\prime}$. By Corollary 31 and Lemma 13 there is a proper subvariety
$V\subset W$ of degree at most $d^{3}n$ on the complement of which the
projection from $W$ to $W^{\prime}$ is at most $(d/d^{\prime})$-to-$1$. Hence,
$N(W;A)\leq N(V;A)+(d/d^{\prime})N(W^{\prime};A)$. As $\dim V\leq m-1$, Lemma
14 yields $N(V;A)\leq d^{3}n\lvert A\rvert^{m-2}$. Thus if
$(d/d^{\prime})N(W^{\prime};A)\leq N(V;A)$, then we are done. Otherwise,
$\lvert A+A\rvert+\frac{\lvert A\rvert^{m+1}}{N(W;A)}\geq\lvert
A+A\rvert+\frac{\lvert A\rvert^{m+1}}{2(d/d^{\prime})N(W^{\prime};A)}$
and the results follows from the previous lemma. ∎
For the case where $A$ is an interval, similar results to Lemma 20 were
obtained by Fujiwara [Fuj88] and Schmidt [Sch86].
The preceding lemma implies Theorem 4 concerning the lower bound on $\lvert
A+A\rvert+\frac{\lvert B\rvert\lvert A\rvert^{4}}{N(f;A,A,B)}$.
###### Proof of Theorem 4.
Let $V\subset\mathbb{A}_{\mathbb{F}_{q}}^{5}$ be the $3$-dimensional variety
$f(x_{1},x_{2},x_{5})=f(x_{3},x_{4},x_{5})=0.$
Note that $V$ is the fiber product of $f(x,y,z)=0$ with itself along the
projection to the last coordinate. By the Cauchy–Schwarz inequality,
$N(V;A,A,A,A,B)\geq\frac{N(f;A,A,B)^{2}}{\lvert B\rvert}$. Let
$U\subset\mathbb{A}_{\mathbb{F}_{q}}^{4}$ be the projection of $V$ onto
$\\{x_{5}=0\\}$, and $W$ be the Zariski closure of $U$. Note that for any
point $u\in U$, one of two things can happen: the preimage of $u$ in $V$
consists of at most $d$ points, or the preimage in $V$ is all of
$\mathbb{A}_{\mathbb{F}_{q}}^{1}$. But the latter can only happen for at most
$d^{2}$ points, by Corollary 29. Thus we arrive at the following upper bound
on $N(f;A,A,B)$
$\frac{N(f;A,A,B)^{2}}{\lvert B\rvert}\leq N(V;A,A,A,A,B)\leq dN(U;A)+d^{2}q.$
Note that if $dN(U;A)\leq d^{2}q$, then the lemma follows. Thus we may assume
$\frac{N(f;A,A,B)^{2}}{\lvert B\rvert}\leq 2N(U;A)\leq 2N(W;A).$
An upper bound on $N(W;A)$ will now follow from the preceding lemma. Since the
$\deg W\leq d^{2}$, the condition $d<q^{1/5n}$ of the lemma holds, and to
apply the lemma it suffices to check that $W$ does not contain a hyperplane.
We argue by contradiction. Suppose a hyperplane
$L=\\{a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+a_{4}x_{4}+a_{5}=0\\}$ is a component
of $W$. Since the surfaces
$S_{c}=\\{f(x_{1},x_{2},c)=f(x_{3},x_{4},c)=0\\}$
fiber $W$, the varieties $S_{c}\cap L$ fiber $L$. So by the dimension count,
$S_{c}\cap L$ is generically $2$-dimensional. Thus for a generic
$c\in\mathbb{A}^{1}$, $S_{c}$ has a component in $L$. This can happen only if
$a_{1}x_{1}+a_{2}x_{2}$ is constant on a component of $f(x_{1},x_{2},c)=0$,
which means that $f(x_{1},x_{2},c)=0$ contains a line with slope
$s=\frac{-a_{1}}{a_{2}}$. But having slope $s$ is a Zariski-closed condition,
and since $V$ is irreducible, all points in $V$ must have slope $s$, which
contradicts the assumption that $f(x,y,z)$ is not of the form
$P(a_{1}x+a_{2}y,z)$.
Therefore we can apply the previous lemma to get the bound
$\lvert A+A\rvert+\frac{\lvert B\rvert\lvert
A\rvert^{4}}{N(f;A,A,B)^{2}}\gg\begin{cases}\lvert A\rvert(\lvert
A\rvert/\sqrt{q})^{1/2}d^{-1},&\text{if }q^{1/2}\leq\lvert A\rvert\leq
d^{4/5}q^{7/10},\\\ \lvert A\rvert(q/\lvert A\rvert)^{1/3}d^{-1},&\text{if
}\lvert A\rvert\geq d^{4/5}q^{7/10}.\end{cases}$
∎
We conclude by presenting the deduction of Theorem 5 on growth of $\lvert
f(A)+f(A)\rvert+\lvert g(A,A)\rvert$ from Theorem 4.
###### Proof of Theorem 5.
Since $f(x)$ has at most $d$ poles, we may assume that $A$ contains none of
them.
The idea in the proof is to create a polynomial having many solutions on
$f(A)$ and apply the previous lemma. This motivates us to define fields
$\mathbb{K}=\mathbb{F}_{q}(x,y)$ and
$\mathbb{K}_{f}=\mathbb{F}_{q}(f(x),f(y))$. It is easy to see that
$\mathbb{K}_{f}\subset\mathbb{K}$ and
$[\mathbb{K}:\mathbb{K}_{f}]=\deg(f)^{2}$.
Let $H(x,y,t)$ be the minimal polynomial for $g(x,y)$ over $\mathbb{K}_{f}$.
Note that $H(x,y,g(x,y))=0$. Write $H(x,y,t)$ = $S(f(x),f(y),t)$. Then
$S(x,y,t)$ is an irreducible polynomial with at least $\lvert f(A)\rvert^{2}$
roots on $f(A)\times f(A)\times g(A,A)$. If $S(x,y,t)$ is not of the form
$P(ax+by,t)$, Theorem 4 yields
$\lvert f(A)+f(A)\rvert+\frac{\lvert g(A,A)\rvert\lvert
f(A)\rvert^{4}}{(\lvert f(A)\rvert^{2})^{2}}\gg\begin{cases}\lvert
A\rvert(\lvert A\rvert/\sqrt{q})^{1/2}D^{-1},&\text{if }q^{1/2}\leq\lvert
A\rvert\leq D^{4/5}q^{7/10},\\\ \lvert A\rvert(q/\lvert
A\rvert)^{1/3}D^{-1},&\text{if }\lvert A\rvert\geq
D^{4/5}q^{7/10},\end{cases}$
where $D=\deg S$. Since an indeterminate $3$-variable polynomial $S^{\prime}$
of degree $D^{\prime}$ has $\binom{D^{\prime}+3}{3}$ coefficients, and the
condition that the rational function $S^{\prime}(f(x),f(y),g(x,y))$ vanishes
is a system of $\binom{4dD^{\prime}+2}{2}$ linear equations, there is a non-
zero polynomial $S^{\prime}$ satisfying $S^{\prime}(f(x),f(y),g(x,y))=0$, of
degree at most $D^{\prime}$, provided
$\binom{D^{\prime}+3}{3}\leq\binom{4dD^{\prime}+2}{2}+1.$
Since $S$ is irreducible, $S$ divides $S^{\prime}$ implying $D\leq
D^{\prime}\leq 12d^{2}$.
If $S(x,y,t)=P(ax+by,t)$, then it would force $P(af(x)+bf(y),g(x,y))=0$,
implying that $g(x,y)$ is algebraic over $\mathbb{F}_{q}(af(x)+bf(y))$. If
that is so, $g(x,y)$ is of the form $G(af(x)+bf(y)+c)$, $G(x)$, or $G(y)$ by
Lemma 34. ∎
## 6 Growth of $f(A,B)$ for very large sets $A,B$
In this section we prove Theorem 6 on growth of $f(A,B)$ without any
assumptions on $A+A$. Excluding several algebraic lemmas, whose role is
auxiliary to the main flow of the argument, the proof is not long. However,
the shape of the proof might be mysterious without further explanations.
In the previous section we saw a way to use the smallness of $A+A$ to reduce
the task of bounding $N(f;A)$ to estimating Fourier transform of the curve
$\\{f=c\\}$. The latter was achieved by invoking the celebrated Weil’s bound.
In the absence of any assumption on $A+A$, we need to dispose of the Fourier
transform. The motivation for our approach comes from Gowers $U^{2}$ norm,
which is a substitute for the Fourier transform in additive combinatorics. If
instead of smallness of $A+A$ we assume smallness of another polynomial
$g(A,A)$, then treating $g$ as ‘addition’ and its ‘inverse’ (which might exist
only implicitly) as ‘subtraction’, we can create an analogue of the $U^{2}$
norm of $g$. Thus we require a polynomial $g$ for which $g(A)$ is small. It
turns out that often the polynomial $f$ itself can fulfill the role of $g$.
Moreover, assuming smallness of $f(A)$ we can create many varieties $V$ for
which $N(V)$ is small using fiber products as in the previous section.
The substitute for the Fourier transform is not enough if there is no analogue
for Weil’s bound. That final ingredient comes from the bound on the number of
points on irreducible varieties. It should be noted that we will not use the
full strength of Deligne’s bound on the number of such points, but rather an
earlier theorem of Lang and Weil [LW54] that is a consequence of Weil’s work
on curves.
An alternative perspective on the argument that follows is that it is about
turning sums over large subsets $A\subset\mathbb{F}_{q}$ into complete sums
over $\mathbb{F}_{q}$, which are easier to study by algebraic means. A
desirable estimate on such a sum is $\left|\sum_{x\in
A}S(x)\right|\leq\left|\sum_{x\in\mathbb{F}_{q}}S(x)\right|$, but in general
it holds only if $S(x)$ is _positive_. The terms $S(x)$ that appear in our
sums are not always positive, but the Cauchy–Schwarz inequality reduces the
problem to a sum of positive terms, which can be completed:
$\left|\sum_{x\in A}S(x)\right|\leq|A|^{\frac{1}{2}}\left|{\sum_{x\in
A}S(x)^{2}}\right|^{\frac{1}{2}}\leq|A|^{\frac{1}{2}}\left|{\sum_{x\in\mathbb{F}_{q}}S(x)^{2}}\right|^{\frac{1}{2}}.$
The sum on the right is now a complete sum.
###### Proof of Theorem 6.
Set $C=f(A,B)$. Suppose $\lvert C\rvert\leq q/2$, for else there is nothing to
prove. We start with the inequality
$\lvert A\rvert^{2}\lvert
B\rvert^{2}\leq\displaystyle\sum_{f(x_{1},y_{2})=z_{1},f(x_{2},y_{1})=z_{2}}A(x_{2})B(y_{2})C(z_{1})C(z_{2})C(f(x_{1},y_{1})),$
where we identify sets with their characteristic functions, that is $A(t)=1$
if $t\in A$, and $0$ otherwise, and likewise for $B$ and $C$.
As in the usual application of Gowers $U^{2}$ norm, we will need to replace
one of the sets by a function of mean zero. Let $S(t)=1$ for $t\in C$, and
$S(t)=-\frac{\lvert C\rvert}{q-\lvert C\rvert}$ if $t\not\in C$. The function
$S(t)$ is clearly of mean $0$. Since $f$ is monic in $y$, for each $x$ and $z$
there are at most $d$ solutions to $f(x,y)=z$. Thus there are at most $d\lvert
A\rvert\lvert C\rvert$ solutions to $f(x,y)=z$ with $x\in A$,
$y\in\mathbb{F}_{q}$ and $z\in C$, and we have
$\lvert A\rvert^{2}\lvert B\rvert^{2}-d^{2}\frac{\lvert A\rvert\lvert
B\rvert\lvert C\rvert^{3}}{q-\lvert
C\rvert}\leq\sum_{f(x_{1},y_{2})=z_{1},f(x_{2},y_{1})=z_{2}}A(x_{2})B(y_{2})C(z_{1})C(z_{2})S(f(x_{1},y_{1}))$
If $\lvert C\rvert^{3}\geq\lvert A\rvert\lvert B\rvert q/4d^{2}$, we are done.
Otherwise, the first term dominates and we obtain
$\lvert A\rvert^{2}\lvert B\rvert^{2}\leq
2\displaystyle\sum_{f(x_{1},y_{2})=z_{1},f(x_{2},y_{1})=z_{2}}A(x_{2})B(y_{2})C(z_{1})C(z_{2})S(f(x_{1},y_{1}))$
We proceed to ‘clone’ variables by applying Cauchy–Schwarz with respect to
$y_{2},x_{2}$ to obtain
$\lvert A\rvert^{4}\lvert B\rvert^{4}\leq
4\sum_{y_{2},x_{2}}\left(A^{2}(x_{2})B^{2}(y_{2})\right)\sum_{y_{2},x_{2}}\left(\sum_{f(x_{1},y_{2})=z_{1},f(x_{2},y_{1})=z_{2}}C(z_{1})C(z_{2})S(f(x_{1},y_{1}))\right)^{2},$
where the inner sum on the right side ranges over $z_{1},z_{2},x_{1},y_{1}$.
Expanding, we get
$\lvert A\rvert^{3}\lvert B\rvert^{3}\leq
4\displaystyle\sum_{\begin{subarray}{c}f(x_{1},y_{2})=z_{1},f(x_{2},y_{1})=z_{2}\\\
f(x_{3},y_{2})=z_{3},f(x_{2},y_{3})=z_{4}\end{subarray}}C(z_{1})C(z_{2})C(z_{3})C(z_{4})S(f(x_{1},y_{1}))S(f(x_{3},y_{3})).$
We come to our final application of Cauchy–Schwarz to this sum, this time with
respect to $z_{1},z_{2},z_{3},z_{4}$:
$\frac{\lvert A\rvert^{6}\lvert B\rvert^{6}}{N^{4}}\leq
16\sum_{\begin{subarray}{c}f(x_{1},y_{2})=z_{1},f(x_{2},y_{1})=z_{2}\\\
f(x_{3},y_{2})=z_{3},f(x_{2},y_{3})=z_{4}\\\
f(x^{\prime}_{1},y^{\prime}_{2})=z_{1},f(x^{\prime}_{2},y^{\prime}_{1})=z_{2}\\\
f(x^{\prime}_{3},y^{\prime}_{2})=z_{3},f(x^{\prime}_{2},y^{\prime}_{3})=z_{4}\end{subarray}}S(f(x_{1},y_{1}))S(f(x_{3},y_{3}))S(f(x^{\prime}_{1},y^{\prime}_{1}))S(f(x^{\prime}_{3},y^{\prime}_{3})).$
Or more succinctly,
$\frac{\lvert A\rvert^{6}\lvert B\rvert^{6}}{N^{4}}\leq
16\sum_{\begin{subarray}{c}f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2}),f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\\\
f(x_{3},y_{2})=f(x^{\prime}_{3},y^{\prime}_{2}),f(x_{2},y_{3})=f(x^{\prime}_{2},y^{\prime}_{3})\end{subarray}}S(f(x_{1},y_{1}))S(f(x_{3},y_{3}))S(f(x^{\prime}_{1},y^{\prime}_{1}))S(f(x^{\prime}_{3},y^{\prime}_{3})).$
We next split the sum into many. Geometrically, the summation is over the
variety
$V=\left\\{\begin{aligned}
f(x_{1},y_{2})&=f(x^{\prime}_{1},y^{\prime}_{2}),f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\\\
f(x_{3},y_{2})&=f(x^{\prime}_{3},y^{\prime}_{2}),f(x_{2},y_{3})=f(x^{\prime}_{2},y^{\prime}_{3})\end{aligned}\right\\}.$
Define $\phi\colon V\to\mathbb{A}^{4}$ by
$\phi(\dotsc)=(f(x_{1},y_{1}),f(x_{3},y_{3}),f(x^{\prime}_{1},y^{\prime}_{1}),f(x^{\prime}_{3},y^{\prime}_{3}))$,
and let $V_{t_{1},t_{2},t_{3},t_{4}}\subset V$ be the variety
$\phi^{-1}(t_{1},t_{2},t_{3},t_{4})$. Then we can rewrite the preceding
inequality as
$\frac{\lvert A\rvert^{6}\lvert B\rvert^{6}}{N^{4}}\leq
16\sum_{t_{1},t_{2},t_{3},t_{4}\in\mathbb{F}_{q}}S(t_{1})S(t_{2})S(t_{3})S(t_{4})N(V_{t_{1},t_{2},t_{3},t_{4}};\mathbb{F}_{q}).$
Heuristically, one expects such a complicated variety as
$V_{t_{1},t_{2},t_{3},t_{4}}$ to be usually irreducible and of dimension $4$,
since it is given by $8$ equations in $12$ variables. If that was indeed the
case, then by [LW54]
$N(V_{t_{1},t_{2},t_{3},t_{4}};\mathbb{F}_{q})=q^{4}+O_{d}(q^{7/2})$. Since
$S$ is of mean zero, that would give the estimate of the theorem.
Formally, we appeal to Lemmas 23 and 24 that tell us that $V$ is
$8$-dimensional, irreducible and there is a Zariski-dense open set
$U\subset\mathbb{A}^{4}$ such that whenever $(t_{1},t_{2},t_{3},t_{4})\in U$,
the variety $V_{t_{1},t_{2},t_{3},t_{4}}$ is $4$-dimensional and irreducible.
Let $Y=\mathbb{A}^{4}\setminus U$. Since $\dim\phi^{-1}(Y)\leq 7$, the variety
$\phi^{-1}(Y)$ contains at most $O_{d}(q^{7})$ points in $\mathbb{F}_{q}^{12}$
and thus
$\displaystyle\frac{\lvert A\rvert^{6}\lvert B\rvert^{6}}{N^{4}}$
$\displaystyle\ll\sum_{t_{1},t_{2},t_{3},t_{4}\in\mathbb{F}_{q}}S(t_{1})S(t_{2})S(t_{3})S(t_{4})N(V_{t_{1},t_{2},t_{3},t_{4}};\mathbb{F}_{q})$
$\displaystyle=\sum_{t_{1},t_{2},t_{3},t_{4}\in
U\cap\mathbb{F}_{q}}S(t_{1})S(t_{2})S(t_{3})S(t_{4})(q^{4}+O_{d}(q^{7/2}))+O_{d}(q^{7})$
$\displaystyle\ll_{d}q^{7/2}\sum_{t_{1},t_{2},t_{3},t_{4}\in U\cap
F_{q}}\lvert
S(t_{1})S(t_{2})S(t_{3})S(t_{3})\rvert+O_{d}(q^{7})\ll_{d}N^{4}q^{7/2},$
where the most important step is using that $S(t)$ is of mean zero to pass
from the second to the third line. Thus $\frac{\lvert A\rvert^{6}\lvert
B\rvert^{6}}{N^{4}}\ll_{d}N^{4}q^{7/2}$, it follows that $N\gg_{d}\lvert
A\rvert^{3/4}\lvert B\rvert^{3/4}q^{-7/16}$. ∎
## 7 Proofs of Lemmas 23 and 24
Before starting, we introduce a piece of terminology.
###### Definition.
An $n$-dimensional variety $V$ over an algebraically closed field $\mathbb{K}$
is said to be _mainly irreducible_ if it has a unique irreducible component of
dimension $n$.
The following lemma will be very useful in showing that many fiber products
are irreducible:
###### Lemma 21.
Let $V,W$ be $n$-dimensional irreducible varieties over $\mathbb{K}$ with
dominant, finite maps $f\colon V\rightarrow\mathbb{A}_{\mathbb{K}}^{n}$ and
$g\colon W\rightarrow\mathbb{A}_{\mathbb{K}}^{n}$ of degrees prime to
$p=\operatorname{char}\mathbb{K}$, and let
$V^{\prime},W^{\prime}\subset\mathbb{A}_{\mathbb{K}}^{n}$ be the subsets over
which $f$, resp. $g$ are unramified. Then if $V^{\prime}\cup
W^{\prime}=\mathbb{A}_{\mathbb{K}}^{n}$, the fiber product
$V\times_{\mathbb{A}_{\mathbb{K}}^{n}}W$ is irreducible.
###### Proof.
Let $\mathbb{F}=\mathbb{K}(V)$ and $\mathbb{G}=\mathbb{K}(W)$ be the function
fields of $V$ and $W$ considered as finite extensions of
$\mathbb{M}=\mathbb{K}(\mathbb{A}_{\mathbb{K}}^{n})$. Now
$V\times_{\mathbb{A}_{\mathbb{K}}^{n}}W$ is irreducible if and only if
$\mathbb{F}$ and $\mathbb{G}$ are linearly disjoint over $\mathbb{M}$. Since
$V^{\prime}\cup W^{\prime}=\mathbb{A}^{n}_{\mathbb{K}}$, the fields
$\mathbb{K}$ and $\mathbb{G}$ have coprime discriminants over $\mathbb{M}$.
Let $\mathbb{F}^{\prime},\mathbb{G}^{\prime}$ be the Galois closures of
$\mathbb{F}$ and $\mathbb{G}$ over $\mathbb{M}$ respectively. Then
$\mathbb{F}^{\prime}$ and $\mathbb{G}^{\prime}$ still have coprime
discriminants, and so their intersection
$\mathbb{F}^{\prime}\cap\mathbb{G}^{\prime}$ is an unramified extension over
$\mathbb{M}$ with degree prime to $p$, which must be $\mathbb{M}$ itself,
since $\mathbb{A}_{\mathbb{K}}^{n}$ has no nontrivial unramified extensions of
degree prime to $p$. But since $\mathbb{F}^{\prime},\mathbb{G}^{\prime}$ are
Galois, this implies that they are linearly disjoint, and hence the subfields
$\mathbb{F}$ and $\mathbb{G}$ are also linearly disjoint, as desired. ∎
Note that by looking at the map between generic points the above proof carries
through over the part of $\mathbb{A}_{\mathbb{K}}^{n}$ where $f$ and $g$ are
finite. We thus have the following easy corollary:
###### Corollary 22.
Let $V,W$ be $n$-dimensional, mainly irreducible varieties over $\mathbb{K}$
with dominant maps $f\colon V\rightarrow\mathbb{A}_{\mathbb{K}}^{n}$ and
$g\colon W\rightarrow\mathbb{A}_{\mathbb{K}}^{n}$ of degree prime to $p,$ and
let $V^{\prime},W^{\prime}\subset\mathbb{A}_{\mathbb{K}}^{n}$ be the subsets
over which $f$, resp. $g$ are unramified. Then if $V^{\prime}\cup
W^{\prime}=\mathbb{A}_{\mathbb{K}}^{n}$, the fiber product
$V\times_{\mathbb{A}_{\mathbb{K}}^{n}}W$ is mainly irreducible of dimension
$n$ with the possible exception of components that do not map dominantly to
$\mathbb{A}_{\mathbb{K}}^{n}$.
From now on we adopt the notation of proof of Theorem 6, so let $V$ be the
variety defined by the equations
$\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2}),f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1}),f(x_{3},y_{2})=f(x^{\prime}_{3},y^{\prime}_{2}),f(x_{2},y_{3})=f(x^{\prime}_{2},y^{\prime}_{3})\right\\}.$
Define also $f_{1}(x,y)$ and $f_{2}(x,y)$ to be the derivatives of $f$ with
respect to the first and second coordinates respectively. Let $\deg_{1}f$ and
$\deg_{2}f$ be degrees of $f$ in the $x$ and $y$ variable, respectively.
###### Lemma 23.
Under the assumptions of Theorem 6, $V$ is an $8$-dimensional mainly
irreducible variety.
###### Proof.
Denote by $W$ the variety
$\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2}),f(x_{3},y_{2})=f(x^{\prime}_{3},y^{\prime}_{2})\right\\},$
and by $W^{\prime}$ the variety
$\left\\{f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1}),f(x_{2},y_{3})=f(x^{\prime}_{2},y^{\prime}_{3})\right\\}.$
Since $V=W\times W^{\prime}$, it suffices to show that $W$ and $W^{\prime}$
are mainly irreducible of dimension $4$. We focus on $W$, the case of
$W^{\prime}$ being symmetric:
Consider the map $\phi\colon\mathbb{A}^{3}\rightarrow\mathbb{A}^{2}$ given by
$\phi(r,s,t)=(f(r,t),f(s,t))$. Notice that $W$ is the fiber product
$\mathbb{A}^{3}\times_{\mathbb{A}^{2}}\mathbb{A}^{3}$ of $\mathbb{A}^{3}$ with
itself with respect to the map $\phi$, so to show that the $W$ is mainly
irreducible it suffices by Lemma 35 to show that the fibers of $\phi$ are
$1$-dimensional and generically irreducible.
The fibers of $\phi$ are $\phi^{-1}(a,b)=\left\\{f(r,t)=a,f(s,t)=b\right\\}$.
If we fix $t$, then by our assumptions on $f$ we get a finite, non-zero number
of solutions in $r$ and $s$, so the fibers are one-dimensional. To see that
they are generically irreducible, denote by $z_{a}$ the curve $f(r,t)=a$ and
let $\pi_{a}\colon z_{a}\rightarrow\mathbb{A}^{1}$ be the projection map onto
the second coordinate, that is onto $t$.
Notice that $q^{-1}(a,b)=z_{a}\times_{\mathbb{A}^{1}}z_{b}$ with respect to
the maps $\pi_{a},\pi_{b}$. Since $f$ is not a composite polynomial, $z_{a}$
is generically irreducible by the Bertini–Krull theorem (Lemma 15). We are now
in a position to apply Corollary 22 By our assumption on $f$, $\pi_{a}$ is a
finite map for all $a$. By the Jacobian criterion, the ramification locus of
$\pi_{a}$ on the base is the set of $t$ for which there exists an $r$ with
$f_{1}(r,t)=0$, and $f(r,t)=a$.
By our assumptions on $f$, for any fixed $r$ there are only finitely many $t$
such that $f_{1}(r,t)=0$ and so finitely many $a$ such that $r$ is in the bad
locus of $\pi_{a}$. This implies that for generic $a,b,$ the maps $\pi_{a}$
and $\pi_{b}$ have disjoint bad loci and so we can apply Corollary 22. This
proves that $W$ is 4-dimensional and mainly irreducible. ∎
Recall that $V_{t_{1},t_{2},t_{3},t_{4}}$ is the variety
$\left\\{\begin{aligned}
f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})&,f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\\\
f(x_{3},y_{2})=f(x^{\prime}_{3},y^{\prime}_{2})&,f(x_{2},y_{3})=f(x^{\prime}_{2},y^{\prime}_{3})\\\
f(x_{1},y_{1})=t_{1}&,f(x_{3},y_{3})=t_{2}\\\
f(x^{\prime}_{1},y^{\prime}_{1})=t_{3}&,f(x^{\prime}_{3},y^{\prime}_{3})=t_{4}\end{aligned}\right\\}.$
###### Lemma 24.
The $4$-dimensional family of varieties $V_{t_{1},t_{2},t_{3},t_{4}}$ is
generically $4$-dimensional and mainly irreducible.
###### Proof.
Define $W$ to be the variety
$\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2}),f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\right\\}$
and $W_{t_{1},t_{3}}$ to be the variety
$\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2}),f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1}),f(x_{1},y_{1})=t_{1},f(x^{\prime}_{1},y^{\prime}_{1})=t_{3}\right\\}.$
We shall use the fact that $W_{t_{1},t_{3}}$ has a canonical map
$\pi_{t_{1},t_{3}}\colon W_{t_{1},t_{3}}\rightarrow\mathbb{A}^{4}$ given by
projecting onto the coordinates $x_{2},x^{\prime}_{2},y_{2},y^{\prime}_{2}$,
and $V_{t_{1},t_{2},t_{3},t_{4}}\cong
W_{t_{1},t_{3}}\times_{\mathbb{A}^{4}}W_{t_{2},t_{4}}$ with respect to these
maps.
###### Lemma 25.
The varieties $W_{t_{1},t_{3}}$ are $4$-dimensional and generically mainly
irreducible.
###### Proof.
First, since $f$ is non-composite, the variety
$\left\\{f(x_{1},y_{1})=t_{1},f(x^{\prime}_{1},y^{\prime}_{1})=t_{3}\right\\}$
is generically irreducible in $x_{1},y_{1},x^{\prime}_{1},y^{\prime}_{1}$ by
the Bertini–Krull theorem. Now if we can show that for generic
$x_{1},y_{2},x^{\prime}_{1},y^{\prime}_{2}$, the varieties
$\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})\right\\}$ and
$\left\\{f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\right\\}$ are
$1$-dimensional and generically irreducible, then we will be done by an
application of Corollary 22 with respect to the projection map onto
$x_{1},y_{1},x^{\prime}_{1},y^{\prime}_{1}$.
We will handle the variety
$A_{x_{1},x^{\prime}_{1}}:=\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})\right\\}$,
the other one being symmetric. For each $x_{1},x^{\prime}_{1},y_{2}$ we have a
non-empty finite set of solutions for $y^{\prime}_{2}$, so
$A_{x_{1},x^{\prime}_{1}}$ is clearly 1-dimensional.
Define a $3$-dimensional variety $A$ by
$A=\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})\\}$. Since $f$ is not
composite, $A$ is mainly irreducible. Consider the map $\phi\colon
A\to\mathbb{A}^{2}$ given by projection onto the $x_{1},x^{\prime}_{1}$
coordinates. The fibers are precisely the $A_{x_{1},x^{\prime}_{1}}$. So by
Lemma 36, $A_{x_{1},x^{\prime}_{1}}$ is generically irreducible if and only if
the variety $A_{\times{\mathbb{A}^{2}}}A$, defined by
$f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2}),\
f(x_{1},b_{2})=f(x^{\prime}_{1},b^{\prime}_{2})$
is mainly irreducible. But this is our $W^{\prime}$ from Lemma 23, which we
have already shown to be mainly irreducible. This completes the proof. ∎
###### Lemma 26.
$\pi_{t_{1},t_{3}}$ is generically dominant.
###### Proof.
This is equivalent to proving that the Jacobian of $\pi_{t_{1},t_{3}}$ does
not vanish identically on $W_{t_{1},t_{3}}.$ The Jacobian is readily computed
to be
$J=f_{1}(x_{1},y_{1})f_{2}(x_{2},y_{1})f_{1}(x^{\prime}_{1},y^{\prime}_{2})f_{2}(x^{\prime}_{1},y^{\prime}_{1})-f_{1}(x_{1},y_{2})f_{2}(x_{1},y_{1})f_{1}(x^{\prime}_{1},y^{\prime}_{1})f_{2}(x^{\prime}_{2},y^{\prime}_{1}).$
We have to show that $J$ does not vanish on $W$. Define
$g(s,t)=\frac{f_{1}(s,t)}{f_{2}(s,t)}.$ Notice that none of the $f_{1}$ or
$f_{2}$ terms are identically $0$ on $W$. Assume $J$ vanishes on $W$ for the
sake of contradiction. Consider $J$ as a polynomial in $y_{2}$,
$y^{\prime}_{2}$. Then the assumption that $J=0$ on $W$ implies that on
$A_{x_{1},x^{\prime}_{1}}:=\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})\right\\}$
the function $\frac{f_{1}(x_{1},y_{2})}{f_{1}(x^{\prime}_{1},y^{\prime}_{2})}$
is a constant $C(x_{1},x^{\prime}_{1})$.
We showed above $A_{x_{1},x^{\prime}_{1}}$ is irreducible, which means that
the function field of $A_{x_{1},x^{\prime}_{1}}$ is generated by
$y_{2},y^{\prime}_{2}$ with $y^{\prime}_{2}$ being of degree $\deg_{2}(f)$
over $\mathbb{K}(y_{2})$. Since
$f_{1}(x_{1},y_{2})-C(x_{1},x^{\prime}_{1})f_{1}(x^{\prime}_{1},y^{\prime}_{2})=0$
and the degree of
$f_{1}(x_{1},y_{2})-C(x_{1},x^{\prime}_{1})f_{1}(x^{\prime}_{1},y^{\prime}_{2})$
in $y_{2}$ is less than $\deg_{2}(f)$, it must be $0$. That implies that
$f(x_{1},y_{2})=P(y_{2})+Q(x_{1})$, which contradicts our original assumptions
on $f$. So $J$ does not vanish identically on $W$. ∎
We are now almost ready to apply Lemma 22, if we can show that
$\pi_{t_{1},t_{3}}\colon W_{t_{1},t_{3}}\to\mathbb{A}^{4}$ has no ‘bad fixed
locus’. That is, there is no divisor $D\in\mathbb{A}^{4}$ such that for all
$t_{1},t_{2}\in\mathbb{K}$, the map $\pi_{t_{1},t_{3}}$ is either ramified
over $D$, or is not finite over any point $d\in D$.
* Case 1:
Suppose there is some divisor $D\in\mathbb{A}^{4}$ such that for all points
$(y_{2},x_{2},y^{\prime}_{2},x^{\prime}_{2})\in D$, $\pi_{t_{1},t_{3}}$ is not
finite over $(y_{2},x_{2},y^{\prime}_{2},x^{\prime}_{2})$. Consider the
projective closure $\bar{W}$ inside the space
$\mathbb{P}^{4}_{x_{1},y_{1},x^{\prime}_{1},y^{\prime}_{1}}\times\mathbb{A}^{4}_{y_{2},x_{2},y^{\prime}_{2},x^{\prime}_{2}}$.
We can extend $\pi$ to a map
$\bar{\pi}\colon\overline{W}\to\mathbb{A}^{4}$
which is now proper. Since $\pi_{t_{1},t_{3}}$ is not finite over
$\vec{x}\in\mathbb{A}^{4}_{y_{2},x_{2},y^{\prime}_{2},x^{\prime}_{2}}$, the
preimage $\bar{\pi}_{t_{1},t_{3}}^{-1}(\vec{x})$ has a point ‘at infinity’.
Since this is true for all $t_{1},t_{2}$, it follows that
$\bar{\pi}^{-1}(\vec{x})$ has a $2$-dimensional component at infinity.
The variety $\bar{\pi}^{-1}(y_{2},x_{2},y^{\prime}_{2},x^{\prime}_{2})$ is cut
out by the projectivized equations
$\left\\{f(x_{1},y_{2},e)=f(x^{\prime}_{1},y^{\prime}_{2},e),f(x_{2},y_{1},e)=f(x^{\prime}_{2},y^{\prime}_{1},e)\right\\},$
where $e$ is the homogenizing variable. Now, the component at infinity is
given by $e=0$. By our assumption on $f$, the defining equations of
$\bar{\pi}^{-1}(y_{2},x_{2},y^{\prime}_{2},x^{\prime}_{2})\cap\\{e=0\\}$
become
$x_{1}^{\deg_{1}(f)}=x^{\prime\deg_{1}(f)}_{1},y_{2}^{\deg_{2}(f)}=y^{\prime\deg_{2}(f)}_{2}.$
This is a one-dimensional projective variety, which is a contradiction.
* Case 2:
Suppose that there is a divisor $D\in\mathbb{A}^{4}$ such that for all
$t_{1},t_{3}$, the projection $\pi_{t_{1},t_{3}}$ is ramified over $D$. That
is equivalent to saying that $\pi^{-1}_{t_{1},t_{3}}(D)$ has a multiple
component in $W_{t_{1},t_{3}}$. Since this is true for all $t_{1},t_{3}$, it
implies that $\pi^{-1}(D)$ has a multiple component on $W.$ Now, $W$ is a
direct product of
$Y=\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})\right\\}$ and
$Z=\left\\{f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\right\\}$. There
are projections maps $\pi_{Y}\colon Y\to\mathbb{A}^{2}_{y_{2},y^{\prime}_{2}}$
and $\pi_{Z}\colon Z\to\mathbb{A}^{2}_{x_{2},x^{\prime}_{2}}$ such that
$\pi=\pi_{Y}\times\pi_{Z}$. This means that over each point
$\vec{x}=(x_{2},x^{\prime}_{2},y_{2},y^{\prime}_{2})$ in $D$, either
$Y_{y_{2},y^{\prime}_{2}}:=\left\\{f(x_{1},y_{2})=f(x^{\prime}_{1},y^{\prime}_{2})\right\\},$
or
$Z_{x_{2},x^{\prime}_{2}}:=\left\\{f(x_{2},y_{1})=f(x^{\prime}_{2},y^{\prime}_{1})\right\\}$
has a multiple component. We treat the case of $Y_{y_{2},y^{\prime}_{2}}$
having a multiple component, the case of $Z_{x_{2},x^{\prime}_{2}}$ being
symmetric.
If $Y_{y_{2},y^{\prime}_{2}}$ has $C$ as a multiple component, then
$f_{1}(x_{1},y_{2})=f_{1}(x^{\prime}_{1},y^{\prime}_{2})=0$ on $C$. But by our
assumptions on $f$ this is only a finite number of points, which is a
contradiction.
Applying Corollary 22, we see that for generic
$(t_{1},t_{2},t_{3},t_{4}),W_{t_{1},t_{3}}\times_{\mathbb{A}^{4}}W_{t_{2},t_{4}}\cong
V_{t_{1},t_{2},t_{2},t_{4}}$ is $4$-dimensional and mainly irreducible
_except_ for possibly over a proper closed subset $Y\subset\mathbb{A}^{4}.$
The following lemma rules out the existence of $Y$ and so completes the proof
of Lemma 24.
###### Lemma 27.
For generic $t_{1},t_{2},t_{3},t_{4}$ there is no proper Zariski-closed subset
$Y\subset\mathbb{A}^{4}$ with
$W_{t_{1},t_{3}}\times_{\mathbb{A}^{4}}W_{t_{2},t_{4}}$ being more than
$3$-dimensional over $Y.$
###### Proof.
Note that the lemma is equivalent to the statement that for any
$Y\subset\mathbb{A}^{4}$ of dimension at most $3$, we have
$\dim\pi_{t_{1},t_{3}}^{-1}(Y)+\dim\pi_{t_{2},t_{4}}^{-1}(Y)-\dim(Y)\leq 4.$
To prove this, first observe that for any point $\vec{y}\in\mathbb{A}^{4}$ the
dimension of $\dim\pi_{t_{1},t_{3}}^{-1}(\vec{y})$ is at most $1$. Moreover,
since $W_{t_{1},t_{3}}$ is $4$-dimensional and mainly irreducible and
$\pi_{t_{1},t_{3}}$ is dominant, the locus in $\mathbb{A}^{4}$ where the
dimension of the fibers jump is at most $2$-dimensional. Therefore, all we
have to exclude is the existence of a $2$-dimensional closed subvariety
$Y\subset\mathbb{A}^{4}$ such that for almost all points $\vec{y}\in Y$ both
$\dim\pi_{t_{1},t_{3}}^{-1}(\vec{y})$ and
$\dim\pi_{t_{2},t_{4}}^{-1}(\vec{y})$ are $1$-dimensional. Since we want the
result for generic $t_{1},t_{2},t_{3},t_{4}$, it suffices to exclude the case
where a single bad variety $Y$ exists for all $t_{1},t_{3}$. If such a $Y$
existed, then for almost all $\vec{p}\in Y,\dim\pi^{-1}(\vec{p})\geq 3$.
However, since by assumption the polynomial $f(x,y)$ is monic in $x$,
$\pi^{-1}(\vec{p})$ is a product of two non-degenerate curves, and so is
$2$-dimensional. ∎
∎
## 8 Algebraic tidbits
Often it is insufficient to know that some algebraic property holds
generically, but one needs a bound on the degree of the exceptional set. The
next lemma and its corollaries take care of this situation.
###### Lemma 28.
Suppose $V\subset\mathbb{A}^{n}$ is a variety of degree $d$, and
$\pi\colon\mathbb{A}^{n}\to\mathbb{A}^{m}$ is the projection map. Let
$U=\\{\vec{x}\in\pi(V):\pi^{-1}(\vec{x})\cap V=\mathbb{A}^{n-m}\\}$. Suppose
$\dim U=\dim V+m-n-r$. Then $U$ is contained in a variety of dimension $\dim
U$ and degree at most $d^{r+1}$.
###### Proof.
Think of $\mathbb{A}^{n}$ as $\mathbb{A}^{m}\times\mathbb{A}^{n-m}$, where
$\pi(\vec{x},\vec{y})=\vec{x}$. Then for every
$\vec{y}_{0}\in\mathbb{A}^{n-m}$ write
$V_{\vec{y}_{0}}=V\cap\\{\vec{y}=\vec{y}_{0}\\}$. Note that for a generic
$\vec{y}_{0}$ the variety $V_{\vec{y}_{0}}$ is proper and of degree $d$. We
define varieties $W_{1},W_{2},\dotsc$ inductively. Let $W_{1}=V_{\vec{y}_{1}}$
for some generic $\vec{y}_{1}\in\mathbb{A}^{n-m}$. Suppose $W_{i}$ has been
defined, then either for a generic $\vec{y}_{i+1}\in\mathbb{A}^{n-m}$ the
inequality $\dim(W_{i}\cap V_{\vec{y}_{i+1}})<\dim W_{i}$ holds or there is an
irreducible component $W^{\prime}$ of $W_{i}$ of dimension $\dim
W^{\prime}=\dim W_{i}$ contained in $V_{\vec{y}_{i+1}}$ for every choice of
$\vec{y}_{i+1}\in\mathbb{A}^{n-m}$. In the former case let $W_{i+1}=W_{i}\cap
V_{\vec{y}_{i+1}}$ for a generic $\vec{y}_{i+1}$, and continue the sequence.
In the latter case, the sequence stops with $W_{i}$. In that case since
$U=\bigcap_{\vec{y}\in\mathbb{A}^{n-m}}V_{\vec{y}}$, we have
$W^{\prime}\subset U\subset W_{i}$. Thus, $\dim U=\dim W_{i}$, and by Bezout’s
theorem $\deg W_{i}\leq\prod\deg V_{\vec{y}_{i}}=d^{i}$. Since
$\vec{y}_{1}\in\mathbb{A}^{n-m}$ is generic, $\dim W_{1}+(n-m)\leq\dim V$.
Finally, from $\dim W_{i+1}<\dim W_{i}$, $\dim W_{1}\leq\dim V+m-n$ and
$U\subset W_{i}$, it follows that the sequence of $W$’s terminates after at
most $r+1$ elements. ∎
###### Corollary 29.
If an irreducible polynomial $f(x,y,z)$ of degree $d$ is not of the form
$g(x,y)$, then there are at most $d^{2}$ pairs $(a,b)$ for which $f(a,b,z)$ is
zero as a polynomial in $z$.
###### Proof.
Let $V=\\{f(x,y,z)=0\\}$ and $\pi$ be the projection on $(x,y)$. Then in
notation of the preceding lemma, $U$ is the set of pairs
$(a,b)\in\mathbb{A}^{2}$ for which $f(a,b,z)=0$. Write
$f(x,y,z)=\sum_{i}f_{i}(x,y)z^{i}$. The set $U$ is infinite if and only if all
the $f_{i}$ share a common factor, which is contrary to the assumption on $f$.
Thus $\dim U=0$, and the result follows from Lemma 28. ∎
###### Corollary 30.
Let $f$ be a polynomial of degree $d$ in $n$ variables, and suppose the
polynomial $f_{c}(x_{1},\dotsc,x_{n-1})=f(x_{1},\dotsc,x_{n-1},c)$ has no
linear factors for a generic $c$. Then there are at most $d^{n}(n-1)$ values
$c$ for which $f_{c}$ does have a linear factor.
###### Proof.
Without loss of generality $f$ depends non-trivially on each of
$x_{1},\dotsc,x_{n}$. Let
$C=\\{c:f_{c}\text{ has a linear factor }\\}.$
If $a_{1}x_{1}+\dotsb+a_{n-1}x_{n-1}+b$ is a factor of
$f_{c}(x_{1},\dotsc,x_{n-1})$, then at least one of $a_{i}$ is non-zero. Thus,
without loss of generality there is $C^{\prime}\subset C$ of size $\lvert
C^{\prime}\rvert\geq\lvert C\rvert/(n-1)$ for which $f_{c}$ has a linear
factor with non-vanishing coefficient $a_{n-1}$. By rescaling, we may assume
that for every $c\in C^{\prime}$ the linear factor is of the form
$a_{1}x_{1}+\dotsb+a_{n-2}x_{n-2}+b-x_{n-1}$. Define polynomial $g$ in $2n-2$
variables by
$g(x_{1},\dotsc,x_{n-2},a_{1},\dotsc,a_{n-2},b,c)=f(x_{1},\dotsc,x_{n-2},a_{1}x_{1}+\dotsb+a_{n-2}x_{n-2}+b,c).$
Since $f$ depends non-trivially on $x_{n-1}$, the polynomial $g$ depends non-
trivially on $b$, thus the variety $V=\\{g=0\\}\subset\mathbb{A}^{2n-2}$ is of
dimension $2n-3$. Let
$U=\\{(a_{1},\dotsc,a_{n-2},b,c):g(x_{1},\dotsc,x_{n-2},a_{1},\dotsc,a_{n-2},b,c)=0\\}$.
Since $(a_{1},\dotsc,a_{n-2},b,c)\in U$ if and only if
$a_{1}x_{1}+\dotsb+a_{n-2}x_{n-2}+b-x_{n-1}$ is a factor of $f_{c}$, and
$f_{c}$ has at least $1$ linear factor, it follows that $\lvert
C^{\prime}\rvert\leq\lvert U\rvert$. As $C$ is finite, $\dim U=0$, and Lemma
28 implies $\lvert C\rvert\leq(n-1)\lvert U\rvert\leq(n-1)d^{n}$. ∎
###### Corollary 31.
Suppose $V\subset\mathbb{A}^{n}$ is an irreducible variety of degree $d$, the
map $\pi\colon\mathbb{A}^{n}\to\mathbb{A}^{m}$ is the projection, and $\dim
V=\dim\pi(V)$. Let $U=\\{x\in V:\dim(\pi^{-1}(\pi(x))\cap V)>0\\}$. Then $U$
is contained in a subvariety of $V$ of codimension $1$ and of degree at most
$d^{3}(n-m)$.
###### Proof.
Factor $\pi$ as $\pi=\sigma_{n-m}\dotsb\sigma_{1}$, where each $\sigma_{i}$ is
a projection collapsing a single coordinate. Let
$\pi_{i}=\sigma_{i}\dotsb\sigma_{1}$ and
$U_{i}=\\{x\in V:\sigma_{i}^{-1}(\pi_{i}(x))\cap V=\mathbb{A}^{1}\\}.$
Since $\dim\pi(V)\leq\dim\pi_{i}(V)\leq\dim V$, it follows $\pi_{i}(V)=\dim
V$. If $\dim U_{i}=\dim V$, then $U_{i}=V$ by irreduciblity of $V$, which
would contradict $\dim\pi_{i}(V)=\dim V$. Thus by Lemma 28 the degree of
$\pi_{i}(U_{i})$ is at most $d^{2}$. Since $U$ is contained both in $V$ and in
the union of $\pi_{i}^{-1}(\pi_{i}(U_{i}))$’s, the corollary follows from
Bezout’s theorem (Lemma 13). ∎
###### Lemma 32.
Suppose $n\geq 3$ and let $f(x_{1},x_{2},..,x_{n})$ be an irreducible
polynomial of degree $d$ with no linear factors over an algebraically closed
field $\mathbb{K}$. Then there is a coordinate $x_{i}$ such that if we fix the
value of $x_{i}$ to an element $c\in\mathbb{K}$, then for all but $d^{n}(n-1)$
values of $c$, the resulting polynomial
$f(x_{1},x_{2},\dots,x_{i-1},c,x_{i+1}\dots,x_{n})$ also has no linear
factors.
###### Proof.
We will in fact show a stronger result that one can take $x_{i}$ to be one of
$x_{1},x_{2},x_{3}$. Since the roles of the variables are not symmetric, it is
convenient to rename them $x,y,z,w_{1},\dotsc,w_{n-3}$.
Assume the conclusion of the lemma is false. Then, by Corollary 30, for all
elements $c\in\mathbb{K}$, $f(c,y,z,w_{1},\dotsc,x_{n-3})$ has a linear factor
as a polynomial in $y,z,w_{1},\dotsc,w_{n-3}$. Likewise for the $y$ and $z$
coordinates.
Moreover, these linear factors must have coefficients that are algebraic over
$\mathbb{K}(x)$, say
$\alpha(x)y+\beta(x)z+\gamma_{1}(x)w_{1}+\dotsb+\gamma_{n-3}(x)w_{n-3}+\delta(x)$.
The function $\alpha(x)$ vanishes only if $\partial_{y}f$ vanishes as well.
Thus if $\alpha(x)$ vanishes infinitely often, then $\partial_{y}f$ vanishes
on a subvariety of $\\{f=0\\}$ of dimension $n$, which by irreducibility of
$f$ implies that $f$ does not depend on $y$. If $f$ does not depend on $y$,
then the lemma is trivially true. Thus we may assume that for a generic $x$
the linear factor is of the form
$-y+\beta(x)z+\gamma_{1}(x)w_{1}+\dotsb+\gamma_{n-3}(x)w_{n-3}+\delta(x)$.
Since $f$ is irreducible,
$f(x,y,z,w_{1},\dotsc,w_{n-3})=\tau(x)\prod_{j}\bigl{(}-y+\beta_{j}(x)z+\gamma_{1,j}(x)w_{1}+\dotsb+\gamma_{n-3,j}(x)w_{n-3}+\delta_{j}(x)\bigr{)}$
where the product is over the conjugates of
$(\beta(x),\gamma_{1}(x),\dotsc,\gamma_{n-3}(x),\delta(x))$.
By the same reasoning applied to $z$ instead of $x$,
$f(x,y,z,w_{1},\dotsc,w_{n-3})=\tau^{\prime}(z)\prod_{j}\bigl{(}-y+\beta_{j}^{\prime}(z)x+\gamma_{1,j}^{\prime}(z)w_{1}+\dotsb+\gamma_{n-3,j}^{\prime}(z)w_{n-3}+\delta_{j}^{\prime}(z)\bigr{)}.$
Thus comparing these two descriptions of the roots of $f$ considered as a
polynomial in $y$ over $\overline{\mathbb{K}(x,z)}(w_{1},\dotsc,w_{n-3})$, we
conclude that $\gamma_{i,j}=\gamma_{i,j}^{\prime}$ are constant, and
$\beta(x)z+\delta(x)=\beta^{\prime}(z)x+\delta^{\prime}(z).$
Thus $\beta,\beta^{\prime},\delta,\delta^{\prime}$ are linear, and $f$ is of
the form $f(x,y,z,w_{1},\dotsc,w_{n-3})=Axz+Bx+Cy+Dz+\sum_{i}E_{i}w_{i}$.
Since for fixed $y=c$ the polynomial $f(x,c,z,\dotsc)$ has a linear factor, it
follows that $f$ is itself linear, a contradiction. ∎
###### Lemma 33.
Let $W$ be a non-linear irreducible variety of dimension $m$ in
$\mathbb{A}^{n}$, with coordinates being $x_{1},x_{2},..,x_{n}$. Then there
are $m+1$ coordinates $x_{i_{1}},x_{i_{2}},...,x_{i_{m+1}}$ such that the
projection of $W$ onto their span is contained in an $m$-dimensional non-
linear hypersurface.
###### Proof.
We induct on $n$. If $n=1$ or $n=2$, there is nothing to prove. Suppose $n\geq
3$, and consider the $n$ coordinate hyperplanes,
$\mathbb{A}^{n-1}_{i}=\\{x_{i}=0\\}$, and let $W_{i}$ be the projection of $W$
onto $\mathbb{A}^{n-1}_{i}$. Suppose there an $i$ such that $\dim W_{i}=m-1$.
Then $W=W_{i}\times\mathbb{A}^{1}_{i}$ because $W$ is irreducible. Thus
$W_{i}$ is non-linear, and by induction there is a projection of $W_{i}$ onto
the span of $\\{x_{j}\\}_{j\in S}$. Then projection of $W$ onto the span of
$\\{x_{j}\\}_{j\in S\cup\\{i\\}}$ is contained in a non-linear hypersurface.
So, we may assume all the $W_{i}$ are of dimension $m$.
Introduce a vector space structure on $\mathbb{A}^{n}$ in such a way that
$0\in W$. Let $L=\operatorname{span}W$ be the vector space spanned by $W$, and
write $L_{i}$ for the projection of $L$ onto $\mathbb{A}^{n-1}_{i}$. Since $W$
is non-linear, $\dim L\geq\dim W+1=m+1$. If $\dim L_{i}=m$, then $i$’th basis
vector $e_{i}$ is in $L$. If $\dim L_{i}=m$ for all $i$, then
$\mathbb{A}^{n}=\operatorname{span}\\{e_{1},\dotsc,e_{n}\\}\subset L$,
implying $n=m+1$, in which case there is nothing to prove. Thus, we can assume
there is an $i$ such that $\dim L_{i}=m+1$. But then $W_{i}$ is non-linear,
and the results follows from the induction hypothesis. ∎
###### Lemma 34.
Let $f(x)$ be a non-constant rational function in $\mathbb{F}_{q}(x)$ of
degree at most $q-1$. Suppose also there are non-zero rational functions
$P(s,t)$ and $g(s,t)$ and constants $a,b\in\mathbb{F}_{q}$ such that
$P(g(x,y),af(x)+bf(y))=0$. Then there exists a rational function $G(x)$, such
that $g(x,y)$ is one of
$G(x),G(y),\text{ or }G(af(x)+bf(y)).$
###### Proof.
Let $\mathbb{K}$ be the subfield of $\mathbb{F}_{q}(x,y)$ consisting of all
elements algebraic over $\mathbb{F}_{q}(af(x)+bf(y))$. Note that
$g(x,y)\in\mathbb{K}$ by assumption. Since $f$ is non-constant, $\mathbb{K}$
has transcendence degree $1$ over $\mathbb{F}_{q}$. We claim that $\mathbb{K}$
is isomorphic as a field to $\mathbb{F}_{q}(t)$. To see this, first note that
$\mathbb{K}$ is finitely generated, since its a subfield of a finitely
generated field. So $\mathbb{K}$ is the function field of a smooth, non-
singular curve $C$ over $\mathbb{F}_{q}$. Also, the embedding
$\mathbb{K}\subset\mathbb{F}_{q}(x,y)$ corresponds to a dominant rational map
from $\mathbb{A}_{\mathbb{F}_{q}}^{2}$ to $C$. But if $C$ was not birational
to $\mathbb{A}^{1}$, then this map would have to be constant on every line,
since a curve cannot map non-trivially to a curve of higher genus. But this
contradicts that the map is dominant. So $\mathbb{K}$ is indeed generated by a
single element.
If $a$ or $b$ are 0, then $\mathbb{K}$ is generated by one of $y$ or $x$, and
we are done. Suppose then that neither $a$ nor $b$ is $0$. Then it remains to
prove that $\mathbb{K}$ is generated by $af(x)+bf(y)$ over $\mathbb{F}_{q}$,
or equivalently that $af(x)+bf(y)$ is a non-composite rational function. That
is, there are no rational functions $Q(t)\in\mathbb{F}_{q}(t)$, and
$r(x,y)\in\mathbb{F}_{q}(x,y)$ such that $Q(r(x,y))=af(x)+bf(y)$ and
$\deg(Q)>1$. Suppose for the sake of contradiction this is the case.
Since $Q(t)$ is a rational function of degree at least 2, $Q$ must be ramified
over at least one finite point, say over $c\in\overline{\mathbb{F}_{q}}$. This
means that $Q(t)-c$ has a double root at some point
$c^{\prime}\in\mathbb{F}_{q}$, so that $Q(r(x,y))-c=0$ has a multiple
component of the form $(r(x,y)-c^{\prime})^{2}$, and $af(x)+bf(y)-c=0$ must
also have a multiple component. But by the Jacobian criterion,
$af(x)+bf(y)-c=0$ is only singular at points $(x_{0},y_{0})$ such that
$f^{\prime}(x_{0})=f^{\prime}(y_{0})=0$. There are only finitely many of these
points, so $af(x)+bf(y)-c=0$ cannot have multiple components. This
contradiction finishes the proof. ∎
###### Lemma 35.
Let $p\colon V\to W$ be a dominant, equidimensional map such that $W$ is
irreducible of dimension $m$, and for a generic point $\vec{w}\in W$,
$p^{-1}(\vec{w})$ is irreducible of dimension $n$. Then $V$ has a unique
irreducible component of dimension $m+n$.
###### Proof.
That the dimension of $V$ is $m+n$ follows from dimension theory, so assume
$V$ has two disjoint components of dimension $m+n$, $V_{1}$ and $V_{2}$ with
$V_{1}\cup V_{2}=V$. For an open set $U\subset W$ we know that whenever
$\vec{u}\in U$, $p^{-1}(\vec{u})$ is irreducible. So $p^{-1}(\vec{u})$ lies in
either $V_{1}$ or in $V_{2}$. This means that $p(V_{1})\cap p(V_{2})$ is of
dimension less than $m$. However, since the fibers of $p$ are of dimension
$n$, dimension theory says that each of $p(V_{1}),p(V_{2})$ are of dimension
at least $m$, and hence exactly $m$. But then $p(V_{1}),p(V_{2})$ are two
distinct components of $W$, contradicting the irreducibility of $W$. ∎
###### Lemma 36.
Let $f\colon X\to Y$ be an equidimensional map with $X,Y$ irreducible,
$\dim(Y)=m$, $\dim(X)=n.$ Then $f^{-1}(\vec{y})$ is generically mainly
irreducible of dimension $n-m$ iff $X\times_{Y}X$ is mainly irreducible of
dimension $2n-m$.
###### Proof.
Let
$\mathbb{L}=\overline{\mathbb{F}_{q}}(X),\mathbb{K}=\overline{\mathbb{F}_{q}}(Y)$.
Since $f^{-1}(\vec{y})$ having at least two maximal reducible components is a
Zariski-closed condition on $Y$, looking over the generic point we see that
the theorem is equivalent to the following statement about fields:
$\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}\textrm{ is a
domain}\Longleftrightarrow\mathbb{L}\otimes_{\mathbb{K}}\overline{\mathbb{K}}\textrm{
is a field}.$
Call the above statements (i) and (ii) respectively, and consider the
following additional statement: (iii) $\mathbb{K}$ is algebraically closed in
$\mathbb{L}$. We will show that both conditions are equivalent to (iii).
(ii)$\Longrightarrow$ (i). To prove $\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}$
is a domain, it is enough to show that
$\mathbb{L}\otimes_{\mathbb{K}}L\otimes_{\mathbb{K}}\displaystyle\overline{\mathbb{K}}$
is a domain, and the latter is
$\mathbb{L}\otimes_{\mathbb{K}}\mathbb{L}\otimes_{\mathbb{K}}\displaystyle\overline{\mathbb{K}}\cong\left(\mathbb{L}\otimes_{\mathbb{K}}\overline{\mathbb{K}}\right)\otimes_{\displaystyle\overline{\mathbb{K}}}\left(\mathbb{L}\otimes_{\mathbb{K}}\overline{\mathbb{K}}\right)$
which is a domain, since the product of geometrically irreducible varieties is
irreducible.
(i)$\Longrightarrow$ (iii). Suppose (iii) fails to hold so that $\mathbb{L}$
contains a finite algebraic extension $\mathbb{M}$ of $\mathbb{K}$ such that
$\mathbb{M}\neq\mathbb{K}$. Then
$\mathbb{L}\otimes_{\mathbb{K}}\overline{\mathbb{K}}$ contains a copy of
$\mathbb{M}\otimes_{\mathbb{K}}\mathbb{M}$ which is not a domain, so that (i)
fails to hold as well.
(iii)$\Longrightarrow$ (ii). Suppose not, so that $\mathbb{K}$ is
algebraically closed in $\mathbb{L}$, but
$\mathbb{L}\otimes_{\mathbb{K}}\overline{\mathbb{K}}$ is not a field. Since
$\overline{\mathbb{K}}$ is a union of finite extensions of $\mathbb{K}$, there
must exist some finite algebraic extension $\mathbb{M}$ of $\mathbb{K}$ such
that $\mathbb{L}\otimes_{\mathbb{K}}\mathbb{M}$ is not a field either. We can
present $\mathbb{M}$ as $\mathbb{M}\cong\mathbb{K}[x]/(P(x))$ for some
irreducible polynomial $P(x),$ so that
$\mathbb{L}\otimes_{\mathbb{K}}\mathbb{M}\cong\mathbb{L}[x]/(P(x)).$
Since $\mathbb{L}[x]/(P(x))$ is not a field, $P(x)$ must factor as
$P(x)=Q(x)R(x)$, where $Q,R$ are polynomials with coefficients in
$\mathbb{L}$. But the coefficients of $Q$ and $R$ can be expressed as
polynomials in the roots of $P$, and are therefore algebraic over
$\mathbb{K}$. This contradicts the fact that $\mathbb{K}$ is algebraically
closed in $\mathbb{L}$. ∎
## 9 Problems and remarks
* •
We expect Theorem 6 to hold without the condition that $f$ is monic. In fact,
the proof presented above holds provided the irreducibility of $V$ and
$V_{t_{1},t_{2},t_{3},t_{4}}$ can be established, and it is there that the
condition that $f$ is monic is invoked. Whereas the monicity condition on $f$
can be relaxed, it cannot be removed completely in view of the following
counterexample due to Vivek Shende:
###### Example.
Let $p(x),$ and $g(x,y)$ be two non-linear polynomials, such that $g(x,y)$
depends non-trivially on $y$, and $p(x)$ is not a square. Then for
$f(x,y)=p(x)g(x,y)^{2},$
the variety $V$ in the proof of Theorem 6 is reducible.
Nonetheless, it seems likely that sum-product phenomenon should persist in the
absence of any ‘group-like structure’. More precisely, we believe in the
following conjecture:
###### Definition.
Let $(G,+)$ be a one-dimensional abelian algebraic group (i.e. $G$ is either
$\mathbb{G}_{m}$, $\mathbb{G}_{a}$, or an elliptic curve), and define $G_{0}$
to be
$G_{0}=\left\\{(g_{1},g_{2},g_{3})\in G^{3}\mid g_{1}+g_{2}+g_{3}=0\right\\}.$
An irreducible surface $V\subset\mathbb{A}^{3}$ is said to be _group-like_
with respect to $G$ if there are irreducible curves $C_{1},C_{2},C_{3}$, and
an irreducible surface $W\subset C_{1}\times C_{2}\times C_{3}$, and rational
maps $f_{i}\colon C_{i}\to\mathbb{A}^{1}$, $g_{i}\colon C_{i}\to G$ such that
the Zariski closure of $(f_{1}\times f_{2}\times f_{3})(W)$ is $V$ and the
Zariski closure of $(g_{1}\times g_{2}\times g_{3})(W)$ is $G_{0}$.
###### Conjecture.
There exists an absolute constant $\delta>0$ such that whenever
$f\in\mathbb{F}_{p}[x,y,z]$ is an irreducible polynomial that depends non-
trivially on all three variables, and $A$ a subset of $\mathbb{F}_{p}$, then
either
$N(f;A,A,A)\ll_{d}\max\left(\lvert A\rvert^{2-\delta},\lvert
A\rvert^{2}(\lvert A\rvert/p)^{\delta}\right)$
or the surface $\\{f=0\\}$ is group-like with respect to some one-dimensional
abelian algebraic group.
Intuitively, the conjecture states that the only examples of surface
$\\{f=0\\}$ containing many points on the Cartesian products are of the form
$f(x,y,z)=F\bigl{(}G_{x}(x)\oplus G_{y}(y)\oplus G_{z}(z)\bigr{)}$
for some group operation $\oplus$ and algebraic functions
$F,G_{x},G_{y},G_{z}$.
* •
The proof of $\deg f=2$ case of Theorem 1 can be modified to show that $\lvert
A+A\rvert+\lvert f(A)+B\rvert\gg\lvert A\rvert\lvert B\rvert^{1/1000}$ for
every quadratic polynomial $f$ whenever $\lvert A\rvert,\lvert
B\rvert\leq\sqrt{p}$. The modification requires a sum-product estimate on
$\lvert A+A\rvert+\lvert A\cdot_{G}B\rvert$ where $G\subset A\times B$ is a
dense bipartite graph. Such an estimate can be established by a simple
modification of the proof in [Gar08a] of the case $G=A\times B$. However, as
the resulting proof is long and lacks novelty, it is omitted from this paper,
but can be found at http://www.borisbukh.org/sumproductpoly_quadratic.pdf. It
remains an interesting problem to show that $\lvert A+A\rvert+\lvert
f(A)+B\rvert\gg\lvert A\rvert\lvert B\rvert^{\varepsilon}$ for some
$\varepsilon=\varepsilon(\deg f)>0$ for polynomials of any degree.
## Acknowledgments
We thank Vivek Shende for many useful discussions concerning the algebraic
part of the paper. The relevance of [Bou05] was brought to our attention by
Pablo Candela-Pokorna, to whom we are grateful. We also thank Emmanuel
Kowalski for a discussion on exponential sums, and Igor Shparlinski for a
careful reading of a preliminary version of this paper.
## References
* [BK03] Jean Bourgain and S. V. Konyagin. Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. C. R. Math. Acad. Sci. Paris, 337(2):75–80, 2003.
* [BKT04] J. Bourgain, N. Katz, and T. Tao. A sum-product estimate in finite fields, and applications. Geom. Funct. Anal., 14(1):27–57, 2004. arXiv:0301343.
* [Bom66] Enrico Bombieri. On exponential sums in finite fields. Amer. J. Math., 88:71–105, 1966.
* [Bou05] J. Bourgain. More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory, 1(1):1–32, 2005.
* [Bou09] J. Bourgain. The sum-product phenomenon and some of its applications. In Analytic number theory, pages 62–74. Cambridge Univ. Press, Cambridge, 2009.
* [Buk08] Boris Bukh. Sums of dilates. Combin. Probab. Comput., 17(5):627–639, 2008. arXiv:0711.1610.
* [ER00] György Elekes and Lajos Rónyai. A combinatorial problem on polynomials and rational functions. J. Combin. Theory Ser. A, 89(1):1–20, 2000.
* [ES83] P. Erdős and E. Szemerédi. On sums and products of integers. In Studies in pure mathematics, pages 213–218. Birkhäuser, Basel, 1983.
* [Fuj88] Masahiko Fujiwara. Distribution of rational points on varieties over finite fields. Mathematika, 35(2):155–171, 1988.
* [Ful98] W. Fulton. Intersection Theory. Springer-Verlag, 1998.
* [Gar08a] M. Z. Garaev. A quantified version of Bourgain’s sum-product estimate in $\mathbb{F}_{p}$ for subsets of incomparable sizes. Electron. J. Combin., 15(1):Research paper 58, 8, 2008.
* [Gar08b] M. Z. Garaev. The sum-product estimate for large subsets of prime fields. Proc. Amer. Math. Soc., 136(8):2735–2739, 2008. arXiv:0706.0702.
* [HH09] Norbert Hegyvári and François Hennecart. Explicit constructions of extractors and expanders. Acta Arith., 140(3):233–249, 2009.
* [HIS07] Derrick Hart, Alex Iosevich, and Jozsef Solymosi. Sum-product estimates in finite fields via Kloosterman sums. Int. Math. Res. Not. IMRN, (5):Art. ID rnm007, 14, 2007. http://www.math.missouri.edu/~iosevich/solymosi.pdf.
* [HLS09] Derrick Hart, Liangpan Li, and Chun-Yen Shen. Fourier analysis and expanding phenomena in finite fields. arXiv:0909.5471v1, Sep 2009.
* [KT99] Nets Hawk Katz and Terence Tao. Bounds on arithmetic projections, and applications to the Kakeya conjecture. Math. Res. Lett., 6(5-6):625–630, 1999. arXiv:9906097.
* [Li09] Liangpan Li. Slightly improved sum-product estimates in fields of prime order. arXiv:0907.2051, Jul 2009.
* [LW54] Serge Lang and Andre Weil. Number of points of varieties in finite fields. American Journal of Mathematics, 76(4):819–827, 1954.
* [Raz07] Alexander Razborov. A product theorem in free groups. http://people.cs.uchicago.edu/~razborov/files/free_group.pdf, 2007\.
* [Rud10] Misha Rudnev. An improved sum-product inequality in fields of prime order. arXiv:1011.2738v1, Nov 2010.
* [Ruz09] Imre Z. Ruzsa. Sumsets and structure. In Combinatorial number theory and additive group theory, Adv. Courses Math. CRM Barcelona, pages 87–210. Birkhäuser Verlag, Basel, 2009.
* [Sch86] Wolfgang M. Schmidt. Small solutions of congruences with prime modulus. In Diophantine analysis (Kensington, 1985), volume 109 of London Math. Soc. Lecture Note Ser., pages 37–66. Cambridge Univ. Press, Cambridge, 1986.
* [Sch00] Andrzej Schinzel. Polynomials with special regards to reducibility, volume 77 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2000.
* [Sol08] József Solymosi. An upper bound on the multiplicative energy. arXiv:0806.1040, Jun 2008.
* [SSV05] B. Sudakov, E. Szemerédi, and V. H. Vu. On a question of Erdős and Moser. Duke Math. J., 129(1):129–155, 2005.
* [Vu08] Van H. Vu. Sum-product estimates via directed expanders. Math. Res. Lett., 15(2):375–388, 2008. arXiv:0705.0715.
* [VWW08] Van H. Vu, Melanie Matchett Wood, and Philip Matchett Wood. Mapping incidences. arXiv:0711.4407, Nov 2008.
|
arxiv-papers
| 2010-02-12T14:13:20 |
2024-09-04T02:49:08.388535
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Boris Bukh, Jacob Tsimerman",
"submitter": "Boris Bukh",
"url": "https://arxiv.org/abs/1002.2554"
}
|
1002.2611
|
# HST Pa$\alpha$ Survey of the Galactic Center – Searching the missing young
stellar populations within the Galactic Center
H. Dong1, Q. D. Wang1, A. Cotera2, S. Stolovy3, M. R. Morris4, J. Mauerhan3,
E. A. Mills4, G. Schneider5, C. Lang6
###### Abstract
We present preliminary results of our HST Pa$\alpha$ survey of the Galactic
Center (GC), which maps the central 0.65$\times$0.25 degrees around Sgr A*.
This survey provides us with a more complete inventory of massive stars within
the GC, compared to previous observations. We find 157 Pa$\alpha$ emitting
sources, which are evolved massive stars. Half of them are located outside of
three young massive star clusters near Sgr A*. The loosely spatial
distribution of these field sources suggests that they are within less massive
star clusters/groups, compared to the three massive ones. Our Pa$\alpha$
mosaic not only resolves previously well-known large-scale filaments into fine
structures, but also reveals many new extended objects, such as bow shocks and
H II regions. In particular, we find two regions with large-scale Pa$\alpha$
diffuse emission and tens of Pa$\alpha$ emitting sources in the negative
Galactic longitude suggesting recent star formation activities, which were not
known previously. Furthermore, in our survey, we detect $\sim$0.6 million
stars, most of which are red giants or AGB stars. Comparisons of the magnitude
distribution in 1.90 $\mu$m and those from the stellar evolutionary tracks
with different star formation histories suggest an episode of star formation
process about 350 Myr ago in the GC.
1 Department of Astronomy, University of Massachusetts, Amherst, MA 01003
2 SETI Institute
3 Spitzer Science Center, California Institute of Technology
4 Department of Physics and Astronomy, University of California, Los Angeles
5 Steward Observatory, University of Arizona
6 Department of Physics and Astronomy, University of Iowa
E-mail: hdong@astro.umass.edu, wqd@astro.umass.edu
## 1\. Motivation
As the closest galactic nucleus, the GC (8kpc, [1]) is a unique lab to study
resolved stellar populations around a supermassive black hole (SMBH). This
topic was triggered by the discovery of three young massive compact star
clusters within 30 pc around Sgr A* in the last two decades ($<$ 6 Myr old,
$\sim 10^{4}~{}M_{\odot}$, [2, 3, 4] and references therein). These clusters
contribute strong UV radiation that illuminates nearby molecular clouds. The
existence of these clusters provides us with a good opportunity to understand
the star formation process within a galactic nucleus, which is very important
to the study of galaxy formation and evolution.
However, the formation mechanism of these clusters ([5,6]) still remains
greatly uncertain, under the hostile GC environment, characterized by high
temperature gas as well as strong magnetic field and tidal force. Existing
studies are focused on the Arches and Center clusters ([7,8]). This small
sample of massive stellar clusters prevents us from making a firm conclusion
about their formation mode. We do not even know whether they were formed in a
single burst or were just part of a continuous star formation in the GC. We
are also not sure about whether or not these extreme massive star clusters
represent the only way in which stars form in the GC. A survey of a fair
sample of massive stars will thus be very helpful.
We also need to determine the long-term star formation history in the GC.
Figer et al 2004 ([9]) first studied the K-band luminosity function toward the
GC obtained by the HST/NICMOS NIC2. They found that a continuous star
formation is consistent with their data. Their conclusion, however, is based
on stars detected in several regions with small fields of view. Existing
large-scale near-IR surveys, such as 2MASS ([11]), are dominated by foreground
stars, plus a limited number of bright red giants and AGB stars in the GC.
Therefore, it is highly desirable to have a large-scale, high-resolution near-
IR survey, which will facilitate a study of differential properties of the
luminosity function across the GC.
We have carried out a HST/NICMOS Pa$\alpha$ survey of the Galactic Center
([10]). This survey used the NIC3, which has a relatively large field of view
(51.2”$\times$51.2”) and an angular resolution ($\sim$0.2”). This combination,
together with the stable PSF, allows us to effectively separate the point
sources contribution from the extended diffuse Pa$\alpha$ emission. Arising
chiefly from photo-ionized warm gas, Pa$\alpha$ line emission (at 1.87$\mu$m)
is brighter than Br$\gamma$ (2.16$\mu$m, accessible from the ground) by a
factor of 3-4, even considering the high extinction toward the GC. In
particular, point-like Pa$\alpha$ emission should be predominantly produced by
evolved massive stars (e.g., O If, LBV and WR) with age younger than $\sim$10
Myr.
## 2\. Observation and Data Reduction
Our survey covered the central 416 square arcminutes of the GC in 144 HST
orbits. Two narrow-band filters, F187N (1.87 $\mu$m, on line) and F190N (1.90
$\mu$m, off line) were used with the same orbit parameters. Each orbit
included four pointing positions. Each was observed with a four-point
dithering pattern. The exposure time is 192 s for each position and filter.
We will present in Dong et al 2010 ([12]) a detailed description of the data
calibration and analysis procedures. Here, we list several key steps. We
remove the relative background offsets among the 576 positions simultaneously
determined from the detected intensity differences in the overlap regions. The
absolute background is determined and removed based on the intensities
observed toward several foreground dark molecular clouds. The relative spatial
offsets among the orbits are corrected for in the same fashion, while the
absolute astrometry is determined from a comparison with the accurate
positions of the SiO masers in Reid et al 2007.
We use the IDL routine ’Starfinder’ developed by Emiliano Diolaiti ([13]) to
detect stars. The 50% completeness limit of the detection is typically
$m_{1.90\mu m}$=17 and is reduced to $m_{1.90\mu m}\sim 15$ magnitude near Sgr
A*, where faint source confusion becomes important. We are also able to
quantify the photometric errors, accounting for the Poisson uncertainty, the
background fluctuation, and the NIC3’s inter-pixel problem. In total, we
detect $\sim$0.6 million sources at 5$\sigma$ confidence in both filters.
We then produce an extinction map calculated adaptively from local flux ratios
of sources in the two filters. We use this extinction map to correct for the
spatial variable extinction of the Pa$\alpha$ emission. We further excise
individual Pa$\alpha$ sources to construct a diffuse Pa$\alpha$ emission map,
which is present in Fig. 1, together with the final F187N and F190N mosaics.
## 3\. Result
### 3.1. Magnitude Distribution
In the left panel of Fig. 2, we present a magnitude distribution of the 0.6
million sources in F190N. This distribution shows two peaks at $\sim$15.5 and
17 magnitudes. While the later peak could be explained by the variable
incompleteness of the source detection, the former one should be real, which
cannot be due to extinction variation across the field, for example. To
understand the nature of the 15.5 magnitude peak, we compare the F190N
magnitude contours with the Padova stellar evolutionary tracks in the right
panel of Fig. 2. The F190N contours are calculated from the ATLAS 9 atmosphere
model and corrected for the distance and extinction ($A_{V}=31.1$). As shown
by the figure, most of the detected sources should be red giants and AGB
stars. Only Main Sequence stars more massive than $\sim 9M_{\odot}$ should be
detected individually in our survey. The peak around 15.5 magnitude is
apparently due to red clump stars with initial mass around 1-3 $M_{\odot}$.
Figure 1.: The mosaics from our survey: F187N (top), F190N (middle) and
diffuse Pa$\alpha$ emission (bottom). The green circles in the Pa$\alpha$
mosaic mark the positions of the detected Pa$\alpha$ emitting sources (see
Section 6).
Figure 2.: Left: The magnitude distribution of f190N sources. The solid
histogram represents the magnitude distributions of all sources, while the
dashed and dash-dotted ones are for the foreground (defined to have the color
H-K$<$1 ) and the remaining (GC) sources, separately. Right: The Padova
stellar evolutionary tracks with the F190 magnitude contours are overlaid.
Fig. 3 presents the magnitude distributions constructed for eight different
regions. The magnitude distributions of the foreground stars in Regions 3, 4
and 8 show shifts to the dim side because of the presence of foreground
molecular clouds. Similarly, the low star number density of Region 5 can be
accounted for by the presence of the 50 km/s molecular cloud (M-0.13-0.08). In
the remaining 4 regions, the peak around 15.5 magnitude appears to become
increasingly more prominent in the regions closer to Sgr A* (cf. Region 1 and
2) and to the Galactic Plane (cf. Region 6 and 7).
Figure 3.: Outlines of the regions in the F190N mosaic that are used to
extract the magnitude distributions shown in the individual panels. The black
line in each panel represents the original magnitude distribution, while the
green and pink lines are for the foreground star and remaining (GC)
contributions. The blue line is for the GC distribution after the detection
incompleteness correction.
### 3.2. Pa$\alpha$ emitting sources
In Fig. 1, we mark the positions of the sources that show significant
Pa$\alpha$ emission. These sources are defined from their large ratios of
fluxes in the F187N and F190N bands, corrected for local extinction ([12]). 79
of the 157 Pa$\alpha$ sources are apparently located in the three clusters,
including most of the evolved massive stars identified spectroscopically
(e.g., O If, LBV and WR stars). The remaining sources appear to be distributed
outside of these three clusters or in the field. Two third of these field
Pa$\alpha$ sources fall in regions between the Arches/Quintuplet and Sgr A*,
while the other one third are located on the negative Galactic longitude side.
A small fraction of the field Pa$\alpha$ sources are clearly associated with
extended HII regions.
### 3.3. Diffuse Pa$\alpha$ emission
Fig. 4 presents three sample regions that show distinct diffuse Pa$\alpha$
emission features at the full resolution. Around Sgr A*, one can clearly see a
bright swirling structure, as was discovered by Scoville et al. 2003 ([14]).
But in addition, Fig. 4a shows low surface brightness filaments which are
nearly perpendicular to the Galactic Plane, which may be an indication for
outflows from the very central region around Sgr A*. The well-known radio-
thermal filaments in the Sickle nebula and Thermal Arc are now resolved into
many thin filaments, which are apparently illuminated by the Arches and
Quintuplet clusters (Fig. 4b,c). The Sickle nebula, in particular, shows
several finger-like structures similar to those seen in M16. But there are
also many fuzzy, low surface brightness structures between the Quintuplet
cluster and the bright rims. On the Galactic west side of the Sickle nebula,
one can also see a new ring-like Pa$\alpha$ nebula. Follow-up spectroscopic
observations (Mauerhan et al 2010, in preparation) show that the central
Pa$\alpha$ source is an LBV star, similar to the Pistol star.
Figure 4.: Close-ups of distinct diffuse P$\alpha$ features: (a) the central
region around Sgr A*, with the overlaid IRAC 8$\mu$m intensity contours at
1,3,10, and 30 $\times 10^{3}{\rm~{}mJy~{}sr^{-2}}$; (b) the Sickle nebula;
(c) Thermal Arched filaments. All are projected in the Galactic coordinates.
Our survey reveals many new Pa$\alpha$ nebulae too. For example, on the
negative Galactic longitude side, we find two new extended HII regions at
(l,b)=(-0.13,0.0) and (-0.28,-0.036). Compared to the Thermal Arc, they are
much dimmer. That is why these regions do not stand out against the strong
background in the Spitzer IRAC 8 $\mu$m PAH image. Another interesting
discovery is the existence of various linear filaments in the two H II
regions. These filaments probably trace local magnetic field. We give the full
resolution close-ups of extended Pa$\alpha$ emission in Fig. 5. The upper row
shows the interaction between the stars and its surrounding ISM. Nebulae in
the middle row are probably due to stellar ejecta. Nebulae in the bottom row
are several structures that are known as thermal radio sources and are now
well resolved. For example, Sgr A-A and Sgr A-C exhibit clear bow shock
structures.
Figure 5.: Close-ups of various compact Pa$\alpha$ nebulae (equatorially
projected). Detected sources have been subtracted, except for identified
bright Pa$\alpha$ emission stars that appear to be central sources of the
nebulae. Approximate Galactic coordinates are used as labels of the nebulae,
except those with well-known names. The bar in each panel marks a 0.2 pc scale
at the GC distance.
## 4\. Discussion
### 4.1. How do the massive stars shape the ISM?
Intense ionizing radiation from the three massive star clusters clearly
illuminates, erodes and destroys nearby molecular clouds. Our Pa$\alpha$
mosaic provides examples for all these three processes. As shown in Section
3.3., the thin filaments are brighter toward the Arches and Quintuplet
clusters, which presents the direct evidence that they are ionizing the
surface of the surrounding giant molecular clouds. The finger-like structures
at the surface of the Sickle nebula hint that the radiation from the
Quintuplet cluster is shaping the ISM, while the fuzzy Pa$\alpha$ features
between the Sickle nebula and Quintuplet clusters probably represent remnants
of molecular clouds that have already been largely dispersed.
### 4.2. What is the origin of the field Pa$\alpha$ sources
We find that many of the Pa$\alpha$-emitting stars are located outside of the
three young massive star clusters. As we mentioned previously, they are
evolved massive stars with ages of a few Myr. Studying their origin can help
us understand the star formation mode within the GC.
Two third of these field Pa$\alpha$ stars are located on the positive Galactic
longitude side of Sgr A* and are close to the three massive star clusters.
Stolte et al 2008 ([7]) have found that the Arches cluster is moving toward
the Galactic east in a direction parallel to the Galactic plane. In Fig. 1,
one can see that there are several Pa$\alpha$ sources that appear in the
opposite direction of the motion. So one possible scenario is that these
Pa$\alpha$ sources are ejected from the clusters. Because of the mass
segregation, massive stars tend to sink into the cluster centers, where three-
body encounters are frequent, leading to ejections of stars and formation of
tightly bound binaries. So some of the Pa$\alpha$ sources could be due to the
dynamic ejection, although detailed simulations are needed to determine the
efficiency of the process and the distribution of such ejected stars. X-ray
observations have further shown the presence of three bright X-ray point
sources in the Arches cluster ([15] and reference therein). These sources have
hard thermal spectra, which are most likely due to colliding winds in
individual binaries [15]. Assuming that these three X-ray sources represent
all tightly bound binaries of massive stars with strong winds, one may
conclude that there should be only a few massive stars at most that may be
kicked out from the cluster. Stolte et al 2008 ([7]) also suggested that at
most 15% of the total mass may have been stripped away from the Arches cluster
by the tidal force in the GC. Due to the mass segregation, the fraction of the
massive stars stripped from the cluster should be even smaller. Therefore, the
cluster may not be a primary source of the field Pa$\alpha$ sources. Most of
the field Pa$\alpha$ sources probably formed in isolation or in small groups,
independent of the Arches or Quintuplet cluster.
### 4.3. Current star formation process
Fig. 1 shows that some of the field Pa$\alpha$ sources are associated with
extended Pa$\alpha$ diffuse nebulae, which represent regions of massive star
formation in small groups. These regions cannot be too young because of the
presence of the evolved massive stars represented by the Pa$\alpha$ sources.
The star groups also cannot be very rich, because of substantially fewer such
sources than in the Arches cluster, which has more than 10 $Pa\alpha$-emitting
sources within 10” radius. Another evidence for the low mass of the groups is
the small sizes and relatively low diffuse Pa$\alpha$ intensities of the
nebulae.
Our survey also for the first time unambiguously reveals the presence of two
large-scale HII complexes on the negative Galactic longitude side, These
complexes, together with the presence of tens of Pa$\alpha$ sources strongly
indicates recent star formation on this side of the Galactic disk. Overall,
however, the star formation is substantially less active than that on the
positive side. One possibility is that the star formation on the negative side
preceded the positive one; i.e., clusters/groups have largely been resolved.
Additional evidence for the star formation on the negative side comes from
radio observations. Law 2010 ([16]) have shown that a giant Galactic Center
lobe of size $\sim$100 pc is not centered at the GC, but shifts to the
negative side. This shift may be due to a strong starburst on this side about
several Myr ago, responsible for the lobe. Clearly, more observations and
detailed modelling are required to further the understanding of the global
recent star formation pattern in the GC.
### 4.4. Star formation history
Fig. 6 compares the magnitude distribution in 1.9$\mu$m with a stellar
population synthesis model. This comparison shows that the overall shape of
the distribution can be reasonably modelled with a constant star formation
rate, plus a major burst about 350 Myr ago. The large deviation of the
distribution from the model at the small magnitude end is probably caused by a
problematic treatment of TP-AGB phase, which has been calibrated with stellar
clusters in the Magellanic Clouds. This evolutionary phase may be sensitive to
the metallicity, which could be substantially different in the GC from the
calibration clusters. The burst in the magnitude distribution model is
required to explain the presence of the m${}_{1.90\mu m}\sim 15.5$ peak in the
data, representing red clump stars with initial mass around 2 $M_{\odot}$. The
magnitude position of this peak varies among different lines-of-sight (Fig.
3), which hints that this component is indeed in the GC, not the spiral arms.
This spatial variation of the peak intensity further suggests that the
distribution of the red clump stars may follow a disk-like structure. While a
detailed investigation of the magnitude distribution is yet to be made, it is
clear that it can be used to shed important insights into the star formation
history in the GC.
Figure 6.: Top: the fitting result. The three lines from bottom to up are the
recent star formation activity which began $\sim$350 Myr ago, the continuous
star formation through the whole history of the Universe and their
combination. Bottom: the fitting residuals
## 5\. Summary
Based on our HST Pa$\alpha$ survey of the GC, we have detected 0.6 million
stars and mapped out many extended Pa$\alpha$ emission features. The main
results from our preliminary analysis of these products are summarized in the
following:
* •
We have identified 157 Pa$\alpha$-emitting sources. They are most likely
evolved massive stars and trace very recent star formation in the GC. About
half of the sources are located outside the three well-known massive star
clusters. These field sources formed probably mostly in small groups. Some of
the sources are still associated with relatively compact H II regions.
* •
The diffuse Pa$\alpha$ map allows us to resolve many fine structures in known
large-scale thermal radio features in the GC. These structures represent
various stages of the interplay between massive stars and their ISM
environment, including stellar mass ejection as well as the ionization and
destruction of nearby molecular clouds. The filamentary morphology of some of
the structures further indicates that magnetic field plays an important role
in shaping the ISM in the GC.
* •
We have clearly detected two HII complexes as well as $\sim$20 Pa$\alpha$
sources on the negative Galactic longitude side, suggesting recent star
formation there, though probably earlier and/or weaker than that on the other
side. The star formation may be linked to be partly responsible for the GC
lobe, as identified by Law et al 2010 ([16], and reference therein).
* •
We have found evidence for a starburst about 350 Myr years ago, as
characterized by an enhanced number of red clump stars evolved from initial
masses $\sim$2 $M_{\odot}$. These stars seem to be primarily located in a
disk-like region around Sgr A*.
These preliminary results demonstrate that near-IR surveys can be used to
significantly advance our understanding of the mode and history of star
formation in the GC, shedding insights into what may occur in nuclear regions
of other galaxies.
## References
* Ghez et al. (2008) [1] Ghez, A. M., Salim, S., Weinberg, N. N., Lu, J. R., Do, T., Dunn, J. K., Matthews, K., Morris, M. R., Yelda, S., Becklin, E. E, et al., 2008, ApJ, 689, 1044G
* Figer et al. (1999a) [2] Figer, Donald F., McLean, Ian S., Morris, Mark 1999, ApJ, 514, 202F
* Figer et al. (1999b) [3] Figer, Donald F., Kim, Sungsoo S., Morris, Mark, Serabyn, Eugene, Rich, R. Michael, McLean, Ian S., 1999, ApJ, 525, 750F
* Genzel et al. (2003) [4] Genzel, R., Sch$\ddot{o}$del, R., Ott, T., Eisenhauer, F., Hofmann, R., Lehnert, M., Eckart, A., Alexander, T., Sternberg, A., Lenzen, R., et al., 2003, ApJ, 594, 812G
* Fujii et al. (2008) [5] Fujii, M., Iwasawa, M., Funato, Y., Makino, J.,2008, ApJ, 686, 1082F
* Nayakshin et al. (2007) [6] Nayakshin, S., Cuadra, J., Springel, V., 2007, MNRAS, 379, 21
* Stolte et al. (2008) [7] Stolte, A., Ghez, A., Morris, M., Lu, J., Matthews, K., 2008, ApJ, 675, 1278S
* Paumard et al. (2006) [8] Paumard, T., Genzel, R., Martins, F., Nayakshin, S., Beloborodov, A. M., Levin, Y., Trippe, S., Eisenhauer, F., Ott, T., Gillessen, S., 2006, ApJ, 643, 1011P
* Figer et al. (2004) [9] Figer, D. F., Rich, R. M., Kim, S. S., Morris, M., Serabyn, E., 2004, ApJ, 601, 319F
* Wang et al. (2009) [10] Wang, Q. D., Dong, H., Cotera, A., Stolovy, S., Morris, M., Lang, C. C., Muno, M. P., Schneider, G., Calzetti, D., 2009, MNRAS, tmp, 1865W
* Skrutskie et al. (2006) [11] Skrutskie, M. F., Cutri, R. M., Stiening, R., Weinberg, M. D., Schneider, S., Carpenter, J. M., Beichman, C., Capps, R., Chester, T., Elias, J. et al., 2006, AJ, 131, 1163S
* Dong et al. (2010) [12] Dong, H., Wang, Q. D., Cotera, A., Stolovy, S., Morris, M., Lang, C. C., Muno, M. P., Schneider, G., Calzetti, in preparation
* Diolaiti et al. (2000) [13] Diolaiti, E., Bendinelli, O., Bonaccini, D., Close, L., Currie, D., Parmeggiani, G., 2000, A&AS, 147, 335D
* Scoville et al. (2003) [14] Scoville, N. Z., Stolovy, S. R., Rieke, M., Christopher, M., Yusef-Zadeh, F. 2003, ApJ, 594, 294
* Wang et al. (2006) [15] Wang, Q. D., Dong, H., Lang, C., 2006, MNRAS, 371, 38
* Law, C. (2010) [16] Law, C. J., 2010, ApJ, 708, 474
|
arxiv-papers
| 2010-02-12T17:58:18 |
2024-09-04T02:49:08.397271
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Dong (1), Q. D. Wang (1), A. Cotera (2), S. Stolovy (3), M. R.\n Morris (4), J. Mauerhan (3), E. A. Mills (4), G. Schneider (5), C. Lang (6)\n ((1)Umass, Amherst, (2) SETI, (3) SSC, Caltech, (4) UCLA, (5) Steward\n Observatory, (6), Univ. of Iowa)",
"submitter": "Hui Dong",
"url": "https://arxiv.org/abs/1002.2611"
}
|
1002.2915
|
# Nonlinear Electromagnetic Waves in Magnetosphere of a Magnetar
Dan Mazur and Jeremy S. Heyl†
Department of Physics and Astronomy, University of British Columbia,
Vancouver, British Columbia, Canada, V6T 1Z1
†Email: heyl@phas.ubc.ca; Canada Research Chair
(Accepted —. Received —; in original form —)
###### Abstract
We compute electromagnetic wave propagation through the magnetosphere of a
magnetar. The magnetosphere is modeled as the QED vacuum and a cold, strongly
magnetized plasma. The background field and electromagnetic waves are treated
nonperturbatively and can be arbitrarily strong. This technique is
particularly useful for examining non-linear effects in propagating waves.
Waves travelling through such a medium typically form shocks; on the other
hand we focus on the possible existence of waves that travel without evolving.
Therefore, in order to examine the nonlinear effects, we make a travelling
wave ansatz and numerically explore the resulting wave equations. We discover
a class of solutions in a homogeneous plasma which are stabilized against
forming shocks by exciting nonorthogonal components which exhibit strong
nonlinear behaviour. These waves may be an important part of the energy
transmission processes near pulsars and magnetars.
††pagerange: Nonlinear Electromagnetic Waves in Magnetosphere of a
Magnetar–References††pubyear: 2010
## 1 Introduction
The magnetosphere of a magnetar provides a particularly interesting medium for
the propagation of electromagnetic waves. Magnetars are characterized by
exceptionally large magnetic fields that can be several times larger than the
quantum critical field strength (Mereghetti, 2008). Because the magnetic
fields are so large, the fluctuations of the vacuum of quantum electrodynamics
(QED) influence the propagation of light. Specifically, the vacuum effects add
nonlinear terms to the wave equations of light in the presence large magnetic
fields. In addition, the magnetosphere of a magnetar contains a plasma which
alters the dispersion relationship for light. Because of the unique optical
conditions in the magnetospheres of magnetars, they provide excellent arenas
to explore nonlinear vacuum effects arising due to quantum electrodynamics.
The influence of QED vacuum effects from strong magnetic fields in the
vicinities of magnetized stars has previously been studied by several authors.
The combined QED vacuum and plasma medium is discussed in detail in the
context of neutron stars in Mészáros (1992). Vacuum effects have been found to
dominate the polarization properties and transport of X-rays in the strong
magnetic fields near neutron stars (Mészáros & Ventura, 1979, 1978; Meszaros &
Bonazzola, 1981; Meszaros et al., 1980; Galtsov & Nikitina, 1983). Detailed
consideration of magnetic vacuum effects is therefore critical to an
understanding of emissions from highly magnetized stars.
Most studies of waves in systems including plasmas or vacuum effects approach
the problem perturbatively, which limits the applicability of their results.
The purpose of the present paper is to examine the combined impact of the QED
vacuum and a magnetized plasma using nonperturbative methods to fully preserve
the nonlinear interaction between the fields. Neutron stars may be capable of
producing very intense electromagnetic waves, comparable to the ambient
magnetic field. For example, a coupling between plasma waves and seismic
activity in the crust could produce an Alfvén wave with a very large amplitude
(Blaes et al., 1989; Thompson & Duncan, 1995). Even if they may not be
produced directly, electromagnetic waves naturally develop through the
interactions between Alfvén waves. To lowest order in the size of the wave,
Alfvén waves do not suffer from shock formation (Thompson & Blaes, 1998)
whereas electromagnetic waves do (Heyl & Hernquist, 1998, 1999a); therefore,
how to stabilise the propagation of the the latter is the focus of this paper.
If such a magnetospheric disturbance results in electromagnetic waves of
sufficiently large amplitude and low frequency, then nonperturbative
techniques are required to characterize the wave. The importance of studying
such a system nonperturbatively is particularly well illustrated by the fact
that some nonlinear wave behaviour is fundamentally nonperturbative, as is
generally the case with solitons (Rajaraman, 1982). In order to handle the
problem nonperturbatively, we choose to study waves whose spacetime dependence
is described by the parameter $S=x-vt$, where $v$ is a constant speed of
propagation through the medium in the $\hat{\bf x}$-direction. In the study of
waves, one normally chooses the ansatz $e^{i(\omega t-{\bf k}\cdot{\bf x})}$.
However, in this picture, a numerical study would typically treat the self-
interactions of the electromagnetic field by summing the interactions of
finitely many Fourier modes. So, this ansatz conflicts with our goal of
studying the nonlinear interactions to all orders. In contrast, a plane wave
ansatz given by $S=x-vt$ allows us to study a simple wave structure to all
orders without any reference to individual Fourier modes.
We model a magnetar atmosphere by including the effects of arbitrarily strong
electromagnetic fields using a QED one-loop effective Lagrangian approach.
These effects are discussed in section 2.2. Plasma effects are included by
assuming free electrons moving under the Lorentz force without any self-
interactions. The model is that of a cold magnetohydrodynamic plasma and is
discussed in section 2.3. We have also assumed that the medium is homogeneous
in agreement with the travelling-wave ansatz. Of course the actual situation
is more complicated with a thermally excited plasma (e.g. Gill & Heyl, 2009)
and inhomogeneities — the latter can result in a whole slew of interesting
interactions between the wave modes (Heyl & Shaviv, 2000, 2002; Heyl et al.,
2003; Lai & Ho, 2003) that are especially crucial to our understanding of the
thermal radiation from their surfaces, but these are beyond the scope of this
paper.
The formation of electromagnetic shocks is expected to be an important
phenomenon for electromagnetic waves in the magnetized vacuum since
electromagnetic waves can evolve discontinuities under the influence of
nonlinear interactions (Lutzky & Toll, 1959; Zheleznyakov & Fabrikant, 1982;
Heyl & Hernquist, 1998). Such shocks can form even in the presence of a plasma
(Heyl & Hernquist, 1999b). In this study, through our explicit focus on
travelling waves, we examine an alternate class of solutions to the wave
equations which do not suffer this fate. Instead, they are stabilized against
the formation of discontinuities by nonlinear features. These waves travel as
periodic wave trains without any change to their form, such as wave steepening
or shock formation. Waves such as these may contribute to the formation of
pulsar microstructures (Chian & Kennel, 1987; Jenet et al., 2001).
## 2 Wave Equations
### 2.1 The Maxwell’s equations
The vacuum of QED in the presence of large magnetic fields can be described as
a non-linear optical medium (Heyl & Hernquist, 1997b). We also choose to treat
the effect of the plasma on the waves through source terms $\rho_{p}$ and
${\bf J}_{p}$; therefore, we begin by considering Maxwell’s equations in the
presence of a medium and plasma sources. In Heaviside-Lorentz units with
$c=1$, Maxwell’s equations can be used to derive the wave equations
$\displaystyle{\nabla}^{2}{\bf D}-\frac{\partial^{2}{\bf D}}{\partial t^{2}}$
$\displaystyle=$ $\displaystyle-{\nabla}\times({\nabla}\times({\bf D}-{\bf
E}))+$ (1) $\displaystyle\frac{\partial}{\partial t}({\nabla}\times({\bf
B}-{\bf H}))+{\bf\nabla}\rho_{p}+\frac{\partial{\bf J_{p}}}{\partial t}$
$\displaystyle{\nabla}^{2}{\bf H}-\frac{\partial^{2}{\bf B}}{\partial t^{2}}$
$\displaystyle=$ $\displaystyle-\nabla(\nabla\cdot({\bf B}-{\bf H}))-$ (2)
$\displaystyle\frac{\partial}{\partial t}({\nabla}\times({\bf D}-{\bf
E}))-{\nabla}\times{\bf J_{p}}.$
For clarity, we will avoid making cancellations or dropping vanishing terms.
We define the vacuum dielectric and inverse magnetic permeability tensors as
follows (Jackson, 1975)
$D_{i}=\varepsilon_{ij}E_{j},~{}~{}H_{i}=\mu^{-1}_{ij}B_{j}.$ (3)
In the next few sections we build a model describing travelling waves in a
magnetar’s atmosphere from these equations.
### 2.2 Vacuum Dielectric and Inverse Magnetic Permeability Tensors
In this section, we describe our model of the QED vacuum in strong background
fields in terms of vacuum dielectric and inverse magnetic permeability
tensors. These are most conveniently described in terms of two Lorentz
invariant combinations of the fields.
$\displaystyle K$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\varepsilon^{\lambda\rho\mu\nu}F_{\lambda\rho}F_{\mu\nu}\right)^{2}=-(4{\bf
E}\cdot{\bf B})^{2}$ (4) $\displaystyle I$ $\displaystyle=$ $\displaystyle
F_{\mu\nu}F^{\mu\nu}=2(|{\bf B}|^{2}-|{\bf E}|^{2})$ (5)
In order to examine the nonlinear effects of the vacuum nonperturbatively, we
wish to use vacuum dielectric and inverse magnetic permeability tensors which
are valid to all orders in the fields. Analytic expressions for these tensors
were derived by Heyl & Hernquist (1997b) for the case of wrenchless fields
($K=-(4{\bf E}\cdot{\bf B})^{2}=0$) from the Heisenberg-Euler-Weisskopf-
Schwinger (Heisenberg & Euler, 1936; Weisskopf, 1936; Schwinger, 1951) one-
loop effective Lagrangian in Heyl & Hernquist (1997a) and expressed in terms
of a set of analytic functions.
$\displaystyle X_{0}(x)$ $\displaystyle=$ $\displaystyle
4\int_{0}^{x/2-1}\ln(\Gamma(v+1)){\rm
d}v+\frac{1}{3}\ln\left(\frac{1}{x}\right)$ (6) $\displaystyle~{}~{}+2\ln
4\pi-4\ln A-\frac{5}{3}\ln 2$ $\displaystyle~{}~{}-\left[\ln
4\pi+1+\ln\left(\frac{1}{x}\right)\right]x$
$\displaystyle~{}~{}+\left[\frac{3}{4}+\frac{1}{2}\ln\left(\frac{2}{x}\right)\right]x^{2}$
$\displaystyle X_{1}(x)$ $\displaystyle=$
$\displaystyle-2X_{0}(x)+xX_{0}^{(1)}(x)+\frac{2}{3}X_{0}^{(2)}(x)-\frac{2}{9}\frac{1}{x^{2}}$
(7) $\displaystyle X_{2}(x)$ $\displaystyle=$
$\displaystyle-24X_{0}(x)+9xX_{0}^{(1)}(x)$ (8)
$\displaystyle~{}~{}+(8+3x^{2})X_{0}^{(2)}(x)+4xX_{0}^{(3)}(x)$
$\displaystyle~{}~{}-\frac{8}{15}X_{0}^{(4)}(x)+\frac{8}{15}\frac{1}{x^{2}}+\frac{16}{15}\frac{1}{x^{4}}$
where
$X_{0}^{(n)}(x)=\frac{{\rm d}^{n}X_{0}(x)}{{\rm d}x^{n}}$ (9)
and
${\rm ln}A=\frac{1}{12}-\zeta^{(1)}(-1)=0.248754477.$ (10)
The tensors we need are derived in Heyl & Hernquist (1997b), except that we
have kept terms up to linear order in the expansion about $K=0$ instead of
dealing with the strictly wrenchless case. Our analysis therefore requires
that $K\ll B_{k}^{4}$.
$\varepsilon^{ij}=\Delta^{ij}-\frac{\alpha}{2\pi}\left[\frac{2}{I}X_{1}\left(\frac{1}{\xi}\right)+\frac{12K}{I^{3}}X_{2}\left(\frac{1}{\xi}\right)\right]B^{i}B^{j}$
(11)
$(\mu^{-1})^{ij}=\Delta^{ij}+\frac{\alpha}{2\pi}\left[\frac{2}{I}X_{1}\left(\frac{1}{\xi}\right)+\frac{K}{12I^{3}}X_{2}\left(\frac{1}{\xi}\right)\right]E^{i}E^{j}$
(12)
where
$\displaystyle\Delta^{ij}$ $\displaystyle=$
$\displaystyle\delta^{ij}\Bigg{[}1+\frac{\alpha}{2\pi}\Bigg{(}-2X_{0}\left(\frac{1}{\xi}\right)+\frac{1}{\xi}X_{0}^{(1)}\left(\frac{1}{\xi}\right)$
(13)
$\displaystyle+\frac{K}{4I^{2}}X_{1}\left(\frac{1}{\xi}\right)+\frac{K}{8I^{2}\xi}X_{1}^{(1)}\left(\frac{1}{\xi}\right)\Bigg{)}\Bigg{]},$
the fine-structure constant is $\alpha=e^{2}/4\pi$ in these units where we
have set $\hbar=c=1$, and
$\xi=\frac{1}{B_{k}}\sqrt{\frac{I}{2}}.$ (14)
Equations (11) and (12) define our model for the QED vacuum in a strong
electromagnetic field. Because we will focus on photon energies much lower
than the rest-mass energy of the electron, we have treated the vacuum as
strictly non-linear. It is not dispersive. The treatment of the dispersive
properties of the vacuum would require an effective action treatment (e.g.
Cangemi et al., 1995a, b) rather than the local effective Lagrangian treatment
used here.
### 2.3 Plasma
To investigate travelling waves, we choose our coordinate system so that the
$\hat{\bf x}$-direction is aligned with the direction of propagation. Then,
the spacetime dependence of the fields and sources is given by a single
parameter $S\equiv x-vt$ where $v$ is the constant phase velocity in the
$\hat{\bf x}$-direction of the travelling wave. At this point, we are choosing
to work in a specific Lorentz frame.
We model the plasma as a free electron plasma which enters the wave equation
through the source terms $\rho_{p}$ and ${\bf J}_{p}$. For electromagnetic
fields obeying the travelling wave ansatz, the sources must also obey the
ansatz. Then, $\rho_{p}$ and ${\bf J}_{p}$ are functions only of $S$. We
therefore treat them as an additional field which is integrated along with the
electromagnetic components of the field.
In order to perform the numerical ODE integration for the currents, we wish to
find expressions for $\frac{{\rm d}\rho_{p}}{{\rm d}S}$ and $\frac{{\rm d}{\bf
J}_{p}}{{\rm d}S}$. For travelling waves, the continuity equation is
$-v\frac{d\rho_{p}}{dS}+\delta^{ix}\frac{dJ_{p}^{i}}{dS}=0.$ (15)
where we are using index notation to label our explicitly cartesian
$\\{x,y,z\\}$ coordinate system. Repeated indices are summed. However,
whenever $x$ or $z$ appears as an index, it is fixed and does not run from 1
to 3.
We use equation (15) to rewrite the source terms from equation (1)
$\partial^{i}\rho_{p}+\frac{\partial{J}^{i}_{p}}{\partial
t}=\frac{1}{v}\frac{dJ_{p}^{i}}{dS}(\delta^{ix}-v^{2}).$ (16)
Similarly, the source term from equation (2) is
$({\nabla}\times
J_{p})^{i}=\varepsilon^{ixk}\left(\frac{dJ_{p}}{dS}\right)^{k}.$ (17)
To find an expression for $\frac{d{\bf J}_{p}}{dS}$ we express the current as
an integral over the phase-space distribution of the electrons and linearize
the plasma density.
${\bf J}_{p}=\int{f({\bf p}_{p})e{\bf v}_{p}d^{3}{\bf
p}_{p}}\approx\bar{\gamma}ne\bar{\bf v}_{p}$ (18)
where $\bar{\bf v}_{p}\equiv\frac{\int f({\bf p}_{p}){\bf v}_{p}d^{3}{\bf
p}_{p}}{\bar{\gamma}n}$ and $n$ is the mean electron density in the plasma in
the reference frame where $\bar{{\bf v}}_{p}=0$. It is important to note the
distinction between the mean plasma speed, $\bar{v}_{p}$, and the propagation
speed of the wave, $v$. The Lorentz factor
$\bar{\gamma}\equiv\frac{1}{\sqrt{1-\bar{v}_{p}^{2}}}$ accounts for a
relativistic increase in the plasma density since $d^{3}{\bf
x}=\gamma^{-1}d^{3}{\bf x}^{\prime}$.
Next, we take a time derivative of the current and express this in terms of a
three-force acting on the plasma.
$\displaystyle\frac{\partial{\bf J}_{p}}{\partial t}$ $\displaystyle=$
$\displaystyle ne\frac{\partial\bar{\gamma}\bar{{\bf v}}_{p}}{\partial t}$
(19) $\displaystyle=$ $\displaystyle\frac{ne}{m}{\bf F}$
Noting that ${\bf J}_{p}$ is a function only of $S$, we insert the Lorentz
force ${\bf F}=e({\bf E}+\bar{{\bf v}}_{p}\times{\bf B})$ and arrive at an
expression that can be substituted into the source terms, equations (16) and
(17), to find ${\bf J}_{p}(S)$.
$\displaystyle\frac{d{\bf J}_{p}}{dS}=-\frac{1}{v}\frac{e}{m}(en{\bf
E}+\bar{\gamma}^{-1}{\bf J}_{p}\times{\bf B})$ (20)
The equations we have given above describe a cold, relativistic,
magnetohydrodynamic plasma. Forces on the plasma arising due to pressure
gradients and gravity are neglected. Moreover, in our simulations, we neglect
the forces on the plasma due to the magnetic field. This approximation is
suitable if the plasma in question is a pair plasma, or for wave frequencies
much less than the cyclotron frequency.
As we will show in section 4 (see figure 6), the field configurations
generated in our simulation vary over timescales similar to the inverse of the
plasma frequency; the latter is about 9 orders of magnitude smaller than the
cyclotron frequency. This observation allows us to justify some aggressive
assumptions regarding the plasma response. As mentioned above, we may neglect
forces on the plasma from the magnetic field, and quantum effects are expected
to be small far away from the cyclotron resonance (Mészáros, 1992).
We also neglect thermal effects since the influence of the electromagnetic
fields will dominate over thermal motion. For strong background magnetic
fields such that $\frac{eB}{m}\gg kT$, the electrons will be confined to the
lowest Landau level, restricting thermal motion perpendicular to the
background magnetic field. As we will elaborate on in section 4, the greatest
nonlinear effects occur for waves with an electric field component oriented
along the background magnetic field. Because we are interested in waves with
amplitudes comparable to $B_{k}$, thermal motion is negligible relative to the
dynamics induced by the wave. We are therefore justified in neglecting thermal
effects in every direction for the cases of greatest interest.
If one combines Eq. (19) with Eq. (1), one sees that any nonlinearity in this
treatment must originate with the dielectric and permeability tensors — any
non-linearity that may originate from the plasma itself has been neglected
(c.f. Kozlov et al., 1979; Cattaert et al., 2005). The plasma is modelled as
strictly dispersive. In section 2.5, we confirm that this method of describing
the plasma is consistent with standard accounts in the weak-field, small-wave
limit.
### 2.4 Travelling Wave ODEs
The wave equations for travelling waves, are found by combining Maxwell’s
equations (1) and (2) with the continuity equation, (15). We also make the
plane-wave approximation, and assume that the fields and sources are described
by the parameter
$S=x-vt.$ (21)
The equations governing travelling wave propagation are
$\frac{d^{2}\psi^{i}(S)}{dS^{2}}=\frac{1}{v}\frac{{\rm d}J^{i}(S)}{{\rm
d}S}(\delta^{ix}-v^{2})$ (22)
$\frac{d^{2}\chi^{i}(S)}{dS^{2}}=-\varepsilon^{ixj}\frac{{\rm d}J^{j}(S)}{{\rm
d}S}$ (23)
The auxiliary vectors $\psi^{i}$ and $\chi^{i}$ are related to the electric
and magnetic fields.
$\displaystyle\psi^{i}(S)$ $\displaystyle=$
$\displaystyle(1-v^{2})\varepsilon^{ij}E^{j}+\delta^{ix}(\varepsilon^{xj}E^{j}-E^{x})-$
(24)
$\displaystyle~{}~{}(\varepsilon^{ij}E^{j}-E^{i})+\varepsilon^{ixk}v(B^{k}-(\mu^{-1})^{kj}B^{j})$
$\displaystyle\chi^{i}(S)$ $\displaystyle=$
$\displaystyle((\mu^{-1})^{ij}B^{j}-v^{2}B^{i})+\delta^{ix}(B^{x}-(\mu^{-1})^{xj}B^{j})-$
(25) $\displaystyle~{}~{}\varepsilon^{ixk}v(\varepsilon^{kj}E^{j}-E^{k})$
Equations (22) through (25) define a set of coupled ordinary differential
equations that can be integrated to solve for the travelling electric and
magnetic fields. Solving these equations requires that we have at hand the
vacuum dielectric and inverse magnetic permeability tensors as well as an
expression for $\frac{d{\bf J}_{p}}{dS}$. These were discussed in sections 2.2
and 2.3 respectively.
### 2.5 Weak field, Small Wave limit
In this section, we would like to demonstrate that our equations reduce to
standard expressions in the case of small background fields and small
electromagnetic waves. To make this comparison, it is also prudent to assume
waves have a spacetime dependence like $e^{i(\omega t-{\bf k}\cdot{\bf x})}$
instead of $x-vt$. By making this change, we can compare our other assumptions
with those made in standard textbook accounts directly.
Under the assumption that the fields and currents have the standard plane-wave
spacetime dependence, we can make the replacements $\frac{\partial}{\partial
t}\rightarrow-i\omega$ and $\nabla\rightarrow i{\bf k}$. We can then write a
second expression for $\frac{\partial{\bf J}}{\partial t}$.
$\frac{\partial{\bf J}_{p}}{\partial t}=-i\omega{\bf J}_{p}$ (26)
Setting equations (26) and (19) equal, we get
$\frac{e}{m}(\varepsilon^{ijk}J^{j}_{p}B^{k})+i\omega
J^{i}_{p}=\frac{e^{2}}{m}nE^{i}$ (27)
where we have used the nonrelativistic approximation, which is appropriate for
small waves. We will now specialize to a background magnetic field pointing in
the $\hat{\bf z}$-direction. We can then write this background field in terms
of the cyclotron frequency $\omega_{c}=\frac{eB}{m}$. The density $n$
determines the plasma frequency $\omega_{p}^{2}=n\frac{e^{2}}{m}$. Then, we
can write equation (27) as
$\left[-\omega_{c}\varepsilon^{ijz}-i\omega\delta^{ji}\right]J_{p}^{j}=\omega_{p}^{2}E^{i}$
(28)
We can solve this for ${\bf J}_{p}$ by writing a matrix equation
${\bf J}_{p}=\left(\begin{array}[]{ccc}-i\omega&-\omega_{c}&0\\\
\omega_{c}&-i\omega&0\\\ 0&0&-i\omega\end{array}\right)^{-1}\omega_{p}^{2}{\bf
E}$ (29)
inverting the matrix gives
${\bf
J}_{p}=\left(\begin{array}[]{ccc}\frac{i\omega}{\omega^{2}-\omega_{c}^{2}}&\frac{-\omega_{c}}{\omega^{2}-\omega_{c}^{2}}&0\\\
\frac{\omega_{c}}{\omega^{2}-\omega_{c}^{2}}&\frac{i\omega}{\omega^{2}-\omega_{c}^{2}}&0\\\
0&0&i/\omega\end{array}\right)\omega_{p}^{2}{\bf E}.$ (30)
Finally, we would like to use equation (30) to write the right hand side of
equation (1) in the small wave limit in a manner we can interpret as a
dielectric tensor.
Ignoring (for now) the contributions from the vacuum, and assuming an
approximately homogeneous plasma density, equation (1) simplifies to
$\displaystyle{\nabla}^{2}{\bf E}-\frac{\partial^{2}{\bf E}}{\partial t^{2}}$
$\displaystyle=$ $\displaystyle\frac{\partial{\bf J}_{p}}{\partial t}$ (31)
$\displaystyle=$ $\displaystyle-i\omega{\bf J}_{p}.$
If our macroscopic field is to obey ${\nabla}^{2}{\bf
D}-\frac{\partial^{2}{\bf D}}{\partial t^{2}}=0$, we can insert equation (30)
into (31) to obtain the following expression for the dielectric tensor due to
plasma effects:
$\varepsilon^{(p)}_{ij}=\left(\begin{array}[]{ccc}1-\frac{\omega_{p}^{2}}{\omega^{2}-\omega_{c}^{2}}&-i\frac{\omega_{c}}{\omega}\frac{\omega_{p}^{2}}{\omega^{2}-\omega_{c}^{2}}&0\\\
i\frac{\omega_{c}}{\omega}\frac{\omega_{p}^{2}}{\omega^{2}-\omega_{c}^{2}}&1-\frac{\omega_{p}^{2}}{\omega^{2}-\omega_{c}^{2}}&0\\\
0&0&1-\left(\frac{\omega_{p}^{2}}{\omega^{2}}\right)\end{array}\right)$ (32)
This expression is in agreement with the cold plasma dielectric tensor given
in Mészáros (1992). As noted above, our analysis neglects the off-diagonal
(Hall) terms, as is appropriate for pair plasmas or waves with frequencies
much less than the cyclotron frequency.
Because the vacuum effects are added explicitly in the form of dielectric and
magnetic permeability tensors, we only need to confirm that the weak field
limits of our expressions agree with the standard results. This confirmation
is done explicitly in Heyl & Hernquist (1997b).
In the weak field limit, the tensors given by equations (11) and (12) are
$\displaystyle\varepsilon_{ij}^{(v)}$ $\displaystyle=$
$\displaystyle\delta_{ij}+\frac{1}{45\pi}\frac{\alpha}{B_{k}^{2}}\left[2(E^{2}-B^{2})\delta_{ij}+7B_{i}B_{j}\right],$
(33) $\displaystyle\mu^{-1(v)}_{ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}+\frac{1}{45\pi}\frac{\alpha}{B_{k}^{2}}\left[2(E^{2}-B^{2})\delta_{ij}-7E_{i}E_{j}\right]$
(34)
In the case of a weak background magnetic field pointing in the $\hat{\bf
z}$-direction, these become
$\displaystyle\varepsilon_{ij}^{(v)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1-2\delta&0&0\\\ 0&1-2\delta&0\\\
0&0&1+5\delta\end{array}\right)$ (38) $\displaystyle\mu_{ij}^{-1(v)}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}1-2\delta&0&0\\\
0&1-2\delta&0\\\ 0&0&1-6\delta\end{array}\right)$ (42)
with
$\delta=\frac{\alpha}{45\pi}\left(\frac{B}{B_{k}}\right)^{2}$ (43)
Again, this result agrees with the vacuum tensors given in Mészáros (1992).
In this limit, we may simply add together the contributions to the dielectric
tensor from the plasma and the vacuum according to
$\varepsilon_{ij}=\delta_{ij}+(\varepsilon_{ij}^{(p)}-\delta_{ij})+(\varepsilon_{ij}^{(v)}-\delta_{ij})$
(44)
with $\mu_{ij}^{-1}$ given entirely by the vacuum contribution.
We have thus recovered the standard result for a medium consisting of a plasma
and the QED vacuum in the weak field, small wave limit.
## 3 Solution Procedure
In total, there are 15 coupled non-linear ODEs which must be integrated to
produce a solution. Equations (22) and (23) define the electric and magnetic
fields as functions of $S$. In addition, we must simultaneously integrate
equation (20) which gives the plasma current as a function of $S$. Each of
these equations has three spatial components. Initial conditions are given for
each of the six field components, and the six derivatives of the field with
respect to $S$. The equations describing the electromagnetic fields do not
depend on the initial values of the current, but in principle these are the
three remaining initial conditions.
Figure 1: A comparison between the $\hat{\bf z}$ components of the electric
(solid) and magnetic fields (dashed) showing the nonorthogonal stabilizing
wave. On larger scales, the $B_{z}$ component is seen to have a periodic
envelope structure as in figure 2
.
Figure 2: The $B_{z}$ wave component as shown in figure 1 has an envelope
structure when viewed at larger scales. In this case, the $E_{z}$ wave
component has a nearly constant amplitude of $0.1B_{k}$.
.
The ODEs are solved using a variable stepsize Runge-Kutta method. In order to
translate between the ${\bf E}$-and-${\bf B}$ picture and the
${\bf\psi}$-and-${\bf\chi}$ picture, equations (24) and (25) must be solved
numerically at each time step, including the first step when the initial
conditions are given.
Tables of numerical values of the functions defined by equations (6) to (8),
as well as their derivatives, were computed in advance and these were used to
interpolate the values needed in the simulation using a standard cubic spline
interpolation algorithm. In producing these tables, expressions for the weak
and strong field limits were used in the appropriate regimes as this reduced
the numerical errors.
Aside from the initial conditions for the fields and derivatives, there is one
parameter in the model which must be selected. The density of the plasma,
given by $n$ in equation(18) is chosen to be
$n=10^{13}\mathrm{cm}^{-3}.$ (45)
This value corresponds to the Goldreich-Julian density for a star of period
$P\sim 1$ s, $\dot{P}\sim 10^{-10}$. The uniform background field is taken to
equal the quantum critical field strength
$B_{k}=\frac{m^{2}}{e}=4.413\times 10^{13}{\hbox{\rm G}}.$ (46)
## 4 Results
This study focuses on the case of waves propagating transverse to a large
background magnetic field. We have already chosen the direction of propagation
to be the $\hat{\bf x}$-direction through our definition of $S$ in Eq. (21).
We now choose the background magnetic field to point in the $\hat{\bf
z}$-direction. In this situation, the largest non-linear effects occur when
there is a large amplitude wave in the $\hat{\bf z}$ component of the electric
field. This is quite natural. The values of both the dielectric tensor of the
plasma (Eq. 32) and the weak-field limit of the vacuum dielectric tensor
(Eq.38) differ most from unity for this component; for strong fields the index
of refraction (as well as its derivative with respect to the field strength)
is largest for vacuum propagation in this mode (Heyl & Hernquist, 1997b).
Furthermore, the dominant three-photon interaction couples photons with the
electric field pointed along the global magnetic field direction with photons
whose magnetic field points along this direction. The three-point interaction
for photons whose electric field is perpendicular to the magnetic field with
parallel photons vanishes by the $CP-$invariance of QED (Adler, 1971).
Therefore, we focus on initial ($S=0$) conditions in which the dominant
component of the electric field points along in the $\hat{\bf z}$-direction
and that of the magnetic field along the $\hat{\bf y}$-direction.
In the classical vacuum, these initial conditions correspond to transverse,
linearly polarized sine-wave solutions that travel at the speed of light.
However, when the wave amplitudes are large, and there is a strongly
magnetized plasma, we find that there is a deviation from normal transverse
electromagnetic waves. In particular, in order to remain stable, a wave with
large $E_{z}$ and $B_{y}$ field components must also excite waves in the
$E_{y}$ and $B_{z}$ fields. These stabilizing wave components exhibit strong
non-linear characteristics (see figures 1 and 2). The symmetries of the wave
equation require a close correspondence between the $E_{y}$ and $B_{z}$
waveforms as well as between the $B_{z}$ and $B_{y}$ waveforms. For
simplicity, only one of each is plotted in the figures. The field strengths
are given in units of the quantum critical field strength, $B_{k}$.
As is apparent from Fig. 1 the dominant electric field along the direction of
the external magnetic field is essentially sinusoidal. Subsequent figures will
show that there is a small harmonic component. The waveform for the dominant
magnetic field component is similar. On the other hand, the magnetic field
along the direction of the electric field (the non-orthogonal component) is
smaller by nearly four orders of magnitude and obviously exhibits higher
harmonics. In particular if one expands the scale of interest (Fig. 2), the
magnetic field exhibits beating between two nearby frequencies with similar
power.
In order to examine the harmonic content of the waveforms, we perform fast
Fourier transforms (FFTs) on the signals produced in the simulations. We
present the results in terms of power spectra normalized by the square
amplitudes of the electric field of the waves. In these plots, the horizontal
axis is normalized by the frequency with the greatest power in the electric
field, so that harmonics can be easily identified.
Fig. 3 depicts the power spectra of the electric and magnetic fields along the
$\hat{z}$-direction for the wave depicted in Figs. 1 and 2. The conclusions
gathered from an examination of the waveforms are born out by the power
spectra. In particular the electric field is a pure sinusoidal variation to
about one part in ten thousand – the power spectrum of a pure sinusoid is
given by the dashed curved. The duration of the simulation is not an integral
multiple of the period of the sinusoid, resulting in a broad power spectrum
even for a pure sinusoid.
Figure 3: The solid curve depicts power spectra of the electric (upper) and
magnetic fields (lower) along the $\hat{\bf z}$-direction for the solution
depicted in Figs. 1 and 2. The inset focusses on the fundamental and the first
harmonic. The dashed curve follows the power spectrum of a single sinusoid for
the electric field and three sinusoids for the magnetic field. Near the peaks
the dashed curve is essentially indistinguishable from the solid one.
The power spectrum of the magnetic field follows the expectations gleaned from
the waveforms. In particular the fundamental and the first harmonic are
dominant, with the first harmonic having about one-third the power of the
fundamental. If one focusses on the fundamental, one sees that two frequencies
are involved. The envelope structure is produced by a beating between the
fundamental and a slightly lower frequency with a similar amount of power as
the first harmonic. Over the course of the simulation the envelope exhibits
two apparent oscillations; the lower frequency differs by two frequency bins,
so it is resolved separately from the fundamental as shown in the inset.
As the amplitude of the electric field increases the non-linear and non-
orthogonal features of the travelling wave increase. Fig. 4 shows that the
strength of the non-orthogonal magnetic field increases as the square of the
electric field, a hallmark of the non-linear interaction between the fields.
For the strongest waves studied with $E_{z}\approx 0.2B_{k}$ (the rightmost
point in the figure), the magnetic field, $B_{z}$, is about $10^{-4}B_{k}$
nearly one-percent of the electric field. The amplitude of the non-orthogonal
magnetic field is given by
$B_{z}=0.008B_{k}\left(\frac{E_{z}}{B_{k}}\right)^{2}$ (47)
for $B_{0}=B_{k}$. The coefficient is coincidentally very close to three-
quarters of the value of the fine-structure constant. It increases with the
strength of the background field.
Figure 4: Amplitude of the $B_{z}$ component plotted against the amplitude of
the $E_{z}$ component for a $B_{0}=B_{k}$ background field. The line is the
best-fitting power-law relation. The slope is consistent with a scaling
exponent equal to two.
For the strongest waves even the non-orthogonal magnetic field is strong, so
it can generate non-linearities in the electric field. Although the strongest
effect is around the fundamental, it is completely swamped by the fundamental
of the electric field. On the other hand, the magnetic field drives a first
and second harmonic in the electric field as seen in Fig. 5. The strength of
these harmonics is approximately given by the formula in Eq. (47) or
equivalently Fig. 4 if one substitutes the value of $B_{z}$ for $E_{z}$ and
uses result for $E_{z}$. This is essentially a six-order correction from the
effective Lagrangian. Because we have used the complete Lagrangian rather than
a term-by-term expansion, all of the corrections up to sixth order (and
further) are automatically included in the calculation.
Figure 5: This power spectrum demonstrates the development of nonlinear
effects in the $\hat{\bf z}$-component of the electric field as the amplitude
of the wave is increased in a $B_{0}=B_{k}$ background field. From top to
bottom the curves follow the solutions whose amplitude of $E_{z}$ equals
$0.01,0.08$ and $0.16B_{k}$. The inset focusses on the first and second
harmonics. Figure 6: The frequency (solid curve) and wavenumber (dashed curve)
of the travelling wave as a function of its phase velocity.
Figure 6 demonstrates how the frequency of the solutions varies with the speed
of propagation. Because there is no mode information stored directly in our
numerical solutions, we take the frequency to be rate that local minima in the
electric field pass a fixed observer. In general, we find that the frequency
of traveling waves increases as the phase velocity approaches the speed of
light, very closely following the formula
$\left(\frac{\omega}{\omega_{p}}\right)^{2}=\frac{v^{2}}{v^{2}-1}.$ (48)
This formula also results from an analysis of the dielectric tensor, Eq. (32).
The vacuum makes a small contribution to the wave velocity in this regime.
## 5 Conclusion
We have discussed techniques for computing electromagnetic waves in a strongly
magnetized plasma which nonperturbatively account for the field interactions
arising from QED vacuum effects. We applied these methods to the case of
travelling waves, which have a spacetime dependence given by the parameter
$S=x-vt$. Travelling waves can be described without any decomposition into
Fourier modes and this is ideal for exploring the nonperturbative aspects of
waves.
The main result from this analysis is the observation that electromagnetic
waves in a strongly magnetized plasma can self-stabilize by exciting
additional nonorthogonal wave components. In the cases studied above, a large
amplitude excitation of the electromagnetic field, for example, from the
coupling between Alfvén waves to starquakes, can induce nonlinear waves which
are stabilized against the formation of shocks. The result is a periodic wave
train with distinctly nonlinear characteristics. Such structures may play a
role in forming pulsar microstructures.
This result demonstrates that shock formation is not a necessary outcome for
waves in a critically magnetized plasma. It is possible that nonlinear
features of a wave can stabilize it against shock formation. The shock-wave
solutions generally decrease in magnitude, dissipating energy along the way
unless they travel into a low-field region, so the self-stabilizing nonlinear
waves are the only ones that keep their shape and energy content intact as
they propagate. It is not yet clear what set of conditions will determine if a
particular wave will self-stabilize or collapse to form a shock or how plasma
inhomogenaities will affect the propagation of a travelling wave train. These
are issues which can be clarified in future work.
## References
* Adler (1971) Adler S. L., 1971, Ann. Phys., 67, 599
* Blaes et al. (1989) Blaes O., Blandford R., Goldreich P., Madau P., 1989, ApJ, 343, 839
* Cangemi et al. (1995a) Cangemi D., D’hoker E., Dunne G., 1995a, Phys. Rev. D, 51, 2513
* Cangemi et al. (1995b) Cangemi D., D’hoker E., Dunne G., 1995b, Phys. Rev. D, 52, 3163
* Cattaert et al. (2005) Cattaert T., Kourakis I., Shukla P. K., 2005, Physics of Plasmas, 12, 012319
* Chian & Kennel (1987) Chian A., Kennel C. F., 1987, NASA STI/Recon Technical Report N, 88, 16622
* Galtsov & Nikitina (1983) Galtsov D. V., Nikitina N. S., 1983, Zhurnal Eksperimental noi i Teoreticheskoi Fiziki, 84, 1217
* Gill & Heyl (2009) Gill R., Heyl J. S., 2009, Phys. Rev E, 80, 036407 (8 pages)
* Heisenberg & Euler (1936) Heisenberg W., Euler H., 1936, Z.PHYS., 98, 714
* Heyl & Hernquist (1998) Heyl J., Hernquist L., 1998, Physical Review D, 58, 43005
* Heyl & Hernquist (1997a) Heyl J. S., Hernquist L., 1997a, Phys. Rev. D, 55, 2449
* Heyl & Hernquist (1997b) Heyl J. S., Hernquist L., 1997b, Journal of Physics A Mathematical General, 30, 6485
* Heyl & Hernquist (1999a) Heyl J. S., Hernquist L., 1999a, Phys. Rev. D, 59, 045005 (5 pages)
* Heyl & Hernquist (1999b) Heyl J. S., Hernquist L., 1999b, Phys. Rev. D, 59, 045005
* Heyl & Shaviv (2000) Heyl J. S., Shaviv N. J., 2000, MNRAS, 311, 555
* Heyl & Shaviv (2002) Heyl J. S., Shaviv N. J., 2002, Phys. Rev. D, 66, 023002 (4 pages)
* Heyl et al. (2003) Heyl J. S., Shaviv N. J., Lloyd D., 2003, MNRAS, 342, 134
* Jackson (1975) Jackson J. D., 1975, Classical electrodynamics. Wiley, New York
* Jenet et al. (2001) Jenet F. A., Anderson S. B., Prince T. A., 2001, ApJ, 558, 302
* Kozlov et al. (1979) Kozlov V. A., Litvak A. G., Suvorov E. V., 1979, Soviet Journal of Experimental and Theoretical Physics, 49, 75
* Lai & Ho (2003) Lai D., Ho W. C., 2003, Physical Review Letters, 91, 071101
* Lutzky & Toll (1959) Lutzky M., Toll J. S., 1959, Phys. Rev., 113, 1649
* Mereghetti (2008) Mereghetti S., , 2008, The strongest cosmic magnets: Soft Gamma-ray Repeaters and Anomalous X-ray Pulsars
* Mészáros (1992) Mészáros P., 1992, High-energy radiation from magnetized neutron stars.. University of Chicago Press, Chicago
* Meszaros & Bonazzola (1981) Meszaros P., Bonazzola S., 1981, ApJ, 251, 695
* Meszaros et al. (1980) Meszaros P., Nagel W., Ventura J., 1980, ApJ, 238, 1066
* Mészáros & Ventura (1978) Mészáros P., Ventura J., 1978, Phys. Rev. Lett., 41, 1544
* Mészáros & Ventura (1979) Mészáros P., Ventura J., 1979, Phys. Rev. D, 19, 3565
* Rajaraman (1982) Rajaraman R., 1982, Solitons and instantons. North-Holland Amsterdam etc.
* Schwinger (1951) Schwinger J., 1951, Phys. Rev., 82, 664
* Thompson & Blaes (1998) Thompson C., Blaes O., 1998, Phys. Rev. D, 57, 3219
* Thompson & Duncan (1995) Thompson C., Duncan R. C., 1995, MNRAS, 275, 255
* Weisskopf (1936) Weisskopf V., 1936, Mathematisk-Fysiske Meddelelser, 14
* Zheleznyakov & Fabrikant (1982) Zheleznyakov V., Fabrikant A., 1982, Name: Zh. Eksp. Teor. Fiz
|
arxiv-papers
| 2010-02-15T17:00:24 |
2024-09-04T02:49:08.405902
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dan Mazur and Jeremy S. Heyl",
"submitter": "Dan Mazur",
"url": "https://arxiv.org/abs/1002.2915"
}
|
1002.3106
|
12010
11institutetext: Instituto de Astrofísica de Andalucía (CSIC), Apdo. de
Correos 3004, 18080 Granada, Spain 22institutetext: National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
later
# Size matters
J. C. del Toro Iniesta
11 jti@iaa.es D. Orozco Suárez 2211
###### Abstract
The new generation of ground-based, large-aperture solar telescopes promises
to significantly increase our capabilities to understand the many basic
phenomena taking place in the Sun at all atmospheric layers and how they
relate to each other. A (non-exaustive) summary of the main scientific
arguments we have to pursue these impressive technological goals is presented.
We illustrate how imaging, polarimetry, and spectroscopy can benefit from the
new telescopes and how several wavelength bands should be observed to study
the atmospheric coupling from the upper convection zone all the way to the
corona. The particular science case of sunspot penumbrae is barely discussed
as a specific example.
###### keywords:
Sun: atmosphere, Instrumentation: science drivers
## 1 Introduction
Understanding the plasma physical processes occurring at the Sun at all scales
has always progressed in parallel to the development and construction of new
solar telescopes, post-focus instrumentation, and data analysis techniques. In
a large measure because of the new facilities, our knowledge has increased
significantly in recent years, but still much effort is needed to address
critical questions. The plasma and magnetic field interactions that take place
in the solar atmosphere involve characteristic spatial and temporal scales
that are too small to be fully resolved with current solar facilities, and
leave tracks in the spectrum of polarized light that can hardly be detected
with current instrumentation. These problems can be alleviated by increasing
our photon flux budget capabilities. Therefore, we need larger solar
telescopes. Indeed, a number of them are now being designed and constructed.
The two biggest projected telescopes are the EST (European Solar Telescope)
and the ATST (Advanced Technology Solar Telescope), each having a photon
collecting area larger than the sum of all other currently available and
planned areas. Both, thus, constitute a solid promise of a significant
qualitative advancement for the solar physics community. In some provocative
way, then, we can say as the title of this paper reads that “size matters”.
Summarizing the main reasons that justify the strong effort of building
4-meter class solar telescopes or, in other words, prospecting for the likely
scientific advances to be reached with them is a difficult endeavor. As a
matter of fact, a great deal of information on that topic can already be found
in the two excellent science requirement documents of the mentioned projects
and on, e.g., Keil et al. (2001, 2003, 2004, 2009), Keller et al. (2002),
Rimmele et al. (2003, 2005), or Collados (2008). An exhaustive search cannot
be expected in this paper. Rather, we try to highlight some special topics and
to give some specific examples that are very important according to our
personal point of view.
Figure 1: Simulated continuum intensity maps corresponding to telescopes with
apertures of 4 (first two columns), 1.5, and 1 m, representative (from left
toward right) of the ATST, EST (4 m with a central obscuration), Gregor, and
Sunrise and SST, respectively, all considered without atmospheric effects. The
three rows correspond to three different wavelengths, namely, 525, 630, and
1560 nm from top to bottom. Text inserts show the corresponding rms contrasts.
The simulation snapshot has an average vertical field of 10 G. Only
diffraction effects are considered.
## 2 Main drivers for large-aperture telescopes
### 2.1 Spatial resolution
Solar features and phenomena can be observed at a vast variety of length
scales, the smallest in the photosphere being probably of the order of
kilometers or less (e.g., de Wijn et al. 2009). Our current facilities are
still far from resolving these tiny details, most of which are presumably
close to the mean free path of photons (a few tens of a km). The first,
straightforward approach implies a significant increase in telescope
apertures. The direct influence of the size is clearly understood after a
glance to Fig. 1, where synthetic images are shown that have been obtained
through radiative transfer calculations with the SIR code (Ruiz Cobo & Del
Toro Iniesta, 1992) on MHD numerical simulations generated with the MURAM code
(Vögler et al. 2005) for a solar zone with an average vertical magnetic field
of 10 G. Just the diffraction effects are taken into account for producing the
images. The aperture diameters range from 4 m (ATST and EST), to 1.5 m
(Gregor), and 1 m (Sunrise and SST). At given wavelengths differences in
contrast are significant. For example, values go from 18.3 % with 4-m
telescopes to 15.1 % with 1-m ones at 525 nm. The slight differences between
the two 4-m telescopes are due to the EST central obscuration. Note that no
atmospheric seeing is included in the synthetic images. Hence, these images
represent theoretical limits for the corresponding instrumentation that,
everybody knows, are hardly reachable on ground. The most likely option to
approximate these limits is adaptive optics (AO; e.g., Berkefeld et al. 2002,
Scharmer et al. 2003, Rimmele et al. 2005, and references therein) or post-
facto restoring techniques like MOMFBD (multi-object, multi-frame, blind
deconvolution; van Noort et al. 2005) or phase diversity (Paxman et al. 1996),
that can even be useful for space- or balloon-borne observations when
corrections for residual jittering or other motions are necessary. Such an
approximation has only been possible for individual images or for not-long-
enough data series. MHD simulations have shown, however, that many magnetic
processes take place at such small scales. But even if the ideal resolution is
not exactly reached, we need to continue exploring whether or not smaller and
smaller scales seem to exist. Such analyses will, in turn, feed back the MHD
modeling. Another clear feature in Fig. 1 is the importance of the observing
wavelength. As soon as we go to the infrared (IR), contrast deteriorates
dramatically. We can be interested in the increased diagnostic potential of
the IR wavelengths. Thus, if we aim at spatial resolutions in these
wavelengths similar to those currently reached in the visible with telescopes
smaller than 1 m, we necessarily have to pursue the use of large-aperture
telescopes.
Spatial resolution is also paramount for spectropolarim-etry in order to fully
characterize the many small-scale magnetic fields that are known to populate
the solar atmosphere. The larger the telescope aperture, the less distorted
the polarization maps. Figure 2 shows an example of Stokes $Q$, $U$, and $V$
polarization maps as seen by a 4-m and a 1-m telescopes (left and right
columns, respectively). Signals at a fixed wavelength (+ 7.7 pm far from line
center) of the Fe i line at 525.02 nm are considered. Right panels show less
polarization signals and magnetic structuring than the left panels. Besides,
tiny details scape from detection in the 1-m maps. In view of this, should we
still speak of filling fractions when trying to model the (currently)
unresolved magnetic structures when observed with larger telescopes? Recent
advances in spatial resolution with accurate instrumentation like the solar
optical telescope (Tsuneta et al. 2008) aboard Hinode (Kosugi et al. 2007) are
bringing about a new paradigm of the quiet Sun: it has been unveiled to be
covered by tiny, mostly horizontal, magnetic structures (Lites 2007, 2008).
From 0 (we had not detected them properly yet) we now estimate a filling
factor about $0.2-0.45$ (Orozco Suárez et al. 2007) of the resolution element.
Is this fraction still growing (and decreasing in other places) with the
increasing size of telescopes? We urgently need an answer to this question:
should the answer be negative at a given resolution we would conclude on
having a homogeneous distribution of magnetic features; a positive answer
would entail the conclusion of a heterogeneous distribution. Elucidating
between these two cases has important consequences about the nature
(homogeneously turbulent according to some authors) of the internetwork
magnetic fields and about the dynamo action behind them.
Spatial resolution is not only important for imaging and polarimetry.
Spectroscopy is also enriched by larger apertures. Although very simple and
easy to understand, little attention has received so far this effect in the
literature (Orozco Suárez et al. 2010). The smaller the resolution element,
the larger the details in the spectrum. Figure 3 shows the effect of telescope
diffraction on the Stokes profiles of the Fe i line at 525.02 nm as emerging
from one point of the MHD simulations. Red lines represent the original
resolution of the numerical simulations (almost equal to that of 4-m,
diffraction-limited telescope), in blue are the profiles as seen by a 1.5-m
telescope, and the observed spectrum by a 1-m telescope is seen in black.
Figure 2: Monochromatic Stokes $Q$ (top), $U$ (middle), and $V$ (bottom)
images at +7.7 pm of the central wavelength of the Fe I line at 525.02 nm.
Left and right columns show the images through a diffraction-limited telescope
of 4 and 1 m, respectively. We have added noise at the level of
$10^{-3}\,I_{\mathrm{QS}}$, where $I_{\mathrm{QS}}$ is the average continuum
of the quiet Sun. Grid units are in arcsec.
### 2.2 Photon budget
We measure nothing but light and, therefore, our measurements rely upon
photometric accuracy, $\delta I/I$. This quantity represents the (relative)
smallest detectable signal and is inversely proportional to the signal-to-
noise ratio: $S/N=1/(\delta I/I)$. If $p$ represents the degree of
polarization (no matter linear, circular, or total), a simple algebra leads us
to $\delta
p/p\,\mbox{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle<}}{{\sim}}$}}\,\sqrt{1+1/p^{2}}/(S/N)$.111The
asymptotic behavior of $\delta p/p$ for $p=0$ simply reflects the fact that
polarimetric accuracy worsens for very small degrees of polarization. This
means that if we want to reliably measure the faint polarization signals that
are detected of the order of $10^{-3}-10^{-4}\,I_{\rm QS}$ with low spatial
and temporal resolution (e.g., López Ariste & Casini 2003, Stenflo 2006,
Trujillo Bueno 2009), then we need $S/N$ values well above $10^{3}-10^{4}$.
Enhancing the $S/N$ at a given bandwidth is only possible by increasing the
telescope aperture or the exposure time. Since we cannot afford the latter
option because the extremely dynamic nature of most of the phenomena under
analysis, it seems that we should pursue the construction of larger
telescopes. This will especially be true in the chromosphere where the rapid
evolution of features escapes detection with long exposure times even in total
intensity in the continuum.
The solar surface shows very dynamic phenomena. Evershed (often supersonic)
flows and moving magnetic features in sunspots, convective collapse of flux
tubes, supersonic horizontal flows in granules, moving spicules and other
features in the chromosphere, high-frequency prominence oscillations, and
magnetic reconnection are some examples. Such processes occur at short time
scales that need be resolved for a meaningful analysis. Even more, as soon as
we increase the resolution, we expect to detect new small-scale magnetic
structures whose timescales would demand even faster-cadence observations.
Therefore, a better photon budget will improve the temporal resolution and
allow better coverage for evolutionary studies.
Figure 3: Synthetic Stokes profiles of the Fe I line at 525.02 nm as seen by a
4-m (red), a 1.5-m (blue), and 1-m telescope (black).
### 2.3 Wavelength coverage
One of the most challenging goals for the near future is to understand and
quantify the magnetic coupling of the whole atmosphere from the upper
convection zone through the corona. We need to study how energy is transported
and dissipated and what is the role of magnetic fields in this process. Multi-
layer probing can only be possible by observing several wavelength regions at
the same time with the same telescope and the most efficient way to do that is
to direct different bands to different instruments. The rapid decrease of the
Planck function towards the ultraviolet and the infrared demands large
apertures in order to reach the best observing conditions in these two regions
of the spectrum. They are crucial for chromospheric and coronal studies.
Nevertheless, less ambitious objectives can also benefit from a broad
wavelength coverage as already demonstrated by, for example, the simultaneous
visible and IR observations by Cabrera Solana et al. (2006) that evidenced how
moving magnetic features have their origin in the Evershed effect, or by
Martínez González et al. (2008) that found hG magnetic fields in the
internetwork. The increased information of the two wavelength bands, improves
the reliability of their conclusions.
### 2.4 Coronography
Measuring coronal magnetic fields is a goal (and a challenge) in itself. The
physical phenomena taking place in the chromosphere and corona are better
observed through polarization by means of the Zeeman effect in emission lines
or of the Hanle measurements in scattered radiation. The low degrees of
polarization on very low brightness structures makes current discoveries
painful. Examples are, e.g., the detection of an extended near-Sun He i cloud
by Kuhn et al. (2007), the measurement of a 4 G magnetic flux density 100
arcsec above an active region by (Lin et al. 2004), or the detection of Alfvén
waves in intensity, line-of-sight velocity and linear polarization images by
Tomczyk (2007). Large-aperture telescopes are needed in order not to be
limited by photon noise. The ATST specific design for coronography is expected
to further help.
## 3 A science case: the sunspot penumbra
We do not want to end this contribution without commenting on a specific
scientific case that is very important to our personal interests and that will
clearly benefit from the advancement in instrumentation. Sunspot penumbrae are
among the oldest observable solar features that still puzzle us. Two are in
our opinion the most important discoveries of the last fifteen years related
to the sunspot penumbra, namely, the discovery of an Evershed downward mass
flux at its outer periphery by Westendorp Plaza et al. (1997) and the
discovery of dark cores along bright penumbral filaments by Scharmer et al.
(2002). These and other observational facts that have been gathered by the
whole community need to be explained by any model trying to give account of
the nature of the penumbra. A list of the most important ones include: 1) the
penumbra is bright; 2) the Evershed flow takes place preferentially in the
dark cores (Bellot Rubio et al. 2005); 3) it returns to the surface at the
middle penumbra and beyond (e.g. Ichimoto et al. 2007); 4) it is often
supersonic (e.g., Wiehr 1995, Del Toro Iniesta et al. 2001, Bellot Rubio et
al. 2004); 5) it is magnetized (e.g. Sánchez Almeida & Lites 1992, Martínez
Pillet 2000, Westendorp Plaza et al. 2001a,b); 6) it is associated to the
weakest and more inclined magnetic fields (see Bellot Rubio 2009 and
Tritschler 2009 for reviews); and 7) it continues beyond the outer penumbral
border, often as moving magnetic features (Sainz Dalda & Martínez Pillet 2005,
Cabrera Solana et al. 2006, Ravindra 2006, Kubo et al. 2007). Two are as well
the competing theoretical models for explaining the nature of the penumbra:
the uncombed model by Solanki and Montavon (1993) that, after the theoretical
calculations of Ruiz Cobo & Bellot Rubio (2008; see a magnetogram and a
Dopplergram resulting from these calculations in Fig. 4) of a hot plasma
flowing through the dark cores of penumbral filaments explains most of the
above observational facts; and the gappy penumbra by Spruit & Scharmer (2006)
and Scharmer & Spruit (2006) where field-free gaps protrude the magnetic
penumbra carrying energy from below and, thus, heating the penumbra through
overturning convection. This mechanism of overturning convection is far from
being firmly established by observations since no velocities are detected
along the borders of penumbral filaments. Spatial resolutions much better than
01 are certainly needed to settle the debate.
Figure 4: Magnetogram and Dopplergram of the dark-cored, bright penumbral
filament model by Ruiz Cobo & Bellot Rubio (2008). Both diagrams are
dimensionless. The first is equal to the Stokes $V/I_{\rm c}$ signal at $-10$
pm from the line center; the second is the normalized difference of Stokes $I$
at $\mp 15$ pm. (Receding velocities are positive.)
## 4 Concluding remarks
After having carried out our personal review, we must agree with the title of
the paper: size of telescopes certainly matters. But the observing wavelength
and coverage matter too, and the seeing conditions for the observation, and
the quality of the AO system, and, although not very much discussed in here,
the performance of instruments and of the analysis techniques is very
important to rely upon the results. Moreover, our prejudices based on the
current paradigm are sometimes determinant for reaching one conclusion or
another. In summary, the ideal situation would be such that a large-aperture
telescope feeds several highly efficient, state-of-the-art instruments working
at several wavelength bands, at an extremely good site where seeing conditions
are nevertheless improved with a very powerful AO system. The data are then
analyzed with sophisticated inversion analysis techniques that may take
several different scenarios (set of hypotheses) into account in order to
discriminate among prejudices of the different researchers. This can only be
accomplished after lots of professional imagination and expertise are put to
the service of the community. Therefore, we should conclude that, fortunately,
brain matters too. Very good examples of how the latter is true are the two
current ATST and EST projects aiming at a close approximation to the described
ideal situation.
###### Acknowledgements.
We have to thank the organizers of the workshop for having given us the
opportunity for preparing this contribution. Thanks are due as well to A.
Vögler and to J.A. Bonet who kindly lend us their simulations and software,
respectively. We are also indebted to L.R. Bellot Rubio who prepared Fig. 4.
This work has been partially funded by the Spanish Ministerio de Educación y
Ciencia, through Projects ESP2006-13030-C06-02, AYA2009-14105-C06-06, and
Junta de Andalucía, through Project P07-TEP-2687, including a percentage from
European FEDER funds.
## References
* [Bellot Rubio(2009)] Bellot Rubio, L. R. 2009, arXiv:0903.3619
* [Bellot Rubio et al.(2004)] Bellot Rubio, L. R., Balthasar, H., & Collados, M. 2004, A&A, 427, 319
* [Bellot Rubio et al.(2005)] Bellot Rubio, L. R., Langhans, K., & Schlichenmaier, R. 2005, A&A, 443, L7
* [Berkefeld et al.(2002)] Berkefeld, T., Soltau, D., & von der Luehe, O. 2002, Proc. of SPIE, 4538, 119
* [Cabrera Solana et al.(2006)] Cabrera Solana, D., Bellot Rubio, L. R., Beck, C., & del Toro Iniesta, J. C. 2006, ApJ, 649, L41
* [Collados(2008)] Collados, M. 2008, Lecture Notes and Essays in Astrophysics, 3, 113
* [Del Toro Iniesta et al.(2001)] del Toro Iniesta, J. C., Bellot Rubio, L. R., & Collados, M. 2001, ApJ, 549, L139
* [de Wijn et al.(2009)] de Wijn, A. G., Stenflo, J. O., Solanki, S. K., & Tsuneta, S. 2009, Space Science Reviews, 144, 275
* [Ichimoto et al.(2007)] Ichimoto, K., et al. 2007, PASJ, 59, 593
* [Keil et al.(2001)] Keil, S. L., Rimmele, T. R., Keller, C. U., & The Atst Team 2001, Advanced Solar Polarimetry – Theory, Observation, and Instrumentation, 236, 597
* [Keil et al.(2003)] Keil, S., Rimmele, T., Keller, C., & The ATST Team 2003, Astronomische Nachrichten, 324, 303
* [Keil et al.(2004)] Keil, S. L., Rimmele, T. R., Oschmann, J., Hubbard, R., Warner, M., Price, R., Dalrymple, N., & The Atst Team 2004, Multi-Wavelength Investigations of Solar Activity, 223, 581
* [Keil et al.(2009)] Keil, S. L., Rimmele, T. R., & Wagner, J. 2009, Earth Moon and Planets, 104, 77
* [Keller et al.(2002)] Keller, C. U., Rimmele, T. R., Hill, F., Keil, S. L., Oschmann, J. M., & the ATST Team 2002, Astronomische Nachrichten, 323, 294
* [Kosugi et al.(2007)] Kosugi, T., et al. 2007, Sol. Phys., 243, 3
* [Kubo et al.(2007)] Kubo, M., Shimizu, T., & Tsuneta, S. 2007, ApJ, 659, 812
* [Kuhn et al.(2007)] Kuhn, J. R., Arnaud, J., Jaeggli, S., Lin, H., & Moise, E. 2007, ApJ, 667, L203
* [Lin et al.(2004)] Lin, H., Kuhn, J. R., & Coulter, R. 2004, ApJ, 613, L177
* [Lites et al.(2007)] Lites, B. W., et al. 2007, Pub. Astron. Soc. Japan, 59, 571
* [Lites et al.(2008)] Lites, B. W., et al. 2008, ApJ, 672, 1237
* [López Ariste & Casini(2003)] López Ariste, A., & Casini, R. 2003, ApJ, 582, L51
* [Martínez González et al.(2008)] Martínez González, M. J., Collados, M., Ruiz Cobo, B., & Beck, C. 2008, A&A, 477, 953
* [Martínez Pillet(2000)] Martínez Pillet, V. 2000, A&A, 361, 734
* [Orozco Suárez et al.(2007)] Orozco Suárez, D., et al. 2007, ApJ, 670, L61
* [Orozco Suárez et al. (2010)] Orozco Suárez, D., Bellot Rubio, L.R., Martínez Pillet, V., Bonet, J.A, Vargas Domínguez, S., & Del Toro Iniesta, J.C. 2010, in preparation
* [Paxman et al.(1996)] Paxman, R. G., Seldin, J. H., Loefdahl, M. G., Scharmer, G. B., & Keller, C. U. 1996, ApJ, 466, 1087
* [Ravindra(2006)] Ravindra, B. 2006, Sol. Phys., 237, 297
* [Rimmele et al.(2003)] Rimmele, T., et al. 2003, Current Theoretical Models and Future High Resolution Solar Observations: Preparing for ATST, 286, 3
* [Rimmele et al.(2005)] Rimmele, T. R., et al. 2005, Proc. of SPIE, 5901, 41
* [Ruiz Cobo & Bellot Rubio(2008)] Ruiz Cobo, B., & Bellot Rubio, L. R. 2008, A&A, 488, 749
* [Ruiz Cobo & del Toro Iniesta(1992)] Ruiz Cobo, B., & del Toro Iniesta, J. C. 1992, ApJ, 398, 375
* [Sainz Dalda & Martínez Pillet(2005)] Sainz Dalda, A., & Martínez Pillet, V. 2005, ApJ, 632, 1176
* [Sánchez Almeida & Lites(1992)] Sánchez Almeida, J., & Lites, B. W. 1992, ApJ, 398, 359
* [Scharmer et al.(2003)] Scharmer, G. B., Dettori, P. M., Lofdahl, M. G., & Shand, M. 2003, Proc. of SPIE, 4853, 370
* [Scharmer et al.(2002)] Scharmer, G. B., Gudiksen, B. V., Kiselman, D., Löfdahl, M. G., & Rouppe van der Voort, L. H. M. 2002, Nature, 420, 151
* [Scharmer & Spruit(2006)] Scharmer, G. B., & Spruit, H. C. 2006, A&A, 460, 605
* [Solanki & Montavon(1993)] Solanki, S. K., & Montavon, C. A. P. 1993, A&A, 275, 283
* [Spruit & Scharmer(2006)] Spruit, H. C., & Scharmer, G. B. 2006, A&A, 447, 343
* [Stenflo(2006)] Stenflo, J. O. 2006, Astronomical Society of the Pacific Conference Series, 358, 215
* [Trujillo Bueno(2009)] Trujillo Bueno, J. 2009, arXiv:0909.4463
* [Tomczyk et al.(2007)] Tomczyk, S., McIntosh, S. W., Keil, S. L., Judge, P. G., Schad, T., Seeley, D. H., & Edmondson, J. 2007, Science, 317, 1192
* [Tritschler(2009)] Tritschler, A. 2009, arXiv:0903.1300
* [Tsuneta et al.(2008)] Tsuneta, S., et al. 2008, Sol. Phys., 249, 167
* [van Noort et al.(2005)] van Noort, M., Rouppe van der Voort, L., Löfdahl, M. G. 2005, Sol. Phys., 228, 191
* [Vögler et al.(2005)] Vögler, A., Shelyag, S., Schüssler, M., Cattaneo, F., Emonet, T., & Linde, T. 2005, A&A, 429, 335
* [Westendorp Plaza et al.(1997)] Westendorp Plaza, C., del Toro Iniesta, J. C., Ruiz Cobo, B., Martinez Pillet, V., Lites, B. W., & Skumanich, A. 1997, Nature, 389, 47
* [Westendorp Plaza et al.(2001)] Westendorp Plaza, C., del Toro Iniesta, J. C., Ruiz Cobo, B., Martínez Pillet, V., Lites, B. W., & Skumanich, A. 2001, ApJ, 547, 1130
* [Westendorp Plaza et al.(2001)] Westendorp Plaza, C., del Toro Iniesta, J. C., Ruiz Cobo, B., & Martínez Pillet, V. 2001, ApJ, 547, 1148
* [Wiehr(1995)] Wiehr, E. 1995, A&A, 298, L17
|
arxiv-papers
| 2010-02-16T15:52:13 |
2024-09-04T02:49:08.414059
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "JC. del Toro Iniesta, D. Orozco Suarez",
"submitter": "Jose Carlos del Toro Iniesta",
"url": "https://arxiv.org/abs/1002.3106"
}
|
1002.3190
|
# A Distributed Sequential Algorithm for Collaborative Intrusion Detection
Networks
Quanyan Zhu Coordinated Science Laboratory
Department of Electrical and Computer Engineering
University of Illinois at Urbana Champaign
USA 61820
Email: zhu31@uiuc.edu Carol Fung School of Computer Science
University of Waterloo
Waterloo, Ontario
Email: j22fung@cs.uwaterloo.ca Quanyan Zhu${}^{\textbf{1}}$, Carol J.
Fung${}^{\textbf{2}}$, Raouf Boutaba${}^{\textbf{2}}$, and Tamer
Başar${}^{\textbf{1}}$ ${}^{\textbf{1}}$Department of Electrical and Computer
Engineering,
University of Illinois at Urbana-Champaign, USA. {zhu31, basar1}@illinois.edu
${}^{\textbf{2}}$David R. Cheriton School of Computer Science,
University of Waterloo, Ontario, Canada, {j22fung, rboutaba}@uwaterloo.ca
Quanyan Zhu, Carol J. Fung, Raouf Boutaba and Tamer Başar Quanyan Zhu and
Tamer Başar are with the Department of Electrical and Computer Engineering,
University of Illinois at Urbana Champaign, 61801
(email:{zhu31,basar1}@illinois.edu). Carol J. Fung and Raouf Boutaba are with
David R. Cheriton School of Computer Science, University of Waterloo, Ontario,
Canada (email: {j22fung, rboutaba}@uwaterloo.ca).The work of the authors from
University of Illinois was in part supported by a grant from Boeing through
the Information Trust Institute. The work of the authors from the University
of Waterloo is supported by the Natural Science and Engineering Research
Council of Canada under its strategic program and in part by WCU (World Class
University) program through the Korea Science and Engineering Foundation
funded by the Ministry of Education, Science and Technology (Project No.
R31-2008-000-10100-0).
###### Abstract
Collaborative intrusion detection networks are often used to gain better
detection accuracy and cost efficiency as compared to a single host-based
intrusion detection system (IDS). Through cooperation, it is possible for a
local IDS to detect new attacks that may be known to other experienced
acquaintances. In this paper, we present a sequential hypothesis testing
method for feedback aggregation for each individual IDS in the network. Our
simulation results corroborate our theoretical results and demonstrate the
properties of cost efficiency and accuracy compared to other heuristic
methods. The analytical result on the lower-bound of the average number of
acquaintances for consultation is essential for the design and configuration
of IDSs in a collaborative environment.
## I Introduction
As computer systems become increasingly complex, the accompanied potential
threats also grow to be more sophisticated. Intrusion detection is the process
of monitoring and identifying attempted unauthorized system access or
manipulation. It is one of the most important tools for a network
administrator to detect security breaches along with firewalls.
An IDS can be categorized as either host-based or network-based. A host-based
IDS (HIDS) is intended primarily to monitor a host, which can be a server,
workstation, or any networked device, whereas a network-based IDS (NIDS) is
used to protect a group of computer hosts by capturing and analyzing network
packets. Even though these two types of IDSs are commonly employed in an
enterprise network, they do not adequately leverage the possible information
exchange between IDSs. The exchange of alert data or decisions between
administrative domains can effectively supplement the knowledge gained by a
single local IDS. In a collaborative environment, an IDS can learn the global
state of network attack patterns from its peers. By augmenting the information
gathered from across the network, an IDS can have a more precise picture of an
attacker’s behavior and hence increase its accuracy and efficiency of
detection.
Collaborative intrusion detection networks (CIDNs) have distinct features from
some other types of social networks such as P2P network and E-commerce
network, where the collaboration is one-time or short-term pattern. The
collaboration in IDN is usually long-term based. Unlike other social networks,
communication in CIDNs is often of “low-cost”, which leads to the possibility
of using test messages (a communication overhead generated on purpose to test
the reliability of the collaborators).
Based on the aforementioned properties, we design a CIDN which utilizes test
messages to learn the reliability of others and consultation requests to seek
diagnosis from collaborators. The architecture design is shown in Figure 1,
where NIDSs and HIDSs are connected into a collaboration network. Each IDS
maintains a list of acquaintances (collaborators) and test messages are sent
to acquaintances periodically to update its belief on peer reliability. When
an IDS receives intrusion alerts and lacks confidence to determine the nature
of the alerted source, alert messages are sent to its acquaintances for
evaluation. An acquaintance IDS analyzes the received intrusion information
and replies with a feedback of positive/negative diagnosis. The ambivalent IDS
collects feedback from its acquaintances and decides whether an alarm should
be raised or not to the administrator. If an alarm is raised, the suspicious
intrusion flow will be suspended and the system administrator investigates the
intrusion immediately.
In this paper, we design an efficient distributed sequential algorithm for
IDSs to make decision based on the feedback from its collaborators. We
investigate four possible outcomes of a decision: false positive (FP), false
negative (FN), true positive (TP), and true negative (TN). Each outcome is
associated with a cost. Our proposed sequential hypothesis testing based
feedback aggregation provides improved cost efficiency as compared to other
heuristic methods, such as the simple average model [1] and the weighted
average model [2, 3]. In addition, the algorithm reduces the communication
overhead as it aggregates feedback until a predefined FP and TP goal is
reached. Our analytical model effectively estimates the number of
acquaintances needed for an IDS to reach its predefined intrusion detection
goal. Such result is crucial to the design of an IDS acquaintance list in
CIDN.
The remainder of this paper is organized as follows. In Section II, we review
some existing CIDNs in the literature and IDS feedback aggregation techniques.
The problem formulation is in Section III, where we use hypothesis testing to
minimize the cost of decisions and sequential hypothesis testing to form
consultation termination policy for predefined goals. In Section IV, we use a
simulation approach to evaluate the effectiveness of our aggregation system
and validate the analytical model. Section V concludes the paper and
identifies directions for future research.
## II Related Work
Many CIDNs were proposed in the literature, such as Indra [4], DOMINO [5], and
NetShield [6]. However, these works did not address the problem that the
system might be degraded by some compromised insiders who are dishonest or
malicious.
Simple majority voting [7] and trust management are commonly used to detect
malicious insiders in CIDNs. Existing trust management models for CIDN are
either linear as in [2], [8], or Bayesian model as in [3]. They are based on
heuristic where the feedback aggregation is either a simple average [1] or a
weighted average [3]. Moreover, no decision cost is considered in these
models. In this paper, we use a sequential hypothesis testing model aiming at
finding cost-minimizing decisions based on collected feedback. Existing work
that applies hypothesis testing for intrusion detection includes[9] and [10],
where a central data fusion center is used to aggregate results from
distributed sensors in a local area network. However, their methodologies are
limited to the context that all participants need to engage in every detection
case. While in our context, IDSs may not be involved in all intrusions
detection and the collected responses may be from different groups of IDSs
each time.
Figure 1: A Collaborative Intrusion Detection Network
## III Problem Formulation
In this section, we formulate the feedback aggregation as a sequential
hypothesis testing problem. Consider a set of $N$ nodes, $\mathcal{N}$,
connected in a network, which can be represented by a graph
$\mathcal{G}=(\mathcal{N},\mathcal{E})$. The set $\mathcal{E}$ contains the
undirected links between nodes, indicating the acquaintances of IDSs in the
network.
Let $Y_{i},i\in\mathcal{N},$ be a random variable denoting the decision of IDS
$i$ observed by its peer IDSs on its acquaintance list $\mathcal{N}_{i}$. The
random variable $Y_{i}$ takes values in $\mathcal{Y}_{i}=[0,1]$. In the
intrusion detection setting, $Y_{i}=0$ says that IDS $i$ decides that there is
no intrusion while $Y_{i}=1$ means that IDS $i$ raises an alarm of possible
detection of intrusion. Each IDS makes its decision based upon its own
experience of the previous attacks and its own sophistication of detection. We
let $p_{i}$ as the probability mass function defined on $\mathcal{Y}_{i}$ such
that $p_{i}(Y_{i}=0)$ and $p_{i}(Y_{i}=1)$ denote the probability of no
intrusion and the probability of intrusion from $i$, respectively.
We let
$\mathbf{Y}^{i}:=[Y_{j}]_{j\in\mathcal{N}_{j}}\in\mathcal{Y}^{i}:=\prod_{j\in\mathcal{N}_{i}}\mathcal{Y}_{i}$
be an observation vector of an IDS $i$ that contains the feedback from its
peers in the acquaintance list. Each IDS has two hypotheses $H_{0}$ and
$H_{1}$. $H_{0}$ hypothesizes that no intrusion is detected whereas $H_{1}$
forwards a hypothesis that intrusion is detected and alarm needs to be raised.
Note that we intentionally drop the superscript $i$ because we assume that
each IDS attempts to make the same decision. Denote by
$\pi_{0}^{i},\pi_{1}^{i}$ the apriori probabilities on each hypothesis such
that $\pi_{0}^{i}=\mathbb{P}[H_{0}],\pi_{1}^{i}=\mathbb{P}[H_{1}]$ and
$\pi_{0}^{i}+\pi_{1}^{i}=1$, for all $i\in\mathcal{N}$. The conditional
probability $p^{i}(\mathbf{Y}^{i}=\mathbf{y}^{i}|H_{l}),l=1,2$ denotes the
probability of a complete feedback being
$\mathbf{y}^{i}\in\prod_{j\in\mathcal{N}_{i}}\mathcal{Y}_{j}$ given the
hypothesis. Assuming peers make decisions independently (this is reasonable if
acquaintances are appropriately selected), we can rewrite the conditional
probability as
$p^{i}(\mathbf{Y}^{i}=\mathbf{y}^{i}|H_{l})=\prod_{j\in\mathcal{N}_{i}}p_{j}(Y_{j}=y_{j}|H_{l}),i\in\mathcal{N},l=0,1.$
(1)
A hypothesis testing problem is one of finding a decision function
$\delta^{i}(\mathbf{Y}^{i}):\mathcal{Y}^{i}\rightarrow\\{0,1\\}$ to partition
the observation space $\mathcal{Y}^{i}$ into two disjoint sets
$\mathcal{Y}_{0}^{i}$ and $\mathcal{Y}_{1}^{i}$, where
$\mathcal{Y}_{0}^{i}=\\{\mathbf{y}^{i}:\delta^{i}(\mathbf{y}^{i})=0\\}$, and
$\mathcal{Y}_{1}^{i}=\\{\mathbf{y}^{i}:\delta^{i}(\mathbf{y}^{i})=1\\}$.
To find an optimal decision function according to some criterion, we introduce
the cost function $C^{i}_{ll^{\prime}},l,l^{\prime}=0,1$, which represents IDS
$i$’s cost of deciding that $H_{l}$ is true when $H_{l^{\prime}}$ holds. More
specifically, $C^{i}_{01}$ is the cost associated with a missed intrusion or
attack and $C^{i}_{10}$ refers to the cost of false alarm, while
$C^{i}_{00},C^{i}_{11}$ are the incurred costs when the decision meets the
true situation. In several situations, it can be shown that decision functions
can be picked as function of the likelihood ratio given by
$L^{i}(\mathbf{y}^{i})=\frac{p^{i}(\mathbf{y}^{i}|H_{1})}{p^{i}(\mathbf{y}^{i}|H_{0})}.$
(see [10, 9])
A threshold Bayesian decision rule is expressed in terms of the likelihood
ratio and is given by
$\delta^{i}_{B}(\mathbf{y}^{i})=\left\\{\begin{array}[]{cc}1&\textrm{~{}if~{}}L^{i}(\mathbf{y}^{i})\geq\tau^{i}\\\
0&\textrm{~{}if~{}}L^{i}(\mathbf{y}^{i})<\tau^{i}\end{array}\right.,$ (2)
where the threshold $\tau^{i}$ is defined by
$\tau^{i}=\frac{(C^{i}_{10}-C^{i}_{00})\pi^{i}_{0}}{(C^{i}_{01}-C^{i}_{11})\pi^{i}_{1}}.$
(3)
If the costs are symmetric and the two hypothesis are equal likely, then the
rule in (2) reduces to the maximum likelihood (ML) decision rule
$\delta^{i}_{ML}(\mathbf{y})=\left\\{\begin{array}[]{cc}1&\textrm{~{}if~{}}p^{i}(\mathbf{y}^{i}|H_{1})\geq
p^{i}(\mathbf{y}^{i}|H_{0})\\\
0&\textrm{~{}if~{}}p^{i}(\mathbf{y}^{i}|H_{1})<p^{i}(\mathbf{y}^{i}|H_{0})\end{array}\right.,$
(4)
### III-A Sequential Hypothesis Testing
In this section, we use sequential hypothesis testing to make decisions with
minimum number of feedback from the peer IDSs, [11], [12]. An IDS asks for
feedback from its acquaintance list until a sufficient number of answers are
collected. Let $\Omega^{i}$ denote all the possible collections of feedback in
the acquaintance list to an IDS $i$ and $\omega^{i}\in\Omega^{i}$ denotes a
particular collection of feedback. Let $N^{i}(\omega^{i})$ be a random
variable denoting the number of feedbacks used until a decision is made. A
sequential decision rule is formed by a pair $(\phi,\delta)$, where
$\phi^{i}=\\{\phi_{n}^{i},n\in\mathbb{N}\\}$ is a stopping rule and
$\delta^{i}=\\{\delta_{n}^{i},n\in\mathbb{N}\\}$ is the terminal decision
rule. Introduce a stopping rule with $n$ feedback,
$\phi^{i}_{n}:\mathcal{Y}_{n}^{i}:=\prod_{j\in\mathcal{N}_{i,n}}\mathcal{Y}_{j}\rightarrow\\{0,1\\}$,
where $\mathcal{N}_{i,n}$ is the set of nodes an IDS $i$ asks up to time $n$.
$\phi^{i}_{n}=0$ indicates that IDS $i$ needs to take more samples after $n$
rounds whereas $\phi^{i}_{n}=1$ means to stop asking for feedback and a
decision can be made by the rule $\delta_{n}^{i}$. The minimum number of
feedbacks is given by
$N^{i}(\omega^{i})=\min\\{n:\phi_{n}^{i}=1,n\in\mathbb{N}\\}.$ (5)
Note that $N^{i}(\omega^{i})$ is the stopping time of the decision rule. The
decision rule $\delta^{i}$ is not used until $N.$ We assume that no cost has
incurred when a correct decision is made while the cost of a missed intrusion
is denoted by $C^{i}_{M}$ and the cost of a false alarm is denoted by
$C_{F}^{i}$. In addition, we assume each feedback incurs a cost $D^{i}$. We
introduce an optimal sequential rule that minimizes Bayes risk given by
$R^{i}(\phi^{i},\delta^{i})=R(\phi^{i},\delta^{i}|H_{0})\pi_{0}^{i}+R(\phi^{i},\delta^{i}|H_{1})\pi_{1}^{i},$
(6)
where $R(\phi^{i},\delta^{i}|H_{l}),l=0,1$, are the Bayes risks under
hypotheses $H_{0}$ and $H_{1}$, respectively:
$\displaystyle
R^{i}(\phi^{i},\delta^{i}|H_{0})=C_{F}^{i}\mathbb{P}[\delta_{N}(Y_{j},j\in\mathcal{N}_{i,N})=1|H_{0}]+D^{i}\mathbb{E}[N|H_{0}],$
$\displaystyle
R^{i}(\phi^{i},\delta^{i}|H_{1})=C_{M}^{i}\mathbb{P}[\delta_{N}(Y_{j},j\in\mathcal{N}_{i,N})=0|H_{1}]+D^{i}\mathbb{E}[N|H_{1}].$
Let $V^{i}(\pi_{0}^{i})=\min_{\phi^{i},\delta^{i}}R^{i}(\phi^{i},\delta^{i})$
be the optimal value function. It is clear that when no feedback are obtained
from the peers, the Bayes risks reduce to
$\displaystyle R^{i}(\phi_{0}^{i}=1,\delta_{0}^{i}=1)$ $\displaystyle=$
$\displaystyle C_{F}^{i}\pi_{0}^{i},$ (7) $\displaystyle
R^{i}(\phi_{0}^{i}=1,\delta_{0}^{i}=0)$ $\displaystyle=$ $\displaystyle
C_{M}^{i}\pi_{1}^{i}.$ (8)
Hence, $H_{1}$ is chosen when $C_{F}^{i}\pi_{0}^{i}<C_{M}^{i}\pi_{1}^{i}$ or
$\pi_{0}<\frac{C_{M}^{i}}{C_{F}^{i}+C_{M}^{i}}$, and $H_{0}$ is chosen
otherwise. The minimum Bayes risk under no feedback is thus obtained as a
function of $\pi_{0}^{i}$ and is denoted by
$T^{i}(\pi_{0}^{i})=\left\\{\begin{array}[]{ll}C_{F}^{i}\pi_{0}^{i}&\textrm{~{}if~{}}\pi_{0}<\frac{C_{M}^{i}}{C_{F}^{i}+C_{M}^{i}},\\\
C^{i}_{M}(1-\pi_{0}^{i})&\textrm{~{}otherwise.~{}}\end{array}\right.$ (9)
The minimum cost function (9) is a piecewise linear function. For $\phi^{i}$
such that $\phi_{0}^{i}=0$, i.e., at least one feedback is obtained, let the
minimum Bayes risk be denoted by
$J^{i}(\pi_{0}^{i})=\min_{\\{(\phi^{i},\delta^{i}):\phi^{i}_{0}=0\\}}R^{i}(\phi^{i},\delta^{i})$.
Hence, the optimal Bayes risk needs to satisfy
$V^{i}(\pi_{0}^{i})=\min\\{T^{i}(\pi_{0}^{i}),J^{i}(\pi_{0}^{i})\\}.$ (10)
Note that $J^{i}(\pi_{0}^{i})$ must be greater than the cost of one sample
$D^{i}$ as a sample request incurs $D^{i}$ and $J^{i}(\pi_{0}^{i})$ is concave
in $\pi_{0}^{i}$ as a consequence of minimizing the linear Bayes risk (6). If
the cost $D^{i}$ is high enough so that
$J^{i}(\pi_{0}^{i})>T^{i}(\pi_{0}^{i})$ for all $\pi_{0}^{i}$, then no
feedback will be requested. In this case,
$V^{i}(\pi_{0}^{i})=T^{i}(\pi_{0}^{i}),$ and the terminal rule is described in
(9). For other values of $D^{i}>0$, due to the piecewise linearity of
$T^{i}(\pi_{0}^{i})$ and concavity of $J^{i}(\pi_{0}^{i})$, we can see that
$J^{i}(\pi_{0}^{i})$ and $T^{i}(\pi_{0}^{i})$ have two intersection points
$\pi_{L}^{i}$ and $\pi_{H}^{i}$ such that $\pi_{L}^{i}\leq\pi_{H}^{i}$. It can
be shown that for some reasonably low cost $D^{i}$ and $\pi_{0}^{i}$ such that
$\pi_{L}^{i}<\pi_{0}^{i}<\pi_{H}^{i}$, an IDS optimizes its risk by requesting
another feedback; otherwise, an IDS should choose to raise an alarm when
$\pi_{0}^{i}\leq\pi^{i}_{L}$ and report no intrusion when
$\pi_{0}^{i}\leq\pi^{i}_{L}$.
Assuming that it takes the same cost $D^{i}$ for IDS $i$ to acquire a
feedback, the problem has the same form after obtaining a feedback from a
peer. IDS $i$ can use the feedback to update its apriori probability. After
$n$ feedback are obtained, $\pi_{0}^{i}$ can be updated as follows:
$\displaystyle\pi_{0}^{i}(n)$ $\displaystyle=$
$\displaystyle\frac{\pi_{0}^{i}}{\pi_{0}^{i}+(1-\pi_{0}^{i})L^{i}_{n}};$ (11)
where
$L_{n}^{i}:=\prod_{j\in\mathcal{N}_{i,n}}\frac{p(y_{j}|H_{1})}{p(y_{j}|H_{0})}.$
We can thus obtain the optimum Bayesian rule captured by Algorithm 1 below,
known as the sequential probability ratio test (SPRT) for a reasonable cost
$D^{i}$.
Algorithm 1 SPRT Rule for an IDS $i$
Step 1: Start with $n=0$. Use (12) as a stopping rule until $\phi_{n}^{i}=1$
for some $n\geq 0$.
$\phi_{n}^{i}=\left\\{\begin{array}[]{ll}0&\textrm{~{}if~{}}\pi_{L}^{i}<\pi_{0}^{i}(n)<\pi_{H}^{i},\\\
1&\textrm{~{}otherwise.~{}}\end{array}\right.$ (12)
or in terms of the likelihood ratio $L_{n}^{i}$, we can use
$\phi_{n}^{i}=\left\\{\begin{array}[]{ll}0&\textrm{~{}if~{}}A^{i}<L_{n}^{i}<B^{i}\\\
1&\textrm{~{}otherwise~{}}\end{array}\right.,$ where
$A^{i}=\frac{\pi_{0}^{i}(1-\pi_{H}^{i})}{(1-\pi_{0}^{i})\pi_{H}^{i}}$ and
$B^{i}=\frac{\pi_{0}^{i}(1-\pi_{L}^{i})}{(1-\pi_{0}^{i})\pi_{L}^{i}}$.
Step 2: Go to Step 3 if $\phi_{n}^{i}=1$ or $n=|\mathcal{N}_{i}|$; otherwise,
choose a new peer from the acquaintance list to request a diagnosis and go to
Step 2 with $n=n+1$.
Step 3: Apply the terminal decision rule as follows to determine whether there
is an intrusion.
$\delta_{n}^{i}=\left\\{\begin{array}[]{ll}1&\textrm{~{}if~{}}\pi_{0}^{i}(n)\leq\pi_{L}^{i}\\\
0&\textrm{~{}if~{}}\pi_{0}^{i}(n)>\pi_{H}^{i}\end{array}\right.$ or
$\delta_{n}^{i}=\left\\{\begin{array}[]{ll}1&\textrm{~{}if~{}}L_{n}^{i}\leq
A^{i}\\\ 0&\textrm{~{}if~{}}L_{n}^{i}>B^{i}\end{array}\right.$
### III-B Prior Probabilities
In the above section, the conditional probabilities
$p^{i}(y_{i}|H_{l}),i\in\mathcal{N},l=\\{0,1\\}$ are assumed to be known. In
this section, we use the beta distribution and its Gaussian approximation to
find the probabilities. We let $p^{i}(y_{i}=0|H_{1}):=p^{i}_{M}$ be the
probability of miss of an IDS $i$’s diagnosis, also known as the false
negative (FN) rate; and let $p^{i}_{F}:=p^{i}(y_{i}=1|H_{0})$ be the
probability of false alarm or false positive (FP) rate. The probability of
detection, or true positive (TP) rate, can be expressed as
$p^{i}_{D}=1-p^{i}_{M}$.
Based on historical data, an IDS $j$ can assess the distributions over its
peer IDS $i$’s probabilities of detection and false alarm as beta functions
parameterized by $\alpha_{i}^{F},\alpha_{i}^{D}$ and
$\beta_{i}^{F},\beta_{i}^{F}$;
$\displaystyle p^{i}_{F}\sim$
$\displaystyle\textrm{Beta}(x^{i}|\alpha^{i}_{F},\beta^{i}_{F})=\frac{\Gamma(\alpha^{i}_{F}+\beta^{i}_{F})}{\Gamma(\alpha_{F}^{i})\Gamma(\beta^{i}_{F})}x_{i}^{\alpha^{i}_{F}-1}(1-x_{i})^{\beta^{i}_{F}-1},$
(13) $\displaystyle p^{i}_{D}\sim$
$\displaystyle\textrm{Beta}(y_{i}|\alpha^{i}_{D},\beta^{i}_{D})=\frac{\Gamma(\alpha^{i}_{D}+\beta^{i}_{D})}{\Gamma(\alpha^{i}_{D})\Gamma(\beta^{i}_{D})}y_{i}^{\alpha^{i}_{D}-1}(1-y_{i})^{\beta^{i}_{D}-1},$
(14)
where $x_{i},y_{i}\in[0,1]$; $\alpha_{i}^{F},\alpha_{i}^{D}$ and
$\beta_{i}^{F},\beta_{i}^{F}$ are beta function parameters that are updated
according to historical data as follows.
$\displaystyle\alpha^{i}_{F}=\sum_{k\in\mathcal{M}_{0}}(\lambda_{F}^{i})^{t_{k}^{i}}r^{i}_{F,k},$
$\displaystyle\beta^{i}_{F}=\sum_{k\in\mathcal{M}_{0}}(\lambda_{F}^{i})^{t_{k}^{i}}(1-r^{i}_{F,k});$
(15)
$\displaystyle\alpha^{i}_{D}=\sum_{k\in\mathcal{M}_{1}}(\lambda_{D}^{i})^{t_{k}^{i}}r^{i}_{D,k},$
$\displaystyle\beta^{i}_{D}=\sum_{k\in\mathcal{M}_{1}}(\lambda_{D}^{i})^{t_{k}^{i}}(1-r^{i}_{D,k}).$
(16)
The introduction of the discount factors
$\lambda_{F}^{i},\lambda_{D}^{i}\in[0,1]$ allows more weights on recent data
from IDS $i$ while less on the old ones. The discount factors on the data can
be different for false negative and false positive rates. The parameter
$t_{k}^{i}$ denotes the time when $k$-th diagnosis data is generated (and sent
to its peer) by IDS $i$. The parameter $r^{i}_{F,k},r^{i}_{M,k}\in[0,1]$ is
the revealed results of the $k$-th diagnosis data: $r^{i}_{F,k}=1$ suggests
that the $k$-th diagnosis data from peer $i$ yields a undetected intrusion
while $r^{i}_{F,k}=0$ means otherwise; similarly, $r^{i}_{D,k}=1$ indicates
the data from the peer $i$ results in a correct detection under intrusion and
$r^{i}_{D,k}=0$ suggests otherwise. The total reported diagnosis data is the
set $\mathcal{M}$ and they are classified into two groups: one is where the
result is either false positive or true negative under no intrusion, denoted
by the set $\mathcal{M}_{0}$; and the other is where the result is either
false negative or true positive under intrusion, denoted by the set
$\mathcal{M}_{1}$. Both sets are disjoint satisfying
$\mathcal{M}_{0}\cup\mathcal{M}_{1}=\mathcal{M}$ and
$\mathcal{M}_{0}\cap\mathcal{M}_{1}=\emptyset$.
Each peer $j$ can assess a peer $i$ using (13) and (15), where we have not
included index $j$ in the expressions for simplicity. However, it is clear
that (13) and (15) are assessed from the perspective of a certain IDS $j$. In
addition, the discount factors in (15) need not be the same for all $j$.
Hence, we can implicitly view (15) dependent on $j$.
When parameters of the beta functions $\alpha$ and $\beta$ in (13) are
sufficiently large, i.e., enough data are collected, beta distribution can be
approximated by a Gaussian distribution as
$\textrm{Beta}(\alpha,\beta)\approx
N\left(\frac{\alpha}{\alpha+\beta},\sqrt{\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}}\right).$
(17)
Note that we have dropped the superscripts and subscripts in (17) for
generality as it can be applied to all $i$ in (13). Hence, using the Gaussian
approximation and (15), the expected $p^{i}_{D}$ and $p^{i}_{M}$ are given by
$\mathbb{E}[p^{i}_{F}]=\frac{\alpha_{F}^{i}}{\alpha_{F}^{i}+\beta_{F}^{i}},~{}~{}\mathbb{E}[p^{i}_{D}]=\frac{\alpha_{D}^{i}}{\alpha_{D}^{i}+\beta_{D}^{i}}.$
(18)
The mean values in (18) under large data can be intuitively interpreted as the
proportion of results of false alarm and detection in the set
$\mathcal{M}_{0}$ and $\mathcal{M}_{1}$, respectively. They can thus be used
in (1) as the assessment of the peer probability distribution $p_{j}$.
### III-C Threshold Approximation
In the likelihood sequential ratio test of Algorithm 1, the threshold values
$A$ and $B$ need to be calculated by finding $\pi^{i}_{L}$ and $\pi^{i}_{H}$
from $J^{i}(\pi_{0}^{i})$ and $T^{i}(\pi_{0}^{i})$ in (10). The search for
these values can be quite involved using dynamic programming. However, in this
subsection, we introduce an approximation method to find the thresholds. The
approximation is based on theoretical studies made in [11] and [12] where a
random walk or martingale model is used to yield a relation between thresholds
and false positive and false negative rates. Let $P^{i}_{D},P^{i}_{F}$ be the
probability of detection and the probability of false alarm of an IDS $i$
after applying the sequential hypothesis testing for feedback aggregation. We
need to point out that these probabilities are different from the
probabilities $p^{i}_{D},p^{i}_{F}$ discussed in the previous subsection,
which are the raw detection probabilities without feedback in the
collaborative network. Let $\bar{P}_{D}^{i}$ and $\bar{P}_{F}^{i}$ be
reasonable desired performance bounds such that
${P}_{F}^{i}\leq\bar{P}_{F}^{i},~{}{P}_{D}^{i}\geq\bar{P}_{D}^{i}.$ Then, the
thresholds can be chosen such that
$A^{i}=\frac{1-\bar{P}_{D}^{i}}{1-\bar{P}_{F}}^{i},~{}B^{i}=\frac{\bar{P}_{D}^{i}}{\bar{P}_{F}^{i}}.$
The next proposition gives a result on the bound of the users that need to be
on the acquaintance list to achieve the desired performances.
###### Proposition III.1
Assume that each IDS makes independent diagnosis on their peers’ requests and
each has the same distribution
$p^{i}_{0}=\bar{p}_{0}:=\bar{p}(\cdot|H_{0}),p^{i}_{1}=\bar{p}_{1}:=\bar{p}(\cdot|H_{1})$,
$\bar{p}_{0}(y_{i}=0)=\theta_{0},\bar{p}_{1}(y_{i}=0)=\theta_{1}$, for all
$i\in\mathcal{N}$.
Let $D_{KL}(\bar{p}_{0}||\bar{p}_{1})$ be the Kullback-Leibler (KL) divergence
defined as follows.
$\displaystyle D_{KL}(\bar{p}_{0}||\bar{p}_{1})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{1}\bar{p}_{0}(k)\ln\frac{\bar{p}_{0}(k)}{\bar{p}_{1}(k)},$
(19) $\displaystyle=$
$\displaystyle\theta_{0}\ln\frac{\theta_{0}}{\theta_{1}}+(1-\theta_{0})\ln\frac{1-\theta_{0}}{1-\theta_{1}}$
(20)
Likewise, the K-L divergence $D_{KL}(\bar{p}_{1}||\bar{p}_{0})$ can be
defined. On average, an IDS needs $N_{i}$ acquaintances such that
$N_{i}\geq\max\left(\lceil-\frac{D_{M}^{i}}{D_{KL}(\bar{p}_{0}||\bar{p}_{1})}\rceil,\lceil\frac{D_{F}^{i}}{D_{KL}(\bar{p}_{1}||\bar{p}_{0})}\rceil\right),$
(21)
where
$D_{M}^{i}=P_{F}\ln\left(\frac{P_{D}^{i}}{P_{F}^{i}}\right)+P_{D}\ln\left(\frac{1-P_{D}^{i}}{1-P_{F}^{i}}\right)$
and
$D_{F}^{i}=P_{F}^{i}\ln\left(\frac{1-P_{D}^{i}}{1-P_{F}^{i}}\right)+P_{D}^{i}\ln\left(\frac{P_{D}^{i}}{P_{F}^{i}}\right)$.
If $P_{F}^{i}\ll 1$ and $P_{M}^{i}\ll 1$, we need approximately $N_{i}$ such
that
$N_{i}\geq\max\left(\lceil\frac{P_{D}^{i}-1}{D_{KL}(\bar{p}_{0}||\bar{p}_{1})}\rceil,\lceil-\frac{P_{F}^{i}}{D_{KL}(\bar{p}_{1}||\bar{p}_{0})}\rceil\right).$
(22)
∎
###### Proof:
The conditional expected number of feedback needed to reach a decision on the
hypothesis in SPRT can be expressed in terms of $P_{F}$ and $P_{D}$, [11],
[12].
$\displaystyle\mathbb{E}[N|H_{0}]=$
$\displaystyle\frac{1}{-D_{KL}(\bar{p}_{0}||\bar{p}_{1})}\left[P_{F}^{i}\ln\left(\frac{P_{D}^{i}}{P_{F}^{i}}\right)+P_{D}^{i}\ln\left(\frac{1-P_{D}^{i}}{1-P_{F}^{i}}\right)\right],$
$\displaystyle\mathbb{E}[N|H_{1}]=$
$\displaystyle\frac{1}{D_{KL}(\bar{p}_{1}||\bar{p}_{0})}\left[P_{F}^{i}\ln\left(\frac{1-P_{D}^{i}}{1-P^{i}_{F}}\right)+P^{i}_{D}\ln\left(\frac{P^{i}_{D}}{P^{i}_{F}}\right)\right],$
Hence, to reach a decision we need to have at least
$\max\\{\mathbb{E}[N|H_{0}],\mathbb{E}[N|H_{1}]\\}$ independent acquaintances.
Under the assumption that both $P_{F}$ and $P_{M}^{i}$ are much less than $1$,
we can further approximate
$\mathbb{E}[N|H_{0}]\sim-\frac{1-P^{i}_{D}}{D_{KL}(\bar{p}_{0}||\bar{p}_{1})},\mathbb{E}[N|H_{1}]\sim-\frac{P^{i}_{F}}{D_{KL}(\bar{p}_{1}||\bar{p}_{0})}.$
These lead us to inequalities (22) and (21). ∎
## IV Experiments and Results
In this section, we use simulations to evaluate the efficiency of the
preceding feedback aggregation scheme and compare it with other heuristic
approaches, such as the simple average aggregation and the weighted average
aggregation. We validate and confirm our theoretical results on the number of
acquaintances needed for consultation. The results presented in this section
are produced by averaging a large number of replications with negligible
confidence intervals. The parameters we use are shown in Table I.
### IV-A Simulation Setup
TABLE I: Experimental parameters Parameter | Value | meaning
---|---|---
$\tau_{SA}$ | 0.5 | decision threshold of the simple average model
$\tau_{WA}$ | 0.5 | decision threshold of the weighted average model
$d$ | 0.5 | difficulty levels of intrusions and test messages
$\lambda_{F},\lambda_{D}$ | 0.9 | discount factors in (15)
$\pi_{0},\pi_{1}$ | 0.5 | probability of no-attack and under-attack
$C_{00},C_{11}$ | 0 | cost of correct decisions
The simulation environment uses an IDN of $N$ nodes. Each IDS is represented
by two parameters, expertise level $l$ and decision threshold $\tau_{p}$. At
the beginning, each peer receives an initial acquaintance list containing all
the other neighbor nodes. In the process of the collaborative intrusion
detection, a node sends out requests to its acquaintances for intrusion
assessments. The feedback collected are used to make a final decision, i.e.,
whether to raise an alarm or not. We implement three different feedback
mechanisms, namely, simple average aggregation, weighted average aggregation,
and hypothesis testing aggregation. We compare their efficiency by the average
cost of false decisions.
#### IV-A1 Simple Average Model
If the average of all feedback exceeds a threshold $\tau_{SA}$, then an alarm
is raised. $\tau_{SA}$ is set to $0.5$ if no cost difference is considered for
making FP and FN decisions. The simple average mechanism to aggregate feedback
is adopted in the literature such as [1].
#### IV-A2 Weighed Average Model
Weights are assigned to feedback from different IDSs to calculate weighted
average. Weighted average is widely used to aggregate feedback, such as [2]
and [3], where weights are the trust values of IDSs and trust values are
calculated based on their past history. If the weighted average is greater
than a threshold $\tau_{WA}$, then an alarm is raised. $\tau_{WA}$ is fixed to
$0.5$ in our experiments because their models do not consider the cost
difference between FP and FN. In this simulation, we adopt trust values from
[3] as the weights of feedback.
Figure 2: FP and FN vs. Expertise Level
Figure 3: FP and FN vs. Threshold $\tau_{p}$
Figure 4: Average Cost vs. Threshold $\tau_{p}$
Figure 5: Cost vs. $C_{01}$ under three models
Figure 6: FP, TP vs. Number of Acquaintances
Figure 7: Number of Acquaintances vs. Expertise
### IV-B Modeling of an Individual IDS
To simulate the intrusion detection capability of each node, we use a Beta
distribution for the decision model of an IDS. A Beta density function is
given by
$\displaystyle
f(\bar{p}|\bar{\alpha},\bar{\beta})=\frac{1}{B(\bar{\alpha},\bar{\beta})}\bar{p}^{\bar{\alpha}-1}(1-\bar{p})^{\bar{\beta}-1},$
(23)
$\displaystyle\bar{\alpha}=1+\frac{l(1-d)}{d(1-l)}r,~{}~{}\bar{\beta}=1+\frac{l(1-d)}{d(1-l)}(1-r).$
where
$B(\bar{\alpha},\bar{\beta})=\int_{0}^{1}t^{\bar{\alpha}-1}(1-t)^{\bar{\beta}-1}dt$,
$\bar{p}\in[0,1]$ is the probability of intrusion assessed by the host IDS.
$f(\bar{p}|\bar{\alpha},\bar{\beta})$ is the probability that a peer with
expertise level $l\in[0,1]$ answers with a value of $\bar{p}$ to an intrusion
assessment of difficulty level $d\in[0,1]$. Higher values of $d$ are
associated with attacks that are difficult to detect, i.e., many peers may
fail to identify them. Higher values of $l$ imply a higher probability of
producing correct intrusion assessment. $r\in\\{0,1\\}$ is the expected result
of detection. $r=1$ indicates that there is an intrusion and $r=0$ indicates
that there is no intrusion.
Let $\tau_{p}$ be the decision threshold of $\bar{p}$. If $\bar{p}>\tau_{p}$,
a peer sends feedback $1$ (i.e., under-attack); otherwise, feedback $0$ (i.e.,
no-attack) is generated.
For a fixed difficulty level, the preceding model assigns higher probabilities
of producing correct intrusion diagnosis to peers with higher level of
expertise. $l=1$ or $d=0$ represent extreme cases where the peer can always
accurately detect the intrusion. This is reflected in the Beta distribution
with $\bar{\alpha},\bar{\beta}\rightarrow\infty$.
Figure 7 shows that both the FP and FN decrease when the expertise level of an
IDS increases. We notice that the curves of FP rate and FN rate overlap. This
is because the IDS detection density distributions are symmetric under $r=0$
and $r=1$. Figure 7 shows that the FP rate decreases with the decision
threshold while the FN rate increases with the decision threshold. When the
decision threshold is $0$, all feedback are positive (under-attack); when the
decision threshold is $1$, all feedback are negative (no-attack).
### IV-C Detection Accuracy and Cost
One of the most important metrics to evaluate a feedback aggregation scheme is
the cost of incorrect decisions. In this experiment, we study the costs of the
three aggregation models using a simulated network. We set $N=10$ and fix the
expertise level $l$ of all nodes to $0.5$ and set $C_{10}=C_{01}=1$ in (3) for
the fairness of comparison, since the simple average and the weighted average
models do not account for the cost difference between FP and FN. We fix the
decision threshold for each IDS ($\tau_{p}$) to $0.1$ for the first batch run
and then increase it by $0.1$ in each subsequent batch run until it reaches
$0.9$. We measure the cost of the three models. As shown in Figure 7, the
costs yielded by the aggregation using hypothesis testing remains the lowest
among the three under all threshold settings. The costs of the weighted
average and the simple average are close to each other. This is because in
this experiment, the weights of all IDSs are the same. Therefore, the
difference between the weighted average and the simple average is not
substantial. We also observe that changing the threshold has a big impact on
the costs of the weighted average and the simple average, while the cost of
the hypothesis testing changes only slightly with the thresholds. All costs
reach a minimum when the threshold is $0.5$ and increase when it deviates from
$0.5$.
In the next experiment, the expertise levels of all nodes remain $0.5$ and
their decision thresholds vary from $0.1$ to $0.9$. We set $C_{10}=C_{01}=1$
in the first batch run and increase $C_{01}$ by $1$ in every subsequent batch
run. We observe the costs under three different models. Figure 7 shows that
the costs of the simple average model and the weighted average model increase
linearly with $C_{01}$ while cost of hypothesis testing model grows the
slowest among the three. This is because the hypothesis testing model has a
flexible threshold to optimize its cost. The hypothesis testing model has
superiority when the cost difference between FP and FN is large.
### IV-D Sequential Consultation
In this experiment, we study the number of acquaintances needed for
consultation to reach a predefined goal. Suppose the TP lower-bound
$\bar{P}_{D}=0.95$ and FP upper-bound $\bar{P}_{F}=0.1$. We observe the change
of FP rate and TP rate with the number of acquaintances consulted ($n$).
Figure 7 shows that FP rate decreases and TP rate increases with $n$.
Consulting higher expertise nodes leads to a higher TP rate and a lower FP
rate. In the next experiment we implement Algorithm 1 on each node and measure
the average number of acquaintances needed to reach the predefined TP lower-
bound and the FP upper-bound. Figure 7 compares the simulation results with
the theoretical results (see (22)), where the former confirms the latter. In
both cases, the number of consultations decreases quickly with the expertise
levels of acquaintances. For example, the IDS needs to consult around $50$
acquaintances of expertise $0.2$, while only $3$ acquaintances of expertise
$0.7$ are needed for the same purpose. This is partly because low expertise
nodes are more likely to make conflicting feedbacks and consequently increase
the number of consultations. The analytical results can be useful for IDSs to
design the size of their acquaintance lists.
## V Conclusion
In this paper, we have presented a sequential hypothesis testing approach to
feedback aggregation in a collaborative intrusion detection network. In this
mechanism, an IDS consults sequentially for peer diagnoses until it is capable
of making an aggregated decision that satisfies Bayes optimal cost criterion.
The decision is made based on a threshold rule leveraging the likelihood ratio
approximated by beta distribution and thresholds by target rates. Our
experimental results show that our proposed feedback aggregation model is
superior to other proposed models in the literature in terms of cost
efficiency. Our simulation results have also corroborated our theoretical
results on the average number of acquaintances needed to reach the predefined
false positive upper-bound and true positive lower-bound. As future work, we
intend to investigate the robustness of the collaboration system against
malicious insiders, especially under collusion attacks. Furthermore, we aim to
extend our results to deal with the case of correlated feedbacks.
## References
* [1] P. Resnick, R. Zeckhauser, J. Swanson, and K. Lockwood, “The value of reputation on eBay: A controlled experiment,” _Experimental Economics_ , vol. 9, no. 2, pp. 79–101, 2006.
* [2] C. Duma, M. Karresand, N. Shahmehri, and G. Caronni, “A trust-aware, p2p-based overlay for intrusion detection,” in _DEXA Workshops_ , 2006.
* [3] C. Fung, J. Zhang, I. Aib, and R. Boutaba, “Robust and scalable trust management for collaborative intrusion detection,” in _11th IFIP/IEEE International Symposium on Integrated Network Management_ , 2009.
* [4] R. Janakiraman and M. Zhang, “Indra: a peer-to-peer approach to network intrusion detection and prevention,” _Proceedings of the 12th IEEE International Workshops on Enabling Technologies_ , 2003.
* [5] V. Yegneswaran, P. Barford, and S. Jha, “Global intrusion detection in the domino overlay system,” in _Proceedings of Network and Distributed System Security Symposium_ , 2004.
* [6] M. Cai, K. Hwang, Y. Kwok, S. Song, and Y. Chen, “Collaborative internet worm containment,” _IEEE Security & Privacy_, vol. 3, no. 3, pp. 25–33, 2005\.
* [7] A. Ghosh and S. Sen, “Agent-based distributed intrusion alert system,” in _Proceedings of the 6th International Workshop on Distributed Computing (IWDC’04)_. Springer, 2004.
* [8] C. Fung, O. Baysal, J. Zhang, I. Aib, and R. Boutaba, “Trust management for host-based collaborative intrusion detection,” in _19th IFIP/IEEE International Workshop on Distributed Systems_ , 2008.
* [9] J. Tsitsiklis, “Decentralized detection,” _Advances in Statistical Signal Processing_ , pp. 297–344, 1993.
* [10] K. Nguyen, T. Alpcan, and T. Başar, “A Decentralized Bayesian Attack Detection Algorithm for Network Security,” in _Proceedings of the 23rd International Information Security Conference_ , 2005.
* [11] A. Wald, _Sequential Analysis_. John Wiley and Sons, 1947.
* [12] B. C. Levy, _Principles of Signal Detection and Parameter Estimation_. Springer-Verlag, 2008.
|
arxiv-papers
| 2010-02-17T00:36:16 |
2024-09-04T02:49:08.420290
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quanyan Zhu, Carol J. Fung, Raouf Boutaba and Tamer Basar",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1002.3190"
}
|
1002.3198
|
[labelstyle=]
# Subcoalgebras and endomorphisms of free Hopf algebras
Alexandru Chirvăsitu University of California, Berkeley, 970 Evans Hall #3480,
Berkeley CA, 94720-3840, USA chirvasitua@gmail.com
###### Abstract.
For a matrix coalgebra $C$ over some field, we determine all small
subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a
bijective antipode on $C$, and the free Hopf algebra with antipode $S$
satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this
information to find the endomorphisms of these free Hopf algebras, and to
determine the centers of the categories of Hopf algebras, Hopf algebras with
bijective antipode, and Hopf algebras with antipode of order dividing $2d$.
###### Key words and phrases:
free Hopf algebra, matrix coalgebra, center of a category
###### 2000 Mathematics Subject Classification:
16T99, 16T15, 18A40
## Introduction
The free Hopf algebra $H(C)$ on a coalgebra $C$ (over some base field $k$) was
introduced by Takeuchi in [Ta], and this construction was used to give the
first example of a Hopf algebra with non-bijective antipode: if $n>1$,
Takeuchi shows that the antipode of $H(M_{n}(k)^{*})$ is not bijective. Later,
Nichols ([Ni]) constructed bases for $H(M_{n}(k)^{*})$, and showed that for
$n>1$ the antipode is injective.
In [Sc], Schauenburg constructed the free Hopf algebra with bijective antipode
on a Hopf algebra. This can be combined with Takeuchi’s construction to yield
a left adjoint for the forgetful functor from the category of Hopf algebras
with bijective antipode to that of coalgebras ([Sc, Lemma 3.1]). In the same
paper, a basis for the free Hopf algebra with bijective antipode (which we
will denote by $H_{\infty}(C)$) on a matrix coalgebra $C$ was constructed, by
methods analogous to those used by Nichols in [Ni], and used to give the first
examples of a Hopf algebra with surjective, non-injective antipode.
In this paper we study the subcoalgebras and endomorphisms of these objects,
and also of the free Hopf algebra $H_{d}(M_{n}(k)^{*})$ with antipode whose
order divides $2d$ ($n>1$). This seems not to have been done in too much
detail in the literature. Further motivation comes from the desire to study
the so-called centers of the categories appearing in the above discussion,
i.e. their monoids of self-natural transformations of the identity functor:
the category ${\rm HopfAlg}$ of Hopf algebras, ${\rm HopfAlg}_{\infty}$ of
Hopf algebras with bijective antipode, and ${\rm HopfAlg}_{d}$ of Hopf
algebras with antipode of a given order $2d$ (for some positive integer $d$).
For a justification for the term “center”, notice that if our category is a
monoid (i.e. a one-object category), then its center is precisely the center
of the monoid. As another example, notice that there is an obvious isomorphism
between the center of the category ${}_{A}\mathcal{M}$ of left modules over a
ring $A$, and the center of $A$.
One would expect, for example, that the center of ${\rm HopfAlg}$ is the free
monoid on one element generated by the square of the antipode, together with a
“multiplicative $0$”, the (natural transformation induced by the) trivial
endomorphism. It is obvious from this statement that antipodes cannot all be
bijective, so it can be regarded as a natural generalization of Takeuchi’s
results in [Ta]. We prove this result and the analogous ones for ${\rm
HopfAlg}_{\infty}$ and ${\rm HopfAlg}_{d}$ below, using the fact that, as will
become apparent, the free Hopf algebras mentioned above on an $n\times n$
matrix coalgebra ($n>1$) have, in a certain sense, “no more endomorphisms than
expected” (in most cases).
The paper is organized as follows:
In Section 1 we introduce the notations and conventions to be used throughout,
and recall a few facts on free Hopf algebras.
In Section 2 the main results are proven. We show that the free Hopf algebra
$H(C)$, the free Hopf algebra with bijective antipode $H_{\infty}(C)$, and the
free Hopf algebra $H_{d}(C)$ with antipode of order $2d\geq 4$ on an $n\times
n$ matrix coalgebra $C$ ($n>1$) contains no subcoalgebras of dimension $\leq
n^{2}$, other than the $1$-dimensional coalgebra $k$ and the iterates of $C$
by the antipode. $d=1$ is a little trickier, so in this case, we prove our
result only for $n>2$ or in characteristic zero. We then use this to prove
that the Hopf algebra endomorphisms of $H(C),\ H_{\infty}(C)$ and $H_{d}(C)$
are precisely those we would expect, in most cases (i.e. apart from $d=1,n=2$,
positive characteristic): the compositions of those induced by the
(anti)endomorphisms of $C$ with the powers of the antipode, and also the
trivial endomorphism (unit composed with counit); note that an
antiendomorphism of $C$ is required if we are to compose with an odd power of
the antipode.
In Section 3, the results outlined above are used to determine the centers of
the categories ${\rm HopfAlg},\ {\rm HopfAlg}_{\infty}$, and ${\rm
HopfAlg}_{d}$. Again, the final result is exactly as expected: in all three
cases we have the (natural transformations induced by) the trivial
endomorphism, and the even powers $S^{2t}$ of the antipode $S$. Of course, in
the three cases in question, $t$ ranges through the appropriate sets: the non-
negative integers, the integers, and $\mathbb{Z}/2d$ respectively.
Finally, in Section 4 we look at the exceptions mentioned before: $d=1,n=2$,
positive characteristic. It is shown that indeed, there are counterexamples to
the results in Section 2 in characteristic $2$ or $3$.
## 1\. Preliminaries
We work over some fixed field $k$. Algebras, coalgebras, Hopf algebras, etc.
are over $k$, and all (co)algebras are (co)unital and (co)associative. We
assume familiarity with basic Hopf algebra theory, as in [Sw, A] or [Mo], for
instance. We denote the categories of Hopf algebras, Hopf algebras with
bijective antipode, and Hopf algebras with antipode $S$ such that $S^{2d}={\rm
id}$ ($d\geq 1$) by ${\rm HopfAlg},\ {\rm HopfAlg}_{\infty}$ and ${\rm
HopfAlg}_{d}$ respectively. We reserve the usual notations for other
categories that might appear (${\rm CoAlg}$ is the category of $k$-coalgebras,
for example). The usual symbols are used for the structure maps of our
coalgebras, bialgebras, etc.: $\Delta,\varepsilon,S$ denote the
comultiplication, counit and respectively the antipode of an appropriate
object. We might use the name of the object as a subscript for the structure
map: $S_{H}$ is the antipode of the Hopf algebra $H$, for example.
Recall ([Ta]) that the forgetful functor ${\rm HopfAlg}\to{\rm CoAlg}$ has a
left adjoint; there is a free Hopf algebra $H(C)$ on any coalgebra $C$, with
the usual universal property. Similarly ([Sc]), there is a free Hopf algebra
$H_{\infty}(C)$ on any coalgebra $C$. Using the exact same techniques as in
those papers, or, alternatively, just factoring $H_{\infty}(C)$ through the
appropriate ideal, we have the following result:
###### Proposition 1.1.
For every positive integer $d$, the forgetful functor ${\rm
HopfAlg}_{d}\to{\rm CoAlg}$ has a left adjoint.
We denote this left adjoint by $H_{d}(-)$. The proof is entirely routine, and
is left to the reader; one simply factors $H_{\infty}(C)$ (or even $H(C)$)
through the appropriate ideal to get $H_{d}(C)$. When we wish to state a
result in a unified manner for $H,H_{\infty}$ and $H_{d}$ all at once, we use
the notation $\tilde{H}(C)$ to stand for either one of them.
In most cases for us (but not always), $C$ will be an $n\times n$ matrix
coalgebra for some $n>1$. This is the dual $M_{n}(k)^{*}$ of the matrix
algebra $M_{n}(k)$; it has a basis $(t_{ij})_{i,j=1}^{n}$, with the coalgebra
structure given by
$\Delta(t_{ij})=\sum_{k}t_{ik}\otimes t_{kj},\
\varepsilon(t_{ij})=\delta_{ij},$ (1)
where $\delta_{ij}$ is the Kronecker delta, as usual.
An important tool for us will be the $k$-basis constructed by Nichols for
$H(M_{n}(k)^{*})$ ([Ni]), and the analogous bases for $H_{\infty}$ ([Sc]) and
$H_{d}$. To my knowledge, the $H_{d}$ case has not appeared in the literature,
but the calculations are mostly parallel to those used for $H$ and
$H_{\infty}$, and we do not repeat those here. The only problem when trying to
adapt the proof in [Ni] to $H_{d}$ arises when $d=1$. We address this briefly
below, again, omitting the verifications.
Recall the notation $\tilde{H}$ introduced above. Below, it is understood that
$r$ ranges through the non-negative integers if $\tilde{H}=H$, through
$\mathbb{Z}$ if $\tilde{H}=H_{\infty}$, and through $\mathbb{Z}/2d$ if
$\tilde{H}=H_{d}$. Consider the set $\mathcal{X}=\\{x_{ij}^{r},\ |\
i,j=\overline{1,n},r\\}$. We now work inside the free algebra
$k\langle\mathcal{X}\rangle$ on $\mathcal{X}$, and seek to write
$\tilde{H}(M_{n}(k)^{*})$ as a quotient of $k\langle\mathcal{X}\rangle$, using
Bergman’s diamond lemma ([Be]). The images of $x^{0}_{ij}$ in
$\tilde{H}(M_{n}(k)^{*})$ are supposed to be matrix generators for
$M_{n}(k)^{*}$, and we want the antipode to act by sending $x^{r}_{ij}$ to
$x^{r+1}_{ji}$. The diamond lemma comes in when trying to factor out the
relations imposed by the condition that this map be an antipode. In [Ni, Sc],
this is done as follows (for $\tilde{H}=H$ and $H_{\infty}$, respectively):
Following [Sc], we say that a monomial $w$ in the free monoid
$\langle\mathcal{X}\rangle$ on $\mathcal{X}$ is less than another monomial
$w^{\prime}$ if either $w$ is shorter than $w^{\prime}$, or if they have the
same length, the same sequence of $r$-indices, and the sequence of $i,j$
indices of $w$ is lexicographically less than that of $w^{\prime}$. Now
consider the reductions
$x_{in}^{r}x_{jn}^{r+1}\to\delta_{ij}-\sum_{a<n}x_{ia}^{r}x_{ja}^{r+1}$ (2)
$x_{ni}^{r+1}x_{nj}^{r}\to\delta_{ij}-\sum_{a<n}x_{ai}^{r+1}x_{aj}^{r}$ (3)
$x_{in}^{r}x_{jn-1}^{r+1}x_{kn-1}^{r+2}\to\delta_{jk}x_{in}^{r}-\delta_{ij}x_{kn}^{r+2}+\sum_{a<n}x_{ia}^{r}x_{ja}^{r+1}x_{kn}^{r+2}-\sum_{a<n-1}x_{in}^{r}x_{ja}^{r+1}x_{ka}^{r+2}$
(4)
$x_{ni}^{r+2}x_{n-1j}^{r+1}x_{n-1k}^{r}\to\delta_{jk}x_{ni}^{r+2}-\delta_{ij}x_{nk}^{r}+\sum_{a<n}x_{ai}^{r+2}x_{aj}^{r+1}x_{nk}^{r}-\sum_{a<n-1}x_{ni}^{r+2}x_{aj}^{r+1}x_{ak}^{r}.$
(5)
It is shown in [Ni] that this data satisfies the hypotheses of the diamond
lemma (although Nichols uses a slightly but not essentially different
semigroup partial order), and hence the irreducible words (i.e. those which
contain no subwords as in the left-hand-sides of the reductions above) form a
$k$-basis for $H(M_{n}(k)^{*})$ (with $r\in\mathbb{N}$). In [Sc] it is claimed
that the same is true for $H_{\infty}(M_{n}(k)^{*})$ with $r\in\mathbb{Z}$,
and we claim here that this holds for $H_{d}$ and $r\in\mathbb{Z}/2d$ as well,
and hence that the irreducible words form a basis for $\tilde{H}$.
As mentioned before, there are some problems when $d=1$: more ambiguities
appear in this situation, which would not appear otherwise. This means that we
need to check that these ambiguities are resolvable (using the language of
[Be]). One obvious example of such an ambiguity is $x_{nn}^{r}x_{nn}^{r+1}$,
since for $d=1$ both (2) and (3) can be used to reduce this word. Similarly,
another example (and the most tedious to resolve) of ambiguity which doesn’t
arise in general is $x_{in}^{r}x_{nn-1}^{r+1}x_{n-1n-1}^{r}x_{n-1j}^{r+1}$.
Although, luckily, the new ambiguities do resolve under the reductions above,
we do not perform the long but entirely straightforward calculations here.
By a slight abuse of notation, we denote the images of the $x_{ij}^{r}$’s in
the quotient $\tilde{H}$ of $k\langle\mathcal{H}\rangle$ by the same symbols.
As noted above, the Hopf algebra structure is given by the fact that for every
$r$, the $x_{ij}^{r}$ behave as the usual generators of an $n\times n$ matrix
coalgebra (as in (1)), and the antipode acts by
$S(x_{ij}^{r})=x_{ji}^{r+1},\ \forall r,i,j.$
The coalgebra $M_{n}(k)^{*}\subset\tilde{H}(M_{n}(k)^{*})$ is identified with
$\\{x_{ij}^{0}\\}_{i,j}$, and hence $\\{x_{ij}^{r}\\}_{i,j}$ are its iterates
through the antipode. Again, we refer to [Ni, Sc] for details.
In order to deal effectively with monomials in the $x_{ij}^{r}$’s, we use the
following notation: bold letters such as ${\bf r}$ and ${\bf i}$ represent
vectors of indices, i.e. ${\bf r}=(r_{1},r_{2},\ldots,r_{t})$. By $x_{{\bf
i}{\bf j}}^{{\bf r}}$ we mean the monomial $x_{i_{1}j_{1}}^{r_{1}}\ldots
x_{i_{t}j_{t}}^{r_{t}}$. Note that the length of a vector ${\bf i}$ may vary,
but in order for $x_{{\bf i}{\bf j}}^{{\bf r}}$ to make sense, ${\bf i}$,
${\bf j}$ and ${\bf r}$ must all have the same length.
Notice also that given a fixed vector ${\bf r}$ as above, the linear span of
the monomials $x_{{\bf i}{\bf j}}^{{\bf r}}$ is a subcoalgebra $C_{\bf r}$ of
$\tilde{H}(M_{n}(k)^{*})$. Moreover, $\tilde{H}(M_{n}(k)^{*})$ is the sum of
all $C_{\bf r}$’s, so every simple subcoalgebra is contained in one of the
$C_{\bf r}$’s.
The notation introduced above will be used freely throughout the rest of the
paper. Here are a few more observations on free Hopf algebras which will be
useful in the sequel:
###### Remark 1.2.
It is shown in [Ta] that the functor $H(-)$ behaves well with respect to
scalar extension to a larger field. More precisely, if $k\to K$ is a field
extension, then $H(C)\otimes K$ is naturally isomorphic to $H(C\otimes K)$.
The analogous results for $\tilde{H}$ are very easy to prove, using the
universal property of $\tilde{H}(C)$.
###### Remark 1.3.
Also in [Ta], it is shown that for any coalgebra $C$, if $C_{0}$ denotes its
coradical and $C=C_{0}\oplus V$ for some vector space $V$, then $H(C)$ is
$H(C_{0})\coprod T(V)$ as an algebra. Here, $T(V)$ is the tensor algebra on
$V$ and the coproduct is in the category of algebras. The same is in fact true
if we replace $H$ with $\tilde{H}$. This can be seen by examining Takeuchi’s
proofs ([Ta, Lemmas 26, 27, 28]) and checking that they work in general.
In particular, it follows easily from this that an inclusion of coalgebras
$C\to D$ induces an inclusion $\tilde{H}(C)\to\tilde{H}(D)$.
## 2\. Main results
As outlined in the introduction, the purpose of this section is to find all
small subcoalgebras and all endomorphisms of $\tilde{H}(M_{n}(k)^{*})$. The
latter will be a consequence of the former, since, by the universality
property of $\tilde{H}$, an endomorphism of $\tilde{H}(C)$ is the same as a
coalgebra map from $C$ to $\tilde{H}(C)$. We will first state the main result;
as before, $\tilde{H}(-)$ is one of $H(-),\ H_{\infty}(-)$ or $H_{d}(-)$ for
some positive integer $d$.
We have mentioned before that the cases $d=1,n=2$, positive characteristic
pose some problems. It will be convenient to have a short phrase which refers
to all other cases; hence, we say that we are in a tame case or situation if
either $d\geq 2$, or $n>2$, or ${\rm char}(k)=0$. Otherwise, we say that we
are in a wild case.
###### Theorem 2.1.
Let $n>1$ be a positive integer. In a tame situation, the only subcoalgebras
of $\tilde{H}(M_{n}(k)^{*})$ of dimension $\leq n^{2}$ are $k$ and the
iterates of $M_{n}(k)^{*}\subset\tilde{H}(M_{n}(k)^{*})$ by the antipode.
###### Remark 2.2.
In fact, it turns out that everything works fine as long as the characteristic
is not $2$ or $3$. However, I’ve chosen to state the theorem as above, in
order to keep the proof shorter (it is long enough as it is), and because it
did not seem worthwhile to insist on the greatest possible generality.
Before going into the proof of the theorem, we record the desired
consequences, namely the determination of the endomorphisms of
$\tilde{H}(M_{n}(k)^{*})$ in a tame case. By the functoriality of $\tilde{H}$,
an (anti)endomorphism of $C$ induces an (anti)endomorphism of $\tilde{H}(C)$.
We identify these, to avoid having to repeat the words “induced by” all the
time.
###### Proposition 2.3.
Let $n>1$ be a positive integer. In a tame case, the endomorphisms of
$\tilde{H}(M_{n}(k)^{*})$ are of one of the following types:
1. (a)
The trivial endomorphism, induced by
$M_{n}(k)^{*}\stackrel{{\scriptstyle\varepsilon}}{{\rightarrow}}k\to\tilde{H}$;
2. (b)
$S^{2t}\circ\alpha$, where $t\geq 0$ and $\alpha$ is an automorphism of
$M_{n}(k)^{*}$;
3. (c)
$S^{2t+1}\circ T\circ\alpha$, where $t\geq 0,\ \alpha$ is an automorphism of
$M_{n}(k)^{*}$, and $T$ is the transposition map $x_{ij}^{0}\mapsto
x_{ji}^{0}$ on $M_{n}(k)^{*}\subset\tilde{H}(M_{n}(k)^{*})$.
###### Remark 2.4.
By the Skolem-Noether Theorem, the automorphisms of $M_{n}(k)^{*}$ are
precisely the conjugations by $GL_{n}(k)$. But it is easily seen that in
general, for a coalgebra $C$, a map $C\to k$ is convolution-invertible if and
only if it factors through some algebra map $\tilde{H}(C)\to k$ (for
$\tilde{H}(-)=H(-)$, for example, this follows immediately from [Ta, $\S$2,
Proposition 4], which characterizes algebra maps out of $H(C)$ in terms of
maps out of $C$). This means that the automorphisms $\alpha$ in the statement
of Proposition 2.3 are precisely the inner automorphisms of
$\tilde{H}(M_{n}(k)^{*})$, in the sense that they are the conjugations (under
convolution) by the algebra maps $\tilde{H}(M_{n}(k)^{*})\to k$.
###### Proof of Proposition 2.3.
As observed before, an endomorphism of $\tilde{H}(C)$ is determined uniquely
by a coalgebra map $C\to\tilde{H}(C)$, so we focus on finding these. Of
course, the image of a coalgebra map $C=M_{n}(k)^{*}\to\tilde{H}(C)$ is a
subcoalgebra of dimension no larger than $n^{2}$, so Theorem 2.1 applies. We
thus find that our maps go either to $k$ (in which case it can only be the
counit of $C$, and we are in situation (a)), or to some iterate $S^{r}(C)$.
Up to an automorphism of $C=\\{x_{ij}^{0}\ |\ i,j\\}$, the map in question is
$x_{ij}^{0}\mapsto x_{ij}^{r}$. This is exactly $S^{r}$ if $r$ is even, and
$S^{r}\circ T$ if $r$ is odd. ∎
For the proof of Theorem 2.1, we’ll need some auxiliary results. The following
lemma is an elementary linear algebra fact, whose proof we leave to the
reader:
###### Lemma 2.5.
Let $V,W$ be vector spaces, and $X\leq V$, $Y\leq W$ vector subspaces. Suppose
we have an element
$\sum_{i=1}^{p}a_{i}\otimes b_{i}\in X\otimes Y,$
where $a_{i}\in V$ are linearly independent, and similarly, $b_{i}\in W$ are
linearly independent. Then, both $X$ and $Y$ have dimension $\geq p$.
We now prove a result in some sense weaker than the statement of Theorem 2.1,
but which holds in wild cases too.
###### Proposition 2.6.
Let $n>1$ be a positive integer. The subcoalgebras of
$\tilde{H}(M_{n}(k)^{*})$ different from $k$ have dimension $\geq n^{2}$.
###### Proof.
By Remark 1.2, it suffices to consider the case when our base field $k$ is
algebraically closed. This is to ensure that simple coalgebras are actually
matrix coalgebras, which will be useful in the proof. Hence, throughout the
rest of the argument, $k$ is assumed to be algebraically closed.
Let $H=\tilde{H}(M_{n}(k)^{*})$, and consider an arbitrary element $x\in H$.
$x$ can be written as a linear combination of irreducible monomials in the
standard algebra generators $x_{ij}^{r}$ introduced before. Let $x_{{\bf
i}{\bf j}}^{{\bf r}}$ be such a monomial, having maximal length $\ell$ among
the monomials appearing in $x$. We assume $\ell\geq 2$.
We look at $\Delta(x)$, using the matrix comultiplication
$\Delta(x_{ij}^{r})=\sum_{a=1}^{n}x_{ia}^{r}\otimes x_{aj}^{r},$
and expanding. To get the final result in reduced monomials, we might have to
reduce some of the monomials we get by expansion.
Now fix a pair $(\alpha,\beta)$ of distinct indices in $\overline{1,n}$, and
consider the vector ${\bf u}=(\alpha,\beta,\alpha,\beta,\ldots)$, having the
same length $\ell$ as ${\bf r}$. The term $x_{{\bf i}{\bf u}}^{{\bf r}}\otimes
x_{{\bf u}{\bf j}}^{{\bf r}}$ will appear in $\Delta(x)$ (after all the
reductions have been made). This follows because on the one hand $x_{{\bf
i}{\bf u}}^{{\bf r}}$ and $x_{{\bf u}{\bf j}}^{{\bf r}}$ are reduced (as a
consequence of the fact that $x_{{\bf i}{\bf j}}^{{\bf r}}$ was reduced and
the form of the reduction rules (2), (3), etc.), and on the other hand because
of the maximality of the length of $x_{{\bf i}{\bf j}}^{{\bf r}}$, which
implies that $x_{{\bf i}{\bf u}}^{{\bf r}}$ and $x_{{\bf u}{\bf j}}^{{\bf r}}$
cannot be non-trivial reductions of some other monomials we run into when
trying to compute $\Delta(x)$.
Applying this argument to all $n(n-1)$ ordered pairs $(\alpha,\beta)$ of
distinct indices in $\overline{1,n}$, we find that $\Delta(x)$, in its unique
form as a linear combination of tensor products of reduced monomials, contains
all $x_{{\bf i}{\bf u}}^{{\bf r}}\otimes x_{{\bf u}{\bf j}}^{{\bf r}}$ (as
before, having fixed the pair $(\alpha,\beta)$, we set ${\bf u}$ to be
$(\alpha,\beta,\alpha,\ldots)$).
Now consider a simple subcoalgebra $C\subset H$. $C$ is a matrix coalgebra,
because $k$ is algebraically closed. Assuming $C$ is neither $k$ nor one of
the coalgebras $\\{x_{ij}^{r}\\}_{i,j}$, all of its elements contain monomials
of length $\geq 2$. In conclusion, the argument above applies to all elements
$x\in C$. If $C$ is, say, an $m\times m$ matrix coalgebra, then we can find an
element $x\in C$ such that $\Delta(x)\in M\otimes N$, where $M,N$ are linear
spaces of dimension $m$ (for example, $x$ can be part of a system of matrix
generators, such as the $t_{ij}$ in (1)). Pick a monomial $x_{{\bf i}{\bf
j}}^{{\bf r}}$ for $x$, using the notations above. Let $P\leq H$ (resp. $Q\leq
H$) be the linear subspace generated by all monomials not of the form $x_{{\bf
i}{\bf u}}^{{\bf r}}$ (resp. $x_{{\bf i}{\bf j}}^{{\bf r}}$), where ${\bf u}$,
as before, ranges through $(\alpha,\beta,\alpha,\ldots)$. Now, applying Lemma
2.5 to $V=H/P$, $W=H/Q$, $X=M+P/P$ and $Y=N+Q/Q$, we conclude that the
dimension of $M+P/P$ (and hence that of $M$) is at least $n(n-1)$. Hence,
$m\geq n(n-1)\geq n$.
This proves that simple subcoalgebras of $H$ are either $k$, one of the
iterates of $\\{x_{ij}^{0}\\}$ through the antipode, or $m\times m$ matrix
subcoalgebras with $m\geq n(n-1)\geq n$. This implies that any subcoalgebra
$C$ of $H$ of dimension $\leq n^{2}$ is either connected with coradical $k$,
or an $n\times n$ matrix coalgebra. The former is impossible, however, unless
$C=k$, because $H$ has no non-zero primitive elements: such a primitive
element $x$ would have to contain a monomial of length $\geq 2$, and the
argument above would show that the image of $\Delta(x)$ in $(H/P)\otimes(H/Q)$
is non-zero; but $1\in P\cap Q$, so the image of $1\otimes x+x\otimes 1$ is
$(H/P)\otimes(H/Q)$ is zero.
This finishes the proof of the proposition. ∎
The argument used in the previous proof is the essential ingredient in Theorem
2.1, and it will appear in various guises throughout the rest of our proof of
the main theorem. As a consequence of this argument, we already have the
following partial result:
###### Corollary 2.7.
The conclusion of Theorem 2.1 holds if $n>2$.
###### Proof.
Again, we may as well assume the base field $k$ is algebraically closed.
We remarked in the last paragraph of the proof for Proposition 2.6 that (a)
the simple subcoalgebras of $\tilde{H}(M_{n}(k)^{*})$ different from $k$ or
$\\{x_{ij}^{r}\\}_{i,j}$ are $m\times m$ matrix coalgebras with $m\geq n(n-1)$
(which is strictly larger than $n$ if $n>2$) and (b) there are no
subcoalgebras with coradical $k$. The conclusion is now clear. ∎
In view of this corollary, we can focus on the case $n=2$, although the
simplification is only notational. The following lemma will also come in
handy:
###### Lemma 2.8.
Suppose $n=2$, and let ${\bf r}=(r_{1},r_{2},\ldots)$ be a vector of length at
least $2$, of elements of $\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Z}/2d$
according as $\tilde{H}(-)$ is $H(-)$, $H_{\infty}(-)$ or $H_{d}(-)$,
respectively. In a tame case, the linear span $D=D_{\bf r}$ of the four
elements $x_{{\bf i}{\bf j}}^{{\bf r}}$ where ${\bf i}$ and ${\bf j}$ are
alternating vectors of the form $(1,2,1,\ldots)$ or $(2,1,2,\ldots)$ is not a
subcoalgebra of $\tilde{H}(M_{2}(k)^{*})$.
###### Proof.
Remember that by our conventions at the beginning of this section, and
considering that $n=2$, being in a tame case means that either $\tilde{H}(-)$
is not $H_{1}(-)$, or that we are working in characteristic zero.
Assume first that $\tilde{H}(-)$ is not $H_{1}(-)$. This means that we can
find consecutive entries $r_{i}$ and $r_{i+1}$ of the vector ${\bf r}$ (which
has length at least $2$, by the hypothesis) such that either $r_{i+1}\neq
r_{i}+1$, or $r_{i+1}\neq r_{i}-1$. To fix ideas, suppose, for example, that
$r_{2}\neq r_{1}+1$; the general case is entirely analogous.
Just as before, in the proof of Proposition 2.6, we are going to try to
compute $\Delta(x_{{\bf i}{\bf j}}^{{\bf r}})$ for some monomial in $D$ by
using the matrix comultiplication rules and expanding. Let ${\bf u}$ be the
vector $(2,2,1,2,\ldots)$ of the same length as ${\bf r}$. ${\bf u}$ has $2$
as its first entry, and then alternates, starting with $2$ again. Because of
our assumption on ${\bf r}$, the monomial $x_{{\bf i}{\bf u}}^{{\bf r}}$ is
reduced (this is easily seen by examining the reduction rules (2)-(5)).
Moreover, the same reduction rules imply that $x_{{\bf i}{\bf u}}^{{\bf r}}$
cannot be obtained as a non-trivial reduction from another monomial appearing
in our computation of $\Delta(x_{{\bf i}{\bf j}}^{{\bf r}})$. It follows then
that after reducing everything in the expression of $\Delta(x_{{\bf i}{\bf
j}}^{{\bf r}})$, there will be at least one term of the form $\pm x_{{\bf
i}{\bf u}}^{{\bf r}}\otimes\bullet$ left. This term is not an element of
$D\otimes D$ (because ${\bf u}$ is not alternating), and we are done.
Now assume $\tilde{H}(-)=H_{1}(-)$, but ${\rm char}(k)=0$. Because the entries
of the vector ${\bf r}$ are elements of $\mathbb{Z}/2$, there’s no difference
now between $r_{i}+1$ and $r_{i}-1$. The previous argument still works if two
consecutive entries of ${\mathbb{r}}$ are equal, but not if ${\mathbb{r}}$ is
an alternating vector (i.e. any two consecutive entries are different).
Nevertheless, we try to apply the same technique, and compute $\Delta(x_{{\bf
i}{\bf j}}^{{\bf r}})$ for some reduced monomial in $D$.
Let ${\bf u}$ be the vector $(1,1,1,\ldots)$, of the same length as ${\bf r}$.
Notice that by the reduction rules (2) - (5), for any vector ${\bf v}$ of the
same length, the coefficient of $x_{{\bf i}{\bf u}}^{{\bf r}}$ in the reduced
form of $x_{{\bf i}{\bf v}}^{{\bf r}}$ is equal to the coefficient of $x_{{\bf
u}{\bf j}}^{{\bf r}}$ in the reduced form of $x_{{\bf v}{\bf j}}^{{\bf r}}$.
It follows then, because we are working in characteristic zero, that after
performing all the reductions, the coefficient of $x_{{\bf i}{\bf u}}^{{\bf
r}}\otimes x_{{\bf u}{\bf j}}^{{\bf r}}$ in $\Delta(x_{{\bf i}{\bf j}}^{{\bf
r}})$ is positive. In particular, $\Delta(x_{{\bf i}{\bf j}}^{{\bf r}})$ does
not belong to $D\otimes D$. ∎
###### Remark 2.9.
We will see below, in Section 4, that the tame case hypothesis is necessary.
Finally, we are ready now to finish the proof of the main theorem.
###### Proof of Theorem 2.1.
As remarked repeatedly before, we can assume the base field is algebraically
closed. We already know, from the proof of Proposition 2.6, that (for the
purpose of our theorem) it suffices, over an algebraically closed field, to
look only at matrix subcoalgebras of $H=\tilde{H}(M_{n}(k)^{*})$. Also, we
assume $n=2$, as permitted by Corollary 2.7. Finally, by an observation made
at the end of Section 1, a matrix subcoalgebra of $H$ is contained in some
$C_{\bf r}$, the linear span of the monomials $x_{{\bf i}{\bf j}}^{\bf r}$ for
some fixed ${\bf r}$.
In line with the previous paragraph, let $C\subseteq C_{\bf r}\subset H$ be an
$m\times m$ matrix subcoalgebra of $H$, with $m\leq n$. We may as well assume
that the length $\ell$ of ${\bf r}$ is at least $2$. Pick an $x\in C$, and let
$x_{{\bf i}{\bf j}}^{{\bf r}}$ be a reduced monomial appearing in $x$. We saw
in the proof for Proposition 2.6 that after performing all the reductions,
$\Delta(x_{{\bf i}{\bf j}}^{{\bf r}})$ contains both terms of the form
$x_{{\bf i}{\bf u}}^{{\bf r}}\otimes x_{{\bf u}{\bf j}}^{{\bf r}}$, where
${\bf u}$ is one of the two alternating vectors of length $\ell$ (either
$(1,2,1,\ldots)$ or $(2,1,2,\ldots)$).
The proof for Proposition 2.6 (more specifically the part of the proof which
used Lemma 2.5, contained in the last two paragraphs of the proof) also shows
that if $\Delta(x_{{\bf i}{\bf j}}^{{\bf r}})$ were to contain $x_{{\bf a}{\bf
b}}^{{\bf s}}\otimes x_{{\bf c}{\bf d}}^{{\bf t}}$ with neither ${\bf b}$ nor
${\bf c}$ alternating of length $\ell$, then $x$ could not be one of the
matrix generators of $C$. It follows that for any such generator, all the
terms of $\Delta(x)$ (after all the reductions have been made) are multiples
either of $x_{{\bf i}{\bf u}}^{{\bf r}}\otimes\bullet$ or $\bullet\otimes
x_{{\bf u}{\bf j}}^{{\bf r}}$, with ${\bf u}$ alternating of length $\ell$.
But because of the maximality of the length of $x_{{\bf i}{\bf j}}^{{\bf r}}$
in $x$, it’s clear that the only possible such terms are the multiples of
$x_{{\bf i}{\bf u}}^{{\bf r}}\otimes x_{{\bf u}{\bf j}}^{{\bf r}}$ (in other
words, if $x_{{\bf i}{\bf j}}^{{\bf r}}\otimes\bullet$ were to appear in
$\Delta(x)$, the only possibility for $\bullet$ would be $x_{{\bf u}{\bf
j}}^{{\bf r}}$).
In conclusion, for matrix generators $x$ of $C$, $\Delta(x)$ is a linear
combination of the two $x_{{\bf i}{\bf u}}^{{\bf r}}\otimes x_{{\bf u}{\bf
j}}^{{\bf r}}$, with ${\bf u}$ alternating of length $\ell$. But by using the
counit identities on $x$, we see that this implies that $x$ is a member of
what in the statement of Lemma 2.8 was denoted by $D_{\bf r}$, and hence that
our $C\subseteq C_{\bf r}$ be $D_{\bf r}$. But Lemma 2.8 says precisely that
in a tame case, $D_{\bf r}$ is not a subcoalgebra (for ${\bf r}$ of length
$\geq 2$). This finishes the proof of the theorem. ∎
As a final remark, we record the following consequence of Theorem 2.1:
###### Corollary 2.10.
Let $n>1$ be a positive integer. In a tame case, the only right comodules over
$H=\tilde{H}(M_{n}(k)^{*})$ of dimension $\leq n$ are (a) the direct sums of
$\leq n$ copies of the trivial comodule, and (b) the iterated duals of the
$n$-dimensional comodule obtained by scalar corestriction from
$M_{n}(k)^{*}\to H$.
###### Proof.
Let $M$ be a right comodule over $H$, of dimension $m\leq n$, with comodule
structure map $\rho:M\to M\otimes H$. If $e_{i},\ i=\overline{1,m}$ is a basis
for $M$, then we get elements $c_{ij}$ of $H$ by
$\rho_{e_{j}}=\sum_{i}e_{i}\otimes c_{ij}.$
It’s easy to see that the $c_{ij}$ satisfy matrix coalgebra-type relations, as
in (1), or, in other words, we have a coalgebra map from $M_{n}(k)^{*}$ to $H$
sending the standard generators $t_{ij}$ to $c_{ij}$. But this means that the
$c_{ij}$ form a subcoalgebra of $H$ of dimension $\leq n^{2}$, and the
conclusion follows immediately from Theorem 2.1. ∎
## 3\. Centers of some categories
Here, as an application of Theorem 2.1, we determine the centers of the
categories ${\rm HopfAlg}$, ${\rm HopfAlg}_{\infty}$ and ${\rm HopfAlg}_{d}$.
Because these centers are all monoids with a “multiplicative zero”, namely the
natural transformation which is given on each Hopf algebra (or Hopf algebra
with bijective antipode, or Hopf algebra $H$ with $S_{H}^{2d}=\\{\rm id\\}$)
by the composition between the unit and the counit, it will be convenient to
have a notation for this phenomenon. Hence, we introduce the following
notation:
For a monoid $M$, denote by $M^{+}$ the monoid which as a set is
$M\cup\\{0\\}$, with multiplication defined by the one in $M$ and by
$0x=x0=0,\ \forall x\in M.$
In other words, $M^{+}$ is obtained from $M$ by appending a multiplicative
zero.
In the following statement, $\mathbb{N},\mathbb{Z}$ and $\mathbb{Z}/2d$ are
monoids with their usual additive structure. Notice that in each of our
categories, there is an endo-natural transformation of the identity functor
given by the square of the antipode on each object of the category. To avoid
cumbersome language, we refer to this natural transformation as being the
square of the antipode.
###### Theorem 3.1.
The centers of ${\rm HopfAlg}$, ${\rm HopfAlg}_{\infty}$ and ${\rm
HopfAlg}_{d}$ are $\mathbb{N}^{+}$, $\mathbb{Z}^{+}$, and
$(\mathbb{Z}/2d)^{+}$, respectively, where $\mathbb{N}$, $\mathbb{Z}$ and
$\mathbb{Z}/2d$ are generated by the square of the antipode. In all three
cases, the multiplicative zero is given by the trivial endomorphism.
###### Proof.
We prove the statement for ${\rm HopfAlg}$; the proofs in the other two cases
are entirely parallel.
First, looking at the action of the antipode on our elements $x_{ij}^{0}$ in
some $H(M_{n}(k)^{*})$, it’s clear that the different powers of the antipode
induce different endo-natural transformations of the identity, and hence the
monoid generated by $S^{2}$ and the trivial endomorphism is indeed
$\mathbb{N}^{+}$. The interesting part is showing that conversely, every
element of the center is either trivial or induced by some even power of the
antipode.
Let $\eta$ be an endo-natural transformation of the identity functor on ${\rm
HopfAlg}$. This means that for every Hopf algebra $H$, we are given an
endomorphism $\eta_{H}$ of $H$ such that commutes for every Hopf algebra map
$f:H\to K$. Let us look at what $\eta_{H}$ might be for $H=H(M_{n}(k)^{*})$
for some fixed $n>1$ (we would take $n>2$ if we were dealing with ${\rm
HopfAlg}_{1}$ instead of ${\rm HopfAlg}$, to make sure we are in a tame
situation). Theorem 2.1 says that there are three cases:
(1) $\eta_{H}$ is of the form $S^{2t}\circ\alpha$ for some automorphism
$\alpha$ of $M_{n}(k)^{*}$. It is clear (for example from the structure of the
basis of $H$ we’ve been working with) that the map ${\rm
Aut}(M_{n}(k)^{*})\to{\rm End}(H(M_{n}(k)^{*}))$ given by $\beta\mapsto
S^{2t}\circ\beta$ is injective. From this and the commutativity of the diagram
above for $\eta_{H}=S^{2t}\circ\alpha$ and $f=\beta\in{\rm Aut}(M_{n}(k)^{*})$
it follows that $\alpha$ is in the center of ${\rm Aut}(M_{n}(k)^{*})$. This
implies $\alpha={\rm id}$, and hence $\eta_{H}$ is an even power of the
antipode.
(2) $\eta_{H}$ is of the form $S^{2t+1}\circ T\circ\alpha$, where $T$ is the
transposition on $M_{n}(k)^{*}$, and $\alpha$ is an automorphism of the matrix
coalgebra. Just as before, consider our commutative diagram with $K=H$ and
$f=\beta$, some automorphism of $M_{n}(k)^{*}$. The same argument as in (1)
(and the fact that every endomorphism of $H$ commutes with the antipode) shows
that $T\alpha\beta=\beta T\alpha$ for arbitrary $\beta$. This is easily seen
to be impossible for $n>2$, and hence (2) is ruled out.
(3) $\eta_{H}$ is the trivial endomorphism of $H$.
Denote $\eta_{H},\ H=H(M_{n}(k)^{*})$ by $\eta_{n}$. We now know that
$\eta_{n}$ is either $S^{2r}$ for some $r$ or trivial (for $n>2$, at least). I
claim that either we have the same $r$ for all $n$, or $\eta_{n}$ is trivial
for all $n$. First, notice that the claim finishes the proof. To see this,
suppose, for example, that $\eta_{n}=S^{2r}$ for every large $n$ (the case
where $\eta_{n}$ are all trivial is similar). Now, by the commutativity of the
square diagram above, $\eta_{K}$ is going to be $S_{K}^{2r}$ for every
quotient of a Hopf algebra of the form $H(M_{n}(k)^{*})$ for large $n$. But on
the one hand, every finite-dimensional coalgebra is a quotient of some
$M_{n}(k)^{*}$, and on the other hand, every Hopf algebra is the union of its
finite-dimensional subcoalgebras; this implies that every Hopf algebra is a
union of quotients of Hopf algebras of the form $H(M_{n}(k)^{*})$, and we are
done.
All that remains is to prove the claim. Say for some fixed $n>2$, $\eta_{n}$
is $S^{2r}$, while $\eta_{n+1}$ is $S^{2s}$ (again, the case when one of
$\eta_{n}$, $\eta_{n+1}$ is trivial is analogous). This means, in terms of our
standard algebra generators $\\{x_{ij}^{r}\\}$ for $H(M_{n}(k)^{*})$ and
$\\{y_{ij}^{r}\\}$ for $H(M_{n+1}(k)^{*})$, that $\eta_{n}$ is the
endomorphism induced by $x_{ij}^{0}\mapsto x_{ij}^{2r}$, while $\eta_{n+1}$ is
induced by $y_{ij}^{0}\mapsto y_{ij}^{2s}$.
Now let $C$ be the quotient of $M_{n+1}(k)^{*}$ by the coideal spanned by
$y_{n+1j}^{0},\ j=\overline{1,n}$. We denote the images of $y_{ij}^{0}$ in $C$
by the same symbols. $\eta_{H}$ will be $S^{2s}$ for $H=H(C)$. At the same
time, however, we have an inclusion $H(M_{n}(k)^{*})\to H(C)$ (Remark 1.3)
given by the inclusion $M_{n}(k)^{*}\to C$ given as $x_{ij}^{0}\mapsto
y_{ij}^{0},\ i,j=\overline{1,n}$. It follows now that $\eta_{n}$ is both
$S^{2r}$ and $S^{2s}$. As the $S^{i}$ are different for different $i$ on
$H(M_{n}(k)^{*})$ (by looking at how the powers of the antipode act on the
$x_{ij}^{0}$), we get $r=s$, as desired. ∎
## 4\. What about $H_{1}(M_{2}(k)^{*})$ in positive characteristic?
The purpose of this short section is to point out that, as mentioned several
times before, the tame case hypothesis in Theorem 2.1 is actually necessary.
More specifically, we have counterexamples in characteristics $2$ and $3$. We
observed in Remark 2.2 that in fact Theorem 2.1 works even for
$H_{1}(M_{2}(k)^{*})$ in positive characteristic as long as it is different
from $2$ or $3$, but we will not prove this here. The proof consists of making
a slightly more detailed analysis of what can go wrong with the arguments in
Section 2, using the same techniques as before.
###### Example 4.1.
Suppose the base field $k$ has characteristic $2$, and let ${\bf r}$ be either
$(0,1)$ or $(1,0)$, where $0,1$ are the elements of $\mathbb{Z}/2$. Then,
using the notation from Lemma 2.8, $D_{\bf r}$ is a $2\times 2$ matrix
subcoalgebra of $H_{1}(M_{2}(k)^{*})$.
###### Proof.
This is a simple verification. Assume for example that ${\bf r}$ is $(0,1)$.
We check that $\Delta(x_{11}^{0}x_{22}^{1})$ does indeed belong to $D_{\bf
r}\otimes D_{\bf r}$, and leave the rest to the reader.
We have
$\displaystyle\Delta(x_{11}^{0}x_{22}^{1})$
$\displaystyle=\Delta(x_{11}^{0})\Delta(x_{22}^{1})=(x_{11}^{0}\otimes
x_{11}^{0}+x_{12}^{0}\otimes x_{21}^{0})(x_{21}^{1}\otimes
x_{12}^{1}+x_{22}^{1}\otimes x_{22}^{1})$
$\displaystyle=x_{11}^{0}x_{21}^{1}\otimes
x_{11}^{0}x_{12}^{1}+x_{11}^{0}x_{22}^{1}\otimes
x_{11}^{0}x_{22}^{1}+x_{12}^{0}x_{21}^{1}\otimes
x_{21}^{0}x_{12}^{1}+x_{12}^{0}x_{22}^{1}\otimes x_{21}^{0}x_{22}^{1}.$ (6)
Now simply notice that because of the two reduction rules (2) and (3), we have
(regardless of the characteristic)
$\displaystyle x_{12}^{0}x_{22}^{1}$ $\displaystyle=-x_{11}^{0}x_{21}^{1}$
$\displaystyle x_{21}^{0}x_{22}^{1}$ $\displaystyle=-x_{11}^{0}x_{12}^{1}.$
Because ${\rm char}(k)=2$, the first and last term in (4) cancel out. ∎
Similarly, we have
###### Example 4.2.
Suppose ${\rm char}(k)=3$, and ${\bf r}$ is one of the alternating vectors
$(0,1,0)$ or $(1,0,1)$ with entries from $\mathbb{Z}/2$. Then, $D_{\bf r}$ is
an $2\times 2$ matrix subcoalgebra of $H_{1}(M_{2}(k)^{*})$.
## Acknowledgement
The author wishes to thank the referee, to whom the observatnion in Remark 2.4
is due, for this and other suggestions on how to improve the manuscript.
## References
* [A] Abe, E. - Hopf algebras, Cambridge University Press 1980
* [Be] Bergman, G. - The diamond lemma for ring theory, Adv. Math. 29 (1978), pp. 178 - 218
* [Mo] Montgomery, S. - Hopf algebras and their actions on rings, vol. 82 of CBMS Regional Conference Series in Mathematics, AMS, Providence, Rhode Island 1993
* [Ni] Nichols, W. D. - Quotients of Hopf algebras, Comm. Algebra 6 (1978), pp. 1789 - 1800
* [Sc] Schauenburg, P. - Faithful flatness over Hopf subalgebras: Counterexamples, appeared in Interactions between ring theory and representations of algebras: proceedings of the conference held in Murcia, Spain, CRC Press (2000), pp. 331 - 344
* [Sw] Sweedler, M. E. - Hopf algebras, Benjamin New York 1969
* [Ta] Takeuchi, M. - Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), pp. 561 - 582
|
arxiv-papers
| 2010-02-17T04:09:15 |
2024-09-04T02:49:08.425724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandru Chirvasitu",
"submitter": "Alexandru Chirv{\\ba}situ L.",
"url": "https://arxiv.org/abs/1002.3198"
}
|
1002.3264
|
# Towards a relevant set of state variables to describe static granular
packings
Luis A. Pugnaloni luis@iflysib.unlp.edu.ar Instituto de Física de Líquidos y
Sistemas Biológicos (CONICET La Plata, UNLP), cc. 565, 1900 La Plata,
Argentina. Iván Sánchez Departamento de Física y Matemática Aplicada,
Facultad de Ciencias, Universidad de Navarra, Irunlarrea S/N, 31080 Pamplona,
Spain. Centro de Física, Instituto Venezolano de Investigaciones Científicas,
Apartado Postal 21827, 1020-A Caracas, Venezuela. Paula A. Gago Instituto de
Física de Líquidos y Sistemas Biológicos (CONICET La Plata, UNLP), cc. 565,
1900 La Plata, Argentina. José Damas Departamento de Física y Matemática
Aplicada, Facultad de Ciencias, Universidad de Navarra, Irunlarrea S/N, 31080
Pamplona, Spain. Iker Zuriguel Departamento de Física y Matemática Aplicada,
Facultad de Ciencias, Universidad de Navarra, Irunlarrea S/N, 31080 Pamplona,
Spain. Diego Maza dmaza@unav.es Departamento de Física y Matemática
Aplicada, Facultad de Ciencias, Universidad de Navarra, Irunlarrea S/N, 31080
Pamplona, Spain.
###### Abstract
We analyze, experimentally and numerically, the steady states, obtained by
tapping, of a 2D granular layer. Contrary to the usual assumption, we show
that the reversible (steady state branch) of the density–acceleration curve is
nonmonotonous. Accordingly, steady states with the same mean volume can be
reached by tapping the system with very different intensities. Simulations of
dissipative frictional disks show that equal volume steady states have
different values of the force moment tensor. Additionally, we find that steady
states of equal stress can be obtained by changing the duration of the taps;
however, these states present distinct mean volumes. These results confirm
previous speculations that the volume and the force moment tensor are both
needed to describe univocally equilibrium states in static granular
assemblies.
_Introduction:_ Finding out the appropriate set of macroscopic variables that
characterizes the equilibrium state of a given system is the first step in any
thermodynamic study callen . What these variables are for a sample of sand in
a box is still under discussion leiden . Twenty years ago, Edwards and
Oakshottedwards put forward the idea that the number of grains $N$ and the
volume $V$ are the basic state variables that suffice to characterize a static
granular sample in equilibrium. The $NV$ granular ensemble was introduced as a
collection of microstates, where the sample is in mechanical equilibrium,
compatible with $N$ and $V$.
Experimentally, such equilibrium states are commonly obtained by tapping the
system at different dimensionless accelerations, $\Gamma$, which is the
standard parameter used to quantify the intensity of the external excitation
nowak ; nowak2 ; richard ; schroter ; ribiere . Hence, under the action of
repeated excitations at constant $\Gamma$, the system properties reach a
steady state where all observables fluctuate around well defined mean values.
Therefore, tapping provides the energy necessary to allow the system to
explore different configurations over which macroscopic observables can be
averaged. In the steady state, the mean volume $V$, or, more commonly, the
mean packing fraction, $\phi$ (defined as the percentage of the space occupied
by the particles), can be measured. It is now well established nowak ; nowak2
; richard ; schroter ; ribiere that different steady states can be reached
reversibly by varying $\Gamma$. These experiments also suggest a monotonic
relation between $\Gamma$ and $\phi$. Nevertheless, $\phi$ does not only
depend on $\Gamma$. In harmonic pulses, if $\Gamma$ is kept constant while the
duration and the amplitude of the pulse are simultaneously modified, the
system evolves to different steady states. This fact was used to show that
equivalent equilibrium states pica ; schroter can be generated by using
different combinations of pulse amplitude and duration. On the other hand, new
theoretical works snoeijer ; edwards2 ; blumenfeld ; henkes ; henkes2 ; tighe
suggest that the force moment tensor, $\Sigma$, ($\Sigma=V\sigma$, where
$\sigma$ is the stress tensor) must be added to the set of extensive
macroscopic variables (i.e., an $NV\Sigma$ ensemble) to describe adequately a
packing of grains.
Recent numerical simulations pugnaloni ; gago have predicted also the
existence of a nonmonotonic relation between $\Gamma$ and $\phi$, if
sufficiently large values of $\Gamma$ are considered. A hint as to this
behavior can already be seen in some packings obtained in the laboratory
sibille ; although this has passed unnoticed by those studying equilibrium
properties. These results imply that equilibrium states of equal $V$ (or
$\phi$) could be obtained not only by changing the pulse duration, as in Ref.
schroter , but also by simply using different values of the excitation
parameter $\Gamma$. In what follows, we will show that these equal volume
equilibrium states are experimentally accessible. We will then use numerical
simulations to show that these states are, nevertheless, different. We have
found that two equilibrium states obtained with different excitations, even if
they have same $V$ and $N$, may present distinct values of $\Sigma$. This
result suggests the incomplete picture upon which research has been based for
over two decades when the properties of static granular matter have been
investigated; and takes us one step closer to completing the definition of the
state variables that are necessary to describe a static packing of grains.
Figure 1: (Color online)(a) Evolution of $\phi$ towards the steady state at
$\Gamma=7.0$ starting from the steady state corresponding to $\Gamma=1.5$. (b)
The evolution when switching back to $\Gamma=1.5$ movie1 (Arrows indicate the
tap where $\Gamma$ is switched). (c) Steady state packing fraction, $\phi$, as
a function of $\Gamma$ (green down triangles: steady state obtained in Exp A
through tapping at constant $\Gamma$ from an initial ordered state; orange up
triangles: steady state in Exp B obtained by an annealing protocol) nowak ;
movie1 . (d) Steady state relative packing fraction $\phi/\phi_{\rm{max}}$ as
a function of tapping intensity $\Gamma$ for the 3D cell ($\phi_{\rm{max}}$
corresponds to the value of $\phi$ obtained at the lowest $\Gamma$ studied).
Error bars represent the standard error of the mean estimated from the
standard deviation.
.
_Experimental results:_ A quasi 2D Plexiglass cell (width: 28 mm, height: 150
mm) was used to study the packing dynamics. The cell was filled with 1000
alumina oxide (Al2O3) spheres of diameter 1.000 $\pm$ 0.005 mm. The separation
between the Plexiglass sheets was 10% larger than the bead diameter in order
to minimize the particle-wall friction and prevent arching in the transversal
direction. The system was tapped with an electromagnetic shaker (Tiravib
52100) with a series of harmonic pulses of variable amplitude and constant
frequency $\nu=30$ Hz every three seconds. The tapping intensity was measured
with a piezoelectric accelerometer attached to the base of the cell. Although
in recent years alternative parameters have been proposed to quantify the
intensity of the taps dijksman ; ludewing , we employ the usual nondimensional
peak acceleration $\Gamma\equiv a_{peak}/g$ (where $g$ is the acceleration of
gravity). High resolution digital images of the packing were taken after each
tap. The center of each sphere was detected with an error of less than 2%. The
packing fraction of each image was obtained by considering each grain as a
disk of the corresponding effective diameter and then calculating the
percentage of the area covered by the disks in a rectangular area 10% smaller
than the size of the packing. Due to the space left between the Plexiglass
sheets, small overlaps can be observed in the 2D projection taken by the
photographs. This effect leads to estimations of $\phi$ above the 2D hexagonal
close packing value for the densest configurations studied. After a relative
short number of taps, the system reaches a steady state characterized by a
plateau in curves of $\phi$ vs. the number of taps. In contrast with the three
dimensional case, any stationary steady state is obtained after a few taps. We
checked that the mean value and standard deviation of the packing fraction
differ less than $0.1\,\%$ if $200$ or $2000$ taps of equilibration are
applied. As an example, in Fig. 1(a), we show that the steady state
corresponding to $\Gamma=7.0$ can be reached in a couple taps, even when the
initial packing is in an ordered configuration. Equilibrating the system back
at $\Gamma=1.5$ takes somewhat longer (about 100 taps) [see Fig. 1(b)] movie1
. Therefore, the sample initially prepared in a highly ordered configuration
was tapped 500 times for any given $\Gamma$ before taking averages over 100
taps, and $10$ independent runs were averaged. Alternatively, an annealing
protocol was used in which $\Gamma$ was decreased in discrete steps from the
highest values and tapped 500 times at each $\Gamma$ value without emptying
the cell.
In Fig. 1(c), we plot $\phi$ in the steady state as a function of $\Gamma$ for
different independent repetitions of the experiment. It can be seen that the
same results are obtained by equilibrating from an initial ordered structure
(Exp. A) or by following an annealing path (Exp. B). Although the fluctuations
of $\phi$ are large [see Fig, 1(a)], its mean value is well defined with a
small confidence interval [see error bars in Fig. 1(c)]. For low excitations,
$\phi$ decreases as $\Gamma$ is increased, in agreement with previous results
reported by several groups. However, beyond a certain value
$\Gamma_{\rm{min}}$, the packing fraction grows movie1 . The same trend is
observed if the tap frequency $\nu$ is changed (Exp. C) [see Fig 2(a)]. An
explanation for this behavior based on the formation of arches has been given
in pugnaloni .
Figure 2: (Color online). Results from the MD simulations of the soft disk
model for two different frequencies of the tapping pulse [blue squares:
$\nu_{1}=0.5(g/d)^{1/2}$, and red circles: $\nu_{2}=0.25(g/d)^{1/2}$]. (a)
Steady state packing fraction, $\phi$, as a function of the drive $\Gamma$. In
order to compare with the quasi 2D experiments (green down triangles, orange
up triangles and black stars) the vertical and horizontal axes have been
scaled with the characteristic values $\phi_{\rm{min}}$ and
$\Gamma_{\rm{min}}$, respectively. (b) Fluctuations, $\sigma_{\phi}$, of the
packing fraction in the steady state (measured by the standard deviation) as a
function of $\phi$. The arrows indicate the direction of increasing tapping
intensity. Error bars correspond to the standard error. (c) The trace,
Tr$(\Sigma)$, of the force moment tensor in the steady state as a function of
$\Gamma/\Gamma_{\rm{min}}$.
In order to assess whether the presence of a minimum in the $\Gamma$–$\phi$
curves is induced by the highly ordered crystal-like structures present in the
quasi 2D cell, we repeated the experiment with a 3D cell. This 3D setup
consists of a cell of the same height and width (as the quasi 2D cell) but 6
mm thick. In this case, the granular sample was made of polydisperse glass
beads 1.0 $\pm$ 0.2 mm in diameter. For the sake of simplicity, in the 3D
cell, the relative packing fraction was estimated from the height of the
granular layer. The stationary regime in the 3D setup was obtained after
$2\times 10^{4}$ taps. In Fig. 1(d) we show the steady state packing fraction
as a function of $\Gamma$ for this three-dimensional setup. Again, the same
trend as in the quasi 2D experiment is observed. This nonmonotonic dependence
challenges the idea that the volume defines the equilibrium state, unless
these equal volume states are proved to be equivalent mehta .
_Numerical evidences:_ In 2005, Edwards suggested that the stress tensor
should be included in the description of the equilibrium state of a static
granular sample along with $N$ and $V$ in the so-called full canonical
ensemble edwards2 . There is some consensus now snoeijer ; blumenfeld ; henkes
; henkes2 ; tighe that the force moment tensor, $\Sigma\equiv V\sigma$ ,
allows for a microcanonical description henkes2 . However, no experiments or
simulations have shown that this variable is necessary to distinguish between
different states of thermodynamic equilibrium.
Since it is difficult to measure the force moment tensor in our experimental
setup, we use a realistic computer model of the quasi 2D setup pugnaloni ;
arevalo . We used a velocity Verlet algorithm to integrate the Newton
equations for 512 monosized disks in a rectangular box. The disk–disk and
disk–wall contact interaction comprises a linear spring–dashpot in the normal
direction and a tangential friction force that implements the Coulomb
criterion to switch between dynamic and static friction. Units are reduced
with the diameter of the disks, $d$, the disk mass, $m$, and the acceleration
of gravity, $g$. Details on the force equations and the interaction parameters
can be found elsewhere pugnaloni ; arevalo . Tapping is simulated by moving
the confining box in the vertical direction following a half sine wave
trajectory. The intensity of the excitation is controlled either through the
amplitude, $A$, or the frequency, $\nu$, of the sinusoidal trajectory; and it
is characterized by the parameter $\Gamma=A(2\pi\nu)^{2}/g$. We consider the
system has reached the steady state whenever a plateau in $\phi$ is observed
(with no visible trend in a plot of $\phi$ vs $\log(taps)$). Averages were
taken over 400 taps in the steady state and over 20 independent simulations
for each value of $A$ and $\nu$.
In Fig. 2(a), we show $\phi$ as a function of $\Gamma$ for this model.
Different values of $\Gamma$ were obtained by varying both $A$ and $\nu$. As
in the experiment, states of equal $\phi$ are generated at both sides of a
minimum packing fraction, $\phi_{\rm{min}}$. States of equal volume at each
side of $\phi_{\rm{min}}$ display slightly different volume fluctuations [Fig.
2(b)] suggesting that these states are not the same. In contrast to the
packing fraction, the trace of $\Sigma$ sigma gently grows as $\Gamma$ is
increased [see Fig. 2(c)]. There is neither a minimum nor a maximum in the
$\Gamma$–$\Sigma$ curves. Therefore, states of equal $\phi$ at each side of
$\phi_{\rm{min}}$ present distinct $\Sigma$. Hence, states of equal volume at
each side of $\Gamma_{\rm{min}}$ are, in fact, different. This finding proves
that the volume alone cannot characterize the equilibrium states of a static
granular sample and suggests that the force moment tensor is a convenient
extra state variable.
Figure 3: (Color online). The loci of the generated equilibrium states in the
$\phi$–$\Sigma$ phase diagram. (a) The stationary packing fraction as a
function of the trace Tr$(\Sigma)$ for $\nu_{1}$ (blue squares) and $\nu_{2}$
(red circles). (b) The fluctuations, $\sigma_{\rm{Tr}(\Sigma)}$, versus
Tr$(\Sigma)$. Error bars as in Fig. 2. The shaded areas indicate the position
of $\phi_{\rm{min}}$.
In Fig. 3(a), the loci defined by the equilibrium states visited during the
numerical tapping experiments are plotted in a hypothetical $\phi$–$\Sigma$
thermodynamic phase space. It is clear that states of equal $\Sigma$ can
present different $\phi$ if prepared at different $\nu$. Thus, $\Sigma$ cannot
define the equilibrium state by itself. Both $V$ and $\Sigma$ have to be
specified to fully identify a given state. This result is confirmed by the
fluctuations of the force moment tensor, $\sigma_{\Sigma}$, [see Fig. 3(b)].
The fluctuations $\sigma_{\Sigma}$ for a given $\Sigma$ are not unique: they
depend on the frequency $\nu$ of the tapping. However, for the states obtained
with different excitations that display equal $\phi$ and $\Sigma$ (note that
such states would actually be a unique state), we find that $\sigma_{\phi}$
and $\sigma_{\Sigma}$ display also very similar values. Hence, the coincidence
of the mean and fluctuations of the state variables suggests that no other
extensive parameter would be necessary to describe such equilibrium states.
It is interesting to mention that the definition of an entropy $S(N,V,\Sigma)$
is particularly convenient since $\Sigma$ is not bounded as $V$ is. This
implies that $S$ can be a monotonic increasing function of $\Sigma$, which
permits a well behaved Legendre transformation to a canonical ensemble callen
. Hence, one can define a non-negative temperature-like quantity: the
angoricity edwards2 ; henkes2 —the proposed name for the inverse of the
conjugate variable to the force moment tensor. Let us finally recall that the
temperature-like quantity associated with the volume (so-called compactivity)
can present negative values due to population inversion brey .
_Conclusions:_ We have shown that steady states of static granular packings
obtained by tapping the system with different pulse strength and duration can
present different mean force moment tensor even if they correspond to the same
mean volume. Moreover, steady states that present the same mean force moment
tensor are distinguishable by their mean volume. To our knowledge, this is the
first experimental/numerical evidence that both extensive variables should be
included in an entropic formulation of the thermodynamics of such steady
states.
###### Acknowledgements.
This work was supported by project FIS2008-06034-C02-01 (Spain), PIUNA (Univ.
Navarra), CONICET (Argentina) and ANPCyT (Argentina).
## References
* (1) H. B. Callen, Thermodynamics and an introduction to thermostatistics, 2nd Edition, Wiley-VCH, New York (1985).
* (2) A special workshop on statistical mechanics of static granular media took place in Lorentz Center (Leiden, The Netherland) in July 2009. The main topic was the definition of state variables for equilibrium states in static samples. Presentations can be seen at http://www.lorentzcenter.nl/lc
/web/2009/340/info.php3?wsid=340
* (3) S. F. Edwards and R. B. S. Oakeshott, Physica A 157, 1080 (1989).
* (4) E. R. Nowak, J. B. Knight, M. L. Povinelli, H. M. Jeager and S. R. Nagel, Powder Tech. 94, 79 (1997).
* (5) E. R. Nowak, J. B. Knight, E. Ben-Naim, H. M. Jaeger and S. R. Nagel, Phys. Rev. E 57, 1971 (1998).
* (6) P. Richard, M. Nicodemi, R. Delannay, P. Ribière and D. Bideau, D, Nat. Mater. 4, 121 (2005).
* (7) Ph. Ribière, P. Richard, P. Philippe, D. Bideau and R. Delannay, Eur. Phys. J. E 22, 249 (2007).
* (8) M. Schröter, D. I. Goldman and H. L. Swinney, Phys. Rev. E 71, 030301(R) (2005).
* (9) M. Pica Ciamarra, A. Coniglio and M. Nicodemi, Phys. Rev. Lett. 97, 158001 (2006).
* (10) J. H. Snoeijer, T. J. H. Vlugt, W. G. Ellenbroek, M. van Hecke and J. M. J. van Leeuwen, Phys. Rev. E. 70, 061306 (2004).
* (11) S. F. Edwards, Physica A 353, 114 (2005).
* (12) R. Blumenfeld and S. F. Edwards, J. Phys. Chem. B 113, 3981 (2009).
* (13) S. Henkes, C. S. O’Hern and B. Chakraborty, Phys. Rev. Lett. 99, 038002 (2007).
* (14) S. Henkes and B. Chakraborty, Phys. Rev. E. 79, 061301 (2009).
* (15) B. P. Tighe, A. R. T. van Eerd and T. J. H. Vlugt, Phys. Rev. Lett. 100, 238001 (2008).
* (16) L. A. Pugnaloni, M. Mizrahi, C. M. Carlevaro and F. Vericat, Phys. Rev. E 78, 051305 (2008).
* (17) P. A. Gago, N. E. Bueno and L. A. Pugnaloni, Gran. Matt. 11, 365 (2009).
* (18) L. Sibille, T. Mullin and P. Poullain, Eur. Phys. Lett. 86, 44003 (2009).
* (19) J. A. Dijksman and. M. van Hecke, Eur. Phys. Lett. 88, 44001 (2009).
* (20) F. Ludewig, S. Dorbolo, T. Gilet, and N. Vandewalle, Eur. Phys. Lett. 84, 44001 (2008).
* (21) EPAPS Document No. [number will be inserted by publisher]. Movie 1: A movie that shows the evolution of the packing during tapping at $\Gamma=7.0$ followed by tapping at $\Gamma=1.5$. Movie 2: A movie that shows the dynamics of a single tap at three very different values of $\Gamma$. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
* (22) R. Arévalo, D. Maza and L. A. Pugnaloni, Phys. Rev. E 74, 021303 (2006).
* (23) We have seen that the two invariants of $\Sigma$ (i.e. the trace and the determinant in 2D) are proportional to one another (i.e. $Det(\Sigma)=aTr(\Sigma)+b$). Moreover $a$ and $b$ are independent of the amplitude and duration of the taps used to reach a given $\Sigma$. Therefore, $Tr(\Sigma)$ suffices to characterize $\Sigma$ for our purposes.
* (24) A. Mehta, Granular Physics, Cambridge University Press, Cambridge (2007), Chap. 13.
* (25) J. J. Brey, A. Prados and B. Sanchez-Rey, Physica A 275, 310 (2000).
|
arxiv-papers
| 2010-02-17T13:26:25 |
2024-09-04T02:49:08.431721
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luis A. Pugnaloni, Iv\\'an S\\'anchez, Paula A. Gago, Jos\\'e Damas and\n Iker Zuriguel and Diego Maza",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1002.3264"
}
|
1002.3267
|
∎
11institutetext: Paula A. Gago: Departamento de Física, Facultad de Ciencias
Exactas, Universidad Nacional de La Plata, Casilla de Correo 67, 1900 La
Plata, Argentina.
Nicolás E. Bueno: Laboratorio de Enseñanza de la Física, Departamento de
Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata,
Casilla de Correo 67, 1900 La Plata, Argentina.
L. A. Pugnaloni: Instituto de Física de Líquidos y Sistemas Biológicos
(CONICET La Plata - UNLP), Casilla de Correo 565, 1900 La Plata, Argentina.
Tel.: +54-221-4233283, Fax: +54-221-4257317, E-mail: luis@iflysib.unlp.edu.ar
# High intensity tapping regime in a frustrated lattice gas model of granular
compaction.
Paula A. Gago Nicolás E. Bueno Luis A. Pugnaloni
(Received: date )
###### Abstract
In the frame of a well established lattice gas model for granular compaction,
we investigate the high intensity tapping regime where a pile expands
significantly during external excitation. We find that this model shows the
same general trends as more sophisticated models based on molecular dynamic
type simulations. In particular, a minimum in packing fraction as a function
of tapping strength is observed in the reversible branch of an annealed
tapping protocol.
###### Keywords:
Frustrated Lattice Gas Granular Compaction Tapping Reversible Branch
††journal: Granular Matter
## 1 Introduction
The phenomenology of compaction of granular matter under vertical tapping
captures the attention of a number of investigations and is not exempt from
controversies [1]. Most studies nowadays focus on the dynamics of
compaction—the evolution of structural properties as a function of the number
of taps applied to the sample. In spite of being of chief importance, much
less work is done on the steady state regime achieved by the sample after a
very large number of taps. At low tapping intensities, the relaxation dynamics
of these systems is extremely slow, which makes the steady state very hard to
reach. In a pioneering work, Nowak et al. [2] showed that the steady state can
be achieved by means of a suitable annealing. Recently, Ribière et al. [3]
argued that the steady state is indeed obtainable, reproducible and may
constitute a true thermodynamic state for granular systems. These experiments
show that the packing fraction $\phi$ in the steady state is a monotonic
decreasing function of the tapping intensity: the so-called “reversible
branch”. A number of studies, from experimental [2; 3; 4] and modeling [5; 6;
7; 8; 9; 10; 11] approaches have confirmed this general trend. One exception
has been found in a model of pentagon packings [12; 13] where the reversible
branch presents a monotonic increase of $\phi$ as tapping intensity is
increased.
Recently, by using a few models of granular deposition, it has been shown [14]
that $\phi$, in the reversible branch, presents a minimum at relatively high
tapping intensities. This has now been observed in the laboratory [15].
Previous experimental studies where unable to observe this feature presumably
due to the high intensity taps required. According to the simulation models,
it may be necessary to transfer enough energy to the packing during a tap to
induce the system to expand up to five times its volume before this minimum
can be observed. This is a rather difficult experiment to carry out with a
setup designed to explore slow relaxation by gentle tapping. In twodimensional
systems, the minimum can be observed at much lower tapping strength; an
effective expansion during the tap of less than twice the bed height. It is
worthmentioning that some of the models used in Ref. [14] where the same
models used by others [6; 8], but focusing in a different region of the
parameters that control tap intensity.
The use of annealed tapping to investigate the hysteresis of granular
assemblies subjected to tapping was also used in the frame of a very simple
frustrated lattice gas model proposed by Nicodemi et al. [9; 10] to
investigate granular compaction. In this work we assess the ability of this
model to display a reversible branch similar to the one observed in
experiments and whether the density minimum predicted by more sophisticated
models is also present. With this aim, we explore a range of parameter values
of the model not yet reported in the literature in order to simulate strong
intensity tapping conditions.
The rest of the paper is organized as follows. In Sect. 2 we review the main
features of the frustrated lattice gas model [9]. In Sect. 3 we describe the
tapping protocol. In Sect. 4 we present the results. In Sect. 5 we draw the
conclusions.
## 2 Frustrated lattice gas model
We consider a model first proposed by Nicodemi et al. [9]. The model consists
of a system of particles that move on a square lattice whose bonds are
characterized by fixed random numbers $\epsilon_{i,j}=\pm 1$. Any site $i$ can
be either empty or filled with a particle. Particles are characterized by an
internal degree of freedom (spin) $S_{i}=\pm 1$ and are subjected to the
constraint that whenever two neighboring sites $i$ and $j$ hold particles with
spin $S_{i}$ and $S_{j}$, the following conditions must be satisfied
$\displaystyle\epsilon_{i,j}S_{i}S_{j}=1.$ (1)
This condition introduces an effective frustration. In such system there will
always be empty sites due to this constraint.
In order to introduce external vibrations and gravity, the square lattice is
tilted by $45$ degree. In this way particles diffusing on the lattice will
always move up or down since there are no neighboring sites at the same
vertical position. The model is used to carry out a Monte Carlo (MC)
simulation. In each MC step, we choose a particle and a move type (spin flip
or particle hopping) at random. A spin flip is accepted with probability one
if there is no violation of Eq. 1. If a particle attempts a hop, one of the
four neighboring sites is chosen with equal probability as the landing site.
If the particle attempts a move upward (either up-right or up-left), the
landing site is empty and the internal degrees of freedom satisfy Eq. 1, we
accept the move with probability $P_{2}$. If the particle attempts to move
downward, we accept the move with probability $P_{1}$ (with $P_{1}+P_{2}=1$),
provided that Eq. 1 holds and that the landing site is available.
In the absence of vibrations, the effect of gravity is simulated by imposing
$P_{2}=0$. When vibrations are switched on, $P_{2}$ becomes finite. The
control parameter is the ratio $x=P_{2}/P_{1}$ which describes the intensity
of the vibration.
The MC simulations of the model described above are done on a tilted lattice
with periodic boundary conditions along the horizontal axis and rigid walls at
bottom and top. The top wall is always high enough to avoid particles to reach
it during vibration. After fixing the random bond matrix $\epsilon_{i,j}$, an
initial configuration is prepared by randomly inserting particles of given
spin one at a time into the box from its top. Each particle falls down
following the described dynamics with $x=0$. After a particle has settled, a
new one is released. We introduce a fixed number $N$ of particles. To obtain
an initial low density configuration, particle spins are not allowed to flip
in this preparation process. The state prepared in this way has a packing
fraction of $\phi=0.563\pm 0.007$. Packing fraction $\phi$ is always measured
as the fraction of occupied sites in a rectangular region that span the system
in the horizontal direction and whose height is equivalent to the height that
the system would have had at $\phi=1$ (i.e., if no voids where present). The
region of measurement is vertically centered with the center of mass of the
system.
## 3 Annealed tapping
Figure 1: Packing fraction $\phi$ as a function of MC time $t$ during the
deposition phase in a tap for $x_{0}=1.0$ and $\tau=10^{5}$. At $t=10^{5}$ the
vibration is switched off and deposition begins. After stabilization, the
vibration is switched back on at $t\approx 1.017\times 10^{5}$. The inset
shows $\phi$ along two full taps. The deposition phases cover a very narrow
slot of time as compared with the long duration of the vibration phases in
this example.
In the studies carried out by Nicodemi et al. [10], tapping is simulated
through different protocols depending on the shape of the function $x(t)$ used
to control vibrations. Here, we use a stepwise function that switches on the
vibration to a constant value of $x$ over a period of time $\tau$ and then
switches it back off until full deposition and equilibration is reached.
Hence, each tap is simulated using
$\displaystyle x(t)=x_{0}[1-\theta(t-\tau)],$ (2)
with $\theta(t)$ the Heaviside step function.
Full deposition and equilibration is defined as the state in which all spins
remain in the same site for at least $200$ unit time even though they may
flip. A unit time is defined as $2N$ MC trials. In average, each particle has
one chance to hop and one chance to flip in this time.
In Fig. 1, we show the evolution of the packing fraction $\phi$ during a
’strong’ tap with $x_{0}=1.0$ and $\tau=10^{5}$. Whenever the vibration is on
[$x(t)=x_{0}$], $\phi$ (measured in a central box, as described in Sect. 2)
fluctuates rapidly but continuously decreases in average. Once vibrations are
turned off [$x(t)=0$], $\phi$ rapidly grows as particles fall until full
deposition is reached and a constant value of $\phi$ is obtained. We have run
some trial simulations in which we wait $1000$ unit time (instead of $200$)
after all particle migrations have ceased before deciding that the system is
fully stable and have found no differences in the results.
Since we are interested in simulating strong taps, we have investigated
different values of the parameters $x_{0}$ and $\tau$ in order to achieve an
effective expansion of the system during vibration. We have found that setting
$x_{0}=1$ and varying $\tau$ allows us to explore a wide range of effective
expansions. It is worth mentioning that Nicodemi et al. [9; 10] have
investigated conditions where the slow relaxation of the system were under
scrutiny and for this reason they were mainly focused on values of $x_{0}$
below $1$. We find that for $x_{0}=1$ and $\tau<100$ the system barely expands
during the vibration phase. However, for $\tau>100$ an effective expansion of
the system is achieved. We measure the dimensionless expansion $A$ as the
ratio between the packing fraction of the system at the time we switch off the
vibration ($t=\tau$) and the packing fraction after the bed is fully deposited
(see Sect. 4).
Figure 2: Packing fraction $\phi$ as a function of the number of taps applied
for $x_{0}=1$ and $\tau=10$ (upper curve), $10^{4}$ (lower curve), and
$1.5\times 10^{5}$ (middle curve).
In Fig. 2, we show, as an example, the evolution of the packing fraction
$\phi$ during tapping with $x_{0}=1$ for three values of $\tau$. These
correspond to independent experiments with an initial low density
configuration prepared as described above. It is clear from the figure that
low values of $\tau$ lead to slow relaxation whereas high values of $\tau$
yield a rapid equilibration.
The annealing protocol is carried out as follows. After the initial
preparation of the pack, $10^{3}$ taps with $x_{0}=1$ are applied at a low
tapping intensity, i.e., with a low value of $\tau$. Then, the tapping
strength is increased by increasing the value of $\tau$ and a new series of
$10^{3}$ taps are applied to the final configuration reached in the previous
series. The process is repeated up to very large values of $\tau$. Then, a
decreasing ramp of $\tau$ values is followed down to the initial low tapping
intensity.
## 4 Results for the high intensity tapping regime
We use a $30\times 4000$ squared lattice where we introduce $N=1000$
particles. Initially, particles are “dropped” one at a time from above (with
random horizontal position and spin) and they move according to the prescribed
dynamics with $x=0$ but spin flips are not allowed. Whenever a particle is
unable to migrate to a neighboring site after $16$ MC trials the particle is
frozen in its site and a new particle is released until all particles are
deposited.
The tapping intensity—controlled by $\tau$—is increased and decreased in
discrete steps form $\tau=1$ up to $\tau=1.5\times 10^{5}$ and back down to
$\tau=1$. At each value of $\tau$, a total of $1000$ taps are applied to the
system. The value of $\phi$ considered for each $\tau$ corresponds to the
average over the last $500$ taps. The entire annealing is repeated from the
initial low density configuration for five different random matrices
$\epsilon_{i,j}$. All values of $\phi$ reported correspond to the average over
these five independent experiments and error bars display the corresponding
standard deviation. We notice that, although the mean value of $\phi$ has a
very low dispersion, the “instantaneous” $\phi$ in any given run may present
larger fluctuations around the mean (see Fig. 2).
Figure 3: Packing fraction $\phi$ as a function of $\tau$ for $x_{0}=1$ along
an annealing protocol. The circles correspond to the “non-reversible branch”
of initial increasing tapping intensities (increasing $\tau$). The squares
correspond to the “reversible branch” of decreasing tapping intensities
(decreasing $\tau$). The system is tapped $10^{3}$ times at each value of
$\tau$. The last $500$ configurations are averaged to estimate the value of
$\phi$ reached. The entire simulation has been repeated five times for
different bond random matrices $\epsilon_{i,j}$. The reported data corresponds
to the average—and the error bars to the standard deviation—over these five
repetitions
In Fig. 3 we show the packing fraction $\phi$ as a function of $\tau$ for the
entire annealing protocol. The increasing (irreversible) and decreasing
(reversible) branches are indicated with different symbols. We have also
increased $\tau$ from the last high density configuration in a second
annealing cycle (data not shown). The same values of $\phi$ obtained during
the decreasing ramp are reproduced. This confirms that the upper curve shown
in Fig. 3 is indeed reversible.
It is clear from Fig. 3 that a minimum in density is present at high values of
$\tau$. Beyond this minimum, $\phi$ increases with $\tau$ rather slowly
(notice the log scale in the horizontal axis). Due to limitations in CPU time
we are unable to explore, at present, this regime beyond this values of
$\tau$. However, we can estimate a limiting value of $\phi$ for very strong
tapping (see below).
The existence of a minimum in the density–tapping intensity curve has been
reported very recently [14]. In Ref. [14], the effect is explained in terms of
the formation of arches. At low tapping intensities, the introduction of free
volume is very limited (small expansion). An increase in effective expansion
promotes the formation of larger arches (and hence larger voids) since
particles need some space in order to arrange in these cooperative structures.
However, if expansion during tapping is too large, falling particles have less
chances to meet each other at the time of reaching their stable positions in
order to cooperate and form arches.
In the present model arches are not clearly defined. A definition of which
particles support a given particle is required to identify arches [11; 16].
However, the frustration introduced by the bond matrix $\epsilon_{i,j}$
induces the particles to interlock in loops that leave voids in the structure.
These loops are the analogous to arches in more realistic models.
In order to compare our results with previous models, we plot in Fig. 4 the
packing fraction as a function of the effective dimensionless expansion $A$
induced by the corresponding value of $\tau$. Results from Ref. [14] for
twodimensional models are also reproduced. We observe that all models present
a minimum even though the position varies in the range $1.10<A<2.10$. The
position of the minimum found in 3D models seems to be in the range
$3.0<A<5.0$ [14]. The frustrated lattice gas model yields rather low packing
fractions when compared with more realistic models. However, it is clear that
the main qualitative features of the reversible branch of the annealed tapping
curve are very well captured by the lattice model.
Since the increase in $\phi$ at large values of $\tau$ seems to be due to the
particles depositing without much chances to interact as they reach the
surface of the growing stable pile, we can simulate this in a extreme
condition. Instead of expanding the system using very long $\tau$ values, we
now fill the box dropping particles one at time so that they reach the pile
individually. This is a process similar to the one used in generating the very
first initial condition. However, we now allow the particles to flip their
spin as they fall. Moreover, the already deposited particles are not freezed
but allowed to diffuse and flip while the pile is being built up. We expect
this filling process to mimic the limiting case of an expansion obtained with
$\tau\rightarrow\infty$. The packing fraction so obtained is shown in Fig. 4.
We argue that this should be the limiting value to which $\phi$ would approach
if we further increased $\tau$.
Figure 4: Packing fraction $\phi$ as a function of the effective dimensionless
expansion $A$ induced by tapping. For each value of $\tau$ the corresponding
expansion is calculated as the ratio between the density at the time when the
vibration is switched off and the density after full deposition. Symbols as in
Fig. 3. We show results taken from Ref. [14] for a molecular dynamics of
dissipative soft disks with static friction (down triangles), a pseudo dynamic
model of inelastic disks (right triangles) and an off lattice MC of hard disks
(up triangles). The horizontal dash-dotted line corresponds to the limit of
very strong taps ($\tau\rightarrow\infty$) in the lattice model (see text).
## 5 Conclusions
We have shown that a well known frustrated lattice gas model of granular
compaction—initially designed to study slow dynamics under gentle tapping—is
able to reproduce the qualitative features of more complex models in which
strong tapping conditions are simulated. In particular, we observe the
existence of a minimum in the reversible branch of the annealed tapping. It is
somewhat surprising that such simple model can display nonmonotonic behavior.
This makes the model a suitable working platform to study this newly
discovered feature of granular compaction. It would be of special interest to
investigate if the steady states of equal mean packing fraction produced at
different tapping intensities (at both sides of the minimum) can be
distinguished through the size of the volume fluctuations, an order parameter,
or the distribution of characteristic structures such as voids or loops. This
model presents the advantage that is of simple implementation and that
simulation are much less CPU time demanding than more realistic models which
show the same general features.
In Ref. [14], the existence of a minimum in the reversible branch of granular
compaction is explained in terms of arching. Since this lattice model does not
consider proper contact dynamics nor realistic volume exclusion, it is clear
that the presence of the minimum is due to an underlying phenomenon that
controls both, arching and packing fraction.
At low tapping intensities, the high packing fractions achieved are due to
repeated tapping that allows the system to relax in search for progressively
more compact (lower potential energy) structures. This is achieved through the
local rearrangements promoted from tap to tap. However, at very high tapping
intensities, the system achieves the steady state in a single tap (see Fig. 2)
since the full pack is arranged _de novo_ after a strong tap in a global
relaxation event. In this last case, in order to achieve higher packing
fractions, the particles need to follow a deposition process that allows them
enough time to search for the lowest potential energy configuration in a
single tap. This is best promoted if the bed is expanded significantly during
the tap so that the longer deposition times required and the larger free
volume introduced give greater chances for the particles to find lower
positions. In summary, the existence of the minimum in the reversible branch
of granular compaction seems to be ultimately related to a competition between
the global and local relaxation promoted at different scales of tapping
strength [8].
###### Acknowledgements.
LAP acknowledge financial support from ANPCyT through project PICT-2007-00882
(Argentina).
## References
* Richard [2005] Richard, P., Nicodemi, M., Delannay, R., Ribière, P., Bideau, D. (2005) Slow relaxation and compaction of granular systems. Nature Materials 4: 121–128
* Nowak [1997] Nowak, E.R., Knight, J.B., Povinelli, M.L., Jaeger, H.M., Nagel, S.R. (1997) Reversibility and irreversibility in the packing of vibrated granular material. Powder Tech. 94: 79–83
* Ribiere [2007] Ribière, Ph., Richard, P., Philippe, P., Bideau, D., Delannay R. (2007) On the existence of stationary states during granular compaction. Eur. Phys. J. E 22: 249–253
* Schroter [2005] Schröter, M., Goldman, D.I., Swinney, H.L. (2005) Stationary state volume fluctuations in a granular medium. Phys. Rev. E 71: 030301(R)
* Stadler [2001] Stadler, P.F., Luck, J.M., Mehta, A. (2001) Shaking a box of sand. Europhys. Lett. 57: 46–52
* Philippe [2001] Philippe, P., Bideau, D. (2001) Numerical model for granular compaction under vertical tapping. Phys. Rev. E 63: 051304
* Cimarra [2006] Ciamarra, M.P., Coniglio, A., Nicodemi, M. (2006) Thermodynamics and statistical mechanics of dense granular media. Phys. Rev. Lett. 97: 158001
* Mehta [1991] Mehta, A., Barker, G.C. (1991) Vibrated powders: A microscopic approach. Phys. Rev. Lett. 67: 394–397
* Nicodemi [1997] Nicodemi, M., Coniglio, A., Herrmann, H.J. (1997) The compaction in granular media and frustrated Ising models. J. Phys. A: Math. Gen. 30: L379–L385
* Nicodemi [1997] Nicodemi, M., Coniglio, A., Herrmann, H.J. (1997) Frustration and slow dynamics of granular packings. Phys. Rev. E 55: 3962–3969
* Pugnaloni [2006] Pugnaloni, L.A., Valluzzi, M.G., Valluzzi, L.G. (2006) Arching in tapped deposits of hard disks. Phys. Rev. E 73: 051302
* Vidales [2008] Vidales, A.M., Pugnaloni, L.A., Ippolito, I. (2008) Pentagon deposits unpack under gentle tapping. Phys. Rev. E 77: 051305
* Vidales [2009] Vidales, A.M., Pugnaloni, L.A., Ippolito, I. (2009) Compaction and arching in tapped pentagon deposits. Gran. Matt. 11: 53–61
* Pugnaloni [2008] Pugnaloni, L.A., Mizrahi, M., Carlevaro, C.M., Vericat, F. (2008) Nonmonotonic reversible branch in four model granular beds subjected to vertical vibration. Phys. Rev. E 78: 051305
* Sanchez [2009] Sánchez, I., et al. (unpublished)
* Pugnaloni [2001] Pugnaloni, L.A., Barker, G.C., Mehta, A. (2001) Multi-particle structures in non-sequentially reorganized hard sphere deposits. Adv. Complex Syst. 4: 289–297
|
arxiv-papers
| 2010-02-17T13:40:22 |
2024-09-04T02:49:08.436138
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paula A. Gago, Nicol\\'as E. Bueno and Luis A. Pugnaloni",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1002.3267"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.